Heteroepitaxy of Semiconductor

Half Title PageHETEROEPITAXY OF SEMICONDUCTORS

THEORY, GROWTH,AND CHARACTERIZATION

7195_book.fm Page i Thursday, December 21, 2006 8:59 AM

© 2007 by Taylor & Francis Group, LLC

Title PageHETEROEPITAXY OF SEMICONDUCTORS

THEORY, GROWTH,AND CHARACTERIZATION

John E. AyersUniversity of Connecticut

Storrs, CT, U.S.A.

7195_book.fm Page iii Thursday, December 21, 2006 8:59 AM


The cover figure represents conduction-band minimum (CBM) wave functions of a 6000-atom (110)x(1–10)x(001) GaAs quantum dot. The wave function amplitude, averaged along the [001] direction, is plotted in the (001) plane. Heteroepitaxial quantum dots are of interest for many appli-cations including lasers and single-electron transistor. Figure printed by permission of the National Renewable Energy Laboratory, Golden, CO.CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742

© 2007 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government worksPrinted in the United States of America on acid-free paper10 9 8 7 6 5 4 3 2 1

International Standard Book Number-10: 0-8493-7195-3 (Hardcover)International Standard Book Number-13: 978-0-8493-7195-0 (Hardcover)

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the conse-quences of their use.

No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.

For permission to photocopy or use material electronically from this work, please access www.

222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.

Library of Congress Cataloging-in-Publication Data

Ayers, John E.Heteroepitaxy of semiconductors : theory, growth, and characterization / John

E. Ayers. p. cm.

Includes bibliographical references and index.ISBN 0-8493-7195-31. Compound semiconductors. 2. Epitaxy. I. Title.

QC611.8.C64A94 2007537.6’22--dc22 2006050560

Visit the Taylor & Francis Web site athttp://www.taylorandfrancis.com

and the CRC Press Web site athttp://www.crcpress.com

7195_book.fm Page iv Thursday, December 21, 2006 8:59 AM


copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC)

http://www.copyright.com



http://www.taylorandfrancis.com

http://www.crcpress.com

http://www.crcpress.com

http://www.taylorandfrancis.com




Dedication

To my wife, Kimberly Dawn Ayers,

and our children, Jacob, Sarah, and Rachel.

7195_book.fm Page v Thursday, December 21, 2006 8:59 AM


Preface

Heteroepitaxy, or the single-crystal growth of one semiconductor on another,has been a topic of intense research for several decades. This effort receiveda significant boost with the advent of metalorganic vapor phase epitaxy(MOVPE), molecular beam epitaxy (MBE), and other advancements in epi-taxial growth. It became possible to grow almost any semiconductor materialor structure, including alloys, multilayers, superlattices, and graded layers,with unprecedented control and uniformity. Researchers embraced thesecapabilities and set out to grow nearly every imaginable combination ofepitaxial layer/substrate. Across this great diversity of materials and struc-tures, there has begun to emerge a general understanding of at least someaspects of heteroepitaxy, especially nucleation, growth modes, relaxation ofstrained layers, and dislocation dynamics. The application of this knowledgehas enabled the commercial production of a wide range of heteroepitaxialdevices, including high-brightness light-emitting diodes, lasers, and high-frequency transistors, to name a few.

Our understanding of heteroepitaxy is far from complete, and the field isevolving rapidly. Here I did not attempt to report all of the results fromevery known heteroepitaxial material combination. Even if this had beenpossible, such a book would become out of date with the next wave ofelectronic journals. Instead, I tried to emphasize the principles underlyingheteroepitaxial growth and characterization, with many examples from thematerial systems that have been studied. I hope that this approach willremain useful for some time to come, as a reference to researchers in thefield and also as a starting point for graduate students.

I am sincerely grateful to Professor Sorab K. Ghandhi, who introduced meto the field of heteroepitaxy. I am also indebted to my graduate students andmy fellow researchers, without whom this book would not be possible.Finally, I thank my family for their unending support and patience through-out this endeavor.

John E. Ayers

June 23, 2006Storrs, CT

7195_book.fm Page vii Thursday, December 21, 2006 8:59 AM


The Author

J.E. Ayers

grew up eight miles froman integrated circuit design and fab-rication facility, where he worked asa technician and first developed hispassion for semiconductors. Afterearning a B.S.E.E. from the Univer-sity of Maine, he began experimentaland theoretical work on heteroepi-taxy while at Rensselaer PolytechnicInstitute, Troy, New York, and Phil-ips Laboratories, Briarcliff Manor,New York. He earned an M.S.E.E. in1987 and a Ph.D.E.E. in 1990, bothfrom Rensselaer Polytechnic Insti-tute. Since that time he has beenemployed in academic research andteaching at the University of Con-necticut, Storrs. His scientific papers

in the area of heteroepitaxy have been cited hundreds of times by researchersworldwide. He is a member of the Institute of Electrical and ElectronicsEngineers, the American Physical Society, Eta Kappa Nu, Tau Beta Pi, andPhi Kappa Phi. He currently lives in Ashford, Connecticut, with his wife andthree children.

7195_book.fm Page ix Thursday, December 21, 2006 8:59 AM


Contents

1

Introduction ..................................................................................... 1

2

Properties of Semiconductors........................................................ 7

2.1 Introduction ....................................................................................................72.2 Crystallographic Properties .........................................................................7

2.2.1 The Diamond Structure....................................................................82.2.2 The Zinc Blende Structure ...............................................................82.2.3 The Wurtzite Structure .....................................................................92.2.4 Silicon Carbide.................................................................................102.2.5 Miller Indices in Cubic Crystals ...................................................122.2.6 Miller–Bravais Indices in Hexagonal Crystals ...........................122.2.7 Orientation Effects...........................................................................14

2.2.7.1 Diamond Semiconductors ...............................................142.2.7.2 Zinc Blende Semiconductors ..........................................152.2.7.3 Wurtzite Semiconductors ................................................162.2.7.4 Hexagonal Silicon Carbide..............................................17

2.3 Lattice Constants and Thermal Expansion Coefficients .......................172.4 Elastic Properties..........................................................................................19

2.4.1 Hooke’s Law ....................................................................................202.4.1.1 Hooke’s Law for Isotropic Materials.............................222.4.1.2 Cubic Crystals ...................................................................222.4.1.3 Hexagonal Crystals...........................................................24

2.4.2 The Elastic Moduli ..........................................................................272.4.2.1 Cubic Crystals ...................................................................282.4.2.2 Hexagonal Crystals...........................................................28

2.4.3 Biaxial Stresses and Tetragonal Distortion..................................302.4.4 Strain Energy....................................................................................31

2.5 Surface Free Energy.....................................................................................322.6 Dislocations...................................................................................................36

2.6.1 Screw Dislocations ..........................................................................372.6.2 Edge Dislocations............................................................................382.6.3 Slip Systems .....................................................................................382.6.4 Dislocations in Diamond and Zinc Blende Crystals .................41

2.6.4.1 Threading Dislocations in Diamond and ZincBlende Crystals..................................................................43

2.6.4.2 Misfit Dislocations in Diamond and Zinc BlendeCrystals ...............................................................................44

2.6.5 Dislocations in Wurtzite Crystals .................................................482.6.5.1 Threading Dislocations in Wurtzite Crystals ...............48

7195_book.fm Page xi Thursday, December 21, 2006 8:59 AM


2.6.5.2 Misfit Dislocations in Wurtzite Crystals .......................492.6.6 Dislocations in Hexagonal SiC......................................................51

2.6.6.1 Threading Dislocations in Hexagonal SiC....................512.6.7 Strain Fields and Line Energies of Dislocations ........................51

2.6.7.1 Screw Dislocation..............................................................522.6.7.2 Edge Dislocation ...............................................................542.6.7.3 Mixed Dislocations ...........................................................552.6.7.4 Frank’s Rule .......................................................................552.6.7.5 Hollow-Core Dislocations (Micropipes) .......................56

2.6.8 Forces on Dislocations....................................................................562.6.9 Dislocation Motion..........................................................................572.6.10 Electronic Properties of Dislocations ...........................................58

2.6.10.1 Diamond and Zinc Blende Semiconductors ................582.7 Planar Defects...............................................................................................61

2.7.1 Stacking Faults.................................................................................612.7.2 Twins .................................................................................................642.7.3 Inversion Domain Boundaries (IDBs)..........................................65

Problems.................................................................................................................67References...............................................................................................................68

3

Heteroepitaxial Growth................................................................ 75

3.1 Introduction ..................................................................................................753.2 Vapor Phase Epitaxy (VPE)........................................................................76

3.2.1 VPE Mechanisms and Growth Rates...........................................763.2.2 Hydrodynamic Considerations.....................................................793.2.3 Vapor Phase Epitaxial Reactors ....................................................813.2.4 Metalorganic Vapor Phase Epitaxy (MOVPE)............................85

3.3 Molecular Beam Epitaxy (MBE)................................................................883.4 Silicon, Germanium, and Si

1–x

Ge

x

Alloys.................................................923.5 Silicon Carbide .............................................................................................943.6 III-Arsenides, III-Phosphides, and III-Antimonides ..............................953.7 III-Nitrides ....................................................................................................973.8 II-VI Semiconductors ..................................................................................983.9 Conclusion ....................................................................................................99Problems...............................................................................................................100References.............................................................................................................100

4

Surface and Chemical Considerations in Heteroepitaxy ....... 105

4.1 Introduction ................................................................................................1054.2 Surface Reconstructions............................................................................106

4.2.1 Wood’s Notation for Reconstructed Surfaces...........................1084.2.2 Experimental Observations .........................................................109

4.2.2.1 Si (001) Surface ................................................................1094.2.2.2 Si (111) Surface ................................................................ 1104.2.2.3 Ge (111) Surface............................................................... 111

7195_book.fm Page xii Thursday, December 21, 2006 8:59 AM


4.2.2.4 6H-SiC (0001) Surface .................................................... 1114.2.2.5 3C-SiC (001) ..................................................................... 1124.2.2.6 3C-SiC (111)...................................................................... 1124.2.2.7 GaN (0001) ....................................................................... 1134.2.2.8 Zinc Blende GaN (001)................................................... 1134.2.2.9 GaAs (001)........................................................................ 1134.2.2.10 InP (001)............................................................................ 1134.2.2.11 Sapphire (0001)................................................................ 114

4.2.3 Surface Reconstruction and Heteroepitaxy .............................. 1144.2.3.1 Inversion Domain Boundaries (IDBs) ......................... 1144.2.3.2 Heteroepitaxy of Polar Semiconductors with

Different Ionicities .......................................................... 1154.3 Nucleation................................................................................................... 117

4.3.1 Homogeneous Nucleation ........................................................... 1174.3.2 Heterogeneous Nucleation ..........................................................120

4.3.2.1 Macroscopic Model for HeterogeneousNucleation........................................................................120

4.3.2.2 Atomistic Model..............................................................1224.3.2.3 Vicinal Substrates............................................................125

4.4 Growth Modes ...........................................................................................1254.4.1 Growth Modes in Equilibrium ...................................................127

4.4.1.1 Regime I: (

f

<

ε

1

)..............................................................1304.4.1.2 Regime II: (

ε

1

<

f

<

ε

2

) ....................................................1314.4.1.3 Regime III: (

ε

2

<

f

<

ε

3

) ...................................................1324.4.1.4 Regime IV: (

f

>

ε

3

) ...........................................................1324.4.2 Growth Modes and Kinetic Considerations .............................132

4.5 Nucleation Layers......................................................................................1384.5.1 Nucleation Layers for GaN on Sapphire ..................................139

4.6 Surfactants in Heteroepitaxy ...................................................................1404.6.1 Surfactants and Growth Mode....................................................1404.6.2 Surfactants and Island Shape......................................................1424.6.3 Surfactants and Misfit Dislocations ...........................................1424.6.4 Surfactants and Ordering in InGaP ...........................................143

4.7 Quantum Dots and Self-Assembly .........................................................1434.7.1 Topographically Guided Assembly of Quantum Dots ...........1444.7.2 Stressor-Guided Assembly of Quantum Dots ..........................1454.7.3 Vertical Organization of Quantum Dots ...................................1474.7.4 Precision Lateral Placement of Quantum Dots........................149

Problems...............................................................................................................150References.............................................................................................................151

5

Mismatched Heteroepitaxial Growth and StrainRelaxation .................................................................................... 161

5.1 Introduction ................................................................................................161

7195_book.fm Page xiii Thursday, December 21, 2006 8:59 AM


5.2 Pseudomorphic Growth and the Critical Layer Thickness ................1635.2.1 Matthews and Blakeslee Force Balance Model ........................1655.2.2 Matthews Energy Calculation.....................................................1665.2.3 van der Merwe Model..................................................................1685.2.4 People and Bean Model ...............................................................1695.2.5 Effect of the Sign of Mismatch....................................................1715.2.6 Critical Layer Thickness in Islands ............................................173

5.3 Dislocation Sources ...................................................................................1755.3.1 Homogeneous Nucleation of Dislocations ...............................1775.3.2 Heterogeneous Nucleation of Dislocations ..............................1795.3.3 Dislocation Multiplication ...........................................................179

5.3.3.1 Frank–Read Source .........................................................1805.3.3.2 Spiral Source ....................................................................1855.3.3.3 Hagen–Strunk Multiplication .......................................187

5.4 Interactions between Misfit Dislocations...............................................1895.5 Lattice Relaxation Mechanisms...............................................................191

5.5.1 Bending of Substrate Dislocations..............................................1915.5.2 Glide of Half-Loops ......................................................................1945.5.3 Injection of Edge Dislocations at Island Boundaries ..............1945.5.4 Nucleation of Shockley Partial Dislocations.............................1965.5.5 Cracking..........................................................................................199

5.6 Quantitative Models for Lattice Relaxation ..........................................1995.6.1 Matthews and Blakeslee Equilibrium Model ...........................2005.6.2 Matthews, Mader, and Light Kinetic Model ............................2015.6.3 Dodson and Tsao Kinetic Model ................................................203

5.7 Lattice Relaxation on Vicinal Substrates: Crystallographic Tilting of Heteroepitaxial Layers ............................................................2055.7.1 Nagai Model ..................................................................................2055.7.2 Olsen and Smith Model ...............................................................2075.7.3 Ayers, Ghandhi, and Schowalter Model ...................................2075.7.4 Riesz Model....................................................................................2155.7.5 Vicinal Epitaxy of III-Nitride Semiconductors.........................2185.7.6 Vicinal Heteroepitaxy with a Change in Stacking

Sequence .........................................................................................2205.7.7 Vicinal Heteroepitaxy with Multilayer Steps ...........................2215.7.8 Tilting in Graded Layers: LeGoues, Mooney, and Chu

Model ..............................................................................................2245.8 Lattice Relaxation in Graded Layers......................................................227

5.8.1 Critical Thickness in a Linearly Graded Layer ........................2275.8.2 Equilibrium Strain Gradient in a Graded Layer......................2285.8.3 Threading Dislocation Density in a Graded Layer .................228

5.8.3.1 Abrahams et al. Model ..................................................2295.8.3.2 Fitzgerald et al. Model...................................................230

5.9 Lattice Relaxation in Superlattices and Multilayer Structures...........2315.10 Dislocation Coalescence, Annihilation, and Removal in

Relaxed Heteroepitaxial Layers...............................................................233

7195_book.fm Page xiv Thursday, December 21, 2006 8:59 AM


5.11 Thermal Strain............................................................................................2385.12 Cracking in Thick Films ...........................................................................239Problems...............................................................................................................242References.............................................................................................................243

6

Characterization of Heteroepitaxial Layers ............................. 249

6.1 Introduction ................................................................................................2496.2 X-Ray Diffraction .......................................................................................250

6.2.1 Positions of Diffracted Beams .....................................................2516.2.1.1 The Bragg Equation........................................................2516.2.1.2 The Reciprocal Lattice and the von Laue

Formulation for Diffraction...........................................2536.2.1.3 The Ewald Sphere...........................................................255

6.2.2 Intensities of Diffracted Beams ...................................................2556.2.2.1 Scattering of X-Rays by a Single Electron ..................2566.2.2.2 Scattering of X-Rays by an Atom.................................2576.2.2.3 Scattering of X-Rays by a Unit Cell.............................2586.2.2.4 Intensities of Diffraction Profiles..................................259

6.2.3 Dynamical Diffraction Theory ....................................................2606.2.3.1 Intrinsic Diffraction Profiles for Perfect

Crystals .............................................................................2616.2.3.2 Intrinsic Widths of Diffraction Profiles .......................2626.2.3.3 Extinction Depth and Absorption Depth....................264

6.2.4 X-Ray Diffractometers ..................................................................2656.2.4.1 Double-Crystal Diffractometer .....................................2676.2.4.2 Bartels Double-Axis Diffractometer.............................2706.2.4.3 Triple-Axis Diffractometer.............................................271

6.3 Electron Diffraction ...................................................................................2726.3.1 Reflection High-Energy Electron Diffraction (RHEED)..........2736.3.2 Low-Energy Electron Diffraction (LEED) .................................274

6.4 Microscopy..................................................................................................2756.4.1 Optical Microscopy .......................................................................2766.4.2 Transmission Electron Microscopy (TEM) ................................2766.4.3 Scanning Tunneling Microscopy (STM) ....................................2796.4.4 Atomic Force Microscopy (AFM) ...............................................281

6.5 Crystallographic Etching Techniques.....................................................2826.6 Photoluminescence ....................................................................................2846.7 Growth Rate and Layer Thickness .........................................................2886.8 Composition and Strain............................................................................290

6.8.1 Binary Heteroepitaxial Layer ......................................................2916.8.2 Ternary Heteroepitaxial Layer ....................................................2936.8.3 Quaternary Heteroepitaxial Layer .............................................297

6.9 Determination of Critical Layer Thickness ...........................................2976.9.1 Effect of Finite Resolution ...........................................................299

7195_book.fm Page xv Thursday, December 21, 2006 8:59 AM


6.9.2 X-Ray Diffraction...........................................................................3016.9.2.1 Strain Method..................................................................3016.9.2.2 FWHM Method...............................................................307

6.9.3 X-Ray Topography ........................................................................3126.9.4 Transmission Electron Microscopy.............................................3136.9.5 Electron Beam-Induced Current (EBIC) ....................................3156.9.6 Photoluminescence........................................................................3156.9.7 Photoluminescence Microscopy..................................................3176.9.8 Reflection High-Energy Electron Diffraction (RHEED)..........3196.9.9 Scanning Tunneling Microscopy (STM) ....................................3216.9.10 Rutherford Backscattering (RBS) ................................................323

6.10 Crystal Orientation....................................................................................3246.11 Defect Types and Densities ......................................................................326

6.11.1 Transmission Electron Microscopy.............................................3276.11.2 Crystallographic Etching .............................................................3296.11.3 X-Ray Diffraction...........................................................................331

6.12 Multilayered Structures and Superlattices ............................................3386.13 Growth Mode .............................................................................................342Problems...............................................................................................................345References.............................................................................................................347

7

Defect Engineering in Heteroepitaxial Layers ........................ 355

7.1 Introduction ................................................................................................3557.2 Buffer Layer Approaches..........................................................................355

7.2.1 Uniform Buffer Layers and Virtual Substrates ........................3557.2.2 Graded Buffer Layers ...................................................................3597.2.3 Superlattice Buffer Layers............................................................367

7.3 Reduced Area Growth Using Patterned Substrates.............................3727.4 Patterning and Annealing ........................................................................3767.5 Epitaxial Lateral Overgrowth (ELO) ......................................................3817.6 Pendeo-Epitaxy ..........................................................................................3897.7 Nanoheteroepitaxy ....................................................................................391

7.7.1 Nanoheteroepitaxy on a Noncompliant Substrate ..................3927.7.2 Nanoheteroepitaxy with a Compliant Substrate .....................395

7.8 Planar Compliant Substrates ...................................................................3997.8.1 Compliant Substrate Theory .......................................................4007.8.2 Compliant Substrate Implementation........................................403

7.8.2.1 Cantilevered Membranes...............................................4047.8.2.2 Silicon-on-Insulator (SOI) as a Compliant

Substrate ...........................................................................4067.8.2.3 Twist-Bonded Compliant Substrates ........................... 411

7.9 Free-Standing Semiconductor Films.......................................................4147.10 Conclusion ..................................................................................................415Problems...............................................................................................................416References.............................................................................................................416

7195_book.fm Page xvi Thursday, December 21, 2006 8:59 AM


Appendix A: Bandgap Engineering Diagrams................................. 421

References.............................................................................................................422

Appendix B: Lattice Constants and Coefficients of Thermal Expansion.............................................................................. 423

References.............................................................................................................426

Appendix C: Elastic Constants .......................................................... 427

References.............................................................................................................430

Appendix D: Critical Layer Thickness ............................................. 431

References.............................................................................................................431

Appendix E: Crystallographic Etches ............................................... 433

References.............................................................................................................434

Appendix F: Tables for X-Ray Diffraction ....................................... 437

7195_book.fm Page xvii Thursday, December 21, 2006 8:59 AM


1

1

Introduction

Heteroepitaxy, or the single-crystal growth of one semiconductor on another,is necessary for the development of a wide range of devices and systems.There are three motivations for semiconductor heteroepitaxy:

substrate engi-neering

,

heterojunction devices

, and

device integration

. Figure 1.1 and Figure 1.2illustrate some of the wide range of semiconductor materials, all havingunique properties that make them interesting for device applications. Ofspecial importance is the energy gap, which determines the emission wave-length in light-emitting diodes and lasers, as well as the suitability for otherdevice applications. In most cases, the combination of materials with differ-ent energy gaps will require mismatched heteroepitaxy due to the differentlattice constants.

Substrate engineering

is necessary because many semiconductors with inter-esting device applications are unavailable in the form of large-area, high-quality, single-crystal wafers. Instead, they must be grown on one of the fewavailable substrates. Only Si, GaAs, InP, 6H-SiC, 4H-SiC, and sapphire (

α

-Al

2

O

3

) crystals are available with acceptable quality and cost for widespreadadoption. Among these, only selected low-index crystal orientations areavailable: Si (001), Si (111), GaAs (001), InP (001), 6H-SiC (0001), 4H-SiC(0001), and sapphire (0001). The development of devices using other mate-rials, especially ternary and quaternary alloys, requires the choice of one ofthese common substrates with (hopefully) chemical and crystallographiccompatibility. III-Nitride devices such as blue and violet light-emittingdiodes (LEDs) and laser diodes are fabricated exclusively by heteroepitaxialgrowth on SiC or sapphire substrates, due to the unavailability of GaNwafers. Apart from necessity, cost is also a driver for substrate engineering.Even though GaAs substrates are readily obtained, Si wafers are availablewith larger diameter and lower cost, so tremendous benefit would derivefrom the placement of GaAs circuits on Si wafers.

Heterojunction devices

are another important application area for heteroepi-taxy. Indeed, many of the devices we take for granted would not be possible(or practical) without the ability to form semiconductor heterojunctions:laser diodes, high-brightness light-emitting diodes, and high-frequencytransistors. Heterojunction devices are now entering mainstream electronicsas well, with the development of SiGe heterojunction transistors. Soon,

7195_book.fm Page 1 Thursday, December 21, 2006 8:59 AM


2

Heteroepitaxy of Semiconductors

heteroepitaxial growth in a Stranski–Krastanov or Volmer–Weber growthmode promises to enable practical quantum dot devices, including lasersand single-electron transistors.

Integrated circuits represent another area where heteroepitaxy is anenabling technology. Many semiconductor materials have become estab-lished in application niche areas, but no one material can simultaneouslysatisfy the needs for high-density digital circuits, sensors, high-power elec-tronics, high-frequency amplifiers, and optoelectronic devices operating overthe range from infrared to ultraviolet, including light-emitting diodes andlasers, modulators, and detectors. Heteroepitaxy presents one approach forthe integration of these various functions, or a subset of them, on a singlechip. Tremendous savings in cost, size, and weight can be expected relativeto the wiring together of many packaged devices at the circuit board level.

The many advancements in the field of heteroepitaxy would not have beenpossible without the development of the epitaxial growth techniques molec-ular beam epitaxy (MBE) and metalorganic vapor phase epitaxy (MOVPE).These two methods afford tremendous flexibility and the ability to depositthin layers and complex multilayered structures with precise control andexcellent uniformity. In addition, the high-vacuum environment of MBEmakes it possible to employ

in situ

characterization tools using electron andion beams, which provide the crystal grower with immediate feedback, andimproved control of the growth process. For these reasons, MBE and MOVPEhave emerged as general-purpose tools for heteroepitaxial research and com-

FIGURE 1.1

Energy gap as a function of lattice constant for cubic semiconductors. Room temperature valuesare given. Dashed lines indicate an indirect gap.

0

1

2

3

5.4 5.6 5.8 6.0 6.2 6.4 6.6Lattice constant a (Å)

Ener

gy g

ap (e

V)

AlP

GaP

Si

ZnS

GaAs

Ge

ZnSe

ZnTe

CdSe

CdTe

InSbHgTe

GaSb

InAsHgSe

AlSb

AlAs

InP



Introduction

3

mercial production. Together, these two epitaxial growth methods accountfor virtually all production of compound semiconductor devices today.

The key challenges in the heteroepitaxy of semiconductors, relative to thedevelopment of useful devices, are the control of the growth morphology,stress and strain, and crystal defects. Chapter 2 reviews the properties ofsemiconductors that bear on these aspects of heteroepitaxy, including crys-tallographic properties, elastic properties, surface properties, and defectstructures. Chapter 3 provides a brief overview of epitaxial growth methods,starting with the principles of MOVPE and MBE and concluding with someexamples from important material systems.

An important distinction between heteroepitaxy and homoepitaxy is theneed to nucleate a new phase on the substrate surface. Therefore, the surfaceand its structure, as well as surface-segregated impurities (surfactants), playimportant roles in determining the usefulness of heteroepitaxial layers forthe fabrication of devices. Chapter 4 provides an in-depth description ofsemiconductor crystal surfaces and their reconstructions, nucleation, growthmodes, and surfactants. Control of the growth mode, through the tailoringof growth conditions or the use of surfactants, is critical to the developmentof devices. Two-dimensional growth is desirable in most cases, for theachievement of flat, abrupt interfaces and surfaces, and is mandated forquantum well devices. For the development of quantum dot devices,

FIGURE 1.2

Energy gap as a function of lattice constant for hexagonal semiconductors. Room temperaturevalues are given. Sapphire, a commonly used substrate material for III-nitrides, has roomtemperature lattice constants of a = 4.7592 Å and c = 12.9916 Å. (From Y.V. Shvyd’ko, M. Lucht,E. Gerdau, M. Lerche, E.E. Alp, W. Sturhahn, J. Sutter, and T.S. Toellner, Measuring wavelengthsand lattice constants with the Mössbauer wavelength standard,

J. Synchrotron Rad.

, 9, 17 (2002).)

0

2

4

6


Ener

gy g

ap (e

V)AlN

GaN

In N

6H-SiC

4H-SiC



4


Volmer–Weber (island growth) or Stranski–Krastanov (growth of a continu-ous wetting layer followed by islanding) is actually desirable. Here thecontrol of the sizes, shapes, and distributions of islands is critical. This aspect,called the self-assembly of quantum dots, is covered in Chapter 4.

Heteroepitaxial growth is rarely lattice matched, so strain relaxation andthe associated creation of crystal defects are of great importance. Under thecondition of moderate lattice mismatch (<2%), it is possible to grow apseudomorphic heteroepitaxial layer, which maintains coherency with thesubstrate crystal in the plane of the interface. But at some thickness thecreation of misfit dislocations becomes energetically favorable. The latticerelaxation process is rather complex and is usually limited by the nucleation,multiplication, or glide of dislocations. Invariably, nonequilibrium thread-ing dislocations are introduced together with the stress-relieving misfitdefects. The presence of dislocations in the material tends to degrade itselectronic properties, affecting device performance and lifetime. The controland elimination of these defects is therefore an area of considerable interest.Chapter 5 provides an in-depth review of mismatched heteroepitaxy andlattice relaxation.

Characterization tools have played a key role in the advancement of thescience of heteroepitaxy. Some of the most commonly used techniques aremicroscopic techniques, x-ray diffraction, photoluminescence, and crystallo-graphic etching. These are covered in detail in Chapter 6, with an emphasison x-ray diffraction, which is the most widely used tool for structural char-acterization of heteroepitaxial layers. Individual sections are also devoted tosome key application areas for these characterization tools, such as the deter-mination of the stress, strain, and composition; the determination of thecritical layer thickness for lattice relaxation; the characterization of the mor-phology and growth mode; and the observation of crystal defects, and thedetermination of their types and configurations.

The broad application of heteroepitaxy to device and circuit fabricationrequires the control of the crystal defect structures, and therefore a numberof defect engineering approaches have emerged. These are described inChapter 7 and include buffer layer approaches, patterned growth, patterningand annealing, epitaxial lateral overgrowth, nanoheteroepitaxy, and compli-ant substrates, to name a few. All of these were designed to reduce thedislocation densities of heteroepitaxial layers to practical levels for deviceapplications. Some are intended to remove existing defects from lattice-relaxed heteroepitaxial layers, such as patterning and annealing, epitaxiallateral overgrowth, or superlattice buffer layers (dislocation filters). Otherswere conceived to prevent lattice relaxation in the first place; these includepatterned growth, nanoheteroepitaxy, and compliant substrates.

The proliferation of defect engineering methods could be taken as anindication that none of them are uniquely suited to the purpose, for allmaterial systems. On the other hand, some of these approaches have beenhighly successful, to the point of being used in commercial devices. Gradedbuffer layers are the most important example of this and have been used in



Introduction

5

commercial GaAs

1–x

P

x

LEDs on GaAs substrates and In

x

Ga

1–x

As high-elec-tron-mobility transistors (HEMTs) on GaAs substrates. Epitaxial lateral over-growth (ELO) is an important method used to reduce the threadingdislocation densities in the active regions of III-nitride lasers. Other defectengineering approaches, such as the use of compliant substrates, show greatpromise, and yet their commercial exploitation is not yet in sight. In orderto tap the great potential of heteroepitaxy, defect engineering approacheswill continue to be important, not only in the applications listed above, butin new ones as well.



7

2

Properties of Semiconductors

2.1 Introduction

The key challenges in the heteroepitaxy of semiconductors, relative to thedevelopment of useful devices, are the control of the growth morphology,stress and strain, and crystal defects. The purpose of this chapter is to reviewthe properties of semiconductors that bear on these aspects of heteroepitaxy,including crystallographic properties, elastic properties, surface properties,and defect structures.

2.2 Crystallographic Properties

Semiconductors in common use today are nearly always single-crystal mate-rials.* A crystal is a periodic arrangement of atoms in space. A space latticeand a basis comprise a crystal structure. The space lattice describes theperiodic arrangement of points on which atoms (or groups of atoms) maybe placed, whereas the basis can be a single atom or an arrangement of atomsplaced at each space lattice point. There are 14 space lattices, called theBravais lattices.

1

Of these, the face-centered cubic and hexagonal space lat-tices are most relevant here.

The technologically important semiconductors exhibit a number of differ-ent crystal structures. Silicon, germanium, and their alloys have the diamondstructure. Many III-VI and II-VI semiconductors, including GaAs and InP,crystallize in the cubic zinc blende structure. GaN and related materials aswell as ZnS and other II-VI crystals exhibit the hexagonal wurtzite structure.Some III-V and II-VI semiconductors can assume either a zinc blende or a

* Notable exceptions include thin-film transistors, made using polycrystalline or amorphous sil-icon, and the gates of metal oxide semiconductor field effect transistors, which are made usingpolycrystalline silicon.



8


wurtzite structure. SiC exhibits over 250 different polytypes, including cubic,hexagonal, and rhombohedral variants.

The wide variety of crystal structures among semiconductors presents bothopportunities and challenges for the crystal grower. It is possible to createunique heterojunction devices and metastable structures by the properchoices of materials. Moreover, it is sometimes possible to determine thecrystal structure of a particular epitaxial layer by the choice of substrate orgrowth conditions, adding another dimension to device design. On the otherhand, it is also possible to end up with mixed phase material, which usuallyhas degraded electronic properties. Therefore, it is the purpose of this sectionto describe the crystal structures exhibited by semiconductor materials ofinterest, as well as their characteristics and behavior relevant to heteroepitaxy.

Some of the most important crystallographic properties of semiconductorsare the crystal structure and lattice constants. Also relevant to the growth ofheteroepitaxial layers is the anisotropic behavior of the crystalline materials,especially the etching, nucleation, growth, and cleavage behavior. In manyof the following subsections, materials will be lumped together in the cubicand hexagonal classes of crystals.

2.2.1 The Diamond Structure

The diamond structure is shared by Si, Ge, Si-Ge alloys, and

α

-Sn, as wellas the diamond form of carbon. This structure belongs to the cubic class,with a face-centered cubic (FCC) lattice and a basis of two atoms at eachlattice point: one at the origin (0, 0, 0) and the other at a (1/4, 1/4, 1/4),where a is the lattice constant. Thus, the structure can be thought of as twointerpenetrating FCC sublattices, one displaced from the other by one quar-ter of the unit cell diagonal. The space group is .

Figure 2.1 shows the cubic unit cell of the diamond structure. The lengthof each side of the cubic unit cell is a, the lattice constant. The atoms aretetrahedrally bonded, and each atom in the structure is covalently bondedto its four nearest neighbors.

2.2.2 The Zinc Blende Structure

A number of semiconductors exhibit the zinc blende* (ZB) structure, includ-ing GaAs, InP, and other III-V semiconductors; CdTe, ZnSe, and other II-VIcrystals; and the cubic form of SiC. It is very similar to the diamond crystalstructure, except that the two FCC sublattices are made of two different typesof atoms. The space group is . The zinc blende structure is shownin Figure 2.2.

* The zinc blende structure is occasionally referred to as the sphalerite structure in the literature.

Fd m Oh3 7( )

F m Td43 2( )




9

Because of the two types of atoms, the zinc blende structure has a lowersymmetry than the diamond structure. This can lead to interesting phenom-ena in the heteroepitaxy of ZB materials on diamond substrates.

2.2.3 The Wurtzite Structure

The wurtzite* (Wz) structure is common among III-nitrides such as GaN andInN, and also some II-VI semiconductors. This structure comprises a hexag-onal close-packed (HCP) lattice with a basis of two atoms; it can therefore

FIGURE 2.1

The diamond crystal structure. All atoms are of the same type (e.g., Si).

FIGURE 2.2

The zinc blende crystal structure. The white and black atoms belong to the two differentsublattices (e.g., Ga and As).

* The wurtzite structure is occasionally called the zincite structure in the literature.

a

a

a

a

As

Ga



10


be considered two interpenetrating HCP lattices. Because the unit cell ishexagonal, there are two lattice constants, a and c (c is the lattice constantin the direction parallel to the axis of six-fold rotational symmetry, as shownin Figure 2.3). The two interpenetrating HCP lattices are made up of twodifferent types of atoms, offset along the c-axis by 5/8 of the cell height (5c/8). The space group is .

As with the zinc blende structure, the wurtzite crystal structure involvestwo types of atoms, A and B. Each atom A is bonded tetrahedrally to itsfour nearest neighbors, which are B. Because the nearest-neighbor configu-ration is the same as in the zinc blende structure, the properties of the twostructures are closely related if the second- and third-nearest-neighbor inter-actions are ignored.

2

2.2.4 Silicon Carbide

SiC is unique among semiconductors in that it exists in many polytypes,the number of which has been reported to be as high as 250.

3

Many ofthese polytypes are hexagonal or rhombohedral, but there exists a cubicpolytype as well. Each polytype is built up by stacking sheets of atoms.Each sheet can be represented as a close-packed two-dimensional arrange-

of SiC, each close-packed sphere represents a silicon atom together with acarbon atom.

The polytypes differ in structure only in the stacking sequence for thebilayers of silicon and carbon atoms. This can be understood by referring to

FIGURE 2.3

The wurtzite crystal structure.

Cd

Sa1

a2

a3

c

a

120°

P mc C v63 64( )



ment of spheres with six-fold symmetry, as shown in Figure 2.4. In the case


11

Figure 2.4. If the centers of the spheres in the first sheet (the basal plane fora hexagonal crystal) are at the positions marked A, the next sheet may beplaced with the centers of the spheres over the points marked B or C. Thevarious polytypes may be built up by stacking A-, B-, and C-type layers invarious sequences.

Regardless of the polytype, the atoms are tetrahedrally bonded with coor-dination number 4. That is, each silicon atom is bonded tetrahedrally withfour carbon atoms, with a Si–C bond length of approximately 1.89 Å. Eachcarbon atom is similarly bonded to its four silicon nearest neighbors. Thetetrahedra have three-fold symmetry, so for each stacking position A, B, orC there are two variants in which the tetrahedra are rotated by 180° withrespect to each other. The rotated variants are called A

′

, B

′

, and C

′

. Often theA, B, and C variants are referred to as untwinned, whereas the A

′

, B

′

, andC

′

variants are called twinned.

4

Considering the stacking of the A, A

′

, B, B

′

, C, and C

′

layers, only certainstacking sequences are allowed if corner sharing is to be maintained betweentetrahedra.

5

Thus, an untwinned bilayer must be stacked on either anuntwinned bilayer of the following letter (AB, BC, or CA) or a twinnedbilayer of the preceding letter (AC

′

, BA

′

, or CB

′

). Similarly, a twinned bilayermust be stacked on either an untwinned bilayer of the following letter (A

′

B,B

′

C, or C

′

A) or a twinned bilayer of the preceding letter (A

′

C

′

, B

′

A

′

, or C

′

B

′

).The resulting polytypes made by stacking bilayers in this way are cubic,

hexagonal, or rhombohedral in structure. For example, the zinc blende poly-type has the stacking sequence … ABCABC …. This polytype is referred toas 3C-SiC using the Ramsdell notation,

6

in which the C indicates a cubicstructure and the 3 indicates the periodicity of the stacking sequence. Alsoof technological importance are the 4H and 6H polytypes. These are bothhexagonal structures, with space group . The stacking sequencesare … ABA

′

CABA

′

C … and … ABCB

′

A

′

C

′

ABCB

′

A

′

C

′

… for the 4H and 6H

FIGURE 2.4

Close-packed spheres in two dimensions. The polytypes of SiC may be constructed by stack-ing such sheets of atoms, in which each sphere represents one silicon atom together with acarbon atom.

B

C

A

B

C

A

B

C

A

B

C

A

B

CA

B

CA

B

CA

B

CA

B

CA

B

CA

B

CA

B

CA

P mc C v63 64( )



12


structures, respectively. Two other notable polytypes are the 2H, which is awurtzite structure, and the 15R, which is rhombohedral.

2.2.5 Miller Indices in Cubic Crystals

In cubic crystals such as the diamond and zinc blende structures, crystalplanes and directions are denoted using Miller indices.

The Miller indices for a plane are obtained as follows. The intercepts ofthe plane with the three orthogonal axes

a

,

b

, and

c

are determined in termsof the lattice constant

a

; this yields three integers that may be positive ornegative. The three smallest integers having the same ratios as the reciprocalsof these intercepts are the Miller indices h, k, and l, and the plane is denoted(hkl). For example, consider the plane intercepting the

a

,

b

, and

c

axes at

∞

,

∞

,

a

. The normalized intercepts are

∞

,

∞

, 1. Taking the reciprocals, we have. These are integers so the plane is denoted (001). It is customary to

indicate negative indices with an over bar rather than a minus sign. Thus,the plane (0–11) would usually be denoted as . Families of planes havingthe same symmetry are denoted by curly brackets, as {hkl}. Thereforethe , , , , , and planes are collectively denotedas the {001} planes.

Directions in a crystal are denoted by the smallest set of integers that havethe same ratios as any vector in the direction. Thus, a direction is denoted[uvw] and a family of directions having the same symmetry is denoted <uvw>.

2.2.6 Miller–Bravais Indices in Hexagonal Crystals

It is possible to use Miller indices, as defined above, for hexagonal crystals.Unfortunately, due to the lack of cubic symmetry, it is no longer true thatthe direction [hkl] is perpendicular to the (hkl) plane in this case. Worse yet,planes with similar indices, such as (100) and (001), do not have the samepacking arrangement. For these reasons, the Miller–Bravais indices are usedalmost exclusively for hexagonal crystal structures.

The Miller–Bravais system uses four indices (hkil) to denote a plane. Here,the indices are obtained as in the cubic case, but the basal plane of thehexagonal unit cell is considered to have three axes, as shown in Figure 2.5.As a consequence, the index i is not independent but is given by

(2.1)

Despite the use of the superfluous index i, the Miller–Bravais system assignssimilar indices to similar types of planes, which has led to its nearly universaladoption.

0 0 1, ,

( )011

( )100 ( )100 ( )010 ( )010 ( )001 ( )001

i h k= − +( )




13

Referring to Figure 2.5, the c-axis (axis of six-fold symmetry) is denoted[0001]. The basal plane is the (0001) plane. The

a

1

,

a

2

, and

a

3

axes aredenoted , , and , respectively.

Names have been given to several of the low-index planes in hexagonalcrystals, as listed in Table 2.1. Examples of these planes are illustrated inFigure 2.6.

FIGURE 2.5

Axes for determination of the Miller–Bravais indices in a hexagonal unit cell.

TABLE 2.1

Planes of Hexagonal Semiconductor Crystals

NameExample Planes in

Miller–Bravais Notation

Basal plane

Prism plane: first order ,

Prism plane: second order ,

Pyramidal plane: first order ,

Pyramidal plane: second order ,

a1

a2

−a3

c

[ ]2110 [ ]1210 [ ]1120

( )0001

( )1100 ( )1100

( )1120 ( )2110

( )1011 ( )1011

( )1122 ( )1122



14


2.2.7 Orientation Effects

Because the semiconductor materials used in devices are single-crystal mate-rials, many of their properties and fabrication processes are different for thevarious crystal faces. Thus, the etching and cleaning, epitaxial growth, cleav-age, electronic properties, and defect structure are orientation dependent.Some of these orientation effects will be described here.

2.2.7.1 Diamond Semiconductors

A number of the orientation effects arising in a crystal may be understoodbased on the packing densities, interplanar spacings, or bond densities ofthe low-index planes. In diamond structure materials such as silicon, thea tom dens i t i e s on the pr inc ipa l p lanes a re in the ra t ios

. Because atoms in these planes have two,one, and one dangling bonds to the next plane, respectively, the interplanarbond ratios are .

The {111} planes generally exhibit the lowest growth rates and the lowestetch rates. This is because a {111} surface will have the lowest density ofdangling (interplanar) bonds (but a high density of intraplanar bonds). Thus,epitaxial growth under kinetically controlled conditions will tend to revealthe slow-growth {111} planes and cause faceting. Similarly, surface-sensitiveetches (crystallographic etches) will tend to delineate the {111} planes.

The natural cleavage planes of silicon are the {111} planes, due to theirweaker interplanar bonding. In the case of Si (001), the {111} cleavage planesmeet the surface at an angle of 54.7° along the directions. Therefore,

FIGURE 2.6

Planes of hexagonal semiconductor crystals.

a2

−a3a1

c

(101–0)first-order

prism plane

(101–0)first-order

pyramidal plane

(0001)basal plane

{ } : { } : { } : . : .100 110 111 1 1 414 1 155=

{ } : { } : { } : . : .100 110 111 1 0 707 0 577=

110




15

the crystal will naturally break apart into rectangular dice if scribed alongthe directions, which meet at 90° angles on the surface.

Cleavage of a Si (111) wafer is complicated by the fact that the three other{111} planes meet the surface along directions that are mutually at 60°to each other. Therefore, if rectangular dice are to be cut, only two sides ofthe rectangle may follow natural cleavage planes. The other two sides willhave jagged edges, the teeth of which will be made up of the natural cleavageplanes. The cleavage planes meet the (111) surface at an angle of 70.53°, asshown in Figure 2.7.

2.2.7.2 Zinc Blende Semiconductors

The orientation effects in crystals such as GaAs are different than those insilicon because of the polar nature of the zinc blende structure. Whereas thebonding in Si or Ge is entirely covalent, GaAs has a partly ionic characterdue to the different atoms on the two FCC sublattices. The Ga and As ionsin the crystal (cations and anions, respectively) take on net positive andnegative electrical charges, respectively. The other zinc blende crystals behavein similar fashion, but GaAs will be discussed for the purpose of specificity.

Owing to this polar nature, the {111} directions are not equivalent in azinc blende crystal such as GaAs. In the [111] direction, the crystal may bebuilt up by stacking alternating layers of Ga and As atoms. However, theselayers are not equally spaced but are stacked as … Ga-As—Ga-As—Ga-As…. Each Ga atom will be tetrahedrally bonded to three As atoms in thelayer directly below and to one As atom in the layer directly above. On theother hand, each As atom will be tetrahedrally bonded to one Ga atom inthe layer directly below and three Ga atoms above. This means that an Asatom on the (111) surface will have three dangling bonds, whereas a Gaatom on the (111) surface would have one dangling bond. For this reason,

FIGURE 2.7

Cleavage planes of (111) and (001) Si wafers.

(b)

Si (001)

(a)

Si (111)

70.53° 54.74°

Scribe lines

110

110



16


this surface will be made up entirely of Ga atoms and is called the (111)Gaface. By the same token, the face will comprise only As atoms and iscalled the (111)As face.

The (111)As face is electronically more active than the (111)Ga face. Thisis because on the (111)As face, pentavalent As atoms are bonded to three Gaatoms in the underlying layer, leaving two free electrons each. In the (111)Gaface, however, the trivalent Ga atoms each participate in bonding with threeAs atoms from the layer below, leaving no free electrons. Because of this,crystal growth and etching both occur rapidly on the (111)As face but slowlyon the (111)Ga face. Thus, the (111)As face can be polished to a mirror finishand allows smooth epitaxial layers with good crystal quality, whereas the(111)Ga face is difficult to polish and epitaxial layers on this orientation tendto have poor morphology.

GaAs does not cleave at {111} planes because of their coulombic attraction.(These planes are alternating layers of Ga and As atoms, with net positiveand negative charges, respectively.) Instead, GaAs cleaves on the neutral{110} planes. This is convenient because a GaAs (001) wafer may be cleavedinto rectangular dice with vertical sides. In this case, the edges of the cleaveddie will have (110), , , and sides. Their intersections with thesurface are along the , [110], , and [110] directions, which are mutu-ally at right angles. The cleavage behavior is useful in the packaging ofdevices and also makes it possible to cleave a laser diode cavity with per-fectly parallel end mirrors.

There are no remarkable differences in growth rates or etching rates amongthe low-index planes of GaAs or similar materials. There is, in fact, no goodselective (anisotropic) etch for GaAs. However, the epitaxial growth andetching rates are often in the order (011) > (111)As > (001) > (111)Ga.

2.2.7.3 Wurtzite Semiconductors

Like the zinc blende materials, wurtzite semiconductors such as GaN arepolar. As such, the [0001] and directions are not equivalent. In the[0001] direction, the crystal may be built up by stacking alternating close-packed layers of Ga and N atoms with unequal spacings, as … Ga-N—Ga-N—Ga-N …. In the [0001] direction, each Ga atom is bonded to three Natoms in the layer below and one N atom in the layer above, but each Natom is bonded to one Ga atom below and three Ga atoms above. Based onthis, a N atom on a (0001) surface will have three dangling bonds, but a Gaatom on the same surface would have one dangling bond. For this reason,the (0001) face is made up entirely of Ga atoms and is called the (0001)Gaface. Following the same arguments, the face is made up entirely ofN atoms and is called the (0001)N face.

The (0001)N face is more electronically active than the (0001)Ga face. Thisis because the pentavalent N atoms on the surface of the (0001)N face havethree electrons bonded to Ga atoms in the layer below, but the other two

( )111

( )110 ( )110 ( )110[ ]110 [ ]110

[ ]0001

( )0001




17

valence electrons are free. On the (0001)Ga face, all three valence electronsfrom each Ga atom are bonded with As atoms in the underlying layer, leavingzero free electrons.

The natural cleavage planes of wurtzite crystals like GaN are the first-order prism planes of type . In the case of GaN (0001), these cleavageplanes are perpendicular to the (c-face) surface and intersect the surfacealong directions, which are mutually at 60° angles.

Up to the present time, III-nitride devices have been fabricated exclusivelyon dissimilar substrates, so the cleavage behavior of these substrates is alsoimportant. The most commonly used substrate for III-nitride heteroepitaxyis sapphire (

α

-Al

2

O

3

). For sapphire, the natural cleavage planes areplanes (the so-called R-faces). For this reason, GaN (0001) is sometimesgrown heteroepitaxially on Al

2

O

3

, “a-face sapphire,” so that the nat-ural cleavage planes of the GaN and sapphire line up approximately.

7–9

2.2.7.4 Hexagonal Silicon Carbide

In 4H- and 6H-SiC, the {0001} planes are not equivalent. The (0001) facecontains only Si atoms, whereas the surface (sometimes called the(0001)C face) is made up entirely of C atoms. This leads to a number ofobservable differences in the chemical behavior of these faces. For example,rates of both oxidation10,11 and vapor phase epitaxial growth12–14 are faster onthe (0001)C face than on the (0001)Si face.

2.3 Lattice Constants and Thermal Expansion Coefficients

In an unstrained cubic crystal, a single lattice constant a defines the lengthof the sides of the cubic unit cell. For a hexagonal crystal, there are two latticeconstants, a and c. The former represents the distance from the six-foldrotation axis to a corner of the hexagonal base, and the latter represents theheight of the unit cell. It is important to know the lattice constants of thesubstrate as well as the epitaxial layer, because they determine the latticemismatch for heteroepitaxy. The lattice constants of elemental and binarysemiconductors may be determined by x-ray diffraction experiments, withparts per million accuracy.

Lattice constants increase with temperature above 300K due to normalthermal expansion. This can be an important effect in heteroepitaxy, whichmay take place at greatly elevated temperatures. This is especially true ifthe substrate and epitaxial layer have greatly different thermal expansioncharacteristics.

The linear thermal coefficient of expansion (TCE), α, is defined as

{ }1100

1120

{ }1102

( )1120

( )0001



18 Heteroepitaxy of Semiconductors

(2.2)

and has units of K–1. Typical values are 10–6 to 10–5 K–1, but the value itselfdepends on the temperature. Table 2.2 provides the lattice constants andTCEs for cubic semiconductor crystals.

The thermal coefficient of expansion is itself a function of temperature.Thus, the experimentally obtained thermal expansion characteristics areoften fit to a polynomial:

(2.3)

where is in percent, with respect to 300K, and T is the absolute tem-perature in Kelvin. Thus, at a temperature T, the relaxed lattice constant forthe crystal is given by

(2.4)

The constants A, B, C, and D for cubic crystals are provided in Table 2.3.For hexagonal crystals such as the III-nitrides, 4H- and 6H-SiC, and sap-

phire, the expansion coefficients are different for the a and c lattice constants.Usually both , the thermal expansion coefficient for the lattice constant aalong the direction, and , the thermal expansion coefficient for thelattice constant c along the [0001] direction, are reported. Occasionally, ,the thermal expansion coefficient along the direction, is also given.

Relatively little information has been published on the thermal expansionof the hexagonal crystals, and in some cases there are great disparitiesbetween the available data. For example, the value of for GaN hasbeen reported to be 3.2 × 10–6 K–1 by Maruska and Tietjen,23 2.8 × 10–6 K–1 byLeszczynski et al.,24 and 5.8 × 10–6 K–1 by Oshima et al.25 This may be due,at least in part, to the different methods of preparation for the crystalsexamined. Also, the lack of experimental data for some materials reflects thedifficulty in preparing bulk crystals for thermal expansion characterization.In light of these challenges, the values in Table 2.4 should be considered onlyas best estimates until more data become available.

The lattice constants of alloyed semiconductors such as SiGe alloys, ter-naries, and quaternaries are often estimated by linear interpolation (Vegard’slaw33). For example, the relaxed lattice constant of InxGa1–xAs may be esti-mated using

α = ∂∂

1a

aT

Δaa

A BT CT DT= + + +2 3

Δa a/

a T a KA BT CT DT

( ) ( )= + + + +⎡

⎣⎢

⎤

⎦⎥300 1

100

2 3

αa

[ ]112 0 αc

αm

[ ]101 0

αc K( )300



Properties of Semiconductors 19

(2.5)

where and are the relaxed lattice constants of InAs and GaAs,respectively. In some cases, bowing parameters must be applied to achievea satisfactory level of accuracy.

2.4 Elastic Properties

Heteroepitaxial semiconductors typically contain elastic strains, due to lat-tice mismatch and thermal expansion mismatch. These strains affect theproperties of semiconductor devices in diverse ways. For example, strain

TABLE 2.2

Lattice Constants and Thermal Expansion Coefficients for Cubic Semiconductor Crystals

a(300K)(Å)

α(300K)(10–6 K–1)

α(600K)(10–6 K–1)

α(1000K)(10–6 K–1)

C 3.5668415 1.016 2.8 4.4Si 5.4310817 2.616 3.7 4.4Ge 5.657618 5.7 6.7 7.6α-Sn 6.489419 4.7 — —SiC (3C) 4.359620,21 — — —BN 3.615 1.8 3.7 5.9BP 4.538* — — —BAs 4.777 — — —AlP 5.467 — — —AlAs 5.660 — — —AlSb 6.1357 4.4 — —GaP 5.4512 4.7 5.8 —GaAs 5.6534 5.7 6.7 —GaSb 6.0960 6.1 7.3 —InP 5.8690 4.75 — —InAs 6.0584 5.19 — —InSb 6.4794 5.0 6.1 —BeS 4.865 — — —BeSe 5.139 — — —BeTe 5.626 — — —ZnS 5.4105 7.1 8.6 10.5ZnSe 5.668722 7.1 10.1 —ZnTe 6.1041 8.8 10.0 —CdTe 6.481 5.0 5.4 —β-HgS 5.851 — — —HgSe 6.084 — — —HgTe 6.461 5.1 — —

* Low temperature.

a In Ga As xa x ax x InAs GaAs( ) ( )1 1− = + −

aInAs aGaAs




can change the band structure of a semiconductor, and the energy gap inparticular. Built-in strains can also promote the motion of dislocations duringthe operation of injection lasers, thus causing catastrophic device failure.This section will introduce the basic theory of how semiconductor crystalsrespond to stresses. Special emphasis will be given to tetragonal distortionand elastic strain energies in mismatched heteroepitaxial layers.

2.4.1 Hooke’s Law

Elastic strains in semiconductor crystals are in response to applied stresses.An arbitrary elastic strain may be specified by six quantities. If α, β, and γ

TABLE 2.3

Temperature Dependence of Thermal Expansion for Cubic Crystals

A B (10–4 K–1) C (10–7 K–2) D (10–10 K–3)

C –0.010 –0.591 3.32 –0.5544 (25–1650K)Si –0.071 1.887 1.934 –0.4544 (293–1600K)Ge –0.1533 4.636 2.169 –0.4562 (293–1200K)α-Sn –0.525 13.54 15.87 –2.896 (100–500K)BN –0.0013 –1.278 4.911 –0.8635 (293–1300K)AlSb –0.049 –2.997 22.43 –22.34 (40–350K)GaP –0.110 2.611 4.445 –2.023 (293–850K)GaAs –0.147 4.239 2.916 –0.936 (200–1000K)GaSb –0.138 3.051 66.02 –3.380 (100–800K)InSb –0.099 1.249 8.773 –5.260 (50–750K)ZnS –0.0863 –3.386 30.18 –29.21 (60–335K)ZnSe –0.170 4.419 5.309 –2.158 (293–800K)ZnTe –0.200 5.104 6.811 –3.104 (100–725K)CdTe –0.0980 1.624 7.176 –4.445 (100–700K)HgTe –0.504 9.772 42.66 –59.22 (50–300K)

Note: Δa/a = A + BT + CT2 + DT3, in percent, where T is the absolute temperature.

TABLE 2.4

Lattice Constants and Thermal Expansion Coefficients for Hexagonal Crystals

a(Å)

b(Å)

αa(300K)(10–6 K–1)

αc(300K)(10–6 K–1)

αa(600K)(10–6 K–1)

αc(600K)(10–6 K–1)

α-Al2O326 4.7592 12.9916 4.3 3.9 5.6 7.4

SiC (2H) 3.076 5.048 — — — —SiC (4H)27 3.0730 10.053 — — — —SiC (6H)20 3.0806 15.1173 — — — —AlN 3.11228 4.978 — — — —GaN29,30 3.1886(5) 5.1860(4) 3.1 2.8 4.7 4.2InN31,32 3.533 5.693 3.4 2.7 5.7 3.7ZnS 3.8140 6.2576 — — — —ZnTe 4.27 6.99 — — — —CdS 4.1348 6.7490 — — — —CdSe 4.299 7.010 — — — —CdTe 4.57 7.47 — — — —




are the angles between the a, b, and c axes in the unstrained crystal, thenone possible set of such quantities is Δα, Δβ, Δγ, Δa, Δb, and Δc. Because ofthe mathematical difficulties imposed by nonorthogonal axes, it is customaryinstead to use the six strains defined as follows.

Three orthogonal axes, f, g, and h, of unit length are chosen within theunstrained crystal with their origins fixed at a particular lattice point. Aftera small deformation of the crystal, the axes are distorted in length andorientation to f ′, g′, and h′ such that

(2.6)

The fractional changes in length of the f, g, and h axes are, to the first order,given by

(2.7)

The shear strains,* or those strains related to the changes in α, β, and γ,are to the first order:

(2.8)

Stresses are deformational forces applied to the crystal, per unit area. Wewill define the stress component as a force applied in the i direction to aplane with its normal in the j direction.

* In some references, the quantities, εxy , εyz , and εzx are referred to as engineering shear strains.They are approximately twice the simple shear strains, eyx, exy , ezy, eyz, ezx, and exz. Thus, εxy = eyx +exy ≈ 2exy , εyz = ezy + eyz ≈ 2eyz, and εzx = ezx + exz ≈ 2ezx. Engineering shear strains will be usedthroughout this book.

ε ij

′ = + + +f e f e g e hxx xy xz( )1

′ = + + +g e f e g e hxy yy yz( )1

′ = + + +h e f e g e hzx zy zz( )1

ε

ε

ε

xx xx

yy yy

zz zz

e

e

e

≈

≈

≈

ε

ε

ε

xy yx xy

yz zy yz

zx

f g e e

g h e e

h f

= ′ ⋅ ′ ≈ +

= ′ ⋅ ′ ≈ +

= ′ ⋅ ′ ≈≈ +e ezx xz

σ ij




2.4.1.1 Hooke’s Law for Isotropic Materials

Hooke’s law states that the strain components are linear combinations of thestress components. In an isotropic material, the physical properties are inde-pendent of direction. Therefore, Hooke’s law takes on a simple form involv-ing only two independent variables. In compliance form, Hooke’s law forthe isotropic medium is

(2.9)

where E is the Young’s modulus and is the Poisson ratio. These may betreated simply as material constants for our purposes; they are described inmore detail in Section 2.4.2.

The stresses may also be written as linear combinations of the strains. Instiffness form, Hooke’s law for an isotropic medium is

(2.10)

2.4.1.2 Cubic Crystals

Cubic crystals are anisotropic in their elastic properties. Nonetheless, it ispossible to greatly simplify Hooke’s law by considerations of cubic symme-

εε

εε

εε

xx

yy

zz

yz

zx

xy

E

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

−

1

1 νν νν νν ν

νν

−− −− −

++

0 0 01 0 0 0

1 0 0 00 0 0 2 2 0 00 0 0 0 2 2 00 0 00 0 0 2 2+

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥ν

σσ

σσ

σ

xx

yy

zz

yz

zx

σσxy

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

ν

σσ

σσ

σσ

ν

xx

yy

zz

yz

zx

xy

E

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

+1(( ) −( )

−−

−−1 2

1 0 0 01 0 0 0

1 0 0 00 0 0 1 2 0 0ν

ν ν νν ν νν ν ν

ν/00 0 0 0 1 2 00 0 0 0 0 1 2

//

−−

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

νν

εxxx

yy

zz

yz

zx

xy

ε

εε

εε

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥




try. If the x, y, and z axes coincide with the [100], [010], and [001] directionsin the cubic crystal, respectively, then Hooke’s law in compliance form maybe written as

(2.11)

or

(2.12)

where the are the elastic compliance constants and S is the compliancematrix. Only three independent constants are needed as a consequence ofthe cubic symmetry.

In stiffness form, Hooke’s law for a crystal with cubic symmetry is

(2.13)

or

(2.14)

where C is the compliance matrix and the are the elastic stiffness con-stants, in units of force per area. Here, too, it is assumed that the x, y, and zaxes coincide with the [100], [010], and [001] directions in the cubic crystal.The matrix equation above applies in the general case. The Poisson ratio andthe Young’s modulus may also be used in heteroepitaxy as long as theirdependence on the crystal direction is taken into account.

For cubic crystals, the compliance and stiffness constants are related by

εε

εε

εε

xx

yy

zz

yz

zx

xy

S S⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

11 112 12

12 11 12

12 12 11

44

0 0 00 0 00 0 0

0 0 0 0 00 0

S

S S S

S S S

S

00 0 00 0 0 0 0

44

44

S

S

xx

yy

z

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

σσ

σ zz

yz

zx

xy

σ

σσ

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

ε σ= S

sij

σσ

σσ

σσ

xx

yy

zz

yz

zx

xy

C C⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

11 112 12

12 11 12

12 12 11

44

0 0 00 0 00 0 0

0 0 0 0 00 0

C

C C C

C C C

C

00 0 00 0 0 0 0

44

44

C

C

xx

yy

z

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

εε

ε zz

yz

zx

xy

ε

εε

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

σ ε= C

Cij




(2.15)

Elastic constants of cubic crystals are often determined from acoustic mea-surements.34 In these experiments ultrasonic pulses are generated in thecrystal by a quartz transducer. The pulse traverses the crystal, is reflectedby the back face, and returns. From the time elapsed the velocity of propa-gation is determined. The measurement of three different wave modes allowscalculation of all three unique elastic constants for a cubic crystal.

Table 2.5 provides the elastic stiffness constants for a number of cubicsemiconductor crystals. Scarce elastic constant data are available in the lit-erature for ternary and quaternary alloy layers. For a lack of a betterapproach, linear interpolation (Vegard’s law) is often applied in these cases.However, there have been theoretical predictions of significant departuresfrom linearity in some cases, including In1–xGaxSb,35 Cd1–xZnxTe,36 andSi1–x–yGexCy.37 Experimental data also suggest significant departures fromlinearity in the dilute nitride semiconductor GaAs1–yNy.38

2.4.1.3 Hexagonal Crystals

For a crystal with hexagonal symmetry (wurtzite semiconductor or hexag-onal SiC), there are six distinct elastic stiffness constants, of which five areindependent. Assuming that the z-axis is aligned with the c-axis of thehexagonal unit cell, Hooke’s law can be written in stiffness form as

CS S

S S S S1111 12

11 12 11 122= +

− +( )( )

CS

S S S S1212

11 12 11 122= −

− +( )( )

CS44

44

1=

SC C

C C C C1111 12

11 12 11 122= +

− +( )( )

SC

C C C C1212

11 12 11 122= −

− +( )( )

SC44

44

1=

Cij




(2.16)

Elastic stiffness constants for hexagonal crystals may be determined fromacoustic measurements as in the case of cubic crystals. In some cases, theresonance method44,45 is used to determine the piezoelectric and elastic stiff-ness constants for piezoelectric hexagonal crystals such as 4H-SiC and 6H-SiC. In these experiments, only a subset of the elastic stiffness constants maybe determined, depending on the orientation of the piezoelectric transducer,which is cut from a single crystal of the material under test.

Table 2.6 through Table 2.9 provide the published elastic stiffness con-stants for some hexagonal semiconductors. It should be noted that only

TABLE 2.5

Elastic Stiffness Constants of Cubic Semiconductor Crystals at Room Temperature, in Units of GPa (1 GPa = 1010 dyn/cm2)

C11 C12 C44

C39 107.6 12.52(23) 57.74(14)Si40 160.1 57.8 80.0Ge 124.0 41.3 68.3α-Sn 69.0 29.3 36.2SiC (3C)41 352 120 232.9AlN (ZB)42 322 156 138AlP 132 63.0 61.5AlAs 125 53.4 54.2AlSb 87.69(20) 43.41(20) 40.76(8)GaN (ZB)42 325 142 147GaP43 140.50(28) 62.03(24) 70.33(7)GaAs 118.4(3) 53.7(16) 59.1(2)GaSb 88.50 40.40 43.30InP 102.2 57.6 46.0InAs 83.29 45.26 39.59InSb 65.92(5) 35.63(6) 29.96(3)ZnS 104.62(5) 65.33(6) 46.50(12)ZnSe 87.2(8) 52.4(8) 39.2(4)ZnTe 71.3 40.7 31.2CdTe 53.3 36.5 20.44β-HgS 81.3 62.2 26.4HgSe 69.0 51.9 23.3HgTe 53.61 36.60 21.23

σσ

σσ

σσ

xx

yy

zz

yz

zx

xy

C C⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

11 112 13

12 11 13

13 13 33

44

0 0 00 0 00 0 0

0 0 0 0 00 0

C

C C C

C C C

C

00 0 00 0 0 0 0

44

66

C

C

xx

yy

z

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

εε

ε zz

yz

zx

xy

ε

εε

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

Cij




TABLE 2.6

Elastic Stiffness Constants of 4H- and 6H-SiC at Room Temperature, in Units of GPa (1 GPa = 1010 dyn/cm2)

Elastic Constants4H-SiC

(Kamitani et al.47)6H-SiC

(Kamitani et al.48)

C11 507(4) 501(4)C12 111(5) 111(5)C13 — 52(9)C33 547(7) 553(4)C44 159(4) 163(4)C66 198 195

TABLE 2.7

Elastic Stiffness Constants of Wurtzite GaN at Room Temperature, in Units of GPa (1 GPa = 1010 dyn/cm2)

ElasticConstants

RecommendedValues

Polianet al.48

Degeret al.49

Deguchiet al.50

V. Yu Davydovet al.51

Savastenko and Shelag52

C11 353 390(15) 370 373 315 296C12 135 145(20) 145 141 118 120C13 104 106(20) 110 80.4 96 158C33 367 398(20) 390 387 324 267C44 91 105(10) 90 93.6 88 24C66 110 123(10) 112 118 99 88

TABLE 2.8

Elastic Stiffness Constants of Wurtzite AlN at Room Temperature, in Units of GPa(1 GPa = 1010 dyn/cm2)

ElasticConstants

RecommendedValues

Degeret al.53

V. Yu Davydovet al.53

McNeilet al.54

Tsubouchiet al.55

S. Yu Davydov et al.42

C11 397 410 419 411 345 369C12 145 140 177 149 125 145C13 113 100 140 99 120 120C33 392 390 392 389 395 395C44 118 120 110 125 125 96C66 128 135 121 131 131 112

TABLE 2.9

Elastic Stiffness Constants of Wurtzite InN at Room Temperature, in Units of GPa (1 GPa = 1010 dyn/cm2)

ElasticConstants

RecommendedValues

Sheleg andSavastenko56

Kimet al.57 Wright58

Marmalyuket al.59

Chisholmet al.60

C11 250 190 271 223 257 297.5C12 109 104 124 115 92 107.4C13 98 121 94 92 70 108.7C33 225 182 200 224 278 250.5C44 54 9.9 46 48 68 89.4C66 70 43 74 54 82 95




the first five of these constants are independent. is not always reportedbut may be calculated from46

(2.17)

2.4.2 The Elastic Moduli

Some elastic properties that are useful in heteroepitaxy are the Young’smodulus E, the biaxial modulus Y, the shear modulus G, the Poisson ratio ,and the biaxial relaxation constant . The Young’s modulus (also called themodulus of elasticity or the elastic modulus) is a measure of the stiffness ofa material. It is defined as the ratio of stress to strain:

(2.18)

Usually, this definition for the Young’s modulus is used with the assump-tion of a stress in one direction (uniaxial stress). For the case of biaxial stress,commonly encountered in mismatched heteroepitaxy, we use the biaxialmodulus, which is the ratio of the stress to strain for the biaxial case:

(2.19)

It should be noted, however, that the biaxial modulus is sometimesreferred to as the Young’s modulus in the literature. The shear modulus(also known as the rigidity modulus) is defined as the ratio of the shearstress to shear strain:

(2.20)

The Poisson ratio is defined as the ratio of the transverse contraction tothe longitudinal extension, for a uniaxial tensile stress in the longitudinaldirection. Thus,

(2.21)

Typical semiconductor crystals have a Poisson ratio of 1/3. The Poissonratio is nearly always positive, because the unit cell volume is approximately

C66

CC C

6611 12

2= −

νRB

Young’s modulus = =Estressstrain

Biaxial modulusbiaxial stres

= =Ystressstrain

ss

Shear modulus = =Gshear stressshear strain

Poisson ratio = = −ν transverse strainlongitudinaal strain

uniaxial stress




conserved in the strained crystal. The biaxial relaxation constant is analogousto the Poisson ratio, for the case of biaxial stress, so that

(2.22)

In the following sections, these elastic moduli will be related to the elasticstiffness constants, and their values will be tabulated for the crystals com-monly used in heteroepitaxy.

2.4.2.1 Cubic Crystals

For diamond and zinc blende crystals, the shear modulus is

(2.23)

If the growth plane is (001), the Young’s modulus is

(2.24)

and the Poisson ratio is

(2.25)

The biaxial modulus is given by

(2.26)

and the biaxial relaxation constant is

(2.27)

Elastic moduli for cubic semidonductors are given in Table 2.10.

2.4.2.2 Hexagonal Crystals61

For wurtzite crystals or hexagonal SiC, the shear modulus is anisotropic, butmay be derived from the tensor form of Hooke’s law. If the growth plane isassumed to be (0001), then the Young’s modulus is

Biaxial relaxation constant -= = −R in plane strB

aainout of plane strain- -

biaxial stress

G C C= −( )/11 12 2

EC C C C

C C( )

( )( )( )

001211 12 11 12

11 12

= + −+

ν( )001 12

11 12

=+

CC C

Y C CCC

E( )

( )001

2 001111 12

122

11

= + − =− ν

RCC

B( )0012 12

11

=




(2.28)

The Poisson ratio is

(2.29)

The biaxial modulus is

(2.30)

and the biaxial relaxation constant is given by

(2.31)

Values for III-nitrides and 6H-SiC are given in Table 2.11.

TABLE 2.10

Elastic Moduli of Cubic Semiconductor Crystals at 300K

G E(001) ν(001) Y(001) RB(001)

C 47 105 0.104 117 0.23Si 51 129 0.265 176 0.72Ge 41 103 0.25 138 0.67α-Sn 19.8 52 0.30 73 0.85SiC (3C) 116 290 0.25 390 0.68AlN (ZB) 83 220 0.33 330 0.97AlP 34 91 0.32 135 0.95AlAs 36 93 0.30 133 0.85AlSb 22 59 0.33 88 0.99GaN (ZB) 92 240 0.30 340 0.87GaP 39 102 0.31 148 0.88GaAs 32 85 0.31 124 0.91GaSb 24 63 0.31 92 0.91InP 22 61 0.36 95 1.13InAs 19.0 51 0.35 79 1.09InSb 15.1 41 0.35 63 1.08ZnS 19.6 54 0.38 88 1.25ZnSe 17.4 48 0.38 77 1.20ZnTe 15.3 42 0.36 66 1.14CdTe 8.4 24 0.41 40 1.37β-HgS 9.6 27 0.43 48 1.53HgSe 8.6 24 0.43 43 1.50HgTe 8.5 24 0.41 40 1.37

E CC

C C( )

( )0001

233

132

11 12

= −+

ν( )0001 13

11 12

=+

CC C

Y C CC

CE

( )00012

111 12132

33

= + − =− ν

RC

CB( )0001

2 13

33

=




2.4.3 Biaxial Stresses and Tetragonal Distortion

For heteroepitaxial growth, we usually assume the case of biaxial stress.Using a Cartesian coordinate system, if growth proceeds along the z directionand the growth plane is the x-y plane, then the in-plane stresses applied bythe substrate are equal:

(2.32)

Also, the out-of-plane stress is assumed to be zero:

(2.33)

(The substrate does not constrain the epitaxial layer in the growth direction.)The shear stresses are assumed to be zero for growth on a low-index planesuch as the (001) on a cubic crystal or the (0001) on a hexagonal crystal. Also,it is usually assumed that the substrate is unstrained, because under mostcircumstances the substrate will be many times thicker than the epitaxial layer.

The stress tensor in the epitaxial layer is therefore given by

(2.34)

In the case of a biaxial stress applied to a (001) cubic crystal, the unit cellof the epitaxial layer becomes tetragonal with an in-plane lattice constant aand an out-of-plane lattice constant c. In this situation, referred to as tetrag-onal distortion,

TABLE 2.11

Elastic Moduli of Hexagonal Semiconductor Crystals at 300K

E(0001) ν(0001) Y(0001) RB(0001)

6H-SiC 540 0.085 602 0.19GaN62 320 0.21 430 0.57AlN 340 0.21 480 0.58InN 171 0.27 270 0.87

σ σ σxx yy= = ||

σ σzz = =⊥ 0

Σ =

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

σ

σ||

||

0 0

0 0

0 0 0

ε|| = −a aa

0

0




(2.35)

where is the relaxed (unstrained) lattice constant for the epitaxial layer.The in-plane strain is related to the biaxial stress by

(2.36)

where the constant of proportionality is the biaxial modulus described inthe previous section. The in-plane and out-of-plane strains are related by

(2.37)

where is the biaxial relaxation constant. The strain tensor is therefore

(2.38)

The equations above may be applied to pseudomorphic or partially relaxedheteroepitaxial layers, regardless of the presence or absence of thermal strain.They are applicable to cubic or hexagonal crystals as long as the correctforms are used for the biaxial modulus and the biaxial relaxation constant.

2.4.4 Strain Energy

A load that produces a stress, acting on a crystal to deform it, does an amountof work per unit volume

(2.39)

Integrating the above expression, we can find the total strain energy perunit volume, which for the case of a cubic crystal is

(2.40)

ε⊥ = −c aa

0

0

a0

σ ε|| ||= Y

Y

ε ε⊥ = −RB ||

RB

Ε =

−

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

σ

σ

σ

||

||

||

/

/

/

Y

Y

R YB

0 0

0 0

0 0

δ σ δε σ δε σ δε σ δε σ δεU xx xx yy yy ZZ ZZ xy xy yz yz= + + + + + σσ δεzx zx

UC

Cxx yy zz xx yy yy zz z= + + + + +11 2 2 2122

( ) (ε ε ε ε ε ε ε ε zz xx

xy yz zxC

ε

ε ε ε

)

( )+ + +44 2 2 2

2




For a biaxially strained heteroepitaxial layer, in which the shear termsvanish,

(2.41)

Therefore, the strain energy per unit area is

(2.42)

where is the in-plane strain, Y is the biaxial modulus, and h is the layerthickness. Early calculations of the critical layer thickness for heteroepitaxyinvolved balancing this strain energy with the energy of a grid of strain-relieving misfit dislocations.

It can be shown that Equation 2.42 applies to hexagonal crystals in thecase of (0001) heteroepitaxy, provided that the appropriate value of thebiaxial modulus is used.

2.5 Surface Free Energy

The growth mode of a nucleating heteroepitaxial layer is determined in largepart by the properties of the surfaces and interfaces involved. The mostimportant physical property is the surface free energy, defined as the revers-ible work done to create new surface area.

The surface free energies have been determined experimentally for only afew semiconductor crystals. In cases for which such data are available, theexperimental errors are often quite large. These follow from the difficultiesinvolved in surface free energy determinations. Most such efforts haveinvolved the use of a fracture technique with natural cleavage planes. Inthese experiments, a precursor crack is introduced either by a steel wedge(double-cantilever beam method63) or by an explosive electrical spark dis-charge (electrical spark discharge method64). The crack is expanded by appli-cation of a tensile force, and the relevant surface energy is determined usingthe Griffith criterion for crack propagation.65 More recently, surface freeenergies have been determined for some semiconductors using the observedequilibrium shapes of facetted crystals (Bonzel method66). However, the

UC

C

C

= + + +

= +

⊥ ⊥11 2 2

122

211

22 2( ) ( )|| || ||

||

ε ε ε ε ε

ε CCCC

Y

12122

11

2

2−⎡

⎣⎢

⎤

⎦⎥

= ε||

E Yhε ε= ||2

ε||




accuracy of the values obtained by the Bonzel method depends on theassumed temperature dependence of the step free energy. Another approach,which is model independent, has recently been developed by Metois andMuller.67 This involves observation of the equilibrium shapes for three-dimensional crystals and two-dimensional islands as well as the statisticalanalysis of the thermal fluctuation for an isolated step. Data from these threeexperiments can be combined to find the surface free energies for the facesof the three-dimensional crystal.

Despite the progress with experimental methods, surface free energies aremost often estimated by theoretical calculations. The most common approachinvolves use of the bond-breaking model applicable to covalent crystals.68

In this model, the surface free energy is assumed to equal the bond strengthB times the areal density of broken bonds for the crystal face. Then indiamond and zinc blende crystals,

(2.43)

where γ(hkl) is the surface free energy of the (hkl) face, B is the bond energy,and a is the lattice constant. Surface free energies for the III-V semiconductorshave been calculated using this model, using the bond strength B determinedfrom the molar-atomic heat of sublimation, :

(2.44)

where is the Avogadro constant, 6.02 × 1023 mol–1.Surface energies of the low-index faces may also be estimated based on

knowledge of the lattice constant and elastic stiffness constants, which aremore readily available in the literature.45 Consider the strength of a cubicsemiconductor rod having its long axis aligned with the [001] crystallo-graphic direction. As the rod is stressed, elastic energy is stored until the rodbreaks, at which time the elastic energy is converted to surface energy bythe creation of two new (001) surfaces. The strength of the rod should notdepend on its length, so we will suppose that all of the elastic energy isstored between two atomic planes.

γ( )0012

2= Ba

γ( )0112

22= B

a

γ( )1112

32= B

a

ΔHS

BHN

S

a

= Δ2

Na




The restraining force per unit area is supposed to have the form shownin Figure 2.8, which may be approximated by a sinusoid,

(2.45)

where is the stress, x is the displacement, and K, x0, and W are constants.x0 is the relaxed (001) plane spacing, at which the restraining force is zero.For a zinc blende crystal, x0 = a/4. W is a measure of the range of interatomicforces. A rough estimate for W is between one and two times the interplanarspacing; the geometric mean will be used somewhat arbitrarily. Theconstant K may be found as follows. If E is the Young’s modulus for the[001] direction, then

(2.46)

Differentiating for small displacements , we find

(2.47)

and the restraining force per unit area is

FIGURE 2.8Restraining force per unit area σ vs. displacement x, approximated by a sinusoid.

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

Displacement x

σ/K

x0 x0 + W

σ

σ π= −⎡

⎣⎢

⎤

⎦⎥K

x xW

sin( )0

σ

a / 8

E xddx

= 0σ

( )x x≈ 0

KE= 2

π




(2.48)

Using this result, the (001) surface energy may be calculated as follows.The work done to break the rod is considered equal to the surface energy ofthe two surfaces created. Then

(2.49)

or

(2.50)

In terms of the elastic stiffness constants, the surface energy for the (001)plane of a cubic semiconductor crystal is given by

(2.51)

The surface energies of the other low-index faces may be estimated usingthis relationship and the bond-breaking model. Thus, for a cubic semicon-ductor crystal,

(2.52)

Table 2.12 summarizes values of the surface energies for low-index facesof cubic semiconductors. The values in parentheses were determinedexperimentally, and the values in square brackets were calculated usingthe heat of sublimation (Equation 2.44). All other values were calculatedusing the elastic stiffness constants and the lattice constant (Equations 2.51and 2.52).

σπ

π=

⎛

⎝⎜

⎞

⎠⎟

−( )⎡

⎣⎢⎢

⎤

⎦⎥⎥

E x x

a2 80sin

σ γdxx

x W

0

0

2 001+

∫ = ( )

γπ

( )0012 2= Ea

γπ

( )( )( )

( )001

22

11 12 11 122

11 12

= − ++

a C C C CC C

γ γ( )

( )011

001

2=

γ γ( )

( )111

001

3=




2.6 Dislocations

For partially relaxed layers that are greater than the critical layer thickness,misfit dislocations are produced at the interface to relieve some of the mis-match strain. Associated with these misfit dislocations are threading dislo-cations, which run through the thickness of the heteroepitaxial layer. Anunderstanding of dislocations and their origin is important for the applica-tion of heteroepitaxy, because these defects tend to degrade the performanceof devices.

Dislocations are linear defects, along which the interatomic bonding isdisturbed relative to the case of a perfect crystal. In the core of the dislocation,along its line, there are dangling bonds and large local strains that exceed thelimits of the continuum elasticity theory. Surrounding the core is a strainedregion, in which the interatomic bonds are distorted by small amounts.

Some of the basic features of dislocations may be illustrated using two-dimensional bubble rafts. Figure 2.9 is a photograph of such a bubble raft,which contains a single dislocation. The dislocation is a point defect in thetwo-dimensional lattice and results in an extra half-line of bubbles in thelower part of the raft. Matthews69 has used lattice-mismatched bubble raftsto create models of misfit dislocations.

TABLE 2.12

Surface Free Energies for the Low-Index Faces of Cubic Crystals at 300K, in erg/cm2

γ(001) γ(011) γ(111)

C 1900 1300 1100Si 3600 2500 (1900) 2100 (1140)Ge 3000 2100 1200α-Sn 1700 1200 1000AlP 2500 [3400] 1800 [2400] 1400 (2000)AlAs 2700 [2600] 1900 [1800] 1600 [1500]AlSb 1800 [1900] 1300 [1300] 1000 [1100]GaP 2800 2000 (1900) 1600GaAs 2400 1700 (860) 1400GaSb 2000 [1600] 1400 [1100] 1200 [910]InP 1800 [1900] 1300 [1300] 1000 [1100]InAs 1600 [1400] 1100 [1000] 900 [840]InSb 1300 [1100] 900 [750] 800 [600]ZnS 1500 1100 900ZnSe 1400 1000 800ZnTe 1300 900 800CdTe 450 320 260HgS 800 600 500HgSe 800 600 500HgTe 800 600 500




In a three-dimensional crystal lattice, the dislocation is a linear defect thatmay be associated with an extra half-plane of atoms. Individual dislocationsmay be imaged using high-resolution transmission electron microscopy(HRTEM) and fast Fourier transforms. Figure 2.10 shows a filtered HRTEMimage of a dislocation in heteroepitaxial AlN/6H-SiC (0001). Here, the misfitdislocation manifests as a linear defect in the plane of the interface. (The lineof the dislocation is into the page.) Associated with the dislocation is an extrahalf-plane of atoms in the SiC.

The overall structure of a dislocation is generally complex. However, adislocation can be understood to be a combination of the two basic types:screw and edge dislocations.

2.6.1 Screw Dislocations

A screw dislocation can be created in a regular crystal lattice by the appli-cation of a shear stress, as shown in Figure 2.11. Consider the plane ABCD,which is one of the regular planes of atoms in the crystal. Suppose a shear

FIGURE 2.9A dislocation in a bubble raft. The dislocation near the center of the photograph is associatedwith an extra half-line of bubbles in the lower portion of the image. It is most easily seen byviewing the page with a shallow angle. (Photo courtesy of the University of Cambridge,Cambridge, U.K. from the DoITPoMS Web site. With permission.154)

FIGURE 2.10Filtered HRTEM image of a misfit dislocation in heteroepitaxial Al/6H-SiC (0001). Associatedwith the dislocation is an extra half-plane of atoms in the 6H-SiC substrate. The -filteredimage was obtained by a fast Fourier transform of the HRTEM cross-sectional image. (Reprintedfrom Huang, X.R. et al., Phys. Rev. Lett., 95, 86101, 2005. With permission. Copyright 2005,American Physical Society.)

a/2

1120




stress is applied to this plane, as shown schematically by the forces in thediagram. If this stress is sufficiently large (beyond the elastic limit for thecrystal), it will cause the atoms on either side of the shear plane to bedisplaced by one atomic spacing. The line of the screw dislocation soformed is AD.

The arrangement of atoms around the screw dislocation forms a singlesurface helicoid, similar to a spiral staircase. Looking down the dislocationline AD, if the helix advances one plane for each clockwise rotation madearound it, the dislocation is a right-handed screw dislocation. If the dislocationhas the opposite sense, it is called a left-handed screw dislocation. The dislo-cation shown is therefore right-handed.

2.6.2 Edge Dislocations

An edge dislocation involves the inclusion of an extra half-plane of atomsABCD in an otherwise perfect crystal, as shown in Figure 2.12. Here the lineof the dislocation AD is the edge of the extra half-plane. Such a dislocationcould be created by the application of a shear stress to the plane EFGH asshown in the diagram. The edge dislocation shown is called a positive edgedislocation and is represented by the symbol because the extra half-planeof atoms has been added above the line AD. In a negative edge dislocation,the extra half-plane would exist below the line AD.

2.6.3 Slip Systems

The geometry of a crystal dislocation is specified by its line vector, Burgersvector, and glide plane. The line vector l is in the direction of the line of thedislocation. It need not be a unit vector, and it is usually expressed as a basiclattice translation or combination of lattice translations. The Burgers vectormay be determined by consideration of a Burgers circuit. A Burgers circuit

FIGURE 2.11Screw dislocation. (Adapted from Ghandhi, S.K., VLSI Fabrication Principles, 2nd ed., John Wiley& Sons, New York, 1994. With permission.)

A B B′

CC′DE

F

⊥




is any atom-to-atom path that forms a closed loop around the dislocationcore. For example, the path MNOPQ shown in Figure 2.13a is a Burgerscircuit around an edge dislocation. (The line of the dislocation is into theplane of the page.) Suppose the same sequence of atom-to-atom jumps ismade in a perfect crystal, as shown in Figure 2.13b. The failure of the Burgerscircuit to close upon itself in the perfect crystal shows the presence of thedislocation, and the closure failure is the Burgers vector:

(2.53)

The character of a dislocation can be specified by the angle between theBurgers vector and the line vector. For an edge dislocation such as the one

FIGURE 2.12Edge dislocation. (Adapted from Ghandhi, S.K., VLSI Fabrication Principles, 2nd ed., John Wiley& Sons, New York, 1994. With permission.)

FIGURE 2.13The Burgers circuit. (a) The Burgers circuit MNOPM starts and ends on the same point M andencloses a positive edge dislocation with its line into the plane of the paper. (b) In the perfectcrystal, the same circuit starting at point M, but failing to close, instead ending on the point Q.The closure failure QM is the Burgers vector.

A

B

C

D

E

F

H

G

b = QM� ��

O

P

N

M

O

P M

N

Q

Burgers vector

(a) (b)




shown in Figure 2.13, the Burgers vector is always perpendicular to the linevector. Therefore, edge dislocations are sometimes referred to as 90° disloca-tions. For a screw dislocation, the line vector and Burgers vector are parallel,resulting in the terminology 0° dislocation. Although pure edge and screwdislocations are encountered in real crystals, dislocations of mixed characterare far more common. For example, 60° dislocations are often observed indiamond and zinc blende crystals. The 60° dislocation exhibits a 60° anglebetween the Burgers vector and the line vector. Its nature and core structurecan therefore be considered part edge and part screw.

The Burgers vector is conserved for any dislocation passing through acrystal. Real dislocations are seldom perfectly straight, but tend to followpaths with sometimes jagged changes in direction. Nonetheless, any Burg-ers circuit enclosing the dislocation will reveal the unique Burgers vector.The interesting implication is that any dislocation that changes directionchanges character (the angle between the Burgers vector and the line vectorchanges along the dislocation). Therefore, a dislocation with screw charac-ter along part of its line may have 60° character or edge character elsewherealong its path.

For the determination of the Burgers vector, a clockwise path is takenaround the Burgers circuit when looking down the line of the dislocation (inthe direction of l). The Burgers vector is taken to run from the finish to thestart of the Burgers circuit. This is the so-called right-hand/finish–start (RH/FS) convention. Note that the direction for the line vector can be arbitrarilyassigned one of two ways. However, reversing the line vector also reversesthe Burgers vector and preserves the angle between them.

The Burgers vector shows the direction and amount of slip associated withthe crystal distortion that created the dislocation. Further distortion of thecrystal in response to applied stresses may cause the dislocation to move bya mechanism called slip.* The slip direction is the same as the Burgers vector.Moreover, the slip plane is the plane containing the Burgers vector and theline vector.†

For a perfect, or unit, dislocation, the Burgers vector is a lattice translationvector. That is, the Burgers vector connects two lattice points in the perfectcrystal. A perfect dislocation may dissociate into two partial dislocations,but the Burgers vector is conserved in the process. Thus, if a perfect dislo-cation with Burgers vector b1 dissociates into partial dislocations with Burg-ers vectors b2 and b3, then

b1 = b2 + b3 (2.54)

* Usually, motion of a single dislocation in this way is called glide, and the term slip is usuallyused to mean the glide of many dislocations.† For screw dislocations, the Burgers vector and line vector are parallel and share infinitely manyplanes. But real dislocations follow curved or jagged paths and change character along theirpath, eliminating this apparent ambiguity.




Reactions between two or more dislocations are possible as long as thetotal Burgers vector is conserved. An interesting special case is the reactionbetween two dislocations with opposite Burgers vectors, which results inthe annihilation of both dislocations.

2.6.4 Dislocations in Diamond and Zinc Blende Crystals

The slip planes in a crystal are usually the planes with the highest densityof atoms (the close-packed planes) because these have the greatest separa-tion. In diamond and zinc blende semiconductors, the usual glide planes arethe {111} planes. The direction of slip usually corresponds to the shortestlattice translation vector. Typically, slip directions (Burgers vectors) in the

cubic semiconductors are of the type .

Cubic semiconductor crystals have four {111} planes with three direc-tions in each. Therefore, there are 12 distinct slip systems in a diamond orzinc blende crystal. Table 2.13 enumerates the 12 slip systems for a cubicsemiconductor. A subset of these slip systems may be active during mis-matched heteroepitaxy, depending on the crystal orientation. For example,eight of these are active for (001) heteroepitaxy.

The line vectors for dislocation cubic semiconductors are typically of the

type . Therefore, dislocations on the 12 slip systems will be pure edge,pure screw, or 60° dislocations. All three types have been observed in hetero-epitaxial zinc blende semiconductors, but 60° dislocations are the most prev-alent. For the purpose of compact notation, the slip system with the Burgers

vector and the (111) glide plane would be called the slip

system. This class of slip systems would be referred to collectively

as .

Other types of slip systems have rarely been observed in zinc blende

semiconductors. V-shaped dislocations on slip systems have

been found in heteroepitaxial layers. Chu and Nakahara70 observed disloca-

tions on an slip system in InGaAsP/InP (001). Cooman and

Carter71 reported dislocations on {100} slip planes in GaAs. In degraded zincblende laser diode structures, numerous workers have identified dislocations

on and slip systems, which appear to be associated

with the dark line defects (DLDs).Perfect dislocations in diamond or zinc blende semiconductors belong to

either the glide or shuffle set. Consider the stacking sequence of (111) planesin either type of crystal. The stacking sequence is given by … AaBbCcAaBbCc

a2

011

110

011

a2

101[ ]a2

101 111[ ]( )

a2

110 111{ }

a2

110 011{ }

a2

100 100{ }

a2

100 100{ }a2

100 100{ }




…, as shown in Figure 2.14. A 60° dislocation of the shuffle set can beimagined as being constructed by making a cut at a shuffle plane, betweenplanes of the same letter, followed by the insertion of an extra half-plane.A dislocation of the glide set can be constructed by a similar operation, withthe cut made between different letter planes. Both types of dislocations areglissile. However, dislocations of the shuffle set have a line of interstitialsor vacancies adjacent to their core. Movement of the row of point defectscan occur only by shuffling, which greatly reduces the mobility of disloca-tions from the shuffle set. Following common practice in the field of hete-roepitaxy, it will be assumed in this book that all dislocations are from theglide set.

In a zinc blende semiconductor, 60° dislocations of the glide set may befurther classified as α and β dislocations according to the chemical makeupof their cores.72,73 In the zinc blende semiconductor AB, the α dislocationshave all A atoms at the core, whereas β dislocations have all B atoms at theircores. α and β dislocations can be expected to behave differently due to theirdifferent core structures. Differences in mobility have been demonstrated forthe two types of dislocations,74 and differences in their dissociation to partialdislocations have also been shown.75 These differences can be expected toaffect the dynamics of lattice relaxation in mismatched heteroepitaxial layers.

TABLE 2.13

Slip Systems in Diamond and Zinc Blende Crystals

Glide Plane Burgers Vector

( )111 a2

101[ ]

( )111 a2

011[ ]

( )111 a2

110[ ]

( )111a2

101[ ]

( )111a2

011[ ]

( )111a2

110[ ]

( )111a2

101[ ]

( )111a2

011[ ]

( )111a2

110[ ]

( )111a2

101[ ]

( )111a2

011[ ]

( )111a2

110[ ]




Note that in elemental semiconductors such as Si or Ge, the two sublatticeshave the same type of atoms, eliminating the distinction between α and βdislocations. The same also goes for SiGe alloys, in which the occupation ofatomic sites is random and not ordered.

2.6.4.1 Threading Dislocations in Diamond and Zinc Blende Crystals

Threading dislocations of the edge, screw, and 60° types are present in bulkdiamond and zinc blende crystals due to thermal and mechanical stressesacting on the crystal boules during growth or cooling. Some of these thread-ing dislocations will intersect the surfaces or wafers cut from the crystalboules. A heteroepitaxial layer grown on such a wafer will typically inheritthe threading dislocations from the substrate, which then propagate throughthe heteroepitaxial layer to a free surface.

The one-to-one relation between substrate and epitaxial layer dislocationshas been established by a TEM study of epitaxial GaAs.76 This study alsoshowed that threading dislocations may cause one-to-n multiplication,whereby n threading dislocations propagate in the epitaxial layer. The impor-tance of dislocation multiplication is also demonstrated by the observationin a number of mismatched heteroepitaxial systems that the epitaxial layershave dislocation densities orders of magnitude higher than the substrates.

The threading dislocation densities in semiconductor wafers vary greatlywith the type of material. Three-hundred-millimeter silicon wafers77 arevirtually dislocation free, with threading dislocation densities of <10 cm–2

(inferred from etch pit densities (EPDs)). For InP wafers,78 the dislocationdensities vary greatly depending on the wafer diameter and doping. Sev-enty-five-millimeter iron-doped semi-insulating InP wafers exhibit EPDs ofabout 105 cm–2, whereas 50-mm zinc-doped p-type wafers have EPDs of

FIGURE 2.14The stacking sequence for (111) planes in a diamond or zinc blende crystal (projection).(Reprinted from Hull, D. and Bacon, D.J., Eds., Introduction to Dislocations, 4th ed., Elsevier,Amsterdam, p. 123. Copyright 2001, Elsevier.)

Ba

Ac

Cb

Ba

Ac

(111) ‘Shuffle’

‘Glide’

( )011




<100 cm–2. In the case of GaAs, 150-mm wafers83 grown by the high-pressureliquid-encapsulated Czochralski process are available with dislocation den-sities of <104 cm–2. II-VI semiconductor substrates such as CdTe and ZnSeare available only in small sizes and tend to exhibit relatively high threadingdislocation densities as a consequence of their low values for the criticalresolved shear stress. In the case of bulk CdTe, the high density of disloca-tions tends to arrange in a subgrain structure.79 However, bulk CdTe hasbeen grown with EPDs of <105 cm–2 by a vapor growth process.80

2.6.4.2 Misfit Dislocations in Diamond and Zinc Blende Crystals

Misfit dislocations form at (or near) the interface to relieve strain in amismatched heteroepitaxial layer, once this layer exceeds the critical layerthickness. In the case of (001) heteroepitaxy of cubic semiconductors, thesemisfit dislocations form along the two orthogonal directions in theplane of the interface.81 Figure 2.15 shows a regular grid of misfit disloca-tions aligned with the directions in a 20-nm-thick layer of In0.2Ga0.8As/GaAs (001).

FIGURE 2.15Plan view TEM micrograph showing a rectangular array of misfit dislocations aligned withthe directions at the interface of a 20-nm-thick layer of In0.2Ga0.8As/GaAs (001). (Reprintedfrom Dixon, R.H. and Goodhew, P.J., J. Appl. Phys., 68, 3163, 1990. With permission. Copyright1990, American Institute of Physics.)

110

110

g220

0.5 μm

A

110




For heteroepitaxial layers with moderate mismatch , most ofthese misfit dislocations have 60° character, with Burgers vectors of the

type . This is because any pure edge dislocation in the interface will

have its Burgers vector (slip direction) in the interface as well. Such an edgedislocation is unable to glide into the interface.

During the early stages of relaxation, misfit dislocations are not created inequal numbers along the two directions,82,83 possibly due to differencesin mobility between the α and β dislocations. This is shown by the series ofmicrographs in Figure 2.16 for heteroepitaxial In0.25Ga0.75As/GaAs (001). Thethicknesses of the layers in (a) to (d) are 20, 30, 40, and 60 nm, respectively.The average linear misfit dislocation density increases with thickness from0.12 × 105 dislocations/cm for the 20-nm sample to 1.7 × 105 dislocations/cm for the 60-nm sample. However, the linear densities of misfit dislocationsare different in the two orthogonal directions, and this is most pro-nounced for the 30-nm sample of Figure 2.16b.

In highly mismatched heteroepitaxial layers, the misfit dislocation struc-ture at the interface is much less regular.88 This is shown by the series ofmicrographs in Figure 2.17 for InxGa1–xAs/GaAs (001) samples of varyingcomposition, and therefore mismatch. In the sample of Figure 2.17a, with

and , the misfit dislocations are predominantly straightand aligned with the directions. With increasing mismatch, however,the misfit dislocation structure becomes increasingly irregular. The misfitdislocations in the sample shown in Figure 2.17c with andexhibit a puzzle-piece structure characteristic of dislocations introduced fol-lowing island growth of the highly mismatched heteroepitaxial layer.

Misfit dislocations with pure edge character can be created by the reactionof two 60° dislocations at the interface of a heteroepitaxial zinc blende semi-conductor. For example, consider two 60° misfit dislocations, both with line

vector [110], and with the Burgers vectors and . They may

combine to form a single edge dislocation by the reaction

(2.55)

Such a reaction is energetically favorable, as can be shown by Frank’s rule.The resulting edge dislocation will relax as much mismatch strain as the two60° dislocations, without introducing the tilt components of the Burgersvectors. The edge dislocations observed by TEM in moderately mismatchedheteroepitaxial zinc blende semiconductors are believed to have formed bysuch a reaction mechanism. On the other hand, a screw dislocation is notexpected to form by such a mechanism because it would have no mismatch-relieving component.

(| | %)f < 1

a2

101

110

110

x = 0 25. f = −1 7. %110

x = 0 40. f = −2 7. %

a2

101[ ]a2

011[ ]

a a a2

1012

0112

110[ ] [ ] [ ]+ →




FIGURE 2.16TEM micrographs showing the misfit dislocation structures in heteroepitaxial In0.25Ga0.75As/GaAs (001) layers of increasing thickness.88 The mismatch is f = –1.3% and the critical layerthickness is hc = 6 nm. The actual thickness and linear density of misfit dislocations are (a) t =20 nm and dislocations/cm, (b) t = 30 nm and dislocations/cm, (c) t= 40 nm and dislocations/cm, and (d) t = 60 nm and dislocations/cm.(Reprinted from Breen, K.R. et al., J. Vac. Sci. Technol. B, 7, 758, 1989. With permission. Copyright1989, American Institute of Physics.)

(a) (b)

(c) (d)

022022

022

022

0.1 μm

ρ = ×0 12 105. ρ = ×0 9 105.ρ = ×1 4 105. ρ = ×1 7 105.




FIGURE 2.17TEM micrographs showing the misfitdislocation structures in heteroepitaxialInxGa1–xAs/GaAs (001) layers of vary-ing composition and mismatch.88 Thecomposition, mismatch, and thicknessare (a) x = 0.25, f = –1.7%, and t = 60 nm;(b) x = 0.30, f = –2%, and t = 30 nm; and(c) x = 0.40, f = –2.7%, and t = 20 nm.(Reprinted from Breen, K.R. et al., J. Vac.Sci. Technol. B, 7, 758, 1989. With permis-sion. Copyright 1989, American Insti-tute of Physics.)

(a)

002

(b)

004

0.1 µm

004

(c)




2.6.5 Dislocations in Wurtzite Crystals

In wurtzite semiconductors such as GaN, the common slip directions (Burg-

ers vectors) are of the type . The (0001) basal plane has the highest

density of atoms, and so slip often occurs in this plane. This is called basal

slip. There is one basal plane with three slip directions, resulting in

three basal slip systems.Nonbasal slip is also possible and occurs on the first-order prism planes,

of type , and on the first-order pyramidal planes, of type . Non-basal slip systems can be important in the lattice relaxation process for (0001)heteroepitaxy of wurtzite semiconductors. This is because the slip planesintersect both the surface and the interface with the substrate, allowingdislocations to glide into the interface to relieve lattice mismatch strain.

2.6.5.1 Threading Dislocations in Wurtzite Crystals

Heteroepitaxial III-nitrides are typically grown on c-plane (0001) sapphire(α-Al2O3) substrates or c-plane (0001) 6H-SiC substrates. Low-temperaturebuffer layers of GaN84 or AlN85 are grown on the substrate prior to thedeposition of device-quality layers. In either case, the initial growth modeis three-dimensional and a continuous heteroepitaxial film forms by thecoalescence of islands. The structural evolution of such layers is rathercomplex, but some studies have indicated that the as-grown buffer (nucle-ation) layers are amorphous and crystallize by solid phase epitaxy duringa subsequent heat treatment. In one study of low-temperature GaN on c-plane sapphire it was found that the initial islands are zinc blende GaN,86

even though the final layer exhibits the wurtzite structure. In either case, itappears that the misfit and threading dislocations are introduced predom-inantly by injection at the edges of the islands formed during recrystalliza-tion, rather than by the bending over and multiplication of substratethreading dislocations, as in the heteroepitaxy of zinc blende crystals withmoderate lattice mismatch.

The most common threading dislocations in these layers are pure edgedislocations, with [0001] line vectors and Burgers vectors of the type

. The (nonbasal) slip for this system occurs on the first-order prism

planes of the type .87 Threading dislocations with screw character arealso common, with line vectors and Burgers vectors of [0001] and c[0001],respectively. Because the screw dislocations have a Burgers vector that isperpendicular to the interface, they are not associated with misfit dislocationsin the c-plane interface. Instead, they may be introduced to relax the mis-match strain at the steps of vicinal (tilted) substrates.88 It has been reportedthat the screw dislocations are suppressed by the use of a low-temperature

a3

1120

a3

1120

{ }1100 { }1011

a3

1120

{ }1100




AlN buffer layer on c-plane sapphire. Mixed dislocations with [0001] line

vectors and Burgers vectors of the type have also been reported.98

Heteroepitaxial layers of GaN grown on c-face SiC or sapphire substratestypically contain a tangle of threading dislocations in the first 0.5 μm ofthickness. Above this there are relatively straight threading dislocations,aligned with the direction, with a relatively constant density of 108–109

cm–2. This general behavior is illustrated in Figure 2.18, which shows thread-ing dislocations in a 7-μm-thick layer of GaN grown on 6H-SiC (0001) withan AlN buffer by MOVPE.98 Similar behavior is also observed in the case ofGaN on sapphire (0001), as shown in Figure 2.19 for a 1.2-μm-thick layergrown by MOVPE.

Hollow-core threading dislocations, called nanopipes, are sometimesobserved in thick GaN layers grown on sapphire by hydride vapor phaseepitaxy (HVPE)89 and have also been found in MOVPE-grown GaN/sap-phire (0001).90 These nanopipes have screw character and are fundamentallysimilar to the micropipes common in SiC. Compared to the micropipes inSiC, the GaN defects are found to have much smaller diameters (3.5 to 50nm), as their name suggests.

2.6.5.2 Misfit Dislocations in Wurtzite Crystals

In the case of (0001) heteroepitaxy of III-nitrides, misfit dislocations areintroduced at the interface along directions.91 These misfit disloca-tions usually have edge character and represent the terminations of

FIGURE 2.18Cross-sectional TEM micrograph of GaN/AlN/6H-SiC (0001) showing the structure of thethreading dislocations. (Reprinted from Chien, F.R. et al., Appl. Phys. Lett., 68, 2678, 1996. Withpermission. Copyright 1996, American Institute of Physics.)

2 μm

AlNSiC

GaN

112–0

a3

1123

0001

1120{ }1120




planes. There are three equivalent directions in the basal plane, andthe misfit dislocations would meet each other at 60° angles if they formed aregular, triangular array.

Plan view TEM micrographs show that, in the case of (almost completelyrelaxed) AlN/sapphire (0001), with f = –13%, the misfit dislocations areevenly spaced along a direction with a spacing of 2.0 nm.103 However,plan view TEM micrographs of GaN/sapphire (0001), with f = –17%, revealan irregular, puzzle-piece structure,92 as shown in Figure 2.20. This isexpected for any highly mismatched heteroepitaxial layer with a three-dimensional growth mode. It is possible that the misfit dislocations at a low-mismatch interface between III-nitrides will assume a regular, triangularpattern, but such results have not yet been reported.

FIGURE 2.19Cross-sectional TEM micrograph of GaN/α-Al2O3 (0001) showing the structure of the threadingdislocations. (Reprinted from Kapolnek, D. et al., Appl. Phys. Lett., 67, 1541, 1995. With permis-sion. Copyright 1995, American Institute of Physics.)

FIGURE 2.20Plan view TEM micrographs showing the misfit dislocations in 0.6-μm-thick GaN/sapphire(0001) grown by MOVPE. (a) V/III ratio = 2100 and dislocation density = 2.4 × 109 cm–2; (b) V/III ratio = 2600 and dislocation density = 3.6 × 109 cm–2. (Reprinted from Schenk, H.P.D. et al.,J. Cryst. Growth, 258, 232, 2003. With permission. Copyright 2003, Elsevier.)

GaN

001 0.5 μmAI2O3

1120

1100

200 nm

(a) (b)




2.6.6 Dislocations in Hexagonal SiC

Micropipes93 are defects unique to SiC that degrade the performance ofdevices. These defects are actually the hollow cores of axial screw disloca-tions with [0001] line vectors. The Burgers vectors of these hollow-coredislocations are multiples of the c lattice parameter.94 In both the 4H and 6Hpolytypes, screw dislocations with Burgers vectors of c[0001] or c[0002] haveclosed cores. Super screw dislocations with Burgers vectors of c[0004] orgreater invariably have hollow cores, or micropipes, associated with them.The 3c screw dislocations, with Burgers vectors of c[0003], have beenobserved with hollow and closed cores. The dissociation of super screwdislocations into multiple screw dislocations can therefore eliminatemicropipes if the resulting dislocations have closed cores.95,96 This process iscalled micropipe sealing.

As with wurtzite semiconductors, we would also expect dislocations with

Burgers vectors of the type ; indeed, pure edge dislocations of this

type have been observed in bulk 4H-SiC crystals,97 with line vectors along[0001]. Molten KOH etching studies have revealed that these threading edgedefects sometimes exist in linear arrays, which constitute low-angle grainboundaries98,99 (also called domain walls).

2.6.6.1 Threading Dislocations in Hexagonal SiC

SiC device layers are often grown homoepitaxially on commercially available4H- and 6H-SiC wafers. These substrates contain 103 to 104 cm–2 screw dis-locations running along the [0001] and 104 to 105 cm–2 edge dislocations,100,101

both of which run parallel to the [0001]. As a consequence, they emerge atthe surface and will replicate in epitaxial layers. Basal plane dislocations(screw dislocations lying within the (0001) plane) are also present and emergeat the top surface in vicinal substrates that are typically miscut by 8° fromthe (0001). In addition, wafers of either polytype typically contain 10 to100 cm–2 micropipes that thread to the surface. Recently, 4H-SiC wafers with“ultra-low micropipe densities” of less than 5 cm–2 have become available.102

2.6.7 Strain Fields and Line Energies of Dislocations

A dislocation line is surrounded by a strain field, which raises the energy ofthe crystal and also interacts with externally applied stresses. The elasticstrain energy is the primary contribution to the dislocation line energy. Strainfield interactions give rise to dislocation motion in stressed crystals and alsocause pairs of dislocations to repel or attract.

In this section the strain fields will be given for pure screw and pure edgedislocations, with the simplifying assumption of an isotropic crystal. Usingthese, the line energies will be derived for the basic types of dislocations byintegration of the elastic strain energy. Then, the total line energy will be

a3

1120




estimated by assigning a core energy term to the elastic line energy. Finally,the line energy of a mixed dislocation will be calculated using the superpo-sition principle.

2.6.7.1 Screw Dislocation

Consider a straight screw dislocation lying along the z-axis in a right-handedCartesian coordinate system with l = [001] and b = [00b], as shown in Figure2.21. In the case of an isotropic crystal, the strain field surrounding the screwdislocation is given by103

(2.56)

Thus, only shear strains are associated with the screw dislocation. Trans-forming to cylindrical coordinates, we have

FIGURE 2.21Model of screw dislocation for calculation of the strain energy.

x

y

z

R

r

r0

θ

ε ε ε ε εxx yy zz xy yx= = = = = 0

ε επxz zx

byx y

= = −+4 2 2( )

ε επyz zy

bxx y

= =+4 2 2( )




(2.57)

and from Hooke’s law,

(2.58)

The bulk elastic strain energy per unit length of dislocation may be calcu-lated by integrating Equation 2.58 over the plane perpendicular to the dis-location. Note that the shear strains are proportional to and approachinfinity as . However, the strains exceed the limits of linear elasticitytheory (Hooke’s law) in the core of the dislocation. To avoid this difficulty,we set the lower limit of integration at , which is the radius of the dislo-cation core. Then, for the screw dislocation in an isotropic crystal,

(2.59)

Calculation of the total energy per unit length of dislocation requiresadding a core energy, which in general will include non-Hookian elasticenergies as well as the energy of dangling bonds. Two approaches to includ-ing the core energy are as follows: (1) the core energy term may be accountedfor by adjusting the cutoff parameter to some value much less than b, thelength of the Burgers vector,104 or (2) a value of b is assigned somewhatarbitrarily to the cutoff parameter , and a core parameter is added to thelogarithm in Equation 2.59.105 The latter approach is often used in semicon-ductor work and will be adopted here as well.

Estimates of the core energy are necessarily very approximate because thecore structure is complex and poorly understood. A discrete elasticity theoryhas been applied to screw dislocations in alkali–halide crystals,106 and theextension of this analysis yields an estimate of 1.4 for the core parameter indiamond-type semiconductors.107 Other estimates of the core parameter forscrew and edge dislocations range from 1 to 2 for cubic semiconductors104,108

and GaN with the wurtzite structure.109,110 However, it is likely that thesevalues are still overestimates because they assume linear elastic behavior foratomic displacements of up to one half the relaxed atomic spacing.112 A coreparameter of 1 will be used throughout this text. Therefore, the line energyof the screw dislocation is

(2.60)

ε επθ θz zb

r= =

4

σ σπθ θz z

Gbr

= =2

1/ rr → 0

r0

E screw rdrGGb dr

rGb

el zr

R

r

R

( ) = = =∫ ∫44 4

22

0 0

π επ πθ lln( / )R r0

r0

r0

E screwGb

R b( ) [ln( / ) ]= +2

41

π




There is a line tension that resists the lengthening of a dislocation dueto its finite line energy. This line tension is just equal to the line energy.This reason is that the work W done to lengthen a dislocation by an amountdl is the line tension times this length, W = Fdl, and F = W/dl is the lineenergy. In the literature, the terms line energy and line tension are often usedinterchangeably.

2.6.7.2 Edge Dislocation

Now consider a straight edge dislocation lying along the z-axis in a right-handed Cartesian coordinate system with l = [001] and b = , as shownin Figure 2.22. In the case of an isotropic crystal, the strain field surroundingthe edge dislocation is given by113

FIGURE 2.22Model of edge dislocation for calculation of the strain energy.

xy

z

r0R

[ ]b00

ε ε ε ε εzz xz zx zy yz= = = = = 0

επ ν πxx

by x yx y

byx y

= −− +

−+

( )( )( ) ( )

2 2

2 2 2 2 24 1 2

ε επ νxy yx

bx x yx y

= = −− +( )

( )( )

2 2

2 2 24 1




(2.61)

where is the Poisson ratio. By a transformation to cylindrical polar coor-dinates it can be shown that the elastic line energy of the edge dislocation is

(2.62)

Adopting a core parameter of 1 as in the case of the screw dislocation, andassuming r0 = b, the total line energy per unit length for the edge dislocation is

(2.63)

2.6.7.3 Mixed Dislocations

For a dislocation of mixed character, the strain field is the superposition ofthe individual strain fields for its edge and screw components. There is nointeraction between the two component strain fields, so the line energy isthe sum of the screw and edge contributions,

(2.64)

where is the angle between the Burgers vector and the line vector.

2.6.7.4 Frank’s Rule

The line energy of a dislocation has a relatively weak dependence on thedislocation character, and for any dislocation,

(2.65)

with 0.5 < C < 1.114 This is the basis for Frank’s rule for dislocation reactions:

a dislocation reaction is energetically favorable if for the products is

less than for the reactants. For example, two dislocations with oppo-

site Burgers vectors may react and annihilate. In this case, for the

επ ν πyy

by x yx y

byx y

= +− +

−+

( )( )( ) ( )

34 1 2

2 2

2 2 2 2 2

ν

E edgeGb

R rel( )( )

ln( / )=−

2

04 1π ν

E edgeGb

R b( )( )

[ln( / ) ]=−

+2

4 11

π ν

E mixedGb

R b( )( cos )

( )[ln( / ) ]= −

−+

2 214 1

1ν α

π ν

α

E CGb≈ 2

b2∑b2∑

b2 0=∑




final configuration, so the annihilation reaction is energetically favored.Frank’s rule can also be applied to the dissociation of perfect dislocationsinto partial dislocations.

2.6.7.5 Hollow-Core Dislocations (Micropipes)

Because the line energy of a closed dislocation increases with the square ofthe Burgers vector length, dislocations with very long Burgers vectors some-times develop hollow cores. This reduces the elastic strain energy associatedwith the dislocation, while introducing the surface energy associated withthe hollow tube. Frank first derived an expression for the diameter of sucha hollow core,112 based on minimization of the free energy. Thus, in equilib-rium the increase of free energy due to the addition of surface area shouldequal the free energy released by expanding the hollow core of the disloca-tion. If r is the radius of the hollow core, then the change in free energy perunit length of dislocation associated with a change in radius dr is given by

(2.66)

where is the surface energy of the crystal and G is the shear modulus. Here,the crystal was assumed to be isotropic and the dislocation core parameterwas neglected. Solving, we find the equilibrium core radius is given by

(2.67)

Hollow-core dislocations are common in the 4H, 6H, and 15R polytypesof SiC. These micropipes are screw dislocations with Burgers vectors alignedwith the six-fold axis and having lengths of c, 2c, 3c, and so on. In 4H-SiC,the experimentally determined diameters for hollow cores can be fit113 withthe assumption of γ/G = 1.2 × 10–12 m. Thus, 3c super screw dislocationshave been observed to have hollow cores approximately 180 nm in diameter,and 4c super screw dislocations have hollow cores of approximately 320 nm.For 6H-SiC, the c parameter is longer, and so the corresponding core diam-eters are expected to be 400 and 710 nm, respectively. It has been reportedthat screw dislocations having Burgers vectors of length c, 2c, or 3c in 4H-or 6H-SiC need not have open cores, but those with 4c or greater are alwaysreported to be open.

2.6.8 Forces on Dislocations

The dislocations in a crystal will move under the influence of an appliedstress. The load producing the applied stress therefore does work on the

dU drGb

rdr≈ − =γ π

π2

40

2

γ

rGb=

2

28π γ




crystal, and the dislocation thus responds as though it experiences a forceequal to the work done divided by the distance moved.114

The force on a dislocation in a crystal with an arbitrary stress tensor σσσσ isgiven by the Peach–Koehler formula:115

(2.68)

where is the vector force per unit length, b is the Burgers vector, and sis the unit vector in the direction of the line of the vector. In scalar form,

(2.69)

where b is the length of the Burgers vector and is the shear stress, resolvedon the slip plane, in the slip direction. If the stress in the crystal is produced bya tensile force F applied to a cross section of area A, then the stress is

and the resolved shear stress is

(2.70)

where is the angle between the applied force and the normal to the slipplane, and is the angle between the applied force and the slip direction.The quantity is called the Schmid factor.

2.6.9 Dislocation Motion

Dislocations move by glide, climb, or a combination of both. Glide is motionin the direction of the Burgers vector and is called conservative motion.Climb is motion out of the glide plane (nonconservative motion). Both pro-cesses are thermally activated because they involve the breaking of crystalbonds, but climb requires long-range diffusion and is only important at veryhigh temperatures.

Dislocation glide velocities have been determined by the double-etch tech-nique in a number of crystals.116 This method involves the use of a crystal-lographic etch before and after stressing the crystal. The etch will revealsharp bottom pits at the places where dislocations emerge at the surface. Ifa dislocation moves while the crystal is stressed, a new, sharp-bottomed pitwill be produced at the new point of emergence for the dislocation. At thesame time, the original pit will enlarge and take on a flat bottom as a resultof the additional etching. Multiple cycles of stressing and etching can there-fore be used to track the motion of individual dislocations. This enables basicstudies of dislocation motion, which reveal the dependence of the dislocationglide velocities on temperature and applied stress.

Dislocation glide velocities have been measured by the double-etchmethod in GaAs,117,118 Ge,119,120 Si,134 and InSb,120,121 and by direct observation

F b s/ ( )L = ⋅ ×σ

F /L

F b= τ

τ

σ = F A/

τ σ φ λ= cos cos

φλ

cos cosφ λ




in GexSi1–x/Si using TEM.121 In these materials it has been found that thestress and temperature dependence of the dislocation glide velocity may bedescribed by the empirical relationship125

(2.71)

where is a characteristic velocity, is a characteristic stress, is theresolved shear stress on the glide plane in the slip direction, U is the activa-tion energy for glide, T is the absolute temperature, and k is the Boltzmannconstant. Over restricted ranges of stress this relationship is commonlyapproximated by

(2.72)

where m and B are constants.In bulk GaAs it has been found experimentally that both m and U depend

on the conductivity type (n-type or p-type) as well as the dislocation char-acter.131 The values reported in the literature are in the range , andoften it is assumed that . It should be emphasized that this equation isempirical in nature, and that it has been reported that the activation energyU is stress dependent123 in SiGe alloys.

2.6.10 Electronic Properties of Dislocations

The electronic properties of dislocations vary greatly among the differentclasses of semiconductor crystals. In diamond and zinc blende crystals, dis-locations have been shown to be detrimental to device performance; in somecases the presence of a single threading dislocation can lead to failure in alaser diode. The III-nitrides seem relatively immune to these same effects,so that light-emitting diodes fabricated in material with high dislocationdensities exhibit little degradation and long lifetimes. In the common poly-types of SiC, the most important defects are micropipes (hollow-core, superscrew dislocations), which cause failure of high-voltage devices.

2.6.10.1 Diamond and Zinc Blende Semiconductors

Dislocations have been found to act as nonradiative recombination centersin the arsenides,124 phosphides,125 nitrides,126 and II-VI semiconductors.127 Ina spatially resolved photoluminescence study of dislocations in GaAs andInP,128 it was found that the overall photoluminescence intensity was reducedsignificantly in a region of 5 to 10 μm around dislocations. This has beenattributed to a reduction of the minority carrier lifetime in the vicinity of thedislocation. This may be due to increased rates of nonradiative bulk recom-

v vUkT

= −⎛⎝⎜

⎞⎠⎟

−⎛⎝⎜

⎞⎠⎟0

0exp expττ

v0 τ0 τ

v B U kTm= −τ exp( / )

1 3< <mm = 1




bination mechanisms as well as increased concentrations of lifetime-killingpoint defects near the dislocations.

The minority carrier lifetime in dislocated GaAs has been modeled byYamaguchi and Amano128 with the assumption that the dislocations act asinfinite sinks for minority carriers. In this model, the one-dimensional con-tinuity equation for the transport of minority carriers to the dislocation issolved with the boundary condition that the excess minority carrier concen-tration is zero at the dislocation core. The solution is129

(2.73)

where is the minority carrier lifetime associated with recombination medi-ated by dislocations, D is the dislocation density, is the mobility of minoritycarriers, k is the Boltzmann constant, T is the absolute temperature, and q isthe electronic charge. The overall minority carrier lifetime is then given by

(2.74)

where is the lifetime associated with minority carrier recombination in thedislocation free material, associated with point defects and intrinsic recom-bination processes.

Figure 2.23 shows the minority carrier lifetime as a function of the dislo-cation density for n-GaAs. The experimental results are from Yamaguchi etal.128 They used photoluminescence decay to determine the minority carrierlifetime and TEM to measure the dislocation density. The theoretical curvewas calculated assuming τ0 = 2 × 10–8 s and μp = 250 cm2 V–1 s–1. It can beseen that the dislocations have a significant effect on the minority carrierlifetime for D > 106 cm–2. In p-GaAs, the minority carriers (electrons) typicallyhave 10 times higher mobility, so this effect is more pronounced. Generally,the minority carrier lifetime will be more affected in materials with highminority carrier mobility.

The Yamaguchi and Amano model has been applied to other dislocatedsemiconductor materials as well. However, in the case of heteroepitaxialGaN on sapphire (0001), it has been found that only dislocations of screwor mixed character give rise to nonradiative recombination.130 Edge dislo-cations, which represent the majority of the threading dislocations in GaNon sapphire (0001), do not appear to contribute to the nonradiative recom-bination. This explains, in part, why nitride light-emitting diodes achievegood performance despite their high threading dislocation densities of 108

to 1010 cm–2.131,132

In the 3C, 4H, and 6H polytypes of SiC, threading dislocations appear tointroduce nonradiative recombination centers as they do in zinc blende

14

2

τπ μ

d

kTDq

=

τd

μ

τ

1 1 1

0τ τ τ= +

d

τ0




crystals. This results in excess reverse leakage and soft reverse breakdowncharacteristics in Schottky diodes.133 This conclusion is reinforced by electronbeam-induced current measurements on 4H-SiC p-n junction diodes, whichshow dark spots associated with dislocations.134 In the same study, however,bright halos seen around the dislocations indicated that impurities had beengettered from the surrounding material. It is possible that the nonradiativerecombination at dislocations is partly or entirely extrinsic (associated withgettered impurities).

It has also been found that threading dislocations degrade carrier mobilityin modulation-doped field effect transistors (MODFETs). Ismail135 studiedthe effect of threading dislocations on the electron mobility in strained SiGe/Si MODFETS grown on Si substrates by ultra-high-vacuum vapor phaseepitaxy. The threading dislocation density was varied systematically bychanging the grading rate in the graded SiGe buffer layer. The dislocationdensities were determined by TEM, and the carrier mobilities were foundusing van der Pauw measurements in the range of temperatures from 0.4 to300K. Ismail found that the low-temperature electron mobility was degradedby threading dislocations when their density exceeded 3 × 108 cm–2, and themobility was decreased by about two orders of magnitude with a dislocationdensity of 1 × 1011 cm–2. These results are shown in Figure 2.24. At roomtemperature, the electron mobility was decreased by 10 and 50%, respec-

FIGURE 2.23Minority carrier lifetime as a function of the threading dislocation density in n-GaAs. Theexperimental results are from Yamaguchi et al.128 They used photoluminescence decay to de-termine the minority carrier lifetime and TEM to measure the dislocation density. The theoreticalcurve was calculated assuming and .

Dislocation density (cm−2)

Min

ority

-car

rier l

ife ti

me (

s)

ExperimentTheory

n-GaAs

103 104 105 106 107 108

10−7

10−8

10−9

10−10

τ082 10= × − s μp = 250 cm V s2 –1 –1




tively, with these two dislocations densities. In the same study, it was alsofound that misfit dislocations in the graded buffer reduce the electron mobil-ity in the Si channel if it is less than 0.4 μm from the buffer.

2.7 Planar Defects

A number of planar crystal defects are encountered in semiconductor het-eroepitaxy. These include stacking faults, twins, and inversion domainboundaries (IDBs).

2.7.1 Stacking Faults

A perfect crystal can be considered a stack of atomic layers occurring in aparticular sequence. As explained in Section 2.2.4, the stacking of the zincblende structure in the [111] direction can be described as … ABCABC ….

FIGURE 2.24The effect of the grading rate in the SiGe buffer on the threading dislocation density and the0.4K electron mobility in a modulation-doped Si/SiGe structure. The structures investigatedwere grown by ultra-high-vacuum (UHV) VPE at 500 to 560°C as follows. On the Si substratewas grown a graded Si1–xGex buffer, with a top composition of x = 0.3, followed by a relaxed1-μm Si0.7Ge0.3 buffer, a strained Si channel 8 to 15 nm thick, an undoped 15-nm Si1–xGex spacer,an n-type Si1–xGex supply layer, and a 4-nm-thick Si cap layer. (Reprinted from Ismail, K., J. Vac.Sci. Technol. B, 14, 2776, 1996. With permission. Copyright 1996, American Institute of Physics.)

0 0.5 1 21.5

1011

1010

109

108

107

106

106

105

105

104

103

Elec

tron

mob

ility

(cm

2 V−

1 s−1

)

Thickness of ramp to 30% Ge (μm)

Graded Si1−xGex/Si (001)D

ensit

y of t

hrea

ding

disl

ocat

ions

(cm

−2)




A stacking fault can occur with an extra plane of atoms inserted into thestacking sequence, as in … ABCBABC …. This is called an extrinsic stackingfault. Another possible type of stacking fault involves the removal of oneplane, as in … ABCBC …, and is called an intrinsic stacking fault. Both typesof stacking faults are illustrated in Figure 2.25.

Stacking faults are planar defects that are bounded on either side by partialdislocations. These are called partial dislocations because the Burgers vectoris not a lattice translation vector. In other words, the Burgers vector does notstart and end on normal lattice sites of the perfect crystal lattice.

Stacking faults are created by the dissociation of perfect dislocationsinto partial dislocations. This occurs naturally during the glide of dislo-cations, as can be shown with the aid of Figure 2.26. Shown are the latticepositions on a (111) plane of a zinc blende crystal, labeled A, along withthe lattice sites of the underlying and overlying planes, labeled B and C,respectively. The unit of slip (Burgers vector) for a perfect dislocation inthe overlying layer is the vector b1. Using the hard sphere model for atoms,this translation takes a sphere in one B position directly to the next Bposition. However, such a hard sphere will more easily slide first to a Cposition and then to a B position, along the valleys between the A spheres.These translations are represented by the vectors b2 and b3, respectively.Thus, the perfect lattice translation b1 is naturally split into two simplertranslations , which are the Burgers vectors associated with twopartial dislocations.

FIGURE 2.25Schematic illustration of stacking faults in a zinc blende crystal. The stacking direction is [111].(a) Extrinsic stacking fault; (b) intrinsic stacking fault.

C

B

A

C

B

A

C

C

B

A

B

A

C

A

C

B

A

C

B

C

A

BAC

AB

(a)

(b)

b b2 3+




In diamond and zinc blende crystals, the perfect 60° dislocation may dis-sociate into two Shockley partial dislocations by the reaction

(2.75)

The total Burgers vector is conserved as required, and the dissociation isenergetically favorable according to Frank’s rule. The two partial dislocationslie on the same glide plane as the perfect dislocation; in the example given,this is the plane. Between the two partial dislocations, a stacking faultexists on this glide plane.

The equilibrium width of a stacking fault (the separation of the two Shock-ley partials) may be estimated as follows. Assuming an isotropic crystal, thepartial dislocations repel one another with a force per unit length F, given by

(2.76)

where G is the shear modulus, b is the length of the Burgers vectors for thepartial dislocations, is the Poisson ratio, and d is their separation.

The stacking fault that exists between the partial dislocations has an arealenergy ξ, which produces an attractive force between the two partials. Equat-ing the two forces, we find the equilibrium width of the stacking fault to be

(2.77)

During plastic deformation two partials making up an extended disloca-tion will glide simultaneously. As the extended dislocation passes a region

FIGURE 2.26Schematic drawing of glide on (111) planes by a perfect dislocation (b1) and by Shockley partialdislocations (b2 and b3).

A A

AA A

b2

b1

b3

C

B B

a a a2

0116

1126

121[ ] [ ] [ ]→ +

( )111

FGb

d= −

−

2 28 1

( )( )

νπ ν

ν

dGb=

2

4πξ




of crystal, the leading partial will create a stacking fault after, while thetrailing partial will remove the stacking fault.

Often it is adequate in theoretical work to treat extended dislocations asperfect dislocations, and this greatly simplifies the mathematics. Thisapproach is justified if the stacking fault energy is high, causing the disso-ciated components of the extended dislocation to be tightly bound. Such anassumption is usually good for diamond and zinc blende crystals. In GaAs,for example, the stacking fault energy is 48 mJ/m2, resulting in an equilib-rium stacking fault width of 14.5 nm. For more ionic crystals, the stackingfault energy is less; for example, the value for ZnSe140 is only 10 mJ/m2 witha corresponding equilibrium stacking fault width of 70 nm. Often, theobserved stacking fault widths are considerably less than the equilibriumvalues, because the creation of stacking faults requires long-range self-dif-fusion, which is slow at typical growth temperatures.

2.7.2 Twins

Another type of planar defect resulting from a change in the stackingsequence is the twin. In diamond and zinc blende crystals, twinning occursalmost exclusively on (111) planes. Using the stacking notation of Section2.2.4, a twin boundary in a diamond or zinc blende crystal may be denotedas … ABCABACBA …. Here the normal crystal and its twin share a singleplane of atoms (the twinning plane or composition plane) and there is reflec-tion symmetry about the twinning plane.

Twinning involves a change in long-range order of the crystal; it thereforecannot result from the simple insertion or removal of an atomic plane, as inthe case of the stacking fault. Therefore, twins cannot be created by the glideof dislocations. Instead, twinning occurs during crystal growth, either bulkgrowth or heteroepitaxy.

There is a change in crystal orientation at the twinning plane. For a dia-mond or zinc blende crystal, twinning occurs about a {111} plane. If theoriginal growth plane was (001), then the surface of the twinned crystal isthe plane.137 Additional twinning may bring the surface to vari-ous planes or back to the (001). The (111) twinning plane is alwaysinclined by 54.7° to the (001) surface and may grow out of the crystal in thecase of Czochralski-grown bulk crystals.

For (111) growth of zinc blende crystals, the twinned crystal may haveeither or {115} orientation, depending on the orientation of the twinningplane. In the case, the twinning plane is the same as the growth planeand will not grow out of the crystal. This is a disadvantage of using the (111)orientation for the growth of bulk crystals.

Twin boundaries are commonly found in heteroepitaxial II-VI crystalsgrown on (111) substrates, including ZnSe/GaAs(111),138 CdTe/GaAs(111),139

and HgxCd1–xTe/CdTe(111).140 Twin boundaries are gross defects thatdegrade device performance, and so (001) is the preferred orientation forheteroepitaxial growth of II-VI materials.

( )221{ }221

( )111( )111




2.7.3 Inversion Domain Boundaries (IDBs)

Inversion domain boundaries (IDBs), also known as antiphase domainboundaries, are an important consideration in the heteroepitaxy of a polarsemiconductor on a nonpolar substrate.145 Examples include GaN/α-Al2O3

(0001), AlN/Si (001), GaAs/Si (001), and InP/Si (001). Due to the lowersymmetry of the polar semiconductor, it can grow with one of two (non-equivalent) crystal orientations on the nonpolar substrate. The boundariesbetween regions having these two orientations are inversion domain bound-aries. Generally, IDBs are expected to introduce states within the energy gapand give rise to nonradiative recombination. They therefore degrade theefficiencies of LEDs and cause excess leakage in p-n junctions. In addition,the charging of IDBs will give rise to scattering of charge carriers and degradethe performance of majority-carrier devices such as FETs.

In the case of heteroepitaxial GaN/α-Al2O3 (0001), inversion domains havebeen found in material grown by MOVPE142 and MBE.143 These manifest ashexagonal domains, 5 to 20 nm in lateral size, which exist through the entirethickness of the epitaxial layer.144 The sidewalls of these domains, which areIDBs, are along first-order prism planes of type ; since these are parallelwith the [0001] growth direction, they will not grow out of the layer as it isincreased in thickness. In the case of MBE-grown GaN on sapphire, conver-gent beam electron diffraction (CBED) has been used to show that the sur-faces of the domains are (0001)N, whereas the surrounding material has(0001)Ga orientation.148

For the heteroepitaxial growth of a zinc blende semiconductor on a (001)surface, IDBs that are inclined to the interface may annihilate one another. Onesuch annihilation reaction can occur by the interaction of IDBs on {111}planes,145 as shown in Figure 2.27. Here, IDB annihilation occurs at the line ofintersection of the two IDB planes, which lies along a direction parallelto the interface. Annihilation reactions can also occur between IDBs on {011}planes,146 which can meet along a direction, as shown in Figure 2.28. Ineither case, the material grown above the annihilation point is free from IDBs.These annihilation mechanisms may be important in 3C-SiC. In zinc blendeIII-V semiconductors, the IDBs usually exhibit very irregular structures so thatthese reaction mechanisms can only occur on a very limited basis.

In zinc blende III-V semiconductors on Si or Ge substrates, inversiondomains typically have irregular shapes.147 The majority of their boundariesdo not correspond to low-index crystalline directions. As an example, Figure2.29 shows IDBs in GaAs on silicon-on-insulator (001) grown by MOVPE,148

revealed by etching149 in 10:1 HF:HNO3 and viewed using scanning electronmicroscopy (SEM). Only small segments of the boundary lines orient alongthe [110], [010], [120], and occasionally [130] and [140] directions. The epi-taxial film was removed from the silicon-on-insulator using an HF etch. Thisallowed viewing of the interface side of the GaAs and revealed that the IDBspass through the entire thickness of the layer. (The inversion domains nucle-ate at the GaAs/Si interface.)

{ }1010

110

010




FIGURE 2.27Annihilation of IDBs on {111} planes in a (001) zinc blende semiconductor. (Reprinted fromIshida, Y. et al., J. Appl. Phys., 94, 4676, 2003. With permission. Copyright 2003, American Instituteof Physics.)

FIGURE 2.28Annihilation of IDBs on {011} planes in a (001) zinc blende semiconductor. (Reprinted fromIshida, Y. et al., J. Appl. Phys., 94, 4676, 2003. With permission. Copyright 2003, American Instituteof Physics.)

Annihilation

[001]

[110]

Annihilation

[100]

[001]




Problems

1. For Si, calculate the interplanar spacing and atomic density (inatoms/cm2) for each of the following types of planes: (a) {001}, (b){011}, and (c) {111}.

2. For GaN, calculate the interplanar spacing and atomic density (inatoms/cm2) for each of the following planes: (a) (0001) basal plane,(b) first-order prism plane, and (c) first-order pyrami-dal plane.

3. Determine the compositions of InxGa1–xAsyP1–y, which are lattice-matched to GaAs.

4. Considering the thermal expansion, find the relaxed lattice constantfor GaAs at the following temperatures: (a) 450°C, (b) 550°C, and (c)650°C.

5. Find the lattice mismatch strain for Ge grown on Si at (a) roomtemperature and (b) 600°C.

6. Calculate the strain energy per unit area for a pseudomorphic layerof Si0.95Ge0.05/Si (001) that is 100 nm thick.

7. Find the line energy per unit length for an edge dislocation in GaAs.8. Repeat Problem 7 for a screw dislocation in GaAs.9. Estimate the minority carrier lifetime in n-InP on Si (001) with a

dislocation density of 108 cm–2, assuming a reasonable value for theminority carrier mobility. Repeat for p-InP/Si (001) having the samedislocation density.

FIGURE 2.29SEM micrographs of IDBs revealed in a 1.9-μm film of GaAs on silicon-on-insulator (SOI). (a)Front side of epitaxial film; (b) back side of epitaxial film; (c) higher-magnification micrographshowing [011]-oriented textures. (Reprinted from Chu, S.N.G. et al., J. Appl. Phys., 64, 2981, 1988.With permission. Copyright 1988, American Institute of Physics.)

(a)5 μm 5 μm 5 μm(b) (c)

( )10 10 ( )10 11




References

1. C. Kittel, Introduction to Solid State Physics, 8th ed., John Wiley & Sons, NewYork, 2004.

2. R.M. Martin, Relation between elastic tensors of wurtzite and zinc-blendestructure materials, Phys. Rev. B, 6, 4546 (1972).

3. G.R. Fisher and P. Barnes, Towards a unified view of polytypism in siliconcarbide, Phil. Mag. B, 61, 217 (1990).

4. K. Jarrendahl and R.B. Davis, in Semiconductors and Semimetals, Vol. 52, SiCMaterials and Devices, Academic Press, New York, 1998, p. 3.

5. J.A. Powell, P. Pirouz, and W.J. Choyke, in Semiconductor Interfaces, Microstruc-tures and Devices: Properties and Applications, Z.C. Feng, Ed., Institute of PhysicsPublishing, Bristol, 1993, pp. 257–293.

6. L.S. Ramsdell, Studies on silicon carbide, Am. Miner., 32, 64 (1947).7. S. Nakamura, M. Senoh, S. Nagahama, N. Iwasa, T. Yamada, T. Matshusita, H.

Kiyoku, and Y. Sugimoto, Ridge-geometry InGaN multi-quantum-well-struc-ture laser diodes, Appl. Phys. Lett., 69, 1477 (1996).

8. S. Nakamura, M. Senoh, S. Nagahama, N. Iwasa, T. Yamada, T. Matshusita, H.Kiyoku, and Y. Sugimoto, InGaN-based multi-quantum-well-structure laserdiodes, Jpn. J. Appl. Phys., 35, L74 (1996).

9. J. Bai, T. Wang, H.D. Li, N. Jiang, and S. Sakai, (0001) oriented GaN epilayergrown on sapphire by MOCVD, J. Cryst. Growth, 231, 41 (2001).

10. W. von Muench and I. Pfaffender, Thermal oxidation and electrolytic etchingof silicon carbide, J. Electrochem. Soc., 122, 642 (1975).

11. A. Suzuki, H. Ashida, N. Furui, K. Maneno, and H. Matsunami, Thermaloxidation of SiC and electrical properties of Al-SiO2-SiC MOS structure, Jpn. J.Appl. Phys., 21, 579 (1982).

12. H.S. Hong, J.T. Glass, and R.F. Davis, Growth rate, surface morphology, anddefect microstructures of β-SiC films chemically vapor deposited on 6H-SiCsubstrates, J. Mater. Res., 4, 204 (1989).

13. H. Matsunami and T. Kimoto, Step-controlled epitaxial growth of SiC: highquality homoepitaxy, Mater. Sci. Eng. Rep., 20, 125 (1997).

14. T. Kimoto and H. Matsunami, in Silicon Carbide: Materials, Processing, and De-vices, Z.C. Feng and J.H. Zhao, Eds., Taylor & Francis, New York, 2004, p. 1.

15. W. Kaiser and W.L. Bond, Nitrogen, a major impurity in common type I dia-mond, Phys. Rev., 115, 857 (1959).

16. Y.S. Touloukian, R.K. Kirby, R.E. Taylor, and P.D. Desai, Eds., ThermophysicalProperties of Matter, Vol. 12, Thermal Expansion, Metallic Elements and Alloys,Plenum, New York, 1975.

17. E.R. Cohen and B.N. Taylor, The 1986 Adjustment of the Fundamental PhysicalConstants, report of the Committee on Data for Science and Technology of theInternational Council of Scientific Unions (CODATA) Task Group on Funda-mental Constants, CODATA Bulletin 63, Pergamon, Elmsford, NY, 1986.

18. J. Donahue, The Structure of the Elements, John Wiley & Sons, New York, 1974.19. J. Thewlis and A.R. Davey, Thermal expansion of grey tin, Nature, 174, 1011

(1959).20. A. Taylor and R.M. Jones, in Silicon Carbide: A High Temperature Semiconductor,

J.R. O’Connor and J. Smiltens, Eds., Pergamon Press, Oxford, 1960, p. 147.

( )112 0




21. Y.M. Tairov and V.F. Tsvetkov, in Crystal Growth and Characterization of PolytypeStructures, P. Krishna, Ed., Pergamon Press, Oxford, 1983, pp. 111–162.

22. B. Greenberg, private communication.23. H.P. Maruska and J.J. Tietjen, The preparation and properties of vapor-depos-

ited single-crystalline GaN, Appl. Phys. Lett., 15, 327 (1969).24. M. Leszczynski, T. Suski, H. Teisseyre, P. Perlin, I. Grzegory, J. Jun, S. Porowski,

and T.D. Moustakas, Thermal expansion of gallium nitride, J. Appl. Phys., 76,4909 (1994).

25. Y. Oshima, T. Suzuki, T. Eri, Y. Kawaguchi, K. Watanabe, M. Shibata, and T.Mishima, Thermal and optical properties of bulk GaN crystals fabricatedthrough hydride vapor phase epitaxy with void-assisted separation, J. Appl.Phys., 98, 103509 (2005).

26. Y.V. Shvyd’ko, M. Lucht, E. Gerdau, M. Lerche, E.E. Alp, W. Sturhahn, J. Sutter,and T.S. Toellner, Measuring wavelengths and lattice constants with the Möss-bauer wavelength standard, J. Synchrotron Rad., 9, 17 (2002).

27. R.W.G. Wyckoff, Crystal Structures, Vol. 1, Wiley, New York, 1963.28. C. Kim, I.K. Robinson, J. Myoung, K. Shim, M.-C. Yoo, and K. Kim, Critical

thickness of GaN thin films on sapphire (0001), Appl. Phys. Lett., 69, 2358 (1996).29. S. Poroski, Bulk and homoepitaxial GaN-growth and characterisation, J. Cryst.

Growth, 189/190, 153 (1998).30. M. Leszczynski, T. Suski, H. Teisseyre, P. Perlin, I. Grzegory, J. Jun, S. Porowski,

and T.D. Moustakas, Thermal expansion of gallium nitride, J. Appl. Phys., 76,4909 (1994).

31. A.U. Shelag and V.A. Savastenko, Izv. Akad. Nauk. BSSR Ser. Fiz. Mat. Nauk., 3,126 (1976).

32. K. Wang and R.R. Reeber, Thermal expansion and elastic properties of InN,Appl. Phys. Lett., 79, 1602 (2001).

33. L. Vegard, The constitution of mixed crystals and the space occupied by atoms,Z. Phys., 5, 17 (1921).

34. H. Huntington, Ultrasonic measurements on single crystals, Phys. Rev., 72, 321(1947).

35. N. Bouarissa, Compositional dependence of the elastic constants and the Pois-son ratio of GaxIn1–xSb, Mater. Sci. Eng. B, 100, 280 (2003).

36. A.E. Merad, H. Aourag, B. Khelifa, C. Mathieu, and G. Merad, Predictions ofthe bonding properties in Cd1–xZnxTe, Superlat. Microstruct., 30, 241 (2001).

37. P.C. Kelires, Microstructural and elastic properties of silicon-germanium-car-bon alloys, Appl. Surf. Sci., 102, 12 (1996).

38. M. Reason, X. Weng, W. Ye, D. Dettling, S. Hanson, G. Obeidi, and R.S. Gold-man, Stress evolution in GaAsN alloy films, J. Appl. Phys., 97, 103523 (2005).

39. M.H. Grimsditch and A.K. Ramdas, Brillouin scattering in diamond, Phys. Rev.B, 11, 3139 (1975).

40. S.P. Nikanorov, Yu. A. Burenkov, and A.V. Stepanov, Elastic properties of silicon,Sov. Phys. Solid State, 13, 2516 (1972) (English translation).

41. K.B. Tolpygo, Optical, elastic, and piezoelectric properties of ionic and valencecrystals with the ZnS type lattice, Sov. Phys. Sol. State, 2, 2367 (1961).

42. S. Yu Davydov and A.V. Solomonov, Elastic properties of gallium and alumi-num nitrides, Tech. Phys. Lett., 25, 601 (1999) (translated from Pis’ma Zh. Tekh.Fiz., 25, 23 (1999)).

43. Y.K. Yogurtcu, A.J. Miller, and G.A. Saunders, Pressure dependence of elasticbehaviour and force constants of GaP, J. Phys. Chem. Solids, 42, 49 (1981).




44. W.G. Cady, Piezoelectricity, McGraw-Hill, New York, 1946.45. R. Bechman, Elastic and piezoelectric constants of alpha-quartz, Proc. IRE, 36,

764 (1958).46. G.L. Harris, in Properties of Silicon Carbide, G.L. Harris, Ed., Institution of Elec-

trical Engineers (INSPEC), London, 1995, p. 9.47. K. Kamitani, M. Grimsditch, J.C. Nipko, C.-K. Loong, M. Okada, and I. Kimura,

The elastic constants of silicon carbide: a Brillouin-scattering study of 4H and6H SiC single crystals, J. Appl. Phys., 82, 3152 (1997).

48. A. Polian, M. Grimsditch, and I. Grzegory, Elastic constants of gallium nitride,J. Appl. Phys., 79, 3343 (1996).

49. C. Deger, E. Born, H. Angerer, O. Ambacher, M. Stutzmann, J. Hornsteiner, E.Riha, and G. Fischerauer, Sound velocity of AlxGa1–xN thin films obtained bysurface acoustic-wave measurements, Appl. Phys. Lett., 72, 2400 (1998).

50. T. Deguchi, D. Ichiryu, K. Toshikawa, K. Segiguchi, T. Sota, R. Matsuo, T.Azuhata, M. Yamaguchi, T. Yagi, S. Chichibu, and S. Nakamura, Structural andvibrational properties of GaN, J. Appl. Phys., 86, 1860 (1999).

51. V. Yu Davydov, Yu. E. Kitaev, I.N. Goncharuk, A.N. Smirnov, J. Graul, O.Semchinova, D. Uffmann, M.B. Smirnov, A.P. Mirgorodsky, and R.A. Evarestov,Phonon dispersion and Raman scattering in hexagonal GaN and AlN, Phys.Rev. B, 58, 12899 (1998).

52. V.A. Savastenko and A.U. Sheleg, Study of the elastic properties of galliumnitride, Phys. Status Solidi A, 48, K135 (1978).

53. C. Deger, E. Born, H. Angerer, O. Ambacher, M. Stutzmann, J. Hornsteiner, E.Riha, and G. Fischerauer, Sound velocity of AlxGa1–xN thin films obtained bysurface acoustic-wave measurements, Appl. Phys. Lett., 72, 2400 (1998).

54. L.E. McNeil, M. Grimsditch, and R.H. French, Vibrational spectroscopy ofaluminum nitride, J. Am. Ceram. Soc., 76, 1132 (1993).

55. K. Tsubouchi, K. Sugai, and N. Mikoshiba, AlN material constants evaluationand SAW properties on AlN/Al2O3 and AlN/Si, in 1981 Ultrasonics Symp. Proc.,B.R. McAvoy, Ed., IEEE, New York, 1981, p. 375.

56. A.U. Shelag and V.A. Savastenko, Izv. Akad. Nauk. BSSR Ser. Fiz. Mat. Nauk., 3,126 (1976).

57. K. Kim, W.R.L. Lambrecht, and B. Segall, Elastic constants and related proper-ties of tetrahedrally bonded BN, AlN, GaN, and InN, Phys. Rev. B, 53, 16310(1996).

58. A.F. Wright, Elastic properties of zinc-blende and wurtzite AlN, GaN, and InN,J. Appl. Phys., 82, 2833 (1997).

59. A.A. Marmalyuk, R.K. Akchurin, and V.A. Gorbylev, Thermal expansion andelastic properties of InN, Inorg. Mat., 34, 691 (1998) (translation of Neorg. Mater.).

60. J.A. Chisholm, D.W. Lewis, and P.D. Bristowe, Classical simulations of theproperties of group-III nitrides, J. Phys. Cond. Mater., 11, L235 (1999).

61. J.-M. Wagner and F. Bechstedt, Properties of strained wurtzite GaN and AlN:Ab initio studies, Phys. Rev. B, 66, 115202 (2002).

62. A. Polian, M. Grimsditch, and I. Grzegory, Elastic constants of gallium nitride,J. Appl. Phys., 79, 3343 (1996).

63. J.J. Gilman, Direct measurement of the surface energies of crystals, J. Appl.Phys., 31, 2208 (1960).

64. D. Hull, P. Beardmore, and A.P. Valentine, Crack propagation in single crystalsof tungsten, Phil. Mag., 12, 1021 (1965).

65. A. Kelly, Strong Solids, Clarendon Press, Oxford, 1966, pp. 45–47.




66. H.P. Bonzel, Equilibrium crystal shapes: towards absolute energies, Prog. Surf.Sci., 67, 45 (2001).

67. J.J. Metois and P. Muller, Absolute surface energy determination, Surf. Sci., 548,13 (2004).

68. R.J. Jaccodine, Surface energy of germanium and silicon, J. Electrochem. Soc.,110, 524 (1963).

69. J.W. Matthews, Defects associated with the accommodation of misfit betweencrystals, J. Vac. Sci. Technol., 12, 126 (1975).

70. S.N.G. Chu and S. Nakahara, 1/2<100>{100} dislocation loops in a zinc blendestructure, Appl. Phys. Lett., 66, 434 (1990).

71. B.C. de Cooman and C.B. Carter, The accommodation of misfit at {100} hetero-junctions in III-V compound semiconductors by gliding dissociated disloca-tions, Acta Met., 37, 2765 (1989).

72. E. Peissker, P. Haasen, and H. Alexander, Anisotropic plastic deformation ofindium antimonide, Phil. Mag., 7, 1279 (1962).

73. R.L. Bell and A.F.W. Willoughby, Etch pit studies of dislocations in indiumantimonide, J. Mater. Sci., 1, 219 (1966).

74. T. Ninomiya, Velocities and internal-friction of dislocations in III-V compounds,Journal de Physique, 40, 132 (1979).

75. J. Petruzzello and M.R. Leys, Effect of the sign of misfit strain on the dislocationstructure at interfaces of GaAsxP1–x films, Appl. Phys. Lett., 53, 2414 (1988).

76. J.L. Weyher and J. van de Ven, Influence of substrate defects on the structureof epitaxial GaAs grown by MOCVD, J. Cryst. Growth, 88, 221 (1988).

77. Wacker-Siltronic, Munich, Germany.78. M/A-Com, Lowell, MA.79. K. Durose and G.J. Russell, Structural defects in CdTe crystals grown by two

different vapour phase techniques, J. Cryst. Growth, 86, 471 (1988).80. N.M. Aitken, M.D.G. Potter, D.J. Buckley, J.T. Mullins, J. Carles, D.P. Halliday,

K. Durose, B.K. Tanner, and A.W. Brinkman, Characterisation of cadmiumtelluride bulk crystals grown by a novel “multi-tube” vapour growth tech-nique, J. Cryst. Growth, 198/199, 984 (1999).

81. G.A. Rozgonyi, P.M. Petroff, and M.B. Panish, Control of lattice parameters anddislocations in the system Ga1–xAlxAs1–yPy/GaAs, J. Cryst. Growth, 27, 106 (1974).

82. K.R. Breen, P.N. Uppal, and J.S. Ahearn, Interface dislocation structures inInxGa1–xAs/GaAs mismatched epitaxy, J. Vac. Sci. Technol. B, 7, 758 (1989).

83. M.J. Matragrano, D.G. Ast, J.R. Shealy, and V. Krishnamoorthy, Anisotropicstrain relaxation of GaInP epitaxial layers in compression and tension, J. Appl.Phys., 79, 8371 (1996).

84. S. Nakamura, GaN growth using GaN buffer layer, Jpn. J. Appl. Phys., 30, L1705(1991).

85. H. Amano, N. Sawaki, I. Akasaki, and Y. Toyoda, Metalorganic vapor phaseepitaxial growth of a high quality GaN film using an AlN buffer layer, Appl.Phys. Lett., 48, 353 (1986).

86. D. Kapolnek, X.H. Wu, B. Heying, S. Keller, B.P. Keller, U.K. Mishra, S.P.DenBaars, and J.S. Speck, Structural evolution in epitaxial metalorganic chem-ical vapor deposition grown GaN films on sapphire, Appl. Phys. Lett., 67, 1541(1995).

87. W. Qian, M. Skowronski, M. De Graef, K. Doverspike, L.B. Rowland, and D.K.Gaskill, Microstructural characterization of α-GaN films grown on sapphire byorganometallic vapor phase epitaxy, Appl. Phys. Lett., 66, 1252 (1995).




88. F.R. Chien, X.J. Ning, S. Stemmer, P. Pirouz, M.D. Bremser, and R.F. Davis, Growthdefects in GaN films on 6H-SiC substrates, Appl. Phys. Lett., 68, 2678 (1996).

89. E. Valcheva, T. Paskova, and B. Monemar, Nanopipes and their relationship tothe growth mode in thick HVPE-GaN layers, J. Cryst. Growth, 255, 19 (2003).

90. W. Qian, G.S. Rohrer, M. Skowronski, K. Doverspike, L.B. Rowland, and D.K.Gaskill, Open-core screw dislocations in GaN epilayers observed by scanningforce microscopy and high-resolution transmission electron microscopy, Appl.Phys. Lett., 67, 2284 (1995).

91. F.A. Ponce, J.S. Major, Jr., W.E. Plano, and D.F. Welch, Crystalline structure ofAlGaN epitaxy on sapphire using AlN buffer layers, Appl. Phys. Lett., 65, 2302(1994).

92. H.P.D. Schenk, P. Vennegues, O. Tottereau, T. Riemann, and J. Christen, Three-dimensionally nucleated growth of gallium nitride by low-pressure metalor-ganic vapour phase epitaxy, J. Cryst. Growth, 258, 232 (2003).

93. W.J. Choyke and G. Pensl, Physical properties of SiC, Mater. Res. Soc. Bull., 22,36 (1997).

94. M. Dudley, S. Wang, W. Huang, C.H. Carter, Jr., V.F. Tsvetkov, and C. Fazi,White beam synchrotron topographic studies of defects in 6H-SiC single crys-tals, J. Phys. D, 28, A63 (1995).

95. B.M. Epelbaum and D. Hofmann, On the mechanisms of micropipe and mac-rodefect transformation in SiC during liquid phase treatment, J. Cryst. Growth,225, 1 (2001).

96. I. Kamata, H. Tsuchida, T. Jikimoto, and K. Izumi, Structural transformationof screw dislocations via thick 4H-SiC epitaxial growth, Jpn. J. Appl. Phys., 39,6496 (2000).

97. S. Ha, N.T. Nuhfer, G.S. Rohrer, M. De Graef, and M. Skowronski, Origin ofthe domain structure in hexagonal silicon carbide boules grown by the physicalvapor transport method, J. Cryst. Growth, 220, 308 (2000).

98. M. Tuominen, R. Yakimova, R.C. Glass, T. Tuomi, and E. Janzen, Crystal im-perfections in 4H SiC grown with a seeded Lely method, J. Cryst. Growth, 144,267 (1994).

99. J. Takahashi, N. Ohtani, and M. Kanaya, Structural defects in α-SiC singlecrystals grown by the modified-Lely method, J. Cryst. Growth, 167, 596 (1996).

100. D. Hobgood, M. Brady, W. Brixius, G. Fetchko, R. Glass, D. Henshall, J. Jenny,R. Leonard, D. Malta, St. G. Muller, V. Tsvetskov, and C. Carter, Jr., Status oflarge diameter SiC crystal growth for electronic and optical applications, Mater.Sci. Forum, 338–342, 3 (2000).

101. M. Dudley and X. Huang, Characterization of SiC using synchrotron whitebeam x-ray topography, Mater. Sci. Forum, 338–342, 431 (2000).

102. Cree Materials, Durham, NC.103. D. Hull and D.J. Bacon, Introduction to Dislocations, 4th ed., Butterworth Hei-

nemann, Oxford, 2001, pp. 65–67.104. F.R.N. Nabarro, Theory of Crystal Dislocations, Dover, New York, p. 201. 105. J.W. Matthews and A.E. Blakeslee, Defects in epitaxial multilayers. I. Misfit

dislocations, J. Cryst. Growth, 27, 118 (1974).106. A.A. Maradudin, Screw dislocations and discrete elastic theory, J. Phys. Chem.

Solids, 9, 1 (1958).107. R.H.M. van der Leur, A.J.G. Schellingerhout, F. Tuinstra, and J.E. Mooij, Critical

thickness for pseudomorphic growth of Si/Ge alloys and superlattices, J. Appl.Phys., 64, 3043 (1988).




108. F. Kroupa, Circular edge dislocation loop, Czech. J. Phys., B10, 284 (1960).109. A. Bere and A. Serra, Atomic structure of dislocation cores in GaN, Phys. Rev.

B, 65, 205323 (2002).110. S.M. Lee, A. Belkhir, X.Y. Zhu, Y.H. Lee, Y.G. Hwang, and Th. Frauenheim,

Electronic structure of GaN edge dislocations, Phys. Rev. B, 61, 16033 (2000).111. A.A Maradudin, Screw dislocations and discrete elastic theory, J. Phys. Chem.

Sol., 9, 1 (1958).112. F.C. Frank, Capillary equilibria of dislocated crystals, Acta Cryst., 4, 497 (1951).113. W.M. Vetter and M. Dudley, Micropipes and the closure of axial screw dislo-

cation cores in silicon carbide crystals, J. Appl. Phys., 96, 348 (2004).114. Hull and Bacon, p. 115. M. Peach and J.S. Koehler, The forces exerted on dislocations and the stress

fields produced by them, Phys. Rev., 80, 436 (1950).116. W.G. Johnston and J.J. Gilman, Dislocation velocities, dislocation densities, and

plastic flow in lithium fluoride crystals, J. Appl. Phys., 30, 129 (1959).117. S.K. Choi, M. Mihara, and T. Ninomiya, Dislocation velocities in GaAs, Jpn. J.

Appl. Phys., 16, 737 (1977).118. S.A. Erofeeva and Yu. A. Osip’yan, Mobility of dislocations in crystals with the

sphalerite lattice, Sov. Phys. Sol. State, 15, 538 (1973) (English translation).119. J.J. Gilman, Dislocation mobility in crystals, J. Appl. Phys., 36, 3195 (1965).120. A.R. Chaudhuri, J.R. Patel, and L.G. Rubin, Velocities and densities of dislocations

in germanium and other semiconductor crystals, J. Appl. Phys., 33, 2736 (1962).121. R. Hull and J.C. Bean, Kinetic barriers to strain relaxation in Ge(x)Si(1-x) epi-

taxy, Mater. Res. Symp. Proc., 160, 23 (1990).122. A. George and J. Rabier, Dislocations and plasticity in semiconductors. I. Dis-

location structures and dynamics, Rev. Phys. Appl., 22, 941 (1987).123. B.W. Dodson, Stress dependence of dislocation glide activation energy in single-

crystal silicon-germanium alloys up to 2.6 GPa, Phys. Rev. B, 38, 12383 (1988).124. P.W. Hutchinson and P.S. Dobson, Defect structure of degraded GaAlAs-GaAs

double heterojunction lasers, Phil. Mag., 32, 745 (1975).125. D.V. Lang and C.H. Henry, Nonradiative recombination at deep levels in GaAs

and GaP by lattice-relaxation multiphonon emission, Phys. Rev. Lett., 35, 1525(1975).

126. T. Hino, S. Tomiya, T. Miyajima, K. Yanashima, S. Hashimoto, and M. Ikeda,Characterization of threading dislocations in GaN epitaxial layers, Appl. Phys.Lett., 76, 3421 (2000).

127. S. Tomiya, E. Morita, M. Ukita, H. Okuyama, S. Itoh, K. Nakano, and A.Ishibashi, Structural study of defects induced during current injection to II-VIblue light emitter, Appl. Phys. Lett., 66, 1208 (1995).

128. M. Yamaguchi and C. Amano, Efficiency calculations of thin-film GaAs solarcells on Si substrates, J. Appl. Phys., 58, 3601 (1985).

129. M. Yamaguchi, C. Amano, Y. Itoh, K. Hane, R.A. Ahrenkiel, and M.M. Al-Jassim,Analysis for high-efficiency GaAs solar cells on Si substrates, in 20th IEEEPhotovoltaic Specialists Conference, 749 (1988).

130. T. Hino, S. Tomiya, T. Miyajima, K. Yanashima, S. Hashimoto, and M. Ikeda,Characterization of threading dislocations in GaN epitaxial layers, Appl. Phys.Lett., 76, 3421 (2000).

131. S.D. Lester, F.A. Ponce, M.G. Craford, and D.A. Steigerwald, High dislocationdensities in high efficiency GaN-based light-emitting diodes, Appl. Phys. Lett.,66, 1249 (1995).




132. W. Qian, M. Skowronski, M. Degraef, K. Doverspike, L.B. Rowland, and D.K.Gaskill, Microstructural characterization of α-GaN films grown on sapphire byorganometallic vapor phase epitaxy, Appl. Phys. Lett., 66, 1252 (1995).

133. P.G. Neudeck, W. Huang, M. Dudley, and C. Fazi, Non-micropipe dislocationsin 4H-SiC devices: electrical properties and device technology, Mater. Res. Soc.Symp. Proc., 512, 107 (1998).

134. S. Maximenko, S. Soloviev, D. Cherednichenko, and T. Sudarshan, Electron-beam induced current observed for dislocations in diffused 4H-SiC p-n diodes,Appl. Phys. Lett., 84, 1576 (2004).

135. K. Ismail, Effect of dislocations in strained Si/SiGe on electron mobility, J. Vac.Sci. Technol. B, 14, 2776 (1996).

136. H. Hartmann, R. Mach, and B. Selle, in Current Topics in Material Science, Vol.9, E. Kaldis, Ed., North-Holland, Amsterdam, 1982, p. 1.

137. W.R. Runyan, Silicon Semiconductor Technology, McGraw-Hill, New York, 1965,pp. 98–101.

138. T. Yao and S. Maekawa, Molecular beam epitaxy of zinc chalcogenides, J. Cryst.Growth, 53, 423 (1981).

139. P.D. Brown, J.E. Hails, G.J. Russell, and J. Woods, Defect structure of epitaxialCdTe layers grown in {100} and {111}B GaAs and on {111}B CdTe by metalor-ganic chemical vapor deposition, Appl. Phys. Lett., 50, 1144 (1987).

140. J.E. Hails, G.J. Russell, A.W. Brinkman, and J. Woods, The effect of CdTe sub-strate orientation on the MOVPE growth of CdxHg1–xTe, J. Cryst. Growth, 79,940 (1986).

141. H. Kroemer, Polar-on-nonpolar epitaxy, J. Cryst. Growth, 81, 193 (1987).142. J.L. Rouviere, M. Arlery, A. Bourret, R. Niebuhr, and K. Bachem, Understanding

the pyramidal growth of GaN by transmission electron microscopy, Mater. Res.Soc. Symp. Proc., 395, 393 (1996).

143. V. Potin, P. Ruterana, M. Benamara, and H.P. Strunk, Inversion domains andpinholes in GaN grown over Si(111), Appl. Phys. Lett., 82, 4471 (2003).

144. P. Ruterana, Convergent beam electron diffraction investigation of inversiondomains in GaN, J. Alloys Compounds, 401, 199 (2005).

145. M. Kawabe and T. Ueda, Self-annihilation of antiphase boundary in GaAs onSi(100) grown by molecular beam epitaxy, Jpn. J. Appl. Phys., Part 2, 26, L944(1987).

146. Y. Li and L.J. Giling, Growth by atmospheric pressure OMVPE and x-rayanalysis of ZnTe epilayers on III-V substrates, J. Cryst. Growth, 163, 203 (1996).

147. K. Morizane, Antiphase domain structures in GaP and GaAs epitaxial layersgrown on Si and Ge, J. Cryst. Growth, 38, 249 (1977).

148. S.N.G. Chu, S. Nakahara, S.J. Pearton, T. Boone, and S.M. Vernon, Antiphasedomains in GaAs grown by metalorganic chemical vapor deposition on silicon-on-insulator, J. Appl. Phys., 64, 2981 (1988).

149. P.N. Uppal and H. Kroemer, Molecular beam epitaxial growth of GaAs on Si(211), J. Appl. Phys., 58, 2195 (1985).

150. www.msm.cam.ac.uk/doitpoms.



http://www.msm.cam.ac.uk

http://www.msm.cam.ac.uk

75

3

Heteroepitaxial Growth

3.1 Introduction

Of the many available epitaxial growth techniques, molecular beam epitaxy(MBE) and metalorganic vapor phase epitaxy (MOVPE) have emerged asgeneral-purpose tools for heteroepitaxial research and commercial produc-tion. This is because these methods afford tremendous flexibility and theability to deposit thin layers and complex multilayered structures with pre-cise control and excellent uniformity. Together, MBE and MOVPE accountfor virtually all production of compound semiconductor devices today.

MBE is an ultra-high-vacuum (UHV) technique that involves the impinge-ment of atomic or molecular beams onto a heated single-crystal substratewhere the epitaxial layers grow. The source beams originate from Knudsenevaporation cells or gas-source crackers. These can be turned on and off veryabruptly by shutters and valves, respectively, providing atomic layer abrupt-ness. Because MBE takes place in a UHV environment, it is possible toemploy a number of

in situ

characterization tools based on electron or ionbeams. These provide the crystal grower with immediate feedback, andimproved control of the growth process. Another advantage of MBE is flex-ibility; nearly all semiconductors can be grown, including III-V and II-VIsemiconductors; Si, Ge, and Si

1–x

Ge

x

alloys; and SiC and Si

1–x–y

Ge

x

C alloys.However, III-phosphides are difficult to grow by MBE, and alloys involvingAs and P are especially troublesome. Other drawbacks of MBE are the initialhigh cost and maintenance requirements of the UHV system and also thelimited throughput.

MOVPE is a vapor phase epitaxial process that is carried out at atmo-spheric or reduced (e.g., 0.1 atm) pressure using metalorganic precursors.Often hydride sources are used in conjunction with the metalorganic chem-icals; occasionally even elemental sources are used. Like molecular beamepitaxy, MOVPE provides excellent control over the growth of thin layersand multilayered structures, including quantum well devices and superlat-tices. However, the lack of UHV conditions precludes the

in situ

use ofelectron or ion beam characterization tools. Nonetheless, optical

in situ

char-



76


acterization methods have been utilized to some extent. Another disadvan-tage of MOVPE is the use of highly toxic source chemicals, especially arsine(AsH

3

) and phosphine (PH

3

). These flammable, explosive, and highly toxicgas sources are stored at high pressure in large quantities, raising a numberof safety concerns in a production environment. In some cases, these hydridesources have been replaced with less toxic liquid sources (such as tertiarybutyl arsine or TBAs

)

contained in low-pressure bubblers.Vapor phase epitaxial (VPE) processes are also used for the growth of

column IV semiconductors, including Si, Ge, and Si

1–x

Ge

x

alloys and SiCand Si

1–x–y

Ge

x

C alloys. These VPE processes utilize similar equipment andshare some of the characteristics of MOVPE, but do not involve metalor-ganic precursors. Instead, hydride and halide sources are used. The use ofall hydride sources leads to an irreversible process with abrupt interfaces.On the other hand, any involvement of halide precursors generally resultsin a reversible process that is less suitable for the growth of multilayeredstructures.

This chapter will provide a brief overview of the important epitaxial pro-cesses of VPE and MBE. MOVPE is considered a special case of VPE. Theremaining sections of the chapter describe the growth of particular materials,from the viewpoint of heteroepitaxy.

3.2 Vapor Phase Epitaxy (VPE)

Vapor phase epitaxial (VPE) growth is accomplished by passing gaseoussource chemicals over a heated single-crystal substrate, where epitaxialgrowth occurs. Atmospheric or reduced (~0.1 atm) pressure may be used.In either case, a carrier gas such as hydrogen usually makes up most of theflow (and therefore pressure) in the reactor.

Vapor phase epitaxial growth is extremely flexible and allows the growthof nearly every semiconductor material of interest. The availability of ultra-pure sources and careful reactor design allow the growth of materials withlevels of purity matching those of all other epitaxial techniques. It has alsoproved possible to design reactors capable of handling multiple wafers inone run, while maintaining excellent uniformity, both across wafers and fromwafer to wafer. This scalability has led to its wide commercial application.

3.2.1 VPE Mechanisms and Growth Rates

The vapor phase epitaxial growth of a crystal involves a series of basic steps:

1. The source chemicals are transported in the vapor phase to theheated substrate.




77

2. Source molecules diffuse to the growing surface, where they arechemisorbed or physisorbed.

3. Adsorbed species on the surface react to form the solid crystal.4. Reaction products diffuse from the surface.5. Reaction products are carried away in the flowing gas stream.

The slowest of these five steps will determine the growth rate. Typically, therate limiter is either step 2 or step 3, and these two situations are called masstransfer limited and reaction rate limited, respectively.

Consider the transport of a single reactant to the growing surface. (Usually,one reactant is provided in excess for the growth of a binary compound sothat this assumption remains useful.) The flux of this species to the surfaceat a particular point is given by Henry’s law:

(3.1)

where is the concentration of the reactant in the gas phase, is theconcentration of the reactant at the surface, and

h

is the gas phase masstransfer coefficient.

Suppose the reaction rate is linear; then

(3.2)

where

k

is the surface reaction rate constant. Usually, this rate is thermallyactivated so that

(3.3)

where is the activation energy for the process. Typical activation energiesare in the range of 25 to 100 kcal/mole (1.1 to 4.3 eV/molecule).

Under steady-state conditions, the two fluxes above may be equated. Com-bining these equations, we can determine the growth rate as

(3.4)

where

n

is the number of atoms (or molecules) per unit volume in thegrowing crystal. For Si, , and for GaAs, .

At low temperatures, so that

(3.5)

j h N Ng= −( )0

N g N0

j kN= 0

k k E kTa= −0 exp( / )

Ea

gjn

N

nhk

h kg= =

+⎛⎝⎜

⎞⎠⎟

n = × −5 1022 3cm n = × −2 2 1022 3. cmk h<<

gkN

ng≈



78


This is referred to as reaction-rate-limited growth. Under these conditions,the growth rate is a strong function of temperature. Also, because the reactionrate is sensitive to the surface conditions, the growth rate depends on theorientation of the crystal substrate. This can result in faceted growth, whichis usually undesirable, but can be advantageous for the implementation ofepitaxial lateral overgrowth (ELO).

At high temperatures, so that

(3.6)

This situation is known as mass-transfer-limited growth (or diffusion-limitedgrowth). Under mass-transfer-limited conditions, the growth rate is inde-pendent of the crystal orientation and nearly independent of temperature.There is a slight temperature variation (with an activation energy of 3 to 8kcal/mole) due to the temperature dependence of the diffusivity. UsuallyVPE reactors are designed to operate in the mass-transfer-limited regime.However, this is not always possible due to constraints imposed by thesubstrate or source chemical.

Thermodynamic considerations may also be important in determining thegrowth rates in the mass-transfer-limited regime. This is illustrated in Figure3.1, which shows the general behavior for the cases of the (a) endothermicand (b) exothermic processes. For the endothermic process, which involvesa positive heat of reaction, the growth rate increases monotonically withincreasing temperature. In the reaction-rate-limited regime, the activationenergy is large, typically 25 to 100 kcal/mole. But in the mass-transfer-limited region, there is only a slight variation of the growth rate with tem-perature (3 to 8 kcal/mole). In contrast, for the exothermic process, which

FIGURE 3.1

Growth rate (log scale) vs. reciprocal of temperature for an (a) endothermic process and an (b)exothermic process.

h k<<

ghN

ng≈

Temperature−1

(b)

Gro

wth

rate

(log

scal

e)

Temperature−1

(a)

Gro

wth

rate

(log

scal

e)




79

is favored at lower temperatures, the growth rate decreases rapidly withincreasing temperature under conditions of mass-transfer-limited growth.Many processes of interest for heteroepitaxy are either endothermic (chlorideVPE of Si) or pyrolytic (hydride VPE of Si or SiC, MOVPE) in nature, andso display the general behavior shown in Figure 3.1a.

3.2.2 Hydrodynamic Considerations

The simplified model of Section 3.2.1 fails to reveal many of the detailsassociated with the fluid dynamics of vapor phase epitaxy. Nor does it giveguidance in the determination of the mass transfer coefficient. However, inthe typical case of laminar flow, simple analytical solutions exist for thedetermination of the growth rate under mass-transfer-limited conditions.

The nature of the gas flow in an epitaxial reactor can be understood basedon a study of gas flow in simple pipes.

1

This flow can be characterized bythe unitless Reynold’s number, given by

(3.7)

where

d

is the pipe diameter,

v

is the gas velocity in the pipe, is the absoluteviscosity, and is the gas density. The viscosity of hydrogen, the most com-monly used carrier gas, varies from about dyn cm

–1

s

–1

(200

μ

Poise)at 700

°

C to about dyn cm

–1

s

–1

(250

μ

Poise) at 1200

°

C. The densityof hydrogen at atmospheric pressure varies from about g cm

–3

at700

°

C to g cm

–3

at 1200

°

C. Empirically, it is found that the transi-tion from laminar flow occurs in the range . Typical epitax-ial reactors operate under laminar flow conditions, with .

Consider the idealized reactor shown in Figure 3.2, with a recessed sus-ceptor and a constant cross-sectional area. The reactor height is

h

. Supposethat a single reactant contributes to the growth and the concentration of thisreactant is at the entrance to the reactor. The growth is assumed to takeplace under mass-transfer-limited conditions so that .

The growth rate will be proportional to the flux of reactant species arrivingat the substrate surface. If this flux is controlled by diffusion, then

(3.8)

where

N

is the actual concentration of the reactant in the gas phase at a pointabove the susceptor and

D

is the diffusivity of the reactant species in thecarrier gas. If the gas velocity is assumed to be constant above the susceptor,then under steady-state conditions (with all time derivatives equal to zero)the two-dimensional continuity equation for the reactant species is

Ndv

R = ρμ

μρ

200 10 6× −

250 10 6× −

2 5 10 5. × −

1 65 10 5. × −

2000 3000< <NR

NR ≈ 30

N g

N0 0≈

j DNy

= − ∂∂



80


(3.9)

If diffusion is neglected in the flow direction, then

(3.10)

The boundary conditions are

(3.11)

Solving, the flux of reactant is found to be

(3.12)

FIGURE 3.2

Concentration of reactants in a horizontal reactor and the growth rate. (Adapted from Ghandhi,S.K.,

VLSI Fabrication Principles

, 2nd ed., John Wiley & Sons, New York, 1994. With permission.)

(b)

x0−

x0+

x1 x2

N

y

Ng

h(c)

g

x

(a)

y

0

x0− x0

+ x1 x2

x

h

Susceptor

∂∂

= ∂∂

+ ∂∂

− ∂∂

=Nt

DN

xD

Ny

vNx

2

2

2

20

02

2= ∂

∂− ∂

∂D

Ny

vNx

N N x y h

N x y

Ny

x y h

g= = < <

= < =∂∂

= < =

0 0

0 0 0

0 0

;

,

,

jDN

hDx r

vhg

r

= − − +⎛

⎝⎜⎞

⎠⎟=

∞

∑2 2 14

2 2

20

exp( )π




81

If the flux of reactant species is envisioned to flow through a diffusionboundary layer in which its concentration varies linearly from to 0, thenthis boundary layer thickness is given by

(3.13)

Under the simplifying assumption that , we obtain

(3.14)

so that the diffusion boundary layer thickness is given by

for (3.15)

and the growth rate may be estimated from

(3.16)

It is apparent from this analysis that the achievement of uniform growthover a large wafer requires a uniform diffusion boundary layer thickness. Thiscan be achieved by tilting the susceptor, so that the cross section of the reactordecreases with x. Sometimes, this is done in conjunction with adjustments tothe total flow and reactor pressure in order to achieve the desired result.

The increased performance of computers has enabled detailed numericalcomputations of the mass and heat flows in epitaxial reactors. These calcu-lations take into account continuity, conservation of momentum(Navier–Stokes equations), conservation of energy, and conservation of massof diffusing species. Thus, the growth rate and uniformity (in thickness andcomposition) can be predicted for a reactor design before it is built and tested.Nonetheless, the simple boundary layer picture allows one to construct astarting point for the reactor design, which can then be fine-tuned by theuse of computation.

3.2.3 Vapor Phase Epitaxial Reactors

A VPE reactor comprises a gas delivery system, a reaction chamber, and aneffluent handling system; a basic setup is shown schematically in Figure 3.3.

N g

δ πD

r

xh Dx r

vh( ) exp

( )= − +⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

=

∞

∑22 1

4

2 2

20 ⎦⎦

⎥⎥

−1

( / )h v D x2 π >

j NDv

xg≈ −π

δ πD x

Dxv

( ) ≈ δD x h( ) ≤

gN D

ng

D

≈δ



82


Welded stainless steel construction with metal gasket fittings is used toachieve the necessary leak integrity.

Typically, ultra-high-purity (UHP) H

2

with 7N purity (seven nines purity,or 99.99999% pure) is used as the carrier gas in a VPE reactor. This gas maybe purchased in UHP form, or commercial grade hydrogen (3N5, or 99.95%pure) may be purified by diffusion through a palladium–silver membraneor by the use of a purifying resin. Other carrier gases may also be used, suchas UHP N

2

, He, or Ar. Although these gases preclude the use of a palladiumcell, resin purifiers are available for them.

The source chemicals may be gaseous, liquid, or solid. Gas sources (suchas SiH

4

and AsH

3

) may be obtained in pure form or diluted in hydrogen, inhigh-pressure cylinders. Liquid sources (such as SiH

2

Cl

2

or TMGa) are typ-ically obtained in ultra-high-purity form, in stainless steel bubblers. Solidsources (such as TEIn) may also be obtained in bubbler vessels. However, itis difficult to obtain good run-to-run repeatability with these. Occasionally,vapor phase sources are created

in situ

, as in the case of GaCl, which wasused in hydride and halide epitaxial processes for GaAs.

For a gaseous source, the flow is metered precisely by an electronic massflow controller (MFC). The MFC has a built-in heated capillary. The mea-surement of the temperature difference across the capillary allows the deter-mination of the mass flow with a precision of ±0.5%. MFCs have built-inclosed-loop control systems and metering valves, so they can maintain theflow at a desired set point. Typically MFCs are calibrated for use with pureH

2

or N

2

. Hydrogen calibration is entirely adequate for a dopant gas that isdiluted to a few 100 ppm in H

2

. In the case of a pure gaseous source suchas AsH

3

, the MFC must be calibrated specifically.

FIGURE 3.3

Epitaxial reactor.

MFC

MFC

MFC

MFC

UHP H2

Bubbler source

Gaseous source

Mixingmanifold

To vent

RF induction heating

Susceptor To vent

Wafer




83

Liquid sources are transported to the reactor by a carrier gas, usually H

2

,which is flowed through a stainless steel bubbler arrangement like the oneshown in Figure 3.4. The carrier gas is metered precisely by an MFC placedupstream of the bubbler. If the mass flow of the carrier gas is , the vaporpressure of the liquid source is , and the total pressure in the bubbler is

, then the mass flow of the source is given by

(3.17)

This expression assumes that the carrier gas bubbles have sufficient residencetime in the liquid to become saturated with its vapor, and closely approxi-mates a real bubbler application. Typically, a three-valve arrangement is usedaround the bubbler so that the carrier gas can be made to bypass the bubblerwhen the source is not in use.

The vapor pressures of liquid sources are usually fit by the expression

(3.18)

where is the vapor pressure over the liquid (in torr),

T

is the absolutetemperature, and

A

and

B

are empirical constants. For convenience, thetemperature of a bubbler source is usually set to yield a source vapor pressureof 5 to 50 torr by means of a temperature-controlled bath. In some cases, thebubbler temperature must be kept above room temperature, which necessi-tates heating of the downstream lines to prevent condensation of the source.

FIGURE 3.4

Liquid source bubbler.

FH2 FH2 + FS

PTOT

FH2

PS

PTOT

F FP

P PS HS

TOT S

=−

⎛⎝⎜

⎞⎠⎟2

log /10 P A B TS = −

PS



84


The carrier gas and vapor phase sources are brought to a mixing manifoldprior to their injection into the reactor chamber. In the mixing manifold, gasflows are switched between pressure-balanced vent and reactor lines. Thus,each flow may be stabilized to the vent line before being switched to thereactor. Precise pressure balancing, using a differential pressure transducerand a control valve, avoids unwanted flow transients.

VPE reactors are of the horizontal, vertical, or barrel types. The horizontalconfiguration is the simplest and is often used in research. The reactionchamber is a quartz tube, flanged at one end to facilitate loading and unload-ing of wafers. The substrate wafers are held by recesses in a graphite sus-ceptor, which is usually tilted at a slope of 7 to 10

°

to improve the thicknessuniformity. The configuration is so named because of the horizontal gas flowin the tube.

The vertical reactor utilizes a vertical flow of gases, perpendicular to thesurface of the wafers. Inherent in this geometry is a stagnation point at thecenter of the susceptor, where the gas velocity is zero. This tends to promoterecirculation unless the susceptor is rotated at a high speed (>1000 rpm).Rotation also serves to improve the axial uniformity of growth. The flow,pressure, and rotation rate must be optimized for radial uniformity. Some-times, complex planetary rotation systems are employed as well.

The barrel reactor can handle many wafers in a single run and achieveshigh throughput. Wafers are held in shallow depressions within the steeplysloped susceptors, and the gas flow is nearly parallel to their surfaces. Thus,the barrel reactor geometry is similar to that of the horizontal reactor, butrotated 90

°

.Heating of an epitaxial reactor may be accomplished by radio frequency

(rf) induction, infrared lamps, or resistive heaters. In cold-wall reactors usedfor endothermic and pyrolytic reactions, rf induction or infrared lamps aretypically used. Internal resistive heaters are occasionally used, but the mate-rials must be chosen carefully to avoid metallic contamination. Externalresistive heaters avoid this problem but are only applicable to hot-wallreactors used for exothermic processes. Temperature control with ±2

°

Cprecision is normally adequate if the growth is mass transfer limited. Mea-surement of the temperature can be achieved using optical pyrometry or,for low-temperature reactors (<900

°

C), thermocouples may be embeddedin the susceptor.

Susceptors are usually made from machined graphite. At temperaturesabove 1300

°, however, the hydrogen carrier gas will react with graphite andetch its surface. For this reason, SiC-coated susceptors are often employed inreactors intended for high-temperature operation. Coatings of SiC or pyro-lytic BN are sometimes used in lower-temperature reactors as well, to avoidthe outgassing affects associated with the porosity of uncoated graphite.

Epitaxial reactors may operate at atmospheric or reduced (~0.1 atm) pres-sure. Low-pressure operation reduces the surface coverage of adsorbed spe-cies, increasing their mobility and allowing high-quality growth at reducedtemperatures (50 to 100°C lower than for atmospheric growth). Reduced



Heteroepitaxial Growth 85

pressure can also decrease the tendency for recirculation under otherwisesimilar conditions. In low-pressure reactors, pressure control is achievedusing a mechanical vacuum pump and a butterfly valve. Pressure measure-ment may be by capacitance or optical manometers.

The effluent is typically treated by activated charcoal absorption units orliquid scrubbers before being vented to the outside. These systems requirefrequent service as well as expensive waste disposal.

3.2.4 Metalorganic Vapor Phase Epitaxy (MOVPE)

The MOVPE process was developed in the late 1960s by Manasevit,2–4 whofirst demonstrated its use for the epitaxy of Ga-V compounds. Subsequentlythe process has been adapted to nearly all III-V and II-VI semiconductors,including the antimonides, arsenides, phosphides, nitrides, sulfides,selenides, and tellurides, and also ternary and quaternary alloys. MOVPE-grown material is of extremely high purity: this epitaxial method has pro-duced the highest purity InP produced by any method and GaAs that is aspure as that grown by any technique. Specially designed reactors have alsomade possible the growth of very abrupt interfaces and multilayered struc-tures of the type necessary for quantum layer devices such as laser diodesand high-speed transistors. These developments, and the ease of scaling theMOVPE process to high throughput, have made it important for commercialproduction as well as laboratory research.

MOVPE goes by a number of names, including organometallic vapor phaseepitaxy (OMVPE), metalorganic chemical vapor deposition (MOCVD), orga-nometallic chemical vapor deposition (OMCVD), and occasionally organo-metallic epitaxy (OME). CVD is a more general term that applies tononcrystalline films; as such, the more specific term VPE should be used torefer to epitaxy. OMVPE is preferred by many researchers because it isconsistent with the normal chemical nomenclature. MOVPE is the namechosen by the international conference and will be used throughout thisbook. It is important to realize that these terms are used interchangeably inthe literature and are not meant to refer to process differences.

MOVPE is carried out in a reactor of the type shown schematically inFigure 3.3. Source chemicals are transported to the reactor by a carrier gas,where they react heterogeneously at the surface of a heated single-crystalsubstrate. In the growth of a binary semiconductor, one or both sourcechemicals may be metalorganic compounds. These are typically liquids atroom temperature and are transported to the reactor by flowing a carriergas through a stainless steel bubbler. Gaseous sources may also be used;these are contained in high-pressure cylinders in either pure or diluted form.Ternary or quaternary alloys may be grown by introducing additional sourcechemicals. Changes in composition can be realized by ramping/switchingthe source flows. Layers may also be doped by the introduction of smallconcentrations of the appropriate sources.




MOVPE processes are pyrolytic in nature. As a consequence, cold-wallreactors are used almost exclusively. Also, the irreversible nature of MOVPEallows the growth of extremely abrupt interfaces.

The variety of metalorganic source chemicals has increased greatly overthe years due to the success of the method and resulting market demand.In general, the sources are molecules of the type MRn, where M representsa metal atom and R represents an organic radical. It is common practice torefer to the organic groups using M, E, NP, IP, NB, IB, TB, A, and Cp formethyl, ethyl, n-propyl, i-propyl, n-butyl, i-butyl, t-butyl, allyl, and cyclo-pentadienal, respectively. M, D, and T are used to denote mono-, di-, andtri-, respectively. Thus, TMGa represents trimethylgallium and DETe refersto diethyltelluride.

The metalorganic precursors are generally liquids at room temperature,contained in stainless steel bubbler vessels. A few are solid at room tem-perature, but these can be used with a bubbler arrangement as well. Themelting points, boiling points, and vapor pressure parameters A and B aregiven in Table 3.1 to Table 3.4 for sources of elements from columns II, III,V, and VI, respectively. As a general rule, the vapor pressures are highestfor the lightest molecules.

The decomposition characteristics of the alkyl source molecules are deter-mined in part by the strength of the metal–carbon bond. This bond energy

TABLE 3.1

Melting Points, Boiling Points, and Vapor Pressure Data for Metalorganic Sources of Column II Elements

PrecursorMelting Point

(°C)Boiling Point

(°C)Vapor Pressure P (torr) @ T

(°C)A B (K)

DMZn –42 46 7.802 1560 124 @ 0°CDEZn –28 118 8.280 2109 3.6 @ 0°CDMCd –2 106 7.764 1850 9.7 @ 0°C

Note: log10 P(torr) = A – B/T.

TABLE 3.2

Melting Points, Boiling Points, and Vapor Pressure Data for Metalorganic Sources of Column III Elements


(°C)Boiling Point


(°C)A B (K)

TMAl 15 126 8.224 2134.83 2.2 @ 0°CTEAl –52.5 186 10.784 3625 0.5 @ 55°CTMGa –15.8 55.8 8.501 1824 66 @ 0°CTEGa –82.5 143 9.172 2532 3.4 @ 20°CDEGaCl –7 — 8.78 2815 0.5 @ 60°CTMIn 88 135.8 10.520 3014 0.3 @ 0°CTEIn –32 184 1.2 @ 40°C





determines the stability of the molecule with respect to decomposition bythe removal of organic radicals (free radical homolysis). Therefore, it oftendetermines the activation energy for reaction-rate-limited growth. In general,the metal–carbon bond strength decreases with the number of carbonsbonded to the central carbon of the molecule (methyl > ethyl > i-propyl > t-butyl > allyl). In some situations, this means that a lower growth temperaturemay be used with an i-propyl source than with a methyl source. Bondstrengths for some of the common alkyl precursors are provided in Table 3.5.

Generally, the alkyls of column II and column III elements are Lewis acids(electron acceptors), whereas the alkyls of column V and column VI atomsare Lewis bases (electron donors). It is possible for a gas phase reaction tooccur between alkyls with Lewis acid–Lewis base character, resulting in anadduct. If the adduct so produced is a low-vapor-pressure molecule, it maynot contribute to epitaxial growth, and in fact, it may give rise to fouling of

TABLE 3.3

Melting Points, Boiling Points, and Vapor Pressure Data for Metalorganic Sources of Column V Elements


(°C)Boiling Point


(°C)A B (K)

TMP –84 38 7.7627 1518 381 @ 20°CTEP –88 127 8.035 2065 46.5 @ 50°CTBP 4 54 7.586 1539 141 @ 10°CTMAs –87.3 50 7.3936 1456 238 @ 20°CTEAs 140 5 @ 20°CDMAs 36.3 7.532 1443 176 @ 0°CDEAs 102 7.339 1680 40 @ 20°CTBAs –1 65 7.243 1509 32 @ –10°CTMSb –87.6 80.6 7.73 1709 48.9 @ 10°CTESb –98 160 7.90 2183 4 @ 25°CTMBi –107.7 110 7.628 1816 27 @ 20°C


TABLE 3.4

Melting Points, Boiling Points, and Vapor Pressure Data for Metalorganic Sources of Column VI Elements


(°C)Boiling Point


(°C)A B (K)

DES –100 91 ± 1 8.184 1907 47 @ 20°CDTBS — 149 ± 2 — — —DMSe 57DESe — 108 7.905 1924 7.2 @ 0°CDMTe –10 92 (82) 7.97 1865 65 @ 30°CDMDTe 220 6.94 2200 0.26 @ 23°CDETe — 137 7.99 2093 7.1 @ 20°CDIPTe — — 8.29 2309 2.6 @ 20°C





the reactor. Such a parasitic reaction is highly undesirable. On the other hand,some adducts can contribute to growth, and these (such as TMIn-TEP) aresometimes used intentionally.

3.3 Molecular Beam Epitaxy (MBE)

MBE is an ultra-high-vacuum (UHV) technique that involves the impingementof atomic or molecular beams onto a heated single-crystal substrate where theepitaxial layers grow.5 The source beams originate from Knudsen evaporationcells or gas-source crackers. These can be turned on and off very abruptly byshutters and valves, respectively, providing atomic layer abruptness.

Because MBE takes place in a UHV environment, it is possible to employa number of in situ characterization tools based on electron or ion beams.These provide the crystal grower with immediate feedback, and improvedcontrol of the growth process.

MBE has been developed to the point where nearly every semiconductorof interest may be grown using the technique, including III-V and II-VIsemiconductors; Si, Ge, and Si1–xGex alloys; and SiC and Si1–x–yGexC alloys.However, III-phosphides are difficult to grow by MBE, and alloys involvingAs and P are especially troublesome. Other drawbacks of MBE are the initialhigh cost and maintenance requirements of the UHV system and also thelimited throughput. These drawbacks are offset to a large extent by theprecise control and in situ characterization, so that MBE is used extensivelyfor commercial device production at this time.

An MBE reactor involves a number of source cells arranged radially infront of a heated substrate holder, as shown in Figure 3.5. The source cells

TABLE 3.5

Bond Strengths for Common Alkyl Precursor Molecules

PrecursorD1

(kcal/mole)D2

(kcal/mole)Dave

(kcal/mole)

DMZn 51 (54) 47 42DMCd 53 46 33TMAl 65 66, 61TEAl 58TMGa 59.5 35.4 59TEGa 57TMIn 47TMP 66, 63TMAs 62.8 55TMSb 57 57 52, 47DES 65




supply all atoms necessary for the growth and doping of the required semi-conductor layers; six or more cells may be required. The simplest type ofsource cell is a thermal evaporator (Knudsen cell), but other, more elaborateschemes have been developed for some atoms. A basic requirement for MBEgrowth is line-of-sight source impingement. This means that the evaporatedsource atoms must have mean free paths greater than the source-to-substratedistance, which is typically 5 to 30 cm. This requirement places an upperlimit on the operating pressure for an MBE reactor.

The mean free path for an evaporated particle (atom or molecule) may beestimated if it is assumed that all other particles in the system are at rest.Suppose the evaporated particle is moving at a velocity c, and all particleshave a round cross section with diameter σ. Two particles that pass at adistance of σ or less will collide. Therefore, each particle can be consideredto have a collision cross section of , and the collision volume swept outby a particle in time dt is . If N is the volume concentration of particles,then the collision frequency will be

(3.19)

and the mean free path will be

(3.20)

FIGURE 3.5MBE reactor. (Reprinted from Henini, M., Thin Solid Films, 306, 331, 1997. With permission.Copyright 1997, Elsevier.)

Beam fluxmonitoring gauge

To substrate heater supplyand variable speed motor

Liquid nitrogencooled panels RHEED gun

Gate valve

Sample transfermechanism

View port

Rotating substrateholder

Quadrupolemass spectrometer

Fluorescentscreen

Effusioncells

Shutters

πσ2

πσ2cdt

f N cdt= πσ2

λ πσ= = −cf

N( )2 1




A more accurate calculation of the mean free path may be made assumingthat all of the particles are in motion. Based on this, the mean free path foran evaporated particle is

(3.21)

where P is the pressure.Typical values of the cross section diameter σ range from 2 to 5 Å, so that

the mean free path is about 103 cm at a pressure of 10–5 torr. This pressuretherefore represents an approximate upper limit for the system pressureduring growth, if the beam nature of the sources is to be maintained.

The requirement on the base pressure is considerably more stringentand is set by purity requirements. If the grown films are to have no morethan 10–5 (10 ppm) contaminants, then the base pressure should be nomore than 10–10 torr. Achievement of the necessary ultrahigh vacuumrequires the use of a stainless steel chamber with metal gaskets. The systemmust be load-locked, so that it is opened to the atmosphere only formaintenance. Any exposure of the chamber to air must be followed by along bake-out to remove adsorbed contaminants. During growth, thechamber walls must be cooled to cryogenic temperatures by means of aliquid nitrogen shroud, in order to further reduce evaporation from thislarge surface area.

Growth of pure layers by MBE also requires the use of oil-free pumpingin the UHV system. Cryogenic sorption pumps, titanium sublimation ionpumps, and turbomolecular pumps are used for this reason.

The simplest source cells are thermal evaporators, called effusion cells orKnudsen cells. High-purity elemental sources are used, and one cell isneeded for each element. Typically, the effusion cells are made of pyrolyticboron nitride with tantalum heat shields. The source temperatures are main-tained precisely (±0.1°C) to control the flux of evaporating atoms. Due tothe inability to rapidly ramp up or down the cell temperature, a shutter isused to turn each beam on and off.

The flux of atoms from such an effusion cell may be calculated using thekinetic theory of gases.6 From this treatment it can be shown that the evap-oration rate from a surface area is given by

(3.22)

where P is the equilibrium vapor pressure of the source at the effusion celltemperature T and m is the mass of the evaporant. In terms of the molecularweight of the species, M, the effusion rate is

λ πσπσ

= =−( )NkT

P2 1

22

2

Ae

dNdt

A P

kTme e=

2π




(3.23)

where is Avogadro’s number ( ). Simplifying,

(3.24)

where P is the pressure in torr. Because the equilibrium vapor pressure Pvaries exponentially with temperature, the effusion cell temperature mustbe controlled to within ±0.1°C in order to keep the effusion rate within a±1% tolerance.

The flux of evaporant arriving at the substrate surface can be calculatedfrom the evaporation rate at the effusion cell by

(3.25)

where l is the distance from the effusion cell to the substrate and is theangle between the beam axis and the normal to the substrate.

The model outlined above assumes a full effusion cell so that evaporationoccurs at its mouth. In practice, the cell depletes with time, and this causesa fall-off of the impingement rate and a change in the beam profile7 at thesubstrate. This effect can be mitigated to some extent by the use of taperedeffusion cells.

Usually the evaporation crucibles have a 1 cm2 evaporation surface andare located 5 to 20 cm from the substrate. Typical source pressures are 10–3

to 10–2 torr, resulting in the delivery of 1015 to 1016 molecules cm–2 s–1. Thiscorresponds to a growth rate on the order of one monolayer per second,assuming a unity sticking coefficient for the impinging atoms.

Thermal effusion sources are switched on and off by means of pneumat-ically controlled shutters. A problem associated with this scheme is thechange in thermal loading on the cell upon opening or closing the shutter.This causes unwanted temperature transients in the cell, which result inrather large variations (up to 50%) in the beam flux immediately after theshutter is opened.

Another disadvantage of thermal effusion sources is the inability to rampthe beam flux rapidly with time. Here the limitation is due to the thermalmass of the effusion cell. This places an upper limit on the rate at which thecomposition may be ramped in a ternary or quaternary alloy. Whereas this

dNdt

A P

kTM Ne e

A

=2π /

NA 6 022 1023 1. × −mole

dNdt

A P

MTe e= ×3 51 1022. molecules/s

jl

dNdt

A P

l MT

e

e

=

= ×

cos

.cos

θπ

θ

2

222

1 117 10 moleculees cm s− −2 1

θ




restriction is important in the growth of graded device structures, it is usuallynot a problem in graded buffer layers. In gas-source MBE (GSMBE), thesources are controlled using mass flow controllers, which allow much morerapid ramping.

Electron beam evaporation sources have been used with some elementssuch as Si. Here the elemental source is contained in a water-cooled crucibleand evaporation occurs locally at the surface by the impingement of anelectron beam. On and off control can be achieved by blanking of the electronbeam. Scanning of the beam allows the realization of an extended-area sourcewith characteristics similar to those of the thermal effusion cells.

3.4 Silicon, Germanium, and Si1–xGex Alloys

Si, Ge, and their alloys may be grown by either VPE or MBE. Si (001)substrates are used almost exclusively for the epitaxy of these materials.Therefore, the in situ removal of the native oxide is a critical step prior toepitaxy. In the case of MBE, this can be achieved by flashing to a temperatureup to 1200°C in the high vacuum. Prior to VPE growth, the oxide layer canbe removed by a bake-out in hydrogen.

A number of sources can be used for Si VPE, including silicon tetrachloride(SiCl4), trichlorosilane (SiHCl3), dichlorosilane (SiH2Cl2), and silane (SiH4);however, only dichlorosilane and silane are in common use at this time. Thisdichlorosilane process is heterogeneous (it requires two molecules of SiCl2)and surface catalyzed (it occurs only in the presence of the silicon surface).It is also reversible and is accompanied by etch-back and autodoping pro-cesses, whereby atoms from the grown crystal are etched and returned tothe gas phase. These processes are undesirable in multilayered epitaxialdevice structures, because they compromise the abruptness of heterojunc-tions and also lead to nonideal doping profiles in p-n junctions. However,they can be suppressed by a reduction of the growth temperature.

The silane process is irreversible due to the absence of chlorine. Comparedwith the chlorosilanes, SiH4 epitaxy can be carried out at a lower tempera-ture but is extremely sensitive to oxidizing impurities. Silane epitaxy there-fore mandates the use of load locks and careful bake-out procedures toavoid the formation of silica dust, which is detrimental to layer morphology.A unique aspect of the silane process is that homogeneous, gas phasenucleation is possible with this source.8 The dusting that results fromhomogenous nucleation can also deteriorate layer quality. However, thisproblem can be minimized by the use of low pressure, high gas velocities,and reduced temperature.

The vapor phase epitaxial growth of Ge has been achieved using a numberof halogenic sources,9–11 including germanium tetrabromide (GeBr4), germa-




nium tetrachloride (GeCl4), and germanium diiodide (GeI2), as well as thehydride, germane (GeH4). The halide processes are reversible, and thereforeaccompanied by undesirable autodoping effects that limit the abruptness ofjunctions. In addition, the iodide process is complex, requiring three separatetemperature zones. For these reasons, the germane process is the preferredmethod of growth for germanium today and is used for the realization ofgermanium-on-insulator (GOI).

Si1–xGex epitaxy may be carried out using a mixture of silicon and germa-nium sources in the vapor phase. The gas phase mole fraction is used tocontrol the resulting solid phase mole fraction x. Practical systems for Si1–xGex

VPE utilize SiH2Cl2 + GeH4 or SiH4 + GeH4. In the case of SiH2Cl2 + GeH4,the solid composition x depends on the ratio of the gas-source flows by12

(3.26)

Selective growth of Si1–xGex may be achieved by the use of SiH2Cl2 + GeH4

+ HCl; growth proceeds on bare silicon surfaces but not on dielectric filmssuch as SiO2 or silicon nitride. This can be utilized in patterned or nanohet-eroepitaxial growth schemes. Commercial Si1–xGex VPE reactors provide forthe use of either combination of sources, to allow either nonselective (blan-ket) or selective growth. However, growth over a dielectric film is polycrys-talline and should properly be referred to as chemical vapor deposition(CVD), not vapor phase epitaxy.

In addition to Si1–xGex, the carbon-containing alloys Si1–yCy and Si1–x–yCyGex

are of interest for bandgap engineering of heteroepitaxial devices on Siwafers. These materials may be grown by the addition of a carbon precursorto the growth chemistry, and practical VPE systems employ monomethylsi-lane (SiCH6) for this purpose. Due to the extremely low solubility of C in Si(<10–6), all practical carbon-containing alloys are necessarily metastable13 andmust be grown at low temperatures.

Ultra-high-vacuum (UHV) vapor phase epitaxy14 has also been used togrow Si1–xGex alloys, with a growth pressure of ~10–3 atm. Under UHVconditions, good-quality layers may be grown with a cold-wall reactor andthe homogeneous nucleation is suppressed. Any combination of the sourcesSiH4, Si2H6, GeH4, and Ge2H6 may be used, but the use of Si2H6 was reportedto give better surface morphology.

Si1–xGex alloys across the entire compositional range may be grown by MBEusing e-beam sources. Typical temperatures range from 500 to 900°C. Usually,films with higher Ge content are grown at lower temperatures, keeping thegrowth temperature at approximately 60 to 70% of the melting temperature.Temperature ramping may be employed during the growth of a graded layer.A unique aspect of MBE growth is the ability to grow Si1–xGex films at verylow temperatures15 (300 to 400°C); the altered kinetics of lattice relaxation

xx

X

XGeH

SiH Cl

2

12 66 4

2 2−

= .




appears to enable the growth of layers with reduced threading dislocationdensities. Kinetic models for lattice relaxation are described in Chapter 5.

In the fabrication of multilayered device structures, there is a tendency forthree-dimensional growth of Si1–xGex on Si, except at low values of x. How-ever, the islanding can be suppressed by reduction of the growth temperature(<600°C for MBE), thus allowing the growth of strained-layer superlattices.16

Modes of growth are discussed in Chapter 4.

3.5 Silicon Carbide

Epitaxial SiC may be grown using various combinations of precursors. Themost commonly used silicon source is SiH4,17 but Si2H6

17 and SiCl419 have

also been used. The most popular carbon source is C3H8, but the sourcesC2H2,20

CH3Cl,21 CH4,22 CCl4, C7H8, and C6H14 have been used as well. Someresearchers have even demonstrated the growth of SiC from a single precur-sor. Sources of this type include CH3SiCl3

23 and (CH3)2SiCl2.24

Usually SiC epitaxy is carried out in the system SiH4 + C3H8 + H2 in thetemperature range of 1200 to 1800°C, with growth rates of 1 to 5 μm/h.25

The quality of the epitaxial SiC is strongly dependent on the C/Si ratio inthe gas phase. Typically this ratio is 3:1, corresponding to a C3H8/SiH4 ratioof 1:1, although this depends on the reactor.

The most common substrate for heteroepitaxial growth of SiC is Si. UsuallySi (111) is used for the heteroepitaxy of 4H-SiC or 6H-SiC, due to the three-fold symmetry of its surface. The cubic polytype 3C-SiC may be grownheteroepitaxially on Si (001), however.

For heteroepitaxy of 3C-SiC on Si (001), it is necessary to use misorientedsubstrates to eliminate inversion domain boundaries. For homoepitaxy of6H-SiC (0001), the growth temperature may be lowered significantly (forexample, from 1800 to 1500°C) by the use of substrates that are misorientedby a few degrees from the (0001) plane toward a direction. Growthon exact (0001) substrates at low temperatures is characterized by mixed 3Cand 6H phases, but misorientation of the substrate by 1° or more towardthe direction eliminates this problem and allows growth of single-phase 6H-SiC at 1500°C. This technique is called step-controlled epitaxy26,27

and is now commonly employed for the fabrication of SiC devices.SiC may also be grown by gas-source MBE (GSMBE) using the sources

SiH4 + C2H428 or Si2H6 + C2H4.29 Using the sources SiH4 + C2H4, and 0.75 sccm

(standard cubic centimeters per minute) flow of each, very low growth ratesare obtained: 3 nm/h at 1000°C to ~50 nm/h at 1500°C. The addition of H2

increases the growth rates dramatically (0.2 μm/h at 1500°C). The growthrate depends on both source flows because neither is in strong excess.

1120

1120




3.6 III-Arsenides, III-Phosphides, and III-Antimonides

GaAs and the ternary AlxGa1–xAs may be grown by MOVPE or MBE, andcommercial production of AlxGa1–xAs lasers is split between these two tech-niques. These materials are most often grown on GaAs substrates. However,heteroepitaxy on Si and InP substrates has been investigated extensivelywith the goal of integrating AlxGa1–xAs devices with those from these othermaterial systems.

MOVPE growth is carried out with the sources TMGa + TMAl + AsH3.Typically a growth rate of ~10 μm/h is achieved using a mole fraction XTMGa

= 10–4 and a growth temperature of 650°C. The growth rates for GaAs andAlAs are proportional to the respective organometallic source mole fractions.The V/III ratio is 5 to 30 for atmospheric pressure, but higher values maybe used for reduced pressure growth.

Truly selective area growth of GaAs is possible using the source combina-tion DEGaCl + AsH3 and a SiO2 mask. The GaAs grows where windowshave been opened in the oxide, but there is no deposition on the oxide itself.Moreover, this approach can be extended to AlxGa1–xAs by using DEGaCl +DEAlCl + AsH3.

In the case of MBE, elemental sources (7N Ga, 6N AS, and 6N Al) are usedin conventional Knudsen cells. A temperature of 550 to 600°C is used, withgrowth rates of 0.1 to 1 μm/h.

GaAs heteroepitaxy on Si (001) substrates raises a number of challengingproblems. The growth mode is three-dimensional (Volmer–Weber), so a low-temperature nucleation layer must be used to obtain a smooth device layer.Inversion domain boundaries (also known as antiphase domain boundaries)are produced if on-axis substrates are used, but this problem can be elimi-nated by the use of Si substrates that are misoriented by 2 to 4°. Inversiondomain boundaries are considered in Chapter 4. The large lattice mismatch(~–4%) results in large threading dislocation densities (~108 to 109 cm–2). Also,GaAs has about twice the thermal expansion coefficient of Si, so a largetensile strain is introduced in the GaAs during cool-down. This causes crack-ing in layers greater than about 4 μm thickness.

InxGa1–xAs is an important material for the channel regions of high-electron-mobility transistors and also detectors for fiber-optic communica-tion systems operating in the range of 1.3 to 1.55 μm. These materials canbe grown by MOVPE using TEIn + TMGa + AsH3 or TMIn + TMGa +AsH3.30,31 The ethyl source participates in a parasitic reaction with arsineunless the growth pressure is reduced to ~0.1 atm. This problem is elimi-nated with TMIn so that high-quality material is obtained with atmosphericgrowth. Usually InxGa1–xAs is grown with the methyl sources at 650°C witha growth rate of ~3 μm/h. Other alloys involving Al can be grown by theaddition of TMAl.




InxGa1–xAs is usually grown heteroepitaxially on InP or GaAs substrates.This material can be lattice-matched to InP with x = 0.53. However, smallprocessing variations in the composition result in the introduction of a largedensity of threading dislocations. InxGa1–xAs grown on GaAs is only lattice-matched with x = 0. Linearly graded buffer layers (such as InxGa1–xP)32 areoften employed to transition from the lattice constant of GaAs to that of theInxGa1–xAs device layer. Here, the threading dislocation is found to be pro-portional to the grading coefficient, and in practical layers, dislocation den-sities as low as ~105 cm–2 may be obtained.

AlxInyGa1–x–yP is an important material for high-brightness visible light-emitting diodes such as those used in street signs, traffic lights, and auto-motive applications, and for solar cells. This material can be lattice-matchedto GaAs and is usually grown heteroepitaxially on this substrate. This mate-rial is grown by MOVPE in the range of 600 to 650°C. Methyl sources in thecombination TMAl + TMIn + TMGa + PH3 growth pressures up to 1 atmcan be used without parasitic reactions.

If the AlxInyGa1–x–yP material is constrained to lattice-match the GaAs sub-strate, then the indium content must be fixed at y = 0.5. The material com-positions that match the lattice constant of GaAs may therefore be writtenas (AlxGa1–x)0.5In0.5P. The energy gap of this material lattice-matched to GaAsis given by

(3.27)

Even though the active layers of an AlxInyGa1–x–yP light-emitting diode(LED) may be lattice-matched to the GaAs substrate, commercial high-brightness devices make use of highly mismatched heteroepitaxial GaP win-dow layers, which spread the current of the top contact and greatly improvethe device efficiency. A GaP substrate could serve to further reduce substrateabsorption, but this approach is not used due to the high threading disloca-tion density it would produce in the active layers of the LED.

In0.xGa1–xP may also be used for visible LEDs in the orange and red portionof the spectrum. These devices are usually fabricated by heteroepitaxy onGaAs (001) substrates. However, In0.5Ga0.5P LEDs have also been demon-strated on Si (001) substrates.33 These devices were grown using GaAs bufferlayers and exhibited stable output at 660 nm despite the very high threadingdislocation density (~107 cm–2).

The III-antimonides are of interest for applications as barrier layers in high-electron-mobility transistors (HEMTs),34 focal-plane detector arrays in the 3-to 5-μm atmospheric window, and for the fabrication of thermophotovoltaicdevices.35 These materials include InSb, AlSb, GaSb, and their alloys andmay be grown by MOVPE36 or MBE.37 InSb and GaSb substrates have rela-tively high threading dislocation densities, so GaAs38,39 or InP substrates areusually used. Typically a growth rate of ~2.5 μm/h is obtained at 600°Cusing the methyl sources TMGa, TMAl, TMIn, and TMSb.

E xg = +1 91 0 61. .




For InSb grown on GaAs (001), the extremely large lattice mismatch strain(|f| ~ 15%) gives rise to very high misfit and threading dislocation densities.This causes a degradation in the carrier mobility near the heterointerface.This is associated with the misfit dislocations or a high-density tangle ofthreading dislocations near this interface, for higher mobility is obtainedfarther from the interface.

3.7 III-Nitrides

GaN, InN, AlN, and their alloys exist in the wurtzite structure and are grownalmost exclusively on hexagonal 6H-SiC (0001) and sapphire (0001) sub-strates. However, growth on Si(111) substrates has also been investigated.40,41

In early work, Maruska and Tietjen63 demonstrated the VPE of GaN in theGa + HCl + NH3 system. This approach has been replaced by MOVPE, usingTMGa + NH3, TMAl + NH3, or TMIn + NH3. Ternary or quaternary layersmay be grown by using any combination of the metalorganic sources. Typ-ically, a high V/III ratio is used, so the alkyl flows determine the growthrate and composition of the epitaxial layer. A relatively high substrate tem-perature must be used for the MOVPE growth of any of these III-nitrides,due to the thermal stability of ammonia.

MBE can also be used to grow the III-nitrides and has the advantage ofallowing lower growth temperatures. Either radio frequency (rf) plasma cellsor compact electron cyclotron resonance (ECR) microwave plasma sourcesof nitrogen are employed.42–44 Here, the growth rate is limited by the supplyof active nitrogen from the plasma source, so that operation with a high Gaflux results in the formation of Ga droplets on the surface.

The III-nitrides grow in a three-dimensional island mode on sapphire(0001) or 6H-SiC (0001) substrates. Therefore, low-temperature (LT) AlNnucleation layers45–47 are commonly used to achieve smooth layers free fromlarge columnar islands. Typically, the AlN nucleation layer is grown at atemperature of 450 to 550°C, whereas single-crystal AlN is grown by MOVPEat ~1000°C. The low-temperature AlN grows as an amorphous layer butcrystallizes during a subsequent heat treatment. The success of the recrys-tallization process depends on a thin nucleation layer, so typically this thick-ness is 50 nm or less. LT GaN nucleation layers48,49 have also been used, withsimilar improvements in the overgrown GaN material. A discussion of low-

For the heteroepitaxy of GaN on sapphire, nitridation of the sapphiresurface prior to growth is a critical step for the attainment of good crystalquality.50 This step serves to replace O atoms by N to form a thin AlN layer.The change in the nucleation surface improves the final threading dislocationdensity in the overgrown material by a factor of 1/50.



temperature nucleation layers is given in Chapter 4.


The growth of III-nitrides on highly mismatched substrates gives rise tovery high threading dislocation densities in the material. For GaN, the latticemismatch is ~16% with sapphire (0001), which has the rhombohedral crystalstructure with a = 4.7592 Å and c = 12.9916 Å.51 Therefore, epitaxial lateralovergrowth (ELO) has been applied to obtain material with low threadingdislocation densities for LEDs and laser diodes. ELO and the related tech-nique of pendeo-epitaxy are described in detail in Chapter 7. Both of theseapproaches depend on the large lateral-to-vertical growth rate ratio obtainedusing MOVPE with the [0001] growth direction.

The III-nitrides must be grown at relatively high temperatures by eithergrowth method (1000 to 1100°C for MOVPE of GaN), so considerable thermalstrain is introduced during temperature changes. Sapphire has a larger coef-ficient of thermal expansion than GaN at room temperature. However, thissituation reverses at higher temperatures so that a tensile strain is introducedin the GaN during the cool-down process. Thermal strain and cracking aredescribed in Chapter 5.

The dilute nitrides GaInNAs and GaInNAsSb have potential applicationsin optoelectronics for high-bit-rate communications systems, such as 1.2- to1.6-μm lasers and optical amplifiers. These materials have been grown het-eroepitaxially on GaAs substrates by MBE.52 Due to the low solubility of Nin these materials, they are susceptible to phase separation; therefore, meta-stable alloys must be grown at low temperatures (~425°C for MBE). Thegrowth mode is SK (two-dimensional growth of a wetting layer followed bythree-dimensional island growth), but the two-dimensional-to-three-dimen-sional transition can be suppressed by the introduction of Sb.

3.8 II-VI Semiconductors

The II-VI semiconductors include all combinations of Zn, Cd, and S, Se, andTe and may be grown by MOVPE or MBE. These materials are usually grownheteroepitaxially on GaAs, InP, or Si. ZnSe substrates are available, but withrelatively small area and high defect densities. However, device structurescan be designed to be lattice-matched to either GaAs or InP substrates.Hg1–xCdxTe is of great interest for infrared devices and may also be grownby MOVPE and MBE. Substrates such as CdTe, InSb, and even CdZnTe havebeen utilized, but GaAs and Si are used more commonly due to their largerarea and better quality.

ZnSe and its alloys must be grown at low temperatures to minimize theinfluence of native defects and their complexes. MOVPE growth at relativelylow temperatures is possible using the hydride sources H2Se, H2S, and H2Te.However, these hydrides give rise to gas phase prereactions with the met-alorganics, degrading the layer quality and fouling the reactor. For thisreason, metalorganic sources such as DMSe, DES, and DETe are commonly




used. These sources require an increase in the growth temperature unlessphotoirradiation from an ultraviolet lamp is used. However, this photoas-sisted growth technique is complicated by heteroepitaxial growth. The mech-anism of photoassisted growth appears to involve photogenerated carriersnear the surface of the ZnSe, which promote the breaking of the alkyl sourcebonds. However, photoassisted carriers in a GaAs substrate do not partici-pate in this process. Therefore, heteroepitaxial growth on GaAs substratesrequires the growth of a high-temperature ZnSe buffer prior to the start ofphotoassisted epitaxy.

A critical problem in the heteroepitaxy of wide bandgap II-VI materials onGaAs substrates is the creation of stacking faults at the interface. It has beenfound that a single such defect can give rise to the rapid degradation andfailure of an LED or laser diode. The nucleation of the stacking faults is relatedto the initial condition of the surface. In MBE growth, the formation of stack-ing faults can be suppressed by Zn stabilization (starting the Zn beam first).

The ternary Hg1–xCdxTe, which is of great interest for infrared detectors inthe 8- to 16-μm range, is usually grown on GaAs substrates by MOVPE53,54

or MBE.55,56 Occasionally, Si or sapphire57 substrates have also been utilized.Hg1–xCdxTe exhibits a very large lattice mismatch (~14%) with GaAs sub-strates over the entire compositional range. As a consequence, it has beenreported that the epitaxial relationship can be either CdTe[001]||GaAs[001]or CdTe[111]||GaAs[001].58,59

Hg1–xCdxTe is typically grown on GaAs substrates by MOVPE with thesources Hg + DMCd + DETe.60 Typical growth temperatures range from 350to 420°C. It is found that these layers contain ~500 cm–2 hillocks, which maybe associated with stacking faults at the interface. However, this problemcan be prevented by the inclusion of a CdTe buffer layer. Hg1–xCdxTe canalso be grown at a lower temperature (~175°C) using the source combinationHg + DMCd + DTBTe.61

In order to reduce the dislocation densities in Hg1–xCdxTe device layersgrown on GaAs substrates, Cd1–xZnxTe buffer layers have been used, boththe graded and constant composition types.62 (Uniform and graded bufferlayer approaches are discussed in Chapter 7.) Also, wide-bandgap barrierlayers are used to reduce the interface recombination velocity.

3.9 Conclusion

Molecular beam epitaxial (MBE) and vapor phase epitaxial (VPE) techniqueshave enabled the realization of a wide range of heteroepitaxial devices andstructures. Both afford tremendous flexibility and the ability to deposit thinlayers and complex multilayered structures with precise control and excel-lent uniformity. This enables the practical realization of advanced devicestructures such as heterojunction, quantum well, and quantum dot devices.




Many heteroepitaxial material combinations with diverse characteristicshave been investigated. However, certain aspects of heteroepitaxy appearrepeatedly among them and will be covered in detail in the following chap-ters. These aspects include nucleation, growth modes, lattice mismatch andstrain relaxation, crystal defects, thermal strain, and cracking.

Problems

1. Calculate the Reynold’s number for an MOVPE reactor operating at0.1 atm and 650°C with 10 standard liters per minute (slm) of H2

carrier gas, if the reaction chamber is a round tube with a diameterof 10 cm. Hence, determine if laminar or turbulent flow conditionsprevail in the reactor.

2. Consider the MOVPE growth of GaAs using TMGa + AsH3 at 650°Cin a round reactor tube with a diameter of 10 cm. 10 slm of H2 carriergas is used. The mole fraction of TMGa in the reactor is 10–4 and thetotal pressure is 1 atm. (a) Estimate the boundary layer thickness ata distance of 1 cm down the susceptor. (b) Estimate the growth rate.Assume the diffusivity of TMGa in hydrogen is 0.31 cm2s–1.

3. Repeat Problem 2 for the case of P = 0.1 atm. Assume that thediffusivity scales as 1/P.

4. Suppose 20 sccm of H2 is bubbled through a DMZn bubbler main-tained at –10°C. (a) Assuming the bubbles become saturated withthe vapor of DMZn, estimate the flow of DMZn to the reactor. (b)Calculate the mole fraction of DMZn in the reactor, if the total flowof H2 carrier gas is 5 slm.

5. Consider MBE growth of GaAs. The Ga effusion cell has a diameterof 2 cm, is located 25 cm from the substrate, and is held at a tem-perature of 1000°C. (a) Calculate the impingement rate for Ga at thesubstrate in atoms cm–2s–1. (b) Estimate the growth rate. Assume thevapor pressure of Ga at 1000°C is 4 × 10–3 torr.

References

1. S. Whitaker, Introduction to Fluid Mechanics, Prentice Hall, Englewood Cliffs,NJ, 1968.

2. H.M. Manasevit, Single-crystal gallium arsenide on insulating substrates, Appl.Phys. Lett., 12, 156 (1968).

3. H.M. Manasevit and W.I. Simpson, The use of metal-organics in the preparationof semiconductor materials. I. Epitaxial gallium-V compounds, J. Electrochem.Soc., 116, 1725 (1969).




4. H.M. Manasevit, The use of metalorganics in the preparation of semiconduc-tor materials: growth on insulating substrates, J. Cryst. Growth, 13/14, 306(1972).

5. M.A. Herman and H. Sitter, Molecular Beam Epitaxy: Fundamentals and CurrentStatus, Springer-Verlag, New York, 1988.

6. J.H. Jeans, An Introduction to the Kinetic Theory of Gases, University Press, Cam-bridge, 1967.

7. B.B. Dayton, Gas flow patterns at entrance and exit of cylindrical tubes, in 1956National Symposium on Vacuum Technology Transactions, E.S. Perry and T.H.Devant, Eds., Pergamon Press, Oxford, 1957, p. 5.

8. T.U.M.S. Murthy, N. Miyamoto, M. Shimbo, and J. Nishizawa, Gas-phase nu-cleation during the thermal decomposition of silane in hydrogen, J. Cryst.Growth, 33, 1, (1976).

9. R.P. Ruth, J.C. Marinace, and W.C. Dunlap, Jr., Vapor-deposited single-crystalgermanium, J. Appl. Phys., 31, 995 (1960).

10. E.F. Cave and B.R. Czorny, Epitaxial Deposition of Silicon and GermaniumLayers by Chloride Reduction, RCA Review, December 1963, p. 523.

11. K.J. Miller and M.J. Grieco, Epitaxial P-type germanium filmsby the hydrogenreduction of GeBr4, SiBr4, and BBr3, J. Electrochem. Soc., 110, 1252 (1963).

12. J.M. Hartmann, Y. Bogumilowicz, F. Andrieu, P. Holliger, G. Rolland, and T.Billon, Reduced pressure-chemical vapor deposition of high Ge content Si1–xGex

and high C content Si1–yCy layers for advanced metal oxide semiconductortransistors, J. Cryst. Growth, 277, 114 (2005).

13. S.S. Iyer, K. Eberl, A.R. Powell, and B.A. Ek, SiCGe ternary alloys: extendingSi-based heterostructures, Microelectronic Eng., 19, 351 (1992).

14. C. Li, S. John, E. Quinones, and S. Banerjee, Cold-wall ultrahigh vacuumchemical vapor deposition of doped and undoped Si and Si1–xGex epitaxial filmsusing SiH2 and Si2H6, J. Vac. Sci. Technol. A, 14, 170 (1996).

15. Yu. B. Bolkhovityanov, A.S. Deryabin, A.K. Gutakovskii, M.A. Revenko, andL.V. Sokolov, Heterostructures GexSi1–x/Si(100) grown by molecular beam epi-taxy at low (350°C) temperature: specific features of plastic relaxation, ThinSolid Films, 466, 69 (2004).

16. J.C. Bean, L.C. Feldman, A.T. Fiory, S. Nakahara, and I.K. Robinson, GexSi1–x/Si strained-layer superlattice grown by molecular beam epitaxy, J. Vac. Sci.Technol. A, 2, 436 (1984).

17. J.A. Powell, L.G. Matus, and M.A. Kuczmarski, Growth and characterizationof cubic SiC single-crystal films on Si, J. Electrochem. Soc., 134, 1558 (1987).

18. S. Nishino and J. Saraie, Heteroepitaxial growth of cubic SiC on a Si substrateusing the Si2H6-C2H2-H2 system, in Amorphous and Crystalline Silicon Carbide,G.L. Harris and C.Y.-W. Yang, Eds., Springer, Berlin, 1989, p. 45.

19. W. Muench, W. Kurzinger, and I. Pfaffender, Epitaxial deposition of siliconcarbide from silicon tetrachloride and hexane, Thin Solid Films, 31, 39(1976).

20. P. Liaw and R.F. Davis, Epitaxial growth and characterization of β-SiC thinfilms, J. Electrochem. Soc., 132, 642 (1985).

21. K. Ikoma, M. Yamanaka, H. Yamaguchi, and Y. Shichi, Heteroepitaxial growthof β-SiC on Si(111) by CVD using a CH3Cl-SiH4-H2 gas system, J. Electrochem.Soc., 138, 3031 (1991).

22. P. Rai-Choudhury and N.P. Formigoni, β-Silicon carbide film, J. Electrochem.Soc., 116, 1440 (1969).




23. S. Nishino and J. Saraie, Heteroepitaxial growth of cubic SiC on a Si substrateusing methyltrichlorosilane, in Springer Proc. Phys., Vol. 43, M.M. Rahman, C.Y.Yang, and G.L. Harris, Eds., Springer, Berlin, 1989, pp. 8–13.

24. P. Rai-Choudhury and N.P. Formigoni, β-Silicon carbide film, J. Electrochem.Soc., 116, 1440 (1969).

25. T. Chassagne, G. Ferro, D. Chaussnde, F. Cauwet, Y. Monteil, and J. Bouix, Acomprehensive study of SiC growth processes in a VPE reactor, Thin Solid Films,402, 83 (2002).

26. N. Kuroda, K. Shibahara, W.S. Yoo, S. Nishino, and H. Matsunami, ExtendedAbstracts of the Thirty-Fourth Spring Meeting of Japan Society Applied Physics andRelated Societies, Tokyo, 1987, p. 135 (in Japanese).

27. N. Kuroda, K. Shibahara, W.S. Yoo, S. Nishino, and H. Matsunami, ExtendedAbstracts of the Nineteenth Conference on Solid State Devices and Materials, Tokyo,1987, p. 227.

28. R.S. Kern, S. Tanaka, L.B. Rowland, and R.F. Davis, Reaction kinetics of siliconcarbide deposition by gas-source molecular-beam epitaxy, J. Cryst. Growth, 183,581 (1998).

29. R.F. Davis, S. Tanaka, L.B. Rowland, R.S. Kern, Z. Sitar, S.K. Ailey, and C. Wang,Growth of SiC and III-V nitride thin films via gas-source molecular beamepitaxy and their characterization, J. Cryst. Growth, 164, 132 (1996).

30. C.P. Kuo, R.M. Cohen, K.L. Fry, and G.B. Stringfellow, OMVPE growth ofGaInAs, J. Cryst. Growth, 64, 461 (1983).

31. C.P. Kuo, J.S. Yuan, R.M. Cohen, J. Dunn, and G.B. Stringfellow, Organometallicvapor phase epitaxial growth of high purity GaInAs using trimethylindium,Appl. Phys. Lett., 44, 550 (1984).

32. K. Yuan, K. Radhakrishnan, H.Q. Zheng, and G.I. Ng, MetamorphicIn0.5Al0.5As/In0.53Ga0.47As high electron mobility transistors on GaAs withInxGa1–xP graded buffer, J. Vac. Sci. Technol. B, 19, 2119 (2001).

33. S. Kondo, S.-I. Matsumoto, and H. Nagai, 660 nm In0.5Ga0.5P light-emittingdiodes on Si substrates, Appl. Phys. Lett., 53, 279 (1988).

34. B.R. Bennett, R. Magno, J.B. Boos, W. Kruppa, and M.G. Ancona, Antimonide-based compound semiconductors for electronic devices: a review, Solid StateElectron., 49, 1875 (2005).

35. L.M. Fraas, R. Ballantyne, J. Samaras, and M. Seal, AIP Conference Proceedings,First NREL Conference on Thermophotovoltaic Generation of Electricity, 321, 44 (1994).

36. Y. Paltiel, A. Sher, A. Raizman, S. Shusterman, M. Katz, A. Zemel, Z. Calahorra,and M. Yassen, Metalorganic vapor phase epitaxy InSb p+nn+ photodiodes withlow dark current, Appl. Phys. Lett., 84, 5419 (2004).

37. T.M. Kerr, T.D. McLean, D.I. Westwood, and J.D. Grunge, Summary Abstract:The growth and doping of GaAsysb1–y by molecular beam epitaxy, J. Vac. Sci:Technol. B, 3, 535 (1985).

38. H. Ehsani, I. Bhat, C. Hitchcock, J. Borrego, and R. Gutmann, Characteristicsof GaSb and GaInSb layers grown by metalorganic vapor phase epitaxy, AIPConf. Proc., 358, 423 (1996).

39. H. Ehsani, I. Bhat, R. Gutmann, and G. Charache, p-type GaSb and Ga0.8In0.2Sblayers grown by metalorganic vapor phase epitaxy using silane as the dopantsource, Appl. Phys. Lett., 69, 3863 (1996).

40. Y. Koide, H. Itoh, N. Sawaki, I. Akasaki, and M. Hashimoto, Epitaxial growthand properties of AlxGa1–xN by MOVPE, J. Electrochem. Soc., 133, 1956 (1986).




41. Y. Dikme, A. Szymakowski, H. Kalisch, E.V. Lutsenko, V.N. Zubialevich, G.P.Yablonskii, H.M. Chern, C. Schaefer, R. Jansen, and M. Heuken, Investigationof GaN on Si(111) for optoelectronic applications, Proc. SPIE, 4996, 57 (2003).

42. J.T. Torvik, M. Leksono, J.I. Pankove, B.V. Zeghbroeck, H.M. Ng, and T.D.Moustakas, Electrical characterization of GaN/SiC n-p heterojunction diodes,Appl. Phys. Lett., 72, 1371 (1998).

43. N. Gogneau, E. Sarigiannidou, E. Monroy, S. Monnoye, H. Mank, and B. Dau-din, Surfactant effect of gallium during the growth of GaN on AlN (0001) byplasma-assisted molecular beam epitaxy, Appl. Phys. Lett., 85, 1421 (2004).

44. E. Monroy, N. Gogneau, F. Enjalbert, F. Fossard, D. Jalabert, E. Bellet-Amalric,L.S. Dang, and B. Daudin, Molecular-beam epitaxial growth and characteriza-tion of quaternary, J. Appl. Phys., 94, 3121 (2003).


46. H. Amano, I. Akasaki, K. Hiramatsu, and N. Sawaki, Effects of the buffer layerin metalorganic vapour phase epitaxy of GaN on sapphire substrates, Thin SolidFilms, 163, 415 (1988).

47. Y. Koide, N. Itoh, X. Itoh, N. Sawaki, and I. Akasaki, Effect of AlN buffer layeron AlGaN/α-Al2O3 heteroepitaxial growth by metalorganic vapor phase epi-taxy, Jpn. J. Appl. Phys., 27, 1156 (1988).


49. N. Kuznia, M.A. Khan, D.T. Olsen, R. Kaplan, and J. Freitas, Influence of bufferlayers on the deposition of high quality single crystal GaN over sapphiresubstrates, J. Appl. Phys., 73, 4700 (1993).

50. S. Keller, B.P. Keller, Y.-F. Wu, B. Heying, D. Kapolnek, J.S. Speck, U.K. Mishra,and S.P. DenBaars, Influence of sapphire nitridation on properties of galliumnitride grown by metalorganic chemical vapor deposition, Appl. Phys. Lett., 68,1525 (1996).

51. Y.V. Shvyd’ko, M. Lucht, E. Gerdau, M. Lerche, E.E. Alp, W. Sturhahn, J. Sutter,and T.S. Toellner, Measuring wavelengths and lattice constants with the Möss-bauer wavelength standard, J. Synchrotron Rad., 9, 17 (2002).

52. J.S. Harris, Jr., The opportunities, successes and challenges for GaInNAsSb, J.Cryst. Growth, 278, 3 (2005).

53. W.E. Hoke, P.J. Lemonias, and R. Traczewski, Metalorganic growth of high-purity HdCdTe films, Appl. Phys. Lett., 45, 1092 (1984).

54. I.B. Bhat, N.R. Taskar, and S.K. Ghandhi, The organometallic heteroepitaxyof CdTe and HgCdTe on GaAs substrates, J. Vac. Sci. Technol. A, 4, 2230(1986).

55. J.P. Faurie, S. Sivanathan, M. Boukerche, and J. Reno, Molecular beam epitaxialgrowth of high quality HgTe and Hg1–xCdxTe onto GaAs (001) substrates, Appl.Phys. Lett., 45, 1307 (1984).

56. K. Nishitani, R. Ohkata, and T. Murotani, Molecular beam epitaxy of CdTe andHg1–xCdxTe on GaAs (100), J. Electron. Mater., 12, 619 (1983).

57. T.H. Myers, Y. Lo, R.N. Bicknell, and J.F. Schetzina, Growth of CdTe films onsapphire by molecular beam epitaxy, Appl. Phys. Lett., 42, 247 (1983).

58. J.T. Cheung and T.J. Magee, Recent progress on LADA growth of HgCdTe andCdTe epitaxial layers, J. Vac. Sci. Technol. A, 1, 1604 (1983).




59. H.A. Mar, K.T. Chee, and N. Salansky, CdTe films on (001) GaAs:Cr by molec-ular beam epitaxy, Appl. Phys. Lett., 44, 237 (1984).

60. S.K. Ghandhi, I.B. Bhat, and N.R. Taskar, Growth and properties of Hg1–xCdxTeon GaAs substrates by organometallic vapor-phase epitaxy, J. Appl. Phys., 59,2253 (1986).

61. K. Yasuda, H. Hatano, T. Ferid, M. Minamide, T. Maejima, and K. Kawamoto,Growth characteristics of (100) HgCdTe layers in low-temperature MOVPEwith ditertiarybutyltelluride, J. Cryst. Growth, 166, 612 (1996).

62. V.S. Varavin, S.A. Dvoretsky, V.I. Liberman, N.N. Mikhailov, and Yu. G. Sidorov,Molecular beam epitaxy of high quality Hg1–xCdxTe films with control of thecomposition distribution, J. Cryst. Growth, 159, 1161 (1996).

63. H.P. Maruska, and J.J. Tietjen, The preparation and properties of vapor-depos-ited single-crystalline GaN, Appl. Phys. Lett., 15, 327 (1969).



105

4

Surface and Chemical Considerations

in Heteroepitaxy

4.1 Introduction

Heteroepitaxy differs from homoepitaxy in that it requires the nucleation ofa new phase A on a foreign substrate B. Because of this, the surface chemistryand physics play important roles in determining the properties of heteroepi-taxial deposits, including structural and electrical characteristics, defect den-sities and structure, and the layer morphology.

At the typical temperatures employed for epitaxial growth, the substratesurface may undergo a reconstruction, which is an atomic-scale change insurface structure. The structure of the reconstructed surface may depend onthe temperature and adsorbed species. Moreover, in some cases it has beenfound that the initial structure of the reconstructed substrate affects thestructural and electrical properties of thick heteroepitaxial films grown onthe surface.

On vicinal (tilted) substrates, the structure of steps and kinks on the surfacecan have an important influence on the heteroepitaxial growth. For example,in growth of a polar semiconductor on a nonpolar substrate, inversiondomain boundaries (antiphase domain boundaries) may develop due to thelower symmetry of the heteroepitaxial crystal. However, it has been foundthat this behavior can be controlled by the proper choice of the substrate tiltand direction.

Either energetics or kinetics may control the nucleation and growth modefor heteroepitaxy. Traditionally, three possible growth modes have been iden-tified as Frank–van der Merwe (FM), Volmer–Weber (VW), and Stran-ski–Krastanov (SK) growth. Recently, another mode of growth (ripening) hasbeen identified in which large islands grow, possibly in the presence of awetting layer or smaller, stable islands. The Frank–van der Merwe modeinvolves layer-by-layer growth, giving smooth interfaces; it is desirable formost device applications and is mandatory for quantum well layers.Volmer–Weber (island growth) and Stranski–Krastanov (islanding on a con-tinuous wetting layer) are undesirable for most applications; however, they



106


can be useful for the fabrication of quantum dot devices. In all of thesesituations, however, understanding the growth mode is important for thedesign of the device fabrication process.

Another important aspect of surface chemistry in heteroepitaxy involvesthe use of surfactants. These are atomic (or molecular) species that are pref-erentially segregated to the surface instead of incorporated into the growingcrystal. Such an adsorbed species may have a dramatic effect on the growtheven after modest exposure of the surface (~1 monolayer). If the incorpora-tion and evaporation of the surfactant are both negligible, then even a shortexposure to the surfactant species can alter the surface chemistry throughoutthe growth of a thick film, affecting the composition, structure, and mor-phology of the resulting material.

This chapter will review some of the important aspects of surfaces andtheir chemical considerations, as they apply to heteroepitaxy. Surface recon-structions will be discussed, including some general principles, but also thespecifics of some important surfaces for semiconductor epitaxy. Inversiondomain boundaries will be reviewed; it will also be shown that the surfacestructure controls the introduction of inversion domains in a growing polarsemiconductor on a nonpolar substrate. Nucleation theory will be reviewed,starting with the classical model based on the macroscopic surface andinterface energies. Then the atomistic model for nucleation will be summa-rized as it applies to heteroepitaxy. Growth modes are closely related tonucleation, but are covered in a separate section. The discussion of growthmodes includes a thermodynamic treatment, which allows the calculationof a growth mode phase diagram for heteroepitaxy. Then a kinetic modelwill be presented, which describes the conditions for the onset of islandgrowth on top of a wetting layer. In the next section, surfactants will bediscussed, starting with some general principles and ending with importantapplications to heteroepitaxy. In the final section of the chapter, severalaspects of quantum dot fabrication will be addressed, including self-assem-bly, self-organization, and precision placement.

4.2 Surface Reconstructions

The surface of a semiconductor substrate will generally take on a structuredifferent from that of a truncated bulk crystal. The driving force for this isenergy minimization. In some cases, the rearrangement is rather subtle,altering neither the periodicity nor the symmetry of the surface. This isreferred to as surface relaxation. In other cases, the rearrangement is suchthat it changes the periodicity, and perhaps the symmetry, of the surface,and is called surface reconstruction. Often such reconstructions have beenattributed to dimerization of the surface atoms, which serves to reduce thenumber of unsaturated dangling bonds. Reconstruction can be readily



Surface and Chemical Considerations in Heteroepitaxy

107

detected by electron diffraction techniques, including low-energy electrondiffraction (LEED) and reflection high-energy electron diffraction (RHEED).

Just as three-dimensional crystal structures belong to one of the fourteenBravais lattices, surface atomic arrangements can be considered to belong toone of five types of surface nets. These are the square, rectangular, centeredrectangular, hexagonal, and oblique nets, and their unit meshes are illus-trated in Figure 4.1.

The assignment of the surface structure involves the identification of itssymmetry (the shape and dimensions of the unit cell) and the determinationof the positions of the atoms within the unit cell. The former is readily foundfrom the positions of diffracted electron beams, but the latter requires ananalysis of the diffracted intensities. This interpretation is not straightfor-ward, because phase information is lost in the diffraction pattern. Thus, it isnecessary to guess the structure, simulate the diffracted intensities, and thencompare the calculated and measured results. Several iterations may be

FIGURE 4.1

Unit cells of the five surface nets. (Reprinted from Wood, E.A.,

J. Appl. Phys.

, 35, 1306, 1964.With permission. Copyright 1964, American Institute of Physics.)

a2s

a1s

γ

Square |a1s| = |a2s|, γ = 90°

Rectangle|a1s| ≠ |a2s|, γ = 90°

a2s

a1sγ

Centered rectangle|a1s| ≠ |a2s|, γ = 90°

a2s

a1sγ

Hexagonal|a1s| = |a2s|, γ = 120°

a2s

a1s

γ

Oblique|a1s| ≠ |a2s|, γ ≠ 90°

a2s

a1s

γ



108


required, but may not yield a unique solution. In most cases, the detailedatomic structures of reconstructed semiconductor surfaces remain unknown.However, recent advances in lateral force microscopy (LFM)

1,2

(a variationof atomic force microscopy) have made it possible to observe reconstructedsurfaces with true atomic resolution.

3

By these measurements, it is possibleto elucidate the detailed structures of the unit meshes in reconstructed sur-faces. Routine electron diffraction measurements can then be used to distin-guish between the types of surface structures without a need for suchdetailed analysis.

4.2.1 Wood’s Notation for Reconstructed Surfaces

Usually reconstructed surfaces are classified using Wood’s notation.

4

Sup-pose is the surface mesh with unit translations and . Further sup-pose that the mesh of an unreconstructed surface (bulk exposed plane) iswith unit translations and . In Wood’s notation, the relationshipbetween the reconstructed mesh and the mesh of the bulk exposed plane isexpressed as , where

R

indicates a rotation of the surfacemesh with respect to the bulk and is followed by the value of this rotationin degrees. (If there is no rotation of the surface mesh,

R

is omitted.) Thenotation is used to denote a centered mesh. Wood’snotation is applicable to clean surfaces, but can also be extended to situationsinvolving an adsorbate.

Two example meshes are shown in Figure 4.2, along with their Wood’sclassification. Figure 4.2a shows adsorbed oxygen on a Ni (110) surface, witha centered mesh and no rotation. The notation for this surface struc-ture is . Figure 4.2b shows adsorbed oxygen on a Pt (100)surface, with a mesh, rotated by 45°. The notation in this case is

.

FIGURE 4.2

Surface structures for the illustration of Wood’s notation. (a) A Ni (110) surface with adsorbedoxygen having a centered (2

×

2) mesh and no rotation, denoted . (b) A Pt(100) surface with adsorbed oxygen having a mesh, rotated by 45°, denoted

. (Adapted from Prutton, M.,

Introduction to Surface Physics

, OxfordUniversity Press, Oxford, 1994. With permission.)

aS a S1 a S2

aB

a B1 a B2

( / / )a a a a RS B S B1 1 2 2×

c a a a a RS B S B( / / )1 1 2 2×

( )2 2×Ni c O( ) ( )110 2 2× −

( )2 2 2×Pt R O( ) ( )100 2 2 2 45× ° −

a1b

a2b

a1s

a2s

Ni (110) c (2 × 2) O(a)

Pt (100) (√2 × √2) R 45° O(b)

a2ba1ba2s

a1s

Ni c O( ) ( )110 2 2× −( )2 2 2×

Pt R O( ) ( )100 2 2 2 45× ° −




109

In general, the structure of a reconstructed surface will depend on thetemperature. Different reconstructions may be observed at different temper-atures during the pregrowth processing in an epitaxial reactor. Also, kineticlimitations may prevent the surface from taking on the equilibrium structure.Thus, a reconstructed structure that is stable only at high temperature maybe “frozen in” at a lower temperature.

The presence of adsorbed species will generally affect the surface structure.Many surface structures that have been observed on Si only exist in thepresence of adsorbed impurities. These could have important implicationsfor nucleation and heteroepitaxy. One application area is surfactant-medi-ated epitaxy (SME). For example, in the surfactant-mediated heteroepitaxyof Ge on Si (111) using Sb as a surfactant, the adsorption of Sb changes thesurface structure

5

from to . In gen-eral, adsorption of species can alter the surface structure during epitaxy, evenin the case of homoepitaxy. For example, in the case of ultra-high-vacuum(UHV) vapor phase epitaxy (VPE) of silicon using silane on Si (111), thesurface is found to convert from the reconstruction to a par-tially hydrogenated structure

6

upon exposure to silane. In thiscase, therefore, the nucleation and growth occur on different surfaces.

It is clear that the surface structure will influence heteroepitaxial growthin a variety of ways, many of which are poorly understood. It is known, forexample, that in VPE growth the growth rate is determined by the concen-tration of adsorbed species on reactive sites, and this is controlled by thesurface atomic structure. Further study in this area is likely to result inprogress in the areas of surfactant-mediated epitaxy, selective epitaxy, andquantum devices.

4.2.2 Experimental Observations

Some of the commonly used substrate surfaces have been studied exten-sively; some of the results of these investigations are summarized in thefollowing subsections.

4.2.2.1 Si (001) Surface

The clean Si (001) surface is found to assume a -type reconstruction

7

ifheated in an ultrahigh vacuum or a hydrogen ambient. In all probability, theatomic structure of the reconstructed surface is consistent with the pairingmodel

8–12

; other models have been proposed, however, including the vacancymodel.

13

However, the c(4

×

2) reconstruction

14

and other structures havealso been observed.

15

The nominal (001) surface is populated with steps, the types and densitiesof which are determined by the miscut direction and angle. In the absenceof an intentional miscut angle, or if the miscut angle is small (<2°), mona-tomic (single-layer) steps exist on the surface. These single-layer steps sep-arate S

A

and S

B

domains, which exhibit (2

×

1) and (1

×

2) reconstructions,

Si ( ) ( )111 7 7× Si R Sb( ) ( )111 3 3 30× ° −

Si ( ) ( )111 7 7×Si ( ) ( )111 1 1×

( )2 1×



110


respectively. Here, the surface meshes have the same structure and symmetrybut are rotated with respect to each other by 90°. Vicinal Si (001) substrateswith miscut angles of 2 to 11° are found to have predominantly double-layersteps if annealed at 600 to 1200°C.

16–19

As a consequence, these surfaces havea single type of surface domain with the (2

×

1) reconstruction. Equilibriumcalculations have been used to determine the temperature vs. miscut anglephase diagram for the Si (001) surface.

20

These results predict that the double-step phase is more stable than the single-step phase at typical epitaxialgrowth temperatures and with typical miscut angles.

Kinks will generally exist along the steps of a nominal Si (001) surface.For a Si (001) surface with a tilt exactly toward the [110] axis, the equilibriumconcentration of such kinks is expected to be small at typical growth tem-peratures. However, the deliberate choice of a different offcut directionintroduces a significant forced kink density. This forced kink densityincreases as the offcut azimuth is moved away from the [110] toward eitherthe [100] or [010].

16

4.2.2.2 Si (111) Surface

The Si (111) surface is known to exhibit several reconstructions. After cleav-age under ultrahigh vacuum at room temperature, a Si (111)(2

×

1) recon-struction is found.

21

Upon heating to a temperature above 350°C, this surfaceis found to make an irreversible phase change to a Si (111)(7

×

7).

22

Detailedstudies of this structure by the specular scattering of atomic He reveal thatthe first two double layers of atoms are exposed and that a large fraction ofthe top layer is missing.

23

Recently, dynamic lateral force microscopy (LFM)with atomic resolution has been applied to image the Si (111)(7

×

7).

24

Suchan image is shown in Figure 4.3.

FIGURE 4.3

(a) Lateral force microscopy (LFM) image of the Si (111)(7

×

7) surface. (b) The same imageoverlaid with white circles (individual atoms) and white diamonds (unit cells). (Reprinted fromKawai, S. et al.,

Appl. Phys. Lett.

, 87, 173105, 2005. With permission. Copyright 2005, AmericanInstitute of Physics.)

(a) (b)




111

Impurity-stabilized Si (111) surfaces have been found to exhibit a numberof reconstructions with periodicities of 1, , , 4, 5, 6, 12, 18, and 24.

25

In the case of nickel stabilization, even has beenobserved.

26

Impurity-stabilized surface reconstructions could have impor-tant implications for nucleation and heteroepitaxial growth.

4.2.2.3 Ge (111) Surface

As with Si (111), this surface of Ge is observed to take on a (2

×

1) configu-ration after cleaving in vacuum. This surface, upon annealing in hydrogen,takes on a Ge (111)(8

×

8) structure.

27

The (2

×

8) structure has often beenreported based on LEED measurements.

28

However, it has been shown thatthe observed LEED pattern corresponds to the (8

×

8) reconstruction, butwith certain structure factor cancellations that are due to distortions withinthe unit mesh. A number of other impurity-induced surface structures havealso been observed, as in the case of Si (111).

4.2.2.4 6H-SiC (0001) Surface

6H-SiC (0001) wafers (Si face) are commonly used as substrates for GaNheteroepitaxy, as well as SiC homoepitaxy. The behavior of this surface isquite complex, and many different reconstructions have been observed. Thesurface structures most commonly found

29

are and (3

×

3)under Si-rich conditions, and the under C-rich conditions.Other reconstructions have also been reported, including (3

×

1),

30

(6

×

6),

31

and even .

32

These various structures exhibit differences inmobilities for diffusing surface species and also behave differently withrespect to surface-adsorbed surfactants. They are therefore expected to influ-ence both nucleation and epitaxial growth. In the case of GaN grown on 6H-SiC (0001) with an AlN buffer layer, it has been shown that the nature of the6H-SiC surface reconstruction has a strong effect on the AlN lattice relax-ation, and therefore the crystalline quality of the GaN overlayer.

33

The tendencies toward these various surface reconstructions may be con-trolled to a great extent by the methods of surface preparation. For example,Hartman et al.

34

investigated various gas phase treatments and the resultingsurface structures. They found that etching at 1500 to 1640°C and 1 atmproduced a (1

×

1) surface. Further annealing in hydrogen at about 1000°Cand 1 atm gave rise to a conversion to the reconstruction.Subsequent exposure to SiH

4

under UHV conditions was found to convertthis surface to a (3

×

3) structure, but conversion to a surface resultedfrom additional annealing. Suda et al.

33

observed a structureafter etching in HCl/H

2

at 1300°C, but found that a (1

×

1) surface resultedif the HCl/H

2

etch was followed by a wet HF treatment. Lu et al.

35

observedthe reconstruction after etching in atomic hydrogen at 650°C.Kim et al.

30

investigated the structure of the 6H-SiC (0001) surface after a Gaflash-off process

37

prior to epitaxial growth. The Ga flash-off, which is done

3 2 3Si Ni111 19 19( ) × −( )

( )3 3 30× °R( )6 3 6 3 30× °R

( )2 3 2 13×

( )3 3 30× °R

1 1×( )( )3 3 30× °R

( )3 3 30× °R



112


in the growth chamber prior to MBE growth, typically consists of exposingthe surface to the Ga source flux at a temperature of 650°C, followed byflashing to 800°C in several cycles.

30

This process serves to convert the oxideto the volatile GaO or GaO

2

, which can be removed by flashing in the UHV.They observed the (3

×

1) reconstruction after a Ga flash-off process, even ifthe oxide removal was incomplete. Often, however, the surface structureexhibits mixed phases rather a single structure.

The detailed atomic configurations of the 6H-SiC (0001) surface reconstruc-tions are incompletely understood. However, recent investigations by scan-ning tunneling microscopy (STM) have revealed some important features,

37

and structural models have been proposed for the ,

38

(3

×

3),

39,40

,

42

and

32

reconstructions. An inter-esting feature of the surface is the presence of atomiccracks,

40

which have been observed by STM. These cracks might induceisland growth or control the formation of boundaries between islands.

4.2.2.5 3C-SiC (001)

The clean (001) surface of zinc blende SiC is found to take on a (3 × 2)reconstruction.42–44 Following the adsorption of hydrogen, this surface retainsits (3 × 2) symmetry but undergoes a transition from semiconductor tometallic character.45–47

4.2.2.6 3C-SiC (111)

There are few reported studies of the (111) surface of zinc blende SiC. How-ever, several groups have investigated the surfaces of thin 3C-SiC (111)produced on Si (111) substrates. These layers are either deposited by CVDor produced by the reaction of the surface with fullerene molecules such asC60 or C70.

Hu et al.48 prepared a 3C-SiC (111) by the thermal reaction of the fullerenemolecule C60 with the clean surface of Si (111). By using high-resolutionelectron energy loss spectroscopy (HREELS) and scanning tunneling micros-copy (STM), they were able to study the surface structure and its relationshipwith the surface chemical composition. In these experiments, the C60 sourcewas evaporated onto the Si substrate at room temperature. A short annealat 250°C served to desorb all by one monolayer of the C60 from the surface.As the temperature was raised, there was no further change in the surfaceup to a temperature of 850°C. At 870°C, however, the C60 reacted with thesurface to form SiC islands. The SiC surface contained a mixture of (2 × 2)and (2 × 3) surface reconstructions, and also a mixture of both the C(111)and Si(111) faces. Finally, annealing at or above 1100°C resulted in the Si-terminated 3C-SiC (111)(3 × 3) reconstruction.

( )3 3 30× °R( )6 3 6 3 30× °R ( )2 3 2 3 30× °R

( )2 3 2 3 30× °R



Surface and Chemical Considerations in Heteroepitaxy 113

4.2.2.7 GaN (0001)

Due to the limited availability of single-crystal GaN substrates, most studiesof the GaN surface have been conducted with heteroepitaxial layers, usuallyon sapphire (0001) or 6H-SiC (0001). The surfaces are usually reported totake on a (2 × 2) reconstruction.

4.2.2.8 Zinc Blende GaN (001)

Zinc blende GaN (001) has been grown heteroepitaxially on a number ofsubstrates, including Si (001), 3C-SiC (001), and GaAs (001). Feuillet et al.49

found that as-grown MBE GaN (001) on cubic SiC (001) exhibited a (4 × 1)reconstruction but underwent an irreversible transformation to a (2 × 2)structure upon exposure to an As beam.

4.2.2.9 GaAs (001)

GaAs (001) substrates, as prepared for heteroepitaxy, typically exhibit a(2 × 4) reconstruction.50,51 This surface reconstruction has been studied exten-sively due to its importance in heteroepitaxy. At an As coverage of 0.75 ml,the GaAs (001) and InAs (001)(2 × 4) reconstructions assume the β2 struc-ture.52–64 The structure contains two As dimers in the first (top) atomic layerand one As dimer in the third layer, per unit cell. At a lower As coverage of0.5 ml, the α structure66,69,70 is observed. The α structure contains two Asdimers in the first layer and two Ga dimers in the second layer, per unit cell.Both reconstructions are observed under the normal conditions of As stabi-lization, but the reconstruction is generally observed at lower tem-peratures, whereas the phase is found at higher temperatures.Galitsyn et al.65 studied the phase transition and deter-mined the critical values of temperature and As overpressure.

4.2.2.10 InP (001)

The InP (001) surface has been found to exhibit a (2 × 1) reconstruction.66–68

This structure has been detected on the surface of MOVPE InP (001) by LEEDand XPS; infrared spectroscopy of the same sample after deuterium exposureshowed that the (2 × 1) structure was hydrogen stabilized.69 Ab initio com-putations have shown that the adsorption of the surfactant Sb can lead to anumber of surface structures on InP (001), including a number of (2 × 3),(2 × 4), (4 × 3), and (4 × 4) variants. A phase diagram has been developedfrom these calculations and shows that the equilibrium surface reconstruc-tion depends on the chemical potentials (and therefore partial pressures) ofboth Sb and P over the surface.70

β2 2 4( )×α( )2 4×

β α2 2 4 2 4( ) ( )× → ×




4.2.2.11 Sapphire (0001)

Al2O3 (sapphire) substrates are of considerable interest due to their use inthe heteroepitaxy of GaN and related III-nitrides and other semiconductors.The sapphire (0001) face is found to exhibit (1 × 2), (2 × 2), (5 × 5), and

reconstructions. Several studies have related the initialsurface structure to the properties of deposited heteroepitaxial layers. Shenet al.71 reported that for plasma-assisted molecular beam epitaxy (PAMBE)GaN layers grown on c-face sapphire, the highest electron mobilities wereobtained when the substrate showed the (1 × 2) pattern.

In summary, there have been extensive modeling and experimental studiesof semiconductor surface structures. It is clear that these surface structurescan play important roles in determining the properties and usefulness ofheteroepitaxial semiconductor layers. However, an understanding of thisrelationship is only beginning to emerge.

4.2.3 Surface Reconstruction and Heteroepitaxy

Heteroepitaxy usually involves the creation of an interface between semi-conductor crystals having different ionicities. As a consequence, the surfacestructure of the substrate crystal can have an important effect on the prop-erties of the epitaxial crystal. This is because, depending on the structure ofthe reconstructed surface, an electrostatic field may develop at the interface,resulting in a rough interface with degraded electrical properties.72,73 In thecase of heteroepitaxy of a polar semiconductor on a nonpolar substrate, anadditional challenge arises: the possibility of inversion domain boundaries(IDBs). These two issues will be discussed in this section.

4.2.3.1 Inversion Domain Boundaries (IDBs)

The Si (001) surface nominally exhibits a reconstruction with two typesof domains that are rotated by 90° with respect to each other. The siliconatoms on the surface form oriented dimers along the [110] direction for onetype of domain (called SA terraces) and along the direction for therotated type of domain (called SB terraces).74 These domain terraces areseparated by monatomic steps of height a/4. The presence of these mona-tomic steps has been shown to lead to inversion domain boundaries (IDBs)in the heteroepitaxial growth of polar semiconductors, including GaAs75 andAlN,76 on Si (001). This behavior can be understood with the aid of Figure4.4, which shows a Si (001) with a monatomic step and the atomic configu-ration of a polar semiconductor grown over the stepped surface. It wasassumed that the growth of the polar semiconductor was initiated by expo-sure of the surface to the anion, so that all Si atoms at the surface bond toAs atoms. As a consequence, the polar semiconductor (zinc blende GaAs)contains an IDB associated with the step in the substrate surface.

( )31 31 9× ± °R

2 1×

[ ]110




The key to eliminating IDBs in heteroepitaxy of a polar semiconductor onSi (001) is the elimination of monatomic surface steps. Double steps do notresult in IDBs, as shown in Figure 4.5. Here, it can be seen that the polarsemiconductor (zinc blende GaAs) grows with the same polarity over bothtypes of terraces.

Si (001) surfaces, completely free from monatomic steps, can be preparedby the hydrogen annealing of vicinal substrates. If the Si (001) substrates areoffcut by ≥4° toward a {110} direction, and are annealed for several minutesin hydrogen at a temperature of 900 to 1100°C, only double steps exist onthe surface.19,77–79,82 These surfaces exhibit only a single type of reconstructionterrace, and IDBs can be suppressed by their use for the growth of CdTe,80

GaAs, InP,81 or AlN.82

4.2.3.2 Heteroepitaxy of Polar Semiconductors withDifferent Ionicities

In the case of heteroepitaxy of a polar semiconductor on another polarsemiconductor, inversion domain boundaries are very unlikely to occurbased on energy considerations. However, if the two semiconductors havedifferent ionicities, an electric field may develop at the interface, resultingin a rough interface with degraded electrical properties.80,81

Farrell et al.83 studied this phenomenon as it applies to the growth of II-VI materials on GaAs (001) substrates. They considered the case of ZnSe onGaAs for specificity; in this case, the electrons contributed to bonding by

FIGURE 4.4Growth of a polar semiconductor (zinc blende GaAs) on a Si (001) surface having monatomicsteps. Inversion domain boundaries (IDBs) are associated with the monatomic steps.

Interface

Inversion domainboundary (IDB)

Si

Ga

As




each atom will be 2/4, 3/4, 5/4, and 6/4 per atom for Zn, Ga, As, and Se,respectively. As an example case, consider ZnSe grown with Se initiation ona GaAs (001) surface terminated exclusively with Ga. In this case, there willbe 1/4 excess electron per bond on the surface. To avoid a built-in electricfield at the interface, an optimal surface structure would result in a 50:50interface bond ratio, with a total of exactly two electrons per bond. Byinterface bond ratio, we mean the (Ga + Se)/(As + Zn) ratio in the interface.Farrell et al. demonstrated that certain GaAs surface reconstructions areoptimal in this regard. In the case of GaAs (001) (2 × 4), they showed thatAs-lean variants of this reconstruction result in a 50:50 interface bond ratio,but the missing row structure for the same surface will give an interfacebond ratio of 25:75. They also showed two other surface structures, GaAs(001) (6 × 4) c and GaAs (001) (2 × 6), which allow the growth of ZnSe witha 50:50 interface bond ratio. There is some experimental evidence that lendssupport to this model. Tamargo et al.84 studied the growth of ZnSe on GaAs(001) and obtained improved crystal quality when the substrate surfacereconstruction was GaAs (001) (2 × 4) rather than GaAs (001) (4 × 2). Thiswas attributed to the tendency for development of the As-lean structures inthe former case, resulting in a 50:50 interface bond ratio.

These same principles should apply generally to heteroepitaxial growthsystems that involve a change in ionicity, although little work has been donealong these lines with other material combinations. However, it is clear thatsurface reconstructions and the resulting interface structures can have far-reaching influence on the properties of heteroepitaxial materials.

FIGURE 4.5Growth of a polar semiconductor (zinc blende GaAs) on a Si (001) surface having double steps.The overgrowth is free from IDBs.

Interface

Si

Ga

As




4.3 Nucleation

An important difference between homoepitaxy and heteroepitaxy is thatheteroepitaxy requires nucleation of a new phase on the substrate surface.Often the nucleation step greatly influences the morphology and structuralproperties of heteroepitaxial layers. The nucleation and growth mode,though different, are closely related. Broadly speaking, there are three modesof heteroepitaxial growth:85 the Frank–van der Merwe86 (FM; two-dimen-sional or layer-by-layer growth), Volmer–Weber87 (VW; three-dimensional orisland growth), and Stranski–Krastanov (SK) mechanisms. In the case of FMgrowth, islands of monolayer height coalesce before a new layer can nucleateon top of them. In VW growth, growth proceeds to many atomic layers atdiscrete islands before these islands merge. In the SK mechanism, the growthis initiated in a layer-by-layer fashion, but islanding commences after thegrowth of a certain thickness. In all but a few situations, layer-by-layergrowth is desirable because of the need for multilayered structures with flatinterfaces and smooth surfaces. This requires that the nucleation occur as asingle event, on the substrate surface, but not in the gas phase. This sectionwill describe the physics and chemistry of nucleation. This treatment willbegin with homogeneous nucleation, which serves as a useful starting pointfor the more relevant situation of heterogeneous nucleation. Then heteroge-neous nucleation will be considered, including the development of a rateequation. Growth modes will be treated separately in the following section.

4.3.1 Homogeneous Nucleation

Homogeneous nucleation corresponds to direct condensation out of the gasphase, in the absence of a substrate surface. Though it is rarely observedin practice,* homogeneous nucleation serves as a useful starting point forthe discussion of the more important case of heterogeneous nucleation ona substrate.

Homogeneous nucleation98 may proceed if a condition of supersaturationexists, meaning that the partial pressure of the nucleating species exceedsthe equilibrium vapor pressure over the solid phase. In this case, the freeenergy difference per atom between the vapor and the solid is

(4.1)

where is the supersaturation.

* A notable exception is the epitaxy of silicon from silane, for which special precautions must betaken to avoid gas phase nucleation.

P0

P∞

G kTdPP

kTPP

kT SvP

P

= = −⎛⎝⎜

⎞⎠⎟

= −∞

∞

∫ ln ln0

0

S P P= ∞0 /




The change in free energy per unit volume of solid is then

(4.2)

where n is the number of atoms per unit volume of the semiconductor.Under a condition of supersaturation (S > 1), embryos will form from the

vapor phase. Some embryos will grow by the inclusion of additional materialfrom the vapor phase, if they are large enough so that their growth reducesthe overall free energy of the system. Other embryos will be less than thecritical embryo size and will shrink by reevaporation. The critical embryosize, above which growth will proceed, may be found by balancing thevolume and surface free energy contributions of the embryo. If the surfacefree energy per unit area of the solid is γ, then the total free energy changefor a spherical embryo is given by

(4.3)

Figure 4.6 shows the behavior of qualitatively, for two different valuesof . With increasing embryo size, first increases, due to the dominantsurface energy term, but then reaches a maximum and finally decreases asthe reduction in the free energy due to the phase change prevails. The criticalembryo size corresponds to the maximum change in free energy; a largervalue of supersaturation (and therefore ) will result in a smaller criticalembryo size, as shown in the figure. The lower curve represents a situationwith a higher supersaturation, and thus a more negative value of , result-ing in a smaller size for the critical nucleus. In other words, sothat and .

FIGURE 4.6Total free energy change ΔG for an embryo of the solid phase, as a function of the embryoradius R, for two values of ΔGv.

ΔG nkT Sv = − ln

Δ ΔGr

G rv= +43

43

2π π γ

ΔGΔGv ΔG

rcrit

ΔGv

ΔGv

Δ ΔG Gv v1 2<r rcrit crit1 2< Δ ΔG Ghomo homo1 2<

R

∆G

∆Ghomo 2

∆Gv2

∆Gv1

∆Gv1 < ∆Gv2

∆Ghomo 1

rcrit 1 rcrit 2




The critical embryo size can be evaluated by setting

(4.4)

which yields

(4.5)

Embryos larger than this size will lower their free energy by continuingto grow, whereas those smaller than this size will reduce their free energyby shrinking. The change in free energy evaluated at the critical embryo sizeis the activation energy for the formation of embryos, which is

(4.6)

If the nucleation process occurs in an ideal gas, then from the kinetic theoryof gases the rate at which atoms (or molecules) will arrive at the surface ofa spherical nucleus having the critical radius will be

(4.7)

where m is the atomic (molecular) mass, k is the Boltzmann constant, and Tis the absolute temperature. Then, assuming Boltzmann statistics, the homo-geneous nucleation rate (nuclei formed per unit time in unit volume) can beestimated as

(4.8)

where n is the atomic (molecular) density of the gas.In the simple model outlined here, the Zeldovich factor was assumed to

be unity and the nuclei were assumed to be spherical in shape. Real crystalnuclei will take on more interesting shapes, so that Equation 4.8 will applyonly roughly. However, this result shows that the nucleation rate will varystrongly with the supersaturation (and therefore reactant partial pressure)and with the temperature, two important growth parameters.

∂∂

==

ΔGr

r rcrit

0

rGcrit

v

= − 2γΔ

ΔΔ

GG

homov

= 163

3

2

πγ

jr P

mkTcrit= 4

2

20π

π

R r P n

mkT

GkThomo

crit homo≈ −⎛

⎝⎜

⎞

⎠⎟

4

2

20π

πexp Δ




4.3.2 Heterogeneous Nucleation

Heterogeneous nucleation takes place in the presence of a surface and is morerelevant to the case of heteroepitaxy. Here, the presence of the surfacechanges the situation significantly and can greatly alter the nucleation rate.Heterogeneous nucleation may be considered from a macroscopic point ofview,98 in a fashion paralleling the treatment of the homogeneous case.Atomistic models88–90 have also been developed, which are applicable tonuclei containing as few as two atoms. In the following subsections, themacroscopic model will be outlined, followed by an atomistic model. Thenthe case of vicinal substrates will be described. Through these discussions,nucleation will be used to refer to the growth of nuclei on a foreign substrate(as in the initiation of heteroepitaxy), but also the nucleation of new clustersof epitaxial material on top of an established layer of this same crystal(second-layer nucleation).

4.3.2.1 Macroscopic Model for Heterogeneous Nucleation

Extending the macroscopic model to the case of heterogeneous nucleation,we find that the presence of a surface tends to increase the nucleation rate.This is because the nuclei may wet the substrate, greatly changing theirgeometry. Suppose and represent the surface free energies of the epi-taxial crystal and substrate, respectively, and is the epitaxial layer–sub-strate interfacial free energy. The epitaxial material will not wet the substrateif , because this would be accompanied by an overall increase inthe free energy of the system. On the other hand, complete wetting isexpected if (the epitaxial deposit will spread out to maximizethe area of the interface). For all other situations, consideration of forcebalance on the boundary of the embryo leads to the expectation of partialwetting with a contact angle θ, where

(4.9)

or

(4.10)

These three types of situations are shown schematically in Figure 4.7.Assuming the embryos to be sphere segments with the appropriate contact

angle α, we can follow a development paralleling that for the homogeneouscase. The free energy change upon formation of the embryo, associated withthe surface and interfacial energies, will be , whereand are the areas of the embryo surface (with the gas phase) and interface

γ e γ s

γ i

γ γ γi e s> +( )

( )γ γ γe i s+ <

γ γ γ θs i e= + cos

θ γ γγ

= −⎛⎝⎜

⎞⎠⎟

−cos 1 s i

e

A Ae e i i sγ γ γ+ −( ) Ae

Ai




(with the substrate), respectively. Therefore, the total free energy changeassociated with creation of the embryo on the substrate surface will be

(4.11)

where r is the radius of curvature for the (truncated sphere) embryo.The free energy change reaches a maximum value of

(4.12)

The rate for heterogeneous nucleation will then be

(4.13)

Here, the prefactor will be different from the case of homogeneousnucleation, due to the reduction of the embryo surface area by wetting.Like before, the nucleation rate will depend very strongly on the supersat-uration and the temperature. However, the change in the critical free energycan drastically increase the nucleation rate for given conditions of super-saturation and temperature, compared to the homogeneous case. This isfortunate, for it allows heteroepitaxial growth to occur under conditionsthat suppress gas phase nucleation. However, layer-by-layer growthrequires that the heterogeneous nucleation proceed at a slow rate of oneevent per monolayer.

FIGURE 4.7Wetting of a flat substrate by an epitaxial deposit. (Adapted from Ghandhi, S.K., VLSI FabricationPrinciples, Silicon and Gallium Arsenide, 2nd ed., Wiley, New York, 1994. With permission.)

θ

No wettingθ = 180°

γi > γs + γe

Partial wetting0 < θ < 180°

γs = γi + γe cos θ

Complete wettingθ = 0

γs > γi + γe

Δ ΔGr

G rv i s= − + + −π θ θ π γ γ θ3

2 2 2

31 2( cos ) ( cos ) ( )sin ++ −2 12π θ γr e( cos )

Δ ΔG Ghet o

e

= + −⎡

⎣⎢

⎤

⎦⎥

=

hom( cos )( cos )2 1

4

16

2

3

θ θ

πγ (( cos )( cos )2 112

2

2

+ −θ θΔGv

R C GkThomo

het≈ −⎛

⎝⎜

⎞

⎠⎟1 exp Δ

C1




4.3.2.2 Atomistic Model

The macroscopic model for heterogeneous nucleation is based on macroscopicproperties such as the surface and interfacial free energies. In some cases,heterogeneous nucleation may occur with nuclei containing as few as twoatoms. In such cases, an atomistic model103–105 for nucleation is more relevant.

In developing an atomistic model for heterogeneous nucleation, it isassumed that atoms arrive at a flat surface with an impingement rate of F(atoms per unit area per unit time*). This incident flux gives rise to a con-centration of adatoms (per unit area) on the surface equal to . There willalso be unstable clusters of two or more atoms on the surface. Here, willbe used to denote the concentration (per unit area) of clusters containing jatoms. Unstable clusters can reduce the free energy of the system by shrink-ing. However, there will also be a concentration of stable clusters on thesurface. These are large clusters that reduce the total free energy of the systemby growing. If the critical cluster size contains i atoms, then all clusterscontaining more than i atoms will be stable. Here, denotes the concentra-tion of stable clusters on the surface and is the average number of atomsin a stable cluster (wx > i).

Suppose atoms arrive at the flat surface with a rate F (atoms per unit areaper unit time). The interaction of the adatoms and surface clusters with thegas phase can be illustrated schematically as in Figure 4.8. Adatoms mayreevaporate (with a time constant ), combine with other adatoms or unsta-ble clusters or be captured by a critical cluster (nucleation, with a timeconstant ), or be captured by a stable cluster (with a time constant ). Anycritical cluster of i atoms that succeeds in capturing one more adatom willbecome a stable cluster. In steady state, adatoms are emitted and acceptedat equal rates by unstable clusters, with no net effect on the populations ofthe adatoms or the unstable clusters.

The rate equations for this system have been derived by Stowell andHutchinson91 and Stowell.92 They are as follows:

(4.14)

(4.15)

and

* In the literature, the incident flux and cluster concentrations are sometimes given in atoms persecond and absolute numbers, respectively. Here, these quantities will be given on a per unitarea basis, so that R has units of atoms per unit area per unit time and n1 has units of adatomsper unit area.

n1

nj

nx

wx

τa

τn τc

dndt

Fn d n w

dta

x x1 1= − −τ

( )

dn

dtj ij = ≤ ≤0 2, ( )




(4.16)

Equation 4.14 gives the time rate of change of the adatom concentration,in which the three terms represent condensation, reevaporation, and diffu-sive capture by stable clusters.* Equation 4.15 stems from the assumptionthat the populations of unstable clusters are constant with time; this is trueif the growth occurs near equilibrium. Equation 4.16 gives the time rate ofchange of the concentration of stable clusters. Here, the first term representsthe creation of new stable clusters (the nucleation rate, in nuclei per unit areaper unit time) by the diffusive capture of adatoms (with concentration )by critical clusters (with population ). D is the diffusivity of adatomsand is the (unitless) capture number for the critical-size clusters. Thesecond term in Equation 4.16 represents the coalescence of stable clusters,and Z is the fraction of the surface covered by stable clusters (0 ≤ Z ≤ 1).

Three other basic relationships are needed to solve Equations 4.14 to 4.16and determine the nucleation rate. First, Equations 4.14 and 4.16 are cou-pled through

FIGURE 4.8Schematic diagram of the interactions between the adatoms and surface clusters with the gasphase. All arriving atoms condense with a rate F. This gives rise to an adatom population n1.Adatoms may reevaporate (with a time constant τa), combine with another adatom or unstablecluster (nucleation, with a time constant τn), or be captured by a stable cluster (with a timeconstant τc). Any critical cluster of i atoms that succeeds in capturing one more adatom willbecome a stable cluster. (Reprinted from Venables, J.A., Phys. Rev. B, 36, 4153, 1987. Withpermission. Copyright 1987, American Physical Society.)

* The rate of nucleation is numerically unimportant here for the purpose of determining n1 andwas neglected.

Arrival F

ni Criticalclusters

Subcriticalclusters2 ≤ j ≤ i

nx Stableclusters

Captureτc

Nucleationτn

n1 Adatoms

Evaporationτa

dndt

Dn n ndZdt

xi i x= −σ 1 2

n1

ni

σ i




(4.17)

Moreover, in steady state, , so that

(4.18)

where and , where is the effective capturenumber (unitless) for stable clusters. Second, the substrate coverage Z isrelated to the number of atoms in the stable clusters by

(4.19)

where is the areal density of atoms in the stable clusters. Assuming mono-layer clusters, , where is the atomic volume. Third, the relation-ship between the populations of critical clusters and adatoms, in the case ofa relatively high supersaturation, is given by

(4.20)

where is the atomic density in the substrate crystal (atoms per unitarea), is the free energy change associated with the critical size cluster,and is a constant. This equation can be generalized to account for morethan one configuration of critical clusters, if there is a low supersaturation.

Upon solution of Equations 4.14 through 4.20, we obtain the normalizeddensity of stable clusters (nuclei), assuming monolayer islands and negligi-ble evaporation, as

(4.21)

where C and η are constants and ν is the effective surface vibration frequency(~1011 to 1013 s–1). The nucleation rate, also assuming monolayer islands andnegligible evaporation, is

(4.22)

d n wdt

n nFZx x

n n

( ) = + +1 1

τ τ

dn dt1 0/ =

n F Z1 1= −τ( )

τ τ τ τ− − − −= + +1 1 1 1a n c τ σc x xDn− =1 σx

dZdt

Nd n w

dtax x= −1 ( )

Na

Na− =1 2 3Ω / Ω

nN

nN

CEkT

i

i

ii

0

1

0

=⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

exp

N0

Ei

Ci

nN

CF

NE iEi kT

x

ii

i d

0 0

2

2=

⎛⎝⎜

⎞⎠⎟

++

⎡

⎣⎢

⎤

⎦⎥

+η

νexp

( )

J Dn ni i= σ 1




The atomistic model outlined here has been applied to numerical calcula-tions of the stable cluster densities in several material systems.105 Thesecalculations provide reasonable agreement to experimental results for sev-eral combinations of metal-on-metal,93 metal-on-Si,94 and rare gases on amor-phous carbon.95 This and similar atomistic models for nucleation have alsobeen applied to the analysis of second-layer nucleation in semiconductorheteroepitaxy.96 This has made it possible to predict surface rougheningassociated with a transition from a layer-by-layer growth mode to a Stran-ski–Krastanov (layer-plus-islands) growth mode.

An atomistic approach to nucleation is appropriate if the critical nucleicomprise only a few atoms, for in this case the macroscopic surface andinterfacial energies are inapplicable. In the limit of large critical cluster size,however, the atomistic and macroscopic models should converge. The proofof this is not trivial, however, and so at the present time no unified modelhas emerged.

The atomistic model described above is convenient in that it is analyticalin nature. But it is also completely deterministic and cannot account for thestatistical nature of surface atomic processes such as diffusion. Completelystochastic atomistic models have also been implemented in the form ofmolecular dynamic (MD)97,98 or kinetic Monte Carlo (KMC)99,100 numericalsimulations, which address this issue but are beyond the scope of this book.

4.3.2.3 Vicinal Substrates

In the case of heteroepitaxy of semiconductors, vicinal (tilted) substrates areoften utilized. An example is GaAs (001) 2° → [110]. The surface of a vicinalsubstrate comprises low-index terraces separated by monolayer or bilayersteps, which have a separation determined by the offcut angle. Generally,there will also be kinks (jogs in the steps) whose density will depend on thedirection of the offcut. Here, heterogeneous nucleation of the dissimilarmaterial of the vicinal substrate will occur preferentially at the kinks orsteps, due to the modified value of compared to the open terraces.Therefore, growth on a vicinal substrate by the advancement of steps (stepflow growth) may be possible at a supersaturation too low to result innucleation on a flat substrate and may allow improved crystal quality andthe suppression of islanding.

4.4 Growth Modes

There are three broad classifications for the growth modes for heteroepi-taxy:98 the Frank–van der Merwe99 (FM; two-dimensional growth),Volmer–Weber100 (VW; three-dimensional growth), and Stranski–Krastanov

ΔGhet




(SK)* mechanisms. FM (two-dimensional) growth on a flat substrate† ischaracterized by the nucleation of a new monolayer and its growth to coverthe substrate, followed by the nucleation of the next layer. This growth modeis therefore referred to as layer-by-layer growth. VW growth involves thedevelopment of isolated islands on the substrate, followed by their growthand coalescence. This coalescence process results in a rough surface, with aroot mean square (rms) roughness comparable to the mean distance betweenislands. In the SK mechanism, the initial growth proceeds in a layer-by-layerfashion but becomes three-dimensional in nature after the growth of a certaincritical layer thickness. (It should be emphasized that this is not the same asthe critical layer thickness for lattice relaxation, although the two may becomparable in size and are sometimes used interchangeably.) The FM, VW,and SK growth modes are illustrated schematically in Figure 4.9.

The important distinction between two-dimensional growth and the othermodes is that in a two-dimensional growth mode either (1) a monolayer

FIGURE 4.9(a) Frank–van der Merwe (FM), (b) Volmer–Weber (VW), and (c) Stranski–Krastanov (SK)growth modes for heteroepitaxy.

* This growth mode was named Stranski–Krastanov by Bauer and Poppa (E. Bauer and H.Poppa, Recent advances in epitaxy, Thin Solid Films, 12, 167 (1972)). This came about as a resultof a calculation made by Stranski and Krastanov (I.N. Stranski and L. Krastanov, Zur Theorie derorientierten Ausscheidung von Ionenkristallen aufeinander, Sitzungsbericht Akademie Wissen-schaften Wien Math.-Naturwiss. Kl. IIb, 146, 797 (1938)). They showed that, for a monovalent ioniccrystal condensing onto a divalent substrate, the second layer of condensate has weaker bondingthan the substrate surface layer, even though the first layer of condensate has stronger bonding.It is possible that this phenomenon could result in the mode we have come to know as SK.† In the case of a vicinal substrate, the surface comprises a number of flat terraces separated bymonolayer steps. Here, the nucleation of new layers is unnecessary. Instead, growth proceeds bythe advancement of steps, and this is called step flow growth. In either case (layer-by-layer orstep flow growth), the epitaxial layer retains the surface smoothness of the starting substrate.

Frank-van der Merwe (FM)

Volmer-Weber (VW)

Stranski-Krastanov (SK)

(a)

(b)

(c)




completes before a new one nucleates or (2) if multiple nucleation eventsoccur on a flat substrate surface, the monolayer islands coalesce beforeadditional growth occurs on top of them. Therefore, VW and SK growth,which involve islanding, are sometimes collectively called multilayergrowth, referring to the fact that islands grow beyond one-monolayer thick-ness before coalescing.

In all but a few situations, two-dimensional growth (either layer-by-layeror step flow growth) is desirable because of the need for multilayered struc-tures with flat interfaces and smooth surfaces. A notable exception is thefabrication of quantum dot devices, which requires three-dimensional or SKgrowth of the dots. Even here, though, it is desirable for the other layers ofthe device to grow in a two-dimensional mode. In all cases of heteroepitaxy,it is important to be able to control the nucleation and growth mode.

The growth modes in heteroepitaxy have been considered extensivelybased on thermodynamic models.102–104 Along these lines, Daruka andBarabási102,103

developed an equilibrium phase diagram that identifies thegrowth mode as a function of the lattice mismatch strain and the averagethickness of the deposit. At the same time, it has also been established thatkinetic factors often play an important role in establishing the growth mode,if the growth proceeds with a large supersaturation.104 Of these, the mostimportant are the surface diffusivity, the energy barrier to diffusion at steps,and the growth rate. Based on kinetic considerations, Tersoff et al.96 showedthat there is a critical island size for the achievement of layer-by-layergrowth. While these aspects of heteroepitaxy are far from completely under-stood, it is becoming clear that kinetic factors provide an opportunity forcontrolling the growth mode. An especially interesting aspect of this involvesthe use of surfactants. While the behavior of surfactants in heteroepitaxy isnot yet entirely clear, it is known that surfactants can in some cases modifythe growth mode. In the following sections, equilibrium considerations willfirst be presented, including the development of a general growth modephase diagram. This will be followed by a brief consideration of a kineticmodel and a development of the conditions necessary for layer-by-layergrowth. The possible roles of surfactants will be considered in Section 4.6.

4.4.1 Growth Modes in Equilibrium

In the classical theory, the mechanism of heterogeneous nucleation is dictatedby the surface and interfacial free energies for the substrate and epitaxialcrystal.85 The energy criteria are stated in terms of , the areal change infree energy associated with covering the substrate with the epitaxial layer,not including the bulk free energy of the epitaxial crystal. Then if andare the surface free energies of the epitaxial layer and substrate, respectively,and is the interfacial free energy for the epitaxial–substrate interface, then

(4.23)

Δγ

γ e γ s

γ i

Δγ γ γ γ= + −e i s




If minimum energy dictates the mode for nucleation and growth, theprevalent mechanism will be two-dimensional for and three-dimen-sional for . Often the interfacial contribution can be neglected in com-parison to the surface energy terms. If this is the case, then two-dimensionalgrowth is expected for (the epitaxial layer will wet the substrate),but three-dimensional growth will occur if . However, even in thecase of a wetting epitaxial layer ( ), the existence of mismatch straincan cause islanding after the growth of a few monolayers. This is becausethe strain energy in the coherent epitaxial layer increases in direct propor-tion to the thickness. At some point, it becomes energetically favorable tocreate islands that can relieve some of the mismatch strain by relaxation atthe sidewalls. Therefore, SK growth can be expected in the case of a wettingepitaxial layer unless the lattice mismatch strain is quite small. Whereas itis clear that the VW growth mode is to be expected for a nonwettingepitaxial layer, the behavior of a wetting deposit is more complex andwarrants further consideration.

In order to elucidate this behavior, Daruka and Barabási102,103 investigatedthe growth of a lattice-mismatched, wetting epitaxial layer on a foreignsubstrate and created an equilibrium phase diagram that can help predictthe growth mode for heteroepitaxy. In the development of their model, theyassumed the growth of a wetting epitaxial layer B on a substrate A, with athickness of H monolayers and a lattice mismatch f . The total deposit isdistributed among the wetting layer with a thickness of monolayers, sta-ble islands with an average thickness of monolayers, and large, ripenedislands having an average thickness of monolayers. Both stableand ripened islands were assumed to be square pyramids with a fixed aspectratio; this aspect ratio corresponds to crystal faces for which the facet energyhas a minimum as determined on the Wulff’s plot.105

In their calculations, Daruka and Barabási neglected evaporation and con-sidered the free energy per interfacial lattice site (effectively per unit area),f = u – Ts, where u is the internal energy density, T is the temperature, ands is the entropy density. Furthermore, they assumed that the entropy contri-bution is negligible, which has been shown to be appropriate for lowertemperatures,103 so that f ≈ u. The average free energy per lattice site for thecombination of wetting layer and islands was calculated to be

(4.24)

where is the energy per monolayer in the strained wetting layer, isthe free energy per monolayer in the pyramidal islands, and is the energyper monolayer in the ripened islands.

The energy density of the coherently strained wetting layer may be calcu-lated to first order as , where C is a constant that depends onthe biaxial modulus and the monolayer thickness and is the energy ofan AA atomic bond. Daruka and Barabási accounted for the energy of the

Δγ < 0Δγ > 0

γ γe s<γ γe s>

γ γe s<

n1

n2

H n n− −1 2

u E n E H n n Eml isl rip= + + − −2 1 2( )

Eml Eisl

Erip

G Cf AA= −2 Φ−ΦAA




interfacial AB bonds, , and also the fact that the binding energies of Aatoms will be modified close to the interface.106,107 Including these contribu-tions, the energy density in the wetting layer was assumed to be

(4.25)

where the parameter a specifies the effective range for the interatomic forces(Daruka and Barabási assumed that a = 1 in their calculations),

, and is the unit step function:

(4.26)

Daruka and Barabási noted that the exact form of would not change thequalitative features of the overall behavior.

The free energy per monolayer in the pyramidal islands was calculated using

(4.27)

where g is a form factor that expresses the reduction in the strain energy ofthe islands compared to the continuous wetting layer and . Thenormalized island size is , where L is the length of a side of thepyramidal island and is a characteristic length as defined by Shchukinand coworkers.108 The three terms in the square brackets arise because of thefaceting of the islands. With nonzero facet surface energy, compressive forceswill develop at the facet edges, resulting in a component of stress energy inthe islands,109 which Daruka and Barabási have called the homoepitaxialstress. The first term in the square brackets corresponds to this homoepitaxialstress contribution. The second term in the square brackets is associated withthe cross-term from the interaction of the lattice mismatch stress and thehomoepitaxial stress, and also the facet energy, which has the same x-depen-dence. Therefore, , where p and γ are material constants. The thirdterm in the square brackets represents the energy from the island–islandstress interaction.

The free energy per monolayer in the ripened islands is

(4.28)

Daruka and Barabási used the model outlined here to calculate phasediagrams using various values of the material parameters. To do this, they

−ΦAB

E G U n U n e dnmln a

n

= + − + − − −∫ { [ ( ) ( ) ]}( )/Δ 1 1 1

0

1

Δ Φ Φ= −AA AB U x( )

U xx

x( )

; ,; .

=<>

⎧⎨⎩⎪0 01 0

Em1

E gCf Ex

x ex xisl AA= − + − + +

⎡

⎣⎢

⎤

⎦⎥

20 2 3 2

2Φ ln( ) /

α β

0 1< <gx L L= / 0

L0

α γ= −p f( )

E gCfrip AA= −2 Φ




minimized the energy with respect to , , and x. The detailed procedurehas been described by Daruka.103

The Daruka and Barabási equilibrium model involves a number of materialparameters (a, C, E0, , , g, p, , and b), some of which are eithersemiempirical or not known with a high degree of accuracy. Therefore, thequestion arises as to whether the choice of material parameters will changethe very nature of the phase diagram. However, Daruka103 showed that forany set of material parameters, the resulting phase diagram would assumeone of four basic topologic forms. Moreover, Daruka and Barabási developeda general phase diagram that incorporates the features of all four classes, asshown in Figure 4.10, using the parameters , , ,

, , , , and . Here, seven distinct phaseregions are seen, corresponding to Frank–van der Merwe (FM), Stran-ski–Krastanov (SK), Volmer–Weber (VW), or ripening (R) behavior. Thereare two Stranski–Krastanov phase regions, SK1 and SK2. In both cases, islandscoexist with a wetting layer; however, as will be explained in the following,the behavior with increasing growth time is different in the two cases. Thereare also three ripening phases: R1, R2, and R3. The R1 phase exhibits ripenedislands along with a wetting layer. The R2 phase exhibits stable islands aswell. The R3 phase is characterized by the presence of both ripened andstable islands, but no wetting layer.

Growth phases involving ripened islands are generally undesirable. Basedon thermodynamic considerations, the ripened islands are predicted to haveinfinite size and vanishing density on the surface. In a real epitaxial growthprocess, ripening islands will have finite size and density, as determined bythe kinetics of their growth. The important distinction is that they will beunstable and there will exist a driving force for their growth with time bythe process of Ostwald ripening.110–112 Therefore, stable islands are neededfor device fabrication if additional high-temperature processing will be usedfollowing their growth.

In a typical heteroepitaxial growth process, the lattice mismatch strain f (εin the notation of Daruka and Barabási) is fixed, but the extent of the depositH (in monolayers) increases with time. Based on the phase diagram con-tained in Figure 4.10, we can understand four cases of such a process, whichwill be outlined in the following.

4.4.1.1 Regime I: (f < ε1)

Suppose the lattice mismatch strain is small (f < ε1, with indicated inFigure 4.10). In this case, the initial growth will occur with a Frank–vander Merwe mode. After the deposition of a certain thickness, however,we expect a transition to the R1 phase, and so ripened islands will growon the initial wetting layer. In this phase region, the wetting layer thick-ness does not increase but stays constant, so that the newly depositedmaterial contributes only to the formation of ripened islands. The energy

n1 n2

ΦAA ΦAB γ

a = 1 C E= 40 0 ΦAA E= 0

ΦAB E= 1 27 0. g = 0 7. p = 4 9. γ = 0 3. b = 10

ε1




is minimized when the ripened islands approach infinite size with zerodensity.*

4.4.1.2 Regime II: (ε1 < f < ε2)

If the lattice mismatch strain is increased somewhat ( , withand indicated in Figure 4.10), the initial growth still exhibits a two-dimen-sional nature. As the average thickness of the deposit is increased, however,we can expect a transition to a Stranski–Krastanov mode (phase SK1) inwhich stable islands with finite size and density grow on top of the initialwetting layer. A further increase in the growth time will give rise to theappearance of ripened islands along with the stable islands (phase region

FIGURE 4.10Equilibrium phase diagram for heteroepitaxy of a wetting material A on a substrate B. H is theaverage thickness in monolayers, and ε is the lattice mismatch strain. The small panels showthe morphology of the growing film for each of the phase regions. The small open trianglesrepresent stable islands, whereas the large shaded triangles denote ripened islands. The param-eters used to calculate the phase diagram were a = 1, C = 40E0, ΦAA = E0, ΦAB = 1.27E0, g = 0.7,p = 4.9, γ = 0.3, and b = 10. (Reprinted from Daruka, I. and Barabási, A.-L., Phys. Rev. Lett., 79,3708, 1997. With permission. Copyright 1997, American Physical Society.)

* Of course, kinetic considerations would predict islands with a finite size and finite density.

R1 R2 R3

R1R2 R3

5

4

3

2

1

0

H

0.0 0.1 0.2ε

ε1 ε2 ε3

FM

FM SK VW

VW

SK1

SK2

ε ε1 2< <f ε1

ε2




R2). As in regime I, equilibrium considerations predict the ripened islandswill have infinite size and zero density.

4.4.1.3 Regime III: (ε2 < f < ε3)

Consider a large value of the lattice mismatch strain ( , withand indicated in Figure 4.10). Here, the lattice mismatch is too large topermit two-dimensional growth. Instead, the initial growth occurs in a VWmode, with separated, stable islands and the absence of a wetting layer.However, with increasing growth time the SK2 phase is encountered. Asexpected, the Stranski–Krastanov phase is characterized by stable islandsand a wetting layer. But in this case, the islands grow first, followed by thewetting layer, which fills in the area separating them. As growth proceeds,the wetting layer increases its thickness but the stable islands remain fixedin size. This continues until the SK1 phase boundary is encountered. Then,in the SK1 phase region, the additional material serves to grow additionalstable islands while the wetting layer remains at constant thickness. Even-tually, the growth proceeds in the R2 phase, in which ripened islands growfrom the additional material.

4.4.1.4 Regime IV: (f > ε3)

For very high values of the lattice mismatch strain ( , with indi-cated in Figure 4.10) the initial growth occurs with a VW mode, followedby the growth of ripened islands (phase region R3). A wetting layer neverforms, and so a continuous heteroepitaxial layer will not be achieved inthis case.

In all of the four regimes of mismatch described above, equilibrium theorypredicts the growth of infinitely large ripened islands, with vanishing den-sity. This cannot occur in a real growth situation, in which the growth andsurface diffusion processes occur at finite rates. So, although the equilibriumconsiderations outlined in this section provide guidance in terms of thedriving forces and the direction in which growth will proceed, the kineticconsiderations will dictate the density and size of ripening islands andperhaps also the stable islands. An important result of this is that the growthmorphology can be influenced by changing the growth temperature orgrowth rate, or by the introduction of surfactant species, which can signifi-cantly modify the surface diffusion.

4.4.2 Growth Modes and Kinetic Considerations

Equilibrium considerations dictate that mismatched heteroepitaxial materialwill usually grow in a VW or SK mode, with a rough surface, unless theepitaxial layer wets the substrate and the lattice mismatch is small. On theother hand, heteroepitaxial growth may occur far from equilibrium (i.e., witha large supersaturation). In such a case, kinetic factors provide an opportu-

ε ε2 3< <f ε2

ε3

f > ε3 ε3




nity to tailor the growth mode and morphology. The most important suchfactors influencing the growth mode and morphology are the surface diffu-sivity and the growth rate (flux). In addition to these controllable factors,the diffusion barrier at step edges (the Ehrlich–Schwoebel barrier113,114) mayalso play a role in determining the growth mode. Consideration of kineticsshows that it is possible to tailor the growth conditions (through the tem-perature, growth rate, or introduction of surfactants) in such a way as toobtain layer-by-layer growth. An intriguing discovery is the existence ofreentrant epitaxy,115,116 in which the growth mode observed at high and lowtemperatures differs from that found at the middle range of temperatures.Just as interesting is the finding by many workers that the inclusion of asurfactant can alter the growth mode, by either inhibiting or promotingisland growth.

It should also be possible to design the growth process in such a way asto control the size and density of islands in the SK or VW growth modes. Inthis section, the condition for layer-by-layer growth will be developed. Thecontrol of islanding in heteroepitaxy, also known as self-assembly, will beconsidered in Section 4.7.

In considering the kinetic factors controlling the growth mode, the problemis to find the conditions that give rise to layer-by-layer growth, rather thanthe growth of isolated three-dimensional clusters. Or, stated differently, theproblem is to find the conditions under which a new layer will nucleate ona monolayer island before coalescence (so-called second-layer nucleation),which will give rise to kinetic roughening.

Tersoff, Denier van der Gon, and Tromp96 (TDT) derived the critical islandsize for layer-by-layer growth by a consideration of this second-layer nucle-ation process.* In their model, TDT assumed the existence of circular mono-layer islands with uniform radius. They found the rate of second-layernucleation on top of these islands using classical atomistic nucleation theory,by solving the diffusion equation for adatoms with a constant growth flux(MBE conditions). This model will be summarized in what follows.

Based on an atomistic approach, TDT assumed the nucleation rate to be

(4.29)

where D is the diffusivity for surface atoms, is the surface atomic density(atoms per unit area), and is the surface concentration of adatoms (perunit area). The normalized (dimensionless) adatom density is .

Consider the growth of a heteroepitaxial layer with an incident flux ofatoms F. This could correspond directly to the case of MBE, but can be

* Here, second-layer nucleation refers generally to the formation of stable nuclei on top of anestablished island.

ω η≈⎛⎝⎜

⎞⎠⎟

=DNnN

DNi

i02 1

002

N0

n1

η = n N1 0/




applied to VPE processes by a simple extension. The diffusion equation forthe resulting adatoms on the surface is

(4.30)

To investigate the possibility of second-layer nucleation, we will considerthe growth on an existing monolayer island having circular geometry. In thiscase, the steady-state solution to the diffusion equation for adatoms on thetop of the island is

(4.31)

where r is the distance from the center of the island. The boundary conditionat the island edge is , where represents the probabilitythat an adatom, upon reaching the island edge, will jump over the edge inunit time, divided by the rate for hops within the area of the terrace. If thereis an energy barrier for hopping over the edge of the island, then

, where i s a cons tant , i s the Ehr-lich–Schwoebel barrier,113,114 and is the activation energy for surface dif-fusion on top of the island. Based on the boundary condition above, theconstant in Equation 4.31 may be evaluated as

(4.32)

where and R is the average island radius. (In this simplemodel, the islands are assumed to have uniform size equal to the average size.)

The rate of nucleation on top of the island, in nuclei per unit time, can befound by integrating over the island area:

(4.33)

TDT considered two limiting cases. In case 1, the Ehrlich–Schwoebel bar-rier can be neglected, so and , giving

(4.34)

ddt

DF

Nη η= ∇ +2

0

η η= −00

2

4R

DNr

d dr Nη ηα/ + =0 0 α

Es

α = − −C E E kTs d1 exp[ ( )/ ] C1 Es

Ed

η α00

2

4= +F

DNR RL( )

L Nα α≡ 2 0/( )

Ω = =+

⎛⎝⎜

⎞⎠⎟

+∫ − +ω π πα2

1 2002 2rdr

Di

FD

N R RLR i

i i[( ) 11 1− +( ) ]RL iα

α ≈ 1 L Rα <<

Ω1 02 2 2

1 2=

+⎛⎝⎜

⎞⎠⎟

− +πDi

FD

N Ri

i i




In case 2, the Ehrlich–Schwoebel barrier is much greater than kT, sothat and

(4.35)

Case 2 is most relevant to the consideration of second-layer nucleation andcan be used to determine the fraction of islands f that have nucleated a secondlayer on their top. The time rate of increase for the fraction of islands expe-riencing second-layer nucleation is given by

(4.36)

TDT assumed that the growth of the islands with time can be described by

(4.37)

where is the area per island (the reciprocal of the areal density of thenuclei), so that is approximately the separation between islands. Then thefraction of islands with nuclei on top will be

(4.38)

where Rc is the critical island size (radius) for the transition from FM to SKgrowth and m is a unitless parameter that depends on the critical cluster size.

For case 1, with a negligible Ehrlich–Schwoebel barrier, and

(4.39)

For case 2, with and

(4.40)

α → 0

Ω2 02 2

4=

⎛⎝⎜

⎞⎠⎟

− +π αDFD

N R Li

i i i

dfdt

f= −Ω( )1

RFLN

tn22

0

=

πLn2

Ln

f R Rcm= − −1 exp[ ( / ) ]

m i= +2 4

R i iL D

FNc

n

ii

1

2 1

031 2 4

2 4= + +⎛

⎝⎜⎞

⎠⎟⎛⎝⎜

⎞⎠⎟

−−( )( )

π

⎡⎡

⎣⎢⎢

⎤

⎦⎥⎥

+1 4/( )i

( ),L Rα >> m i= + 4

R iL D

FL Nc

n

ii i

2

2 1

034

2 4= +⎛

⎝⎜⎞

⎠⎟⎛⎝⎜

⎞⎠⎟

⎡

⎣

−− −( )

π α⎢⎢⎢

⎤

⎦⎥⎥

+1 4/( )i




Thus, once the critical island size has been established based on Equation4.39 or 4.40, the growth mode for a wetting layer with an island size R ispredicted as follows:

(4.41)

and

(4.42)

Essentially, if there will be new nucleation on the islands before theycoalesce. This will give rise to undesirable surface roughening in the case ofhomoepitaxy or heteroepitaxy. This is illustrated in Figure 4.11, which showsthe fraction of islands experiencing second-layer nucleation vs. the normal-ized island size , with m as a parameter.

The TDT model may be used to understand the temperature dependenceof the growth mode for heteroepitaxial growth. This is based on the interplayof three characteristic lengths: Ln, Ls, and . Here, is the separationbetween nucleating islands and is an increasing function of temperature. Ls

is the separation between steps on the vicinal substrate,* ,where h is the step height and is the angle of the substrate miscut, anddoes not depend on temperature. is a length that characterizes the diffu-sion barrier at the island edges and decreases with increasing temperature.With a sufficiently high temperature or step density, ; in this case,adatoms can diffuse to the surface steps before nucleating new islands, and

FIGURE 4.11Fraction of islands exhibiting second-layer nucleation vs. the normalized island size R/Rc , withm as a parameter. Rc is the critical island size. (Reprinted from Tersoff, J. et al., Phys. Rev. Lett.,72, 266, 1994. With permission, Copyright 1994, American Physical Society.)

* Even substrates with an “exact” low-index orientation will typically have a miscut of up to 0.5°,and therefore a finite density of surface steps.

0.0

0.5

1.0

0 0.5 1 1.5R/Rc

f

m = 24m = 9m = 6

Rc

R R FMc< →

R R SKc> →

( ),R Rc>

R Rc/

Lα Ln

L hs = / tan θθ

Lα

L Ln s<




the result will be step flow (SF) growth. At a lower temperature, , sothe growth mode will be layer by layer (FM), whereby monolayer islandsnucleate and then coalesce. At a still lower temperature, so that mul-tilayer (SK) growth will result.

The case of reentrant layer-by-layer growth at still lower temperatures hasbeen attributed to a reduction in the diffusion barrier associated with rough-ening of the island shapes.146 TDT offered another explanation for this reentrantbehavior. Suppose the islands take on a dendritic shape with arms of charac-teristic width , and this width stays roughly constant as the islands grow.Then in this third case of dendritic growth, the critical island size is given by

(4.43)

Layer-by-layer growth will occur with , and this can occur at a lowtemperature if the characteristic width decreases strongly with temperature.

Figure 4.12 illustrates the expected regimes of growth for various temper-atures and substrate miscut angles. If the miscut angle is sufficiently large,the progression from high temperature to lower temperature is as follows:step flow (SF) growth, layer-by-layer (FM) growth, multilayer (SK) growth,and finally reentrant layer-by-layer (RFM) growth. For the miscut anglerepresented by the dotted horizontal line, the transitions occur at the tem-peratures , , and , respectively.

The model described here is capable of explaining, at least qualitatively,most of the available experimental evidence in InAs/GaAs (001),118 Si1–xGex/Si (001),118 and InGaN/GaN (0001).119

FIGURE 4.12Regimes of kinetically controlled growth modes, for various values of temperature T andsubstrate miscut angle θ. (Reprinted from Tersoff, J. et al., Phys. Rev. Lett., 72, 266, 1994. Withpermission, Copyright 1994, American Physical Society.)

Temperature

0 Tr

SF

SKRFM

Tα

ln θ

Ts

FM

L Ln s>

L Lnα >

2W

RDF

L NW

c

i i i

i3

1

03

2

2 2=⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦

− − −

+πα ⎥⎥

⎥

1 2/

R Rc 3 >W

Ts Tα Tr




As in the case of nucleation, kinetic Monte Carlo (KMC) simulations havebeen applied to predict the growth mode, and specifically surface roughen-ing, in heteroepitaxy. Recently, Mandreoli et al.121 have also reported a hybridapproach that combines elements from the rate equation formulation andthe kinetic Monte Carlo approach.

4.5 Nucleation Layers

In highly mismatched heteroepitaxial growth, the equilibrium growth modewill tend to be either VW or SK, depending on the relevant surface andinterfacial energies. The deposition process therefore involves the growthand coalescence of islands. Layers produced in this way tend to have roughsurfaces due to the rounded morphology of the islands; therefore, the surfaceroughness is comparable to the island size. Moreover, films grown by thecoalescence of larger, irregular islands may contain pinholes. These featuresof three-dimensional nucleation are undesirable in the fabrication of devices,but fortunately, they may be suppressed by the use of an appropriate nucle-ation layer.

The growth conditions for a nucleation layer of this type must be designedto give a layer with a smooth surface, as determined by kinetic limitations.According to the discussion of the previous section, this should be achievablein either a high-temperature or low-temperature regime. In practice, how-ever, other factors usually make it necessary to grow such a nucleation layerat a low growth temperature or a high growth rate. Under these conditions,the resulting material will typically exhibit a fine polycrystalline or amor-phous structure. The nucleation layer must cover the substrate uniformly,but it must also be thin enough so that it can be crystallized by annealingafter deposition. Therefore, a low growth temperature is favored over a highgrowth rate. After deposition of the nucleation layer, an appropriate heattreatment may be used for its crystallization. The avoidance of large islandsdramatically improves the surface smoothness of the nucleation layer andalso the device layer grown on top of it.

Often, the nucleation layer is made of the same material as the device layerto be grown above it. In this type of situation, a smooth layer may bepromoted by growing the nucleation layer at a significantly reduced tem-perature. Examples of the use of low-temperature (LT) nucleation layersinclude GaN/LT GaN/Al2O3 (0001), GaAs/LT GaAs/Si (001), and InP/LTInP/Si (001). Sometimes, a third material is used as the nucleation layer, asin GaN/AlN/Al2O3 (0001).

The as-grown crystal quality of such a nucleation layer is necessarily verypoor. Polycrystalline or even amorphous growth may occur. However, ashort annealing treatment can promote crystallization of the nucleation layer




if it is sufficiently thin. The end result is hopefully a single crystal that coversthe substrate and serves as a template for epitaxy.

4.5.1 Nucleation Layers for GaN on Sapphire

Nucleation layers of AlN are commonly used in the heteroepitaxy of nitridesemiconductors on sapphire substrates. These AlN buffers avoid the growthof columnar islands and improve the crystal quality of the overgrown GaN.The resulting benefits include an improvement in the electrical and opticalproperties of the GaN top layer. An additional, but unrelated, benefit of theAlN buffer layer is the compensation of the tensile thermal strain introducedby the sapphire substrate during cool-down. Therefore, the AlN buffer layercan help prevent cracking in thick nitride layers grown on sapphire.

Yoshida et al.121,122 first used an AlN buffer layer for the MBE growth ofGaN on Al2O3 (0001), basal-plane sapphire. Here, the AlN buffer was actuallygrown at a higher temperature (1000°C) than the GaN (700°C). It was foundthat inclusion of the buffer layer improved the Hall electron mobility by afactor of three, compared to the case of growth directly on sapphire. More-over, GaN grown with the AlN buffer exhibited up to two orders of magni-tude improvement in cathodoluminescence intensity at 360 nm. Amano etal.123,124 and Koide et al.125 investigated the structural properties of low-temperature AlN buffer layers used for the growth of GaN on sapphire. Theyfound that the AlN buffer grew as an amorphous layer, thereby suppressingthe growth of columnar islands. Further, they observed that heating to thegrowth temperature for GaN led to the crystallization of the AlN buffer,apparently resulting in a single-crystal surface for heteroepitaxy.

Nakamura126 applied a GaN low-temperature nucleation layer for thegrowth of GaN on basal-plane sapphire by MOVPE. The low-temperatureGaN nucleation layer was grown at a temperature of 450 to 600°C, whereasthe thick top layer of GaN was grown at 1000 to 1030°C. Based on Hall effectmeasurements of the carrier mobility, the optimum thickness for the nucle-ation layer was determined to be 200 Å. The GaN grown on an optimizednucleation layer exhibited mirror-smooth morphology over an entire 2-inchwafer. In contrast, GaN grown directly on sapphire without a nucleationlayer grew by the coalescence of large hexagonal islands and exhibited arough surface.

Kuznia et al.127 compared the use of GaN and AlN nucleation layers forthe MOVPE growth of GaN on sapphire (0001). The nucleation layers usedin this study were all grown at 550°C and varied in thickness from 100 to900 Å. The crystallinity of each nucleation layer was investigated by low-energy electron diffraction (LEED) directly after growth and also after a 1-h anneal at 1000°C. It was found that as-grown nucleation layers exhibitedno LEED pattern, indicating their amorphous nature. After annealing for 1h at 1000°C, however, there was a well-defined LEED pattern indicative ofa single-crystal layer. Based on electrical measurements (Hall mobility and




carrier concentration) made on thick GaN grown upon various nucleationlayers, it was found that the optimum nucleation layer thickness was approx-imately 250 Å (500 Å) for a GaN (AlN) nucleation layer. The use of a nucle-ation layer greatly improved the crystalline quality (as measured by the x-ray rocking curve full-width half maximum (FWHM)), increased the carriermobility, and decreased the background doping concentration compared tothe case of growth directly on a sapphire substrate. However, either type ofnucleation layer (AlN or GaN) gave similar results for all three materialparameters. If optimum thickness nucleation layers were used, the AlNnucleation layer gave slightly better results than the GaN nucleation layer.

4.6 Surfactants in Heteroepitaxy

Surfactants, or surface-segregated impurities, have a number of applicationsin heteroepitaxy and engineered heterostructures. The nucleation andgrowth mode can be modified by the presence of a surfactant.128 Surfactantsmay also change the surface reconstruction129,130 or the misfit dislocationstructure in partially relaxed heteroepitaxial layers.131 In the growth ofIn0.5Ga0.5P, the introduction of a surfactant can suppress the CuPt orderingthat normally occurs in this alloy.132

4.6.1 Surfactants and Growth Mode

Surfactants may alter the growth mode for heteroepitaxy by modification ofthe surface energies for the substrate or epitaxial layer, if the growth modeis determined by thermodynamics. Alternatively, a surfactant may changethe surface diffusivities or energy barrier at the edge of the islands, if thegrowth mode is determined by kinetics.

Along the line of thermodynamic considerations, the nucleation andgrowth mode for heteroepitaxy is determined by Δγ, the areal change in freeenergy associated with covering the substrate with the epitaxial layer. Ifand are the surface free energies of the epitaxial layer and substrate,respectively, and is the interfacial free energy for the epitaxial–substrateinterface, then

(4.44)

Often, may be neglected so that the growth mode will be two-dimen-sional (FM) if and the deposit wets the substrate. On the other hand,a three-dimensional (VW) mode will result if the deposit does not wet thesubstrate Even if the deposit wets the substrate, the presence oflattice mismatch strain is expected to result in an SK growth mode (layer-

γ e

γ s

γ i

Δγ γ γ γ= + −e i s

γ i

γ γe s<

( ).γ γe s>




by-layer growth followed by islanding) because the creation of islands willpartially relieve the mismatch strain.

This has important implications for the growth of heterostructures involv-ing two semiconductors A and B. If A wets B, then B will not wet A, andvice versa. Therefore, in an ABA heterostructure or an ABAB … superlattice,one of the two materials will grow in a three-dimensional mode, causing adeterioration in the overall structure and its electrical properties.

Copel et al.128 have proposed that this difficulty may be overcome usinga properly chosen surfactant, on the basis of energy considerations. Thiscould be the case if the surfactant atoms have negligible incorporation in agrowing crystal of either A or B, and if the surfactant atoms satisfy danglingbonds and reduce the surface energy of crystal A or crystal B. Then thesurfactant atoms will “float” on the surface during epitaxy and may suppressislanding due to surface energy considerations.

On the other hand, surfactants can also modify the growth mode bychanging kinetic factors, such as the surface diffusivity for adatoms and theenergy barrier for adatoms hopping off the edge of an island (the Ehr-lich–Schwoebel barrier).

The effect of surfactants on the growth mode has been most studied in theheteroepitaxial system Ge/Si. Voigtländer and Zinner133 studied the surfac-tant-mediated epitaxy of Ge on Si (111) using Sb as the surfactant. The normalgrowth mode for this heteroepitaxial combination is SK. However, Voigt-länder and Zinner found that the Sb surfactant could suppress island for-mation at a growth temperature of 600°C. However, for growth temperaturesof >620°C, the Sb was ineffective in suppressing islanding. They invokedkinetic considerations to explain this result, whereby the Sb surfactant sup-presses the surface diffusion of the Ge, thus suppressing island formation ifthe temperature is sufficiently low. Other studies of surfactant-mediatedgrowth of Ge on Si have shown that group V and VI surfactants decreasethe surface diffusion and suppress islanding. However, group III and IVsurfactants enhance the surface mobility and have the opposite effect. Hibinoet al.134 studied the surfactant-mediated growth of Ge on Si using Pb. Theyfound that the Pb preadsorption changed the surface structurefrom to . Also, in the growth temperaturerange 300 to 450°C, the onset of islanding occurred at a thickness of 6 mlwithout Pb, but occurred at a lower thickness of 4 ml using the Pb surfactant.This result and other work suggest that kinetic considerations are importantin determining the growth mode, as well as energetics.

Surfactants have also been investigated as a means of controlling thegrowth mode in dilute nitride semiconductors such as GaNAs and InGaNAsgrown on GaAs (001) substrates. Tixier et al.136 studied the use of Bi as asurfactant in the MBE growth of GaNAs and InGaNAs. They found that theBi suppressed islanding, and step flow growth could be obtained inGaN0.004As0.996 at substrate temperatures as low as 460°C. The Bi alsoenhanced nitrogen incorporation in the films, though the incorporation ofthe Bi surfactant was negligible under all conditions studied. Wu et al.137

Si( )111 7 7− × Si Pb( )111 3 3− × −




studied the use of Sb as a surfactant in this material, also using MBE. TheSb surfactant was found to suppress islanding and improve the photolumi-nescence intensity of the resulting material. The prevention of islanding wasattributed to a reduction of the surface diffusivity for adatoms.137

Relatively little work has been reported on surfactant-mediated epitaxy ofhexagonal nitride semiconductors on sapphire or SiC substrates. Gupta etal.138 reported the use of Si as an antisurfactant in the MOVPE growth ofGaN/AlN/sapphire (0001). Structures grown without the Si surfactantexhibited two-dimensional growth when grown at 850°C with a V/III ratioof 4.5. Samples grown with the Si surfactant and ramped up to 970°C aftergrowth exhibited an island morphology. Widmann et al.139 and Fong et al.140

reported the use of In as a surfactant for the growth of GaN/sapphire (0001)by MBE. They found that the use of an In flux during growth promoted two-dimensional growth and improved the surface roughness and crystallinityof the resulting GaN.

4.6.2 Surfactants and Island Shape

In the Volmer–Weber (three-dimensional) growth of a heteroepitaxial semi-conductor, fractal islands are favored at low growth temperature or highincident flux, but compact islands are expected at high temperature or lowflux. This behavior has been explained by a diffusion-limited aggregate(DLA) theory, which was proposed by Witten and Sander141 and has beendiscussed extensively in the literature.104,142 However, the opposite behaviorhas been observed in the case of surfactant-mediated growth of Ge/Si (111)using Pb as the surfactant. In this case, fractal islands form at high temper-atures, whereas low growth temperatures result in compact islands.143

Chang et al.144 explained this behavior by invoking a model of reaction-limited aggregation. In the general case, it appears that surfactants couldalter the shapes and sizes of islands by changing diffusion or reaction rates.However, much work remains to clarify the mechanisms and applicabilityof this approach.

4.6.3 Surfactants and Misfit Dislocations

In a study of the growth of Ge/Si (111), Filimonov et al.131 found that theuse of Bi as a surfactant changed the structure and density of misfit dislo-cations at the interface. In their study, Ge was grown on Si (111) by MBE at500°C. In the case of Bi surfactant-mediated epitaxy (Bi-SME), 1 ml of Bi wasevaporated from a Knudsen cell prior to epitaxy. The configurations of theinterfacial misfit dislocations were inferred from the surface undulationsobserved in STM micrographs. For the case of Bi-SME, the Ge islands exhib-ited a regular triangular network of misfit dislocations. On the other hand,for conventional growth, the misfit dislocations formed a disordered honey-comb network, except near the center of the islands where the triangular




network was observed. This difference could come about because of differ-ences in the evolution of the islands. In both cases, the misfit dislocationswere found to be 90° Shockley partial dislocations, but the density of misfitdislocations was 25% larger in the case of Bi-SME. The change in the dislo-cation density could be an indirect effect of the surfactant, caused by thesuppression of the Si–Ge intermixing. The conventional growth, character-ized by greater intermixing, would result in a lower lattice mismatch.

4.6.4 Surfactants and Ordering in InGaP

Several III-V alloys are found to exhibit spontaneous CuPt ordering on the(111) planes when grown by MOVPE.145 This effect is of practical interestbecause it alters the bandgap of the material for a given composition. Thepseudobinary semiconductor InGaP (InxGa1–xP with x = 0.5) exhibits a strongtendency for CuPt ordering, with a corresponding change in the bandgapof up to 160 meV.146 However, the surfactant-mediated epitaxy of this alloycan dramatically suppress the ordering, using either Bi, Sb, or As as thesurfactant.132 This effect has been attributed to the elimination of P dimerson the surface, due to a change in the surface reconstruction. With increasingSb/Group III ratio, the surface structure changes from to and,at still higher Sb source flows, to a non-(2 × 4) structure.129

The examples described above reveal surfactants to be a powerful tool inmodifying the surface structure, growth mode, morphology, and defectstructure in heteroepitaxial layers. The field of surfactant-mediated epitaxyis still in its infancy, however, and much theoretical and experimental workremains to be done, especially with the III-nitride materials.

4.7 Quantum Dots and Self-Assembly

Semiconductor quantum dots (QDs) are of great interest for applications,including single-electron transistors,147,148 lasers,149–151 infrared photodetec-tors,152–154 and quantum dot cellular automata (QCA).155 In all of these appli-cations, the quantum dots may be fabricated by heteroepitaxial growth in aVolmer–Weber or Stranski–Krastanov growth mode. In many cases, theresulting dots may have a random distribution on the growth surface, andthis is entirely adequate for some device applications. On the other hand,some applications require the precise positioning of quantum dots, or regulararrays of dots, either one-dimensional or two-dimensional. Self-assemblyprocesses have emerged that appear capable of satisfying these needs, atleast to some extent.

The term self-assembly has been used extensively in the literature withvarious meanings. In some cases, the term is used to describe the growth of

( )2 × n β2 2 4( )×




islands with uniform size, even though their spacial distribution may bequite random. In other cases, self-assembly is used to describe the growth ofquantum dot islands in a regular geometric pattern, either one-dimensionalor two-dimensional in nature. (This type of self-assembly has also been calledself-organization.)

Practical self-assembly processes serve to alter the surface geometry, chem-istry, stress, or perhaps other properties in such a way as to create preferrednucleation sites for islands grown in a Volmer–Weber or Stranski–Krastanovmode. These processes should therefore be referred to as guided assembly.

4.7.1 Topographically Guided Assembly of Quantum Dots

Kamins and Williams156 demonstrated the guided assembly of Ge islands onSi (001) using VPE. In their work, a local oxidation of silicon (LOCOS) processwas used to create lines of Si surrounded by silicon dioxide. Some of the Silines created in this way had submicron width. Next, selective Si epitaxy,using SiH2Cl2 and HCl at 850°C and 20 torr, was utilized to produce Siplateaus over the exposed Si lines. These Si plateaus exhibited {311} sidewallsalong <110> directions and {110} sidewalls along <100> directions. Ge islandswere next deposited on the Si plateaus using GeH4 at 600°C and 10 torr. TheGe deposition was carried out for either 60 s at a GeH4 partial pressure of5 × 10–4 torr or 120 to 240 s at a GeH4 partial pressure of 2.5 × 10–4 torr.

Kamins and Williams found that for the narrowest silicon lines directedalong a <100> direction, the Ge islands grew in two rows near the cornersof the plateaus, with an island width of about 75 nm and a regular spacingof 80 nm. Figure 4.13 shows a three-dimensional atomic force microscopy(AFM) micrograph of ordered Ge islands on a Si plateau that was 450 nmwide and had its long axis directed along a <100> direction. Figure 4.14shows a two-dimensional AFM micrograph of the same ordered Ge islands,showing the uniformity of the size and spacing of the islands. This resultclearly demonstrates lithographic demagnification, whereby the self-assem-bled islands have predictable dimensions and spacings that are much lessthan the scale of the lithographic features used for their fabrication.

Kamins and Williams also studied Ge island growth on Si lines withdifferent orientations or widths. Wider plateaus exhibited more than twolines of Ge islands, and the ordering of the islands diminished with distancefrom the plateau edge. Examples are shown in Figure 4.15 for plateaus thatwere 670 to 1700 nm wide. The Si plateaus with their long axes orientedalong a <110> direction exhibited even less order.

An understanding of the mechanism for this guided assembly techniqueis of great importance for its application to other geometries or materials.One possible mechanism is based on the kinetics of diffusion for adsorbedspecies. If there is an Ehrlich–Schwoebel type energy barrier for the diffusionof Ge adatoms down the sidewall, then reflection of adatoms from thisbarrier can give rise to enhanced nucleation near the plateau edges. This




mechanism would be enhanced for lines along the <100>, which exhibitsteeper sidewalls. It should be possible to predict the spacing of the islandsbased on the atomistic nucleation theory. Another possible mechanism forthe ordering is related to strain relief. According to this explanation, nucle-ation of islands near the plateau edges is favored because the Si lattice isunconstrained at the sidewall and can distort to reduce the mismatch strainin the islands. This mechanism is also expected to be more effective for theSi lines with the steeper sidewalls, so it is impossible to distinguish betweenthese two mechanisms on this basis.

4.7.2 Stressor-Guided Assembly of Quantum Dots

It has been found that quantum dot nucleation can be strongly influencedby stress in the substrate. In principle, the stress field could be produced byseveral means. An example of this behavior is the case of Ge QDs grown ona partially relaxed GeSi buffer layer on a Si (001) substrate. Here, a buriedarray of misfit dislocations exists at the interface between the substrate andthe SiGe buffer layer. The Ge dots nucleate preferentially along the lines

FIGURE 4.13(a) Three-dimensional AFM micrograph of ordered Ge islands on a Si plateau that was 450 nmwide and had its long axis directed along a <100> direction. The Ge was grown for 120 s at aGeH4 partial pressure of 2.5 × 10–4 torr. (b) Schematic cross section of the sample. (Reprintedfrom Kamins, T.I. and Williams, R.S., Appl. Phys. Lett., 71, 1201, 1997. With permission. Copyright1997, American Institute of Physics.)

300 nm

0

0.5

1.0 μm

Ge islands

Si(001) planeselective Si

Si substrate

(b)

(a)

SiO2

(110) planes




where the dislocation glide planes intersect the surface of the bufferlayer.157,158 (These are along <110>-type directions for the case of SiGe grownon a Si (001) substrate.)

Xie et al.158 studied the use of a relaxed SiGe layer as a template forfabricating Ge quantum dot arrays. In their work, a relaxed SiGe layer andthin Si cap layer were grown by MBE at 400 to 500°C. Following this, Gequantum dots were deposited at 750°C. The AFM micrograph of Figure 4.16shows the resulting geometry of the Ge QDs after the growth of 1.0-nm Gecoverage (average thickness). The QDs form a rectangular array, with linesof dots parallel to the <110> directions. The positions of the dots correspondclosely to the intersections of misfit dislocations at the SiGe/Si interface. Thishas been attributed to the local strain fields of the dislocations, which reducethe mismatch strain energy in nucleating quantum dots.

Several aspects of the behavior shown in Figure 4.16 remain incompletelyunderstood at the present time. First, the islands did not organize in thisway at lower growth temperatures, though the reason is not clear. Second,the islands observed by Xie et al. exhibited {105} facets when grown at 750°C,even though Mo et al.166 showed that {105}-facetted Ge huts are a metastablephase that converted to other structures at this temperature. Third, the Geislands position themselves offset from, instead of directly over, the placeswhere the dislocations intersect in the relaxed buffer layer. It does not appear

FIGURE 4.14(a) Two-dimensional AFM micrograph of ordered Ge islands on a Si plateau that was 450 nmwide and had its long axis directed along a <100> direction. The Ge islands grow in a regularpattern with a period of about 80 nm along the <100> direction. The Ge was grown for 120 sat a GeH4 partial pressure of 2.5 × 10–4 torr. (Reprinted from Kamins, T.I. and Williams, R.S.,Appl. Phys. Lett., 71, 1201, 1997. With permission. Copyright 1997, American Institute of Physics.)

75

0

0 0.25 0.50 0.75μm

−75

Hei

ght (

nm)




that this behavior can be explained on the basis of strain energy alone, butmay be controlled in part by the kinetics of surface diffusion.160 Finally, thismethod of stressor-guided assembly is ineffective for InAs islands grown onrelaxed SiGe buffer layers on Si.159 While the reason is not clear, it may berelated to the difficulty of growing dislocation-free islands of this material,due to the larger lattice mismatch.

4.7.3 Vertical Organization of Quantum Dots

Another application of stressor-guided assembly is the fabrication of verti-cally assembled quantum dots. Here, quantum dots in a multilayer stack alignin vertical columns. In simple terms, the mechanism could be related to themodulation of the stress field by the quantum dots in one layer, which causes

FIGURE 4.15AFM micrographs of Ge islands on Si plateaus of various widths, which had their long axesdirected along a <100> direction. The Ge was grown for 240 s at a GeH4 partial pressure of 2.5× 10–4 torr. The plateau width was (a) 670 nm, (b) 1000 nm, and (c) 1700 nm. (Reprinted fromKamins, T.I. and Williams, R.S., Appl. Phys. Lett., 71, 1201, 1997. With permission. Copyright1997, American Institute of Physics.)

0 0.5 1.0 0 0.5 1.0

1.0

0.5

0

(a) (b)

0 0.5 1.0

(c)μm

μm μm




preferential nucleation of quantum dots in the next layer. The actual detailedmechanism may be much more complex, involving the kinetics of diffusionas well as the stress field.

Xie et al.160 demonstrated the vertical self-organization of InAs QDs grownon GaAs (001) substrates by MBE. In this work, 2 ml of InAs was depositedon GaAs (001) at 500°C and a growth rate of 0.25 ml/s. Then a spacer layerwas grown, typically consisting of 10 ml of GaAs, a 3-ml AlAs marker, and20 ml of GaAs, at 480°C and 0.25 ml/s. This sequence of layers was grownrepeatedly. Figure 4.17 shows a representative cross-sectional TEM micro-graph of five sets of vertically organized InAs QDs grown on a GaAs (001)substrate using 36-ml spacer layers.

FIGURE 4.16AFM micrograph showing a regular array of Ge islands on a partially relaxed SiGe layer ona Si (001) substrate. The Ge coverage is 1.0 nm. The Ge QDs nucleate preferentially over theintersections of misfit dislocations in the partially relaxed SiGe layer. (Reprinted from Xie,Y.H. et al., Appl. Phys. Lett., 71, 3567, 1997. With permission. Copyright 1997, American Instituteof Physics.)

FIGURE 4.17Cross-sectional TEM micrograph of five sets of vertically organized InAs QDs grown on a GaAs(001) substrate using 36-ml spacer layers. (Reprinted from Xie, Q. et al., Phys. Rev. Lett., 75, 2542,1995. With permission. Copyright 1995, American Physical Society.)

10.0

7.5

5.0

2.5

00 2.5 7.5 10.05.0

μm

140.0 nm

70.0 nm

0.0 nm

50 nm




Mukhametzhanov et al.161 showed that in the InAs/GaAs (001) heteroepi-taxial system, vertical assembly of InAs quantum dots could be used toindependently manipulate the density and size of QDs. Here, InAs dots weregrown at 500°C with a growth rate of 0.22 ml/s. A GaAs spacer was grownby migration-enhanced epitaxy at 400°C, followed by the growth of anotherlayer of InAs QDs, and finally a GaAs cap. It was shown that the quantumdots in the second (top) layer aligned with the QDs in the first (bottom) layer.Therefore, the density and size of the QDs could be controlled independently:the deposition time for the first layer of dots controlled the density, and thedeposition time for the second layer determined the QD size in that layer.

In the SiGe material system, Teichert et al.162 demonstrated vertical orga-nization of Ge dots in SiGe/Si multilayer films. Mateeva et al.163 furtherstudied the vertical organization of Ge dots in SiGe/Si multilayers. Usingcross-sectional TEM characterization, they showed that the merging ofislands of different sizes led to a uniform size distribution after the growthof many periods in these multilayered structures.

4.7.4 Precision Lateral Placement of Quantum Dots

Some device applications require precise placement of quantum dots ratherthan the fabrication of dots with uniform size or distribution. In the case ofGe quantum dots grown on Si (001), this has been achieved by focused-ion-beam micropatterning by Hull et al.164 and Kammler et al.165

In the work of Hull et al. and Kammler et al., clean Si (001) surfaces wereirradiated with a Ga+ focused ion beam, using a beam energy of 25 keV anda beam current of 10 pA (6.2 × 107 ions/s). AFM images of the surfacesrevealed that each irradiated spot contained amorphous material sur-rounded by a ring of sputtered material. The ring diameter increased from90 nm for 0.1 ms of irradiation to 320 nm for 10 ms of irradiation. Followingirradiation, Si (001) was annealed in the range of 600 to 750°C to recover itscrystallinity. Following annealing, Ge QDs were deposited by VPE usingdigermane at a temperature of 600°C.

Kammler et al. found that for an irradiation time of 0.01 ms the Ge islandsformed randomly over the surface and the focused-ion-beam pattern hadno influence over their placement. For higher irradiation times (>620 ions/spot), every irradiated spot was occupied by one Ge island, whereas noislands nucleated elsewhere. Figure 4.18 shows Ge quantum dots that werepatterned in this way and demonstrates the remarkable control that is pos-sible. The technique appears to be unaffected by the fill factor or specificpattern to be transferred.

The mechanism underlying this method of precise QD placement is notentirely clear. It could be related to a modification of the surface propertiesby the implanted Ga, which could diffuse to the surface during the annealingstep. Kammler et al. found that the islands formed on the irradiated andannealed areas were smaller and had a larger aspect ratio than those on




unirradiated Si, implying a surfactant effect and lending support to thistheoretical model. Another possibility, however, is that the implanted Gaions introduce strain, resulting in stressor-guided assembly. Topography canbe ruled out as the mechanism because the irradiated and annealed spotsdid not develop any topographic relief.

Problems

1. Sketch the following surface structures, showing the dimensions ineach case: , , and

.2. For epitaxial growth of Si at 1000°C, estimate the critical nucleus

size for gas phase (homogeneous) nucleation. Assume the equilib-rium vapor pressure for Si is ~10–3 Pa in order to estimate thesupersaturation.

3. Consider the epitaxial growth of Si0.5Ge0.5/Si (001) superlattices.Estimate the contact angle for each type of interface. Use Vegard’slaw to estimate the surface energy of the alloy and neglect theinterfacial energy.

4. For the Si0.5Ge0.5/Si superlattices described in Problem 3, describethe expected growth modes for the Si layers and the Si0.5Ge0.5 layers.

FIGURE 4.18In situ TEM images of Ge islands on a Si (001) substrate, precisely patterned using a Ga+ focused-ion beam. All patterns were created using a 10-pA Ga+ beam, but the irradiation times weredifferent for the different regions of the surface, as indicated in (a). Micrograph (b) shows anenlargement of the pattern fabricated using a 100-μs irradiation time. (Reprinted from Hull, R.et al., Mater. Sci. Eng. B, 101, 1, 2003. With permission. Copyright 2003, Elsevier.)

(a) (b)

100 us

10 us10 ms

1ms

2 μm ?

GaAs( )( )001 2 4× Si( )( )111 7 7× 6 0001 3H SiC− ×( )(3 30R× °)




References

1. C.M. Mate, G.M. McClelland, R. Erlandsson, and S. Chiang, Atomic-scale fric-tion of a tungsten tip on a graphite surface, Phys. Rev. Lett., 59, 1942 (1987).

2. G. Binnig, C.F. Quate, and Ch. Gerber, Atomic force microscope, Phys. Rev. Lett.,56, 930 (1986).

3. S. Kawai, S. Kitamura, D. Kobayashi, and H. Kawakatsu, Dynamic lateral forcemicroscopy with true atomic resolution, Appl. Phys. Lett., 87, 173105 (2005).

4. E.A. Wood, Vocabulary of surface crystallography, J. Appl. Phys., 35, 1306 (1964).5. Th. Schmidt, R. Kroger, T. Clausen, J. Falta, A. Janzen, M. Kammler, P. Kury, P.

Zahl, and M. Horn-von Hoegen, Surfactant-mediated epitaxy of Ge on Si (111):beyond the surface, Appl. Phys. Lett., 86, 111910 (2005).

6. L. Masson and F. Thibaudau, Role of steps in deposition rate in silane chemicalvapor deposition on Si (111), Phys. Rev. B, 71, 85314 (2005).

7. J.J. Lander and J. Morrison, Low Energy electron diffraction study of siliconsurface structures, J. Chem. Phys., 37, 729 (1962).

8. J.A. Appelbaum, G.A. Baraff, and D.R. Hamann, The Si (100) surface. III. Sur-face reconstruction, Phys. Rev. B, 14, 588 (1976).

9. R.E. Schlier and H.E. Farnsworth, Structure and adsorption characteristics ofclean surfaces of germanium and silicon, J. Chem. Phys., 30, 917 (1959).

10. J. Levine, Structural and electronic model of negative electron affinity on theSi/Cs/O surface, Surf. Sci., 34, 90 (1973).

11. J.A. Appelbaum, G.A. Baraff, and D.R. Hamann, Si (100) surface reconstruction:spectroscopic selection of a structural model, Phys. Rev. Lett., 35, 729 (1975).

12. J.A. Appelbaum, G.A. Baraff, and D.R. Haman, The Si (100) surface: furtherstudies of the pairing model, Phys. Rev. B, 15, 2408 (1977).

13. J.C. Phillips, Excitonic instabilities, vacancies, and reconstruction of covalentsurfaces, Surf. Sci., 40, 459 (1973).

14. R.A. Wolkow, Direct observation of an increase in buckled dimers on Si(001)at low temperature, Phys. Rev. Lett., 68, 2636 (1992).

15. Y.J. Li, H. Nomura, N. Ozaki, Y. Naitoh, M. Kageshima, and Y. Sugawara, Originof p(2x1) phase on Si(001) by noncontact atomic force microscopy at 5K, Phys.Rev. Lett., 96, 106104 (2006).

16. T. Nakayama, Y. Tanishiro, and K. Takayanagi, Biatomic layer-high steps onSi(001)2×1 surface, Jpn. J. Appl. Phys., Part 2, 26, L280 (1987).

17. R. Kaplan, LEED study of the stepped surface of vicinal Si (100), Surf. Sci., 93,145 (1980).

18. B.Z. Olshanetsky and V.I. Mushanov, LEED studies of clean high Miller indexsurfaces of Si, Surf. Sci., 111, 414 (1981).

19. R.D. Bringans, R.I.G. Uhrberg, M.A. Olmstead, and R.Z. Bachrach, Surfacebands for single-domain 2×1 reconstructed Si(100) and Si(100):As. Photoemis-sion results for off-axis crystals, Phys. Rev. B, 34, 7447 (1986).

20. R.A. Budiman, Interacting dimer rows on terraces: reconstructed Si (001) sur-faces, Phys. Rev. B., 72, 35322 (2005).

21. D. Haneman, Electron paramagnetic resonance from clean single-crystal cleav-age surfaces of silicon, Phys. Rev., 170, 705 (1968).

22. W. Monch, Advances in Solid State Physics, Vol. 13, Pergamon-Viewig, Braun-schweig, 1973, p. 241.




23. M.J. Cardillo, Nature of the Si (111) reconstruction, Phys. Rev. B, 23, 4279 (1981).24. S. Kawai, S. Kitamura, D. Kobayashi, and H. Kawakatsu, Dynamic lateral force

microscopy with true atomic resolution, Appl. Phys. Lett., 87, 173105 (2005).25. J.J. Lander, Solid State Chemistry, Vol. 2, H. Reiss, Ed., Pergamon, Oxford, 1965,

p. 26.26. A.J. Van Bommel and F. Meyer, Low energy electron diffraction studies on Ge

and Na-covered Ge, Surf. Sci., 6, 57 (1967).27. M. Taubenblatt, E. So, P. Sih, A. Kahn, and P. Mark, Atomic structure of the

annealed Ge (111) surface, J. Vac. Sci. Technol., 15, 1143 (1978).28. J.J. Lander and J. Morrison, Structures of clean surfaces of germanium and

silicon: I, J. Appl. Phys., 34, 1403 (1963).29. R. Kaplan, Surface structure and composition of β- and 6H-SiC, Surf. Sci., 215,

111 (1989).30. T.-H. Kim, S. Choi, M. Morse, P. Wu, C. Yi, A. Brown, M. Losurdo, M.M.

Giangregorio, and G. Bruno, Impact of unintentional and intentional nitrida-tion of the 6H-SiC(0001)Si substrate on GaN epitaxy, J. Vac. Sci. Technol. B, 23,1181 (2005).

31. X.N. Xie and K.P. Loh, Observation of a 6×6 superstructure on 6H-SiC (0001)by reflection high energy electron diffraction, Appl. Phys. Lett., 77, 3361 (2000).

32. M. Naitoh, J. Takami, S. Nishigaki, and N. Toyama, A surfacephase in the 6H-SiC (0001) surface studied by scanning tunneling microscopy,Appl. Phys. Lett., 75, 650 (1999); Appl. Phys. Lett., 75, 2692 (1999).

33. J. Suda, K. Miura, M. Honaga, Y. Nishi, N. Onojima, and H. Matsunami, Effectsof 6H-SiC surface reconstruction on lattice relaxation of AlN buffer layers inmolecular beam epitaxial growth of GaN, Appl. Phys. Lett., 81, 5141 (2002).

34. J.D. Hartman, A.M. Roskowski, Z.J. Reitmeier, K.M. Tracy, R.F. Davis, and R.J.Nemanich, Characterization of hydrogen etched 6H-SiC (0001) substrates andsubsequently grown AlN, J. Vac. Sci. Technol. A, 21, 394 (2003).

35. J. Lu, L. Haworth, D.I. Westwood, and J.E. Macdonald, Initial stages of molecular-beam epitaxy growth of GaN on 6H-SiC (0001), Appl. Phys. Lett., 78, 1080 (2001).

36. K. Jeganathan, M. Shimizu, T. Ide, and H. Okumura, The effect of galliumadsorbate on SiC (0001) surface for GaN by MBE, Phys. Status Solidi B, 240, 326(2003).

37. F. Amy, P. Soukiassian, and C. Brylinski, Atomic cracks andreconstruction at 6H-SiC(0001) surface, Appl. Phys. Lett.,

85, 926 (2004).38. V. Ramachandran and R.M. Feenstra, Scanning tunneling spectroscopy of

Mott–Hubbard states on the 6H-SiC(0001) surface, Phys. Rev. Lett., 82,1000 (1999).

39. U. Starke, J. Schardt, J. Bernhardt, M. Franke, K. Reuter, H. Wedler, K. Heinz,J. Furthmuller, P. Kackell, and F. Bechstedt, Novel reconstruction mechanismfor dangling-bond minimization: Combined method surface structure deter-mination of SiC(111)–(3×3), Phys. Rev. Lett., 80, 758 (1998).

40. J. Schardt, J. Bernhardt, U. Starke, and K. Heinz, Crystallography of the (3×3)surface reconstruction of 3C-SiC(111), 4H-SiC(0001), and 6H-SiC(0001) surfacesretrieved by low-energy electron diffraction, Phys. Rev. B, 62, 10335 (2000).

41. I. Forbeaux, J.M. Themlin, and J.M. Debever, Heteroepitaxial graphite on 6Hinterface formation through SiC(0001): conduction-band electronic structure,Phys. Rev. B, 58, 16396 (1998).

( )2 3 2 13×

( )2 3 2 3 30× − °R

3 3×




42. V. Bermudez, Structure and properties of cubic silicon carbide (100) surfaces:a review, Phys. Status Solidi B, 202, 447 (1997).

43. A. Catellani and G. Galli, Theoretical studies of silicon carbide surfaces, Prog.Surf. Sci., 69, 101 (2002).

44. W. Lu, P. Krüger, and J. Pollmann, Atomic and electronic structure of β-SiC(001)-(3×2), Phys. Rev. B, 60, 2495 (1999).

45. V.M. Bermudez, Surface science: a metallic semiconductor surface, Nat. Mater.,2, 218 (2003).

46. V. Derycke, P.G. Soukiassian, F. Amy, Y.J. Chabal, M.D. D’Angelo, H.B. Enriqu-ez, and M.G. Silly, Nanochemistry at the atomic scale revealed in hydrogen-induced semiconductor surface metallization, Nat. Mater., 2, 253 (2003).

47. F. Amy and Y.J. Chabal, Interaction of H, O2, and H2O with 3C-SiC surfaces, J.Chem. Phys., 119, 6201 (2003).

48. C.-W. Hu, A. Kasuya, S. Suto, A. Wawro, and Y. Nishina, Surface structure of3C-SiC(111) fabricated by C60 precursor: a scanning tunneling microscopy andhigh-resolution electron energy loss spectroscopy study, J. Vac. Sci. Technol. B,14, 938 (1996).

49. G. Feuillet, H. Hamaguchi, K. Ohta, P. Hacke, H. Okumura, and S. Yoshida,Arsenic mediated reconstructions on cubic (001) GaN, Appl. Phys. Lett., 70, 1025(1997).

50. J.R. Arthur and J.J. LePore, GaAs, GaP, and GaAsxP1–x epitaxial films grown bymolecular beam deposition, J. Vac. Sci. Technol., 6, 545 (1969).

51. A.Y. Cho, GaAs epitaxy by a molecular beam method: observations of surfacestructure on the (001) face, J. Appl. Phys., 42, 2074 (1971).

52. J.E. Northrup and S. Froyen, Structure of GaAs (001) surfaces: the role ofelectrostatic interactions, Phys. Rev. B, 50, 2015 (1994).

53. T. Hashizume, Q.-K. Xue, J. Zhou, A. Ichimiya, and T. Sakurai, Structures ofAs-rich GaAs (001)-(2×4) reconstructions, Phys. Rev. Lett., 73, 2208 (1995).

54. T. Hashizume, Q.-K. Xue, A. Ichimiya, and T. Sakurai, Determination of thesurface structures of the GaAs (001)-(2×4) As-rich phase, Phys. Rev. B, 51, 4200(1995).

55. H. Yamaguchi and Y. Horikoshi, Surface structure transitions on InAs and GaAs(001) surfaces, Phys. Rev. B, 51, 9836 (1995).

56. W.G. Schmidt and F. Bechstedt, Geometry and electronic structure ofGaAs(001)(2×4) reconstructions, Phys. Rev. B, 54, 16742 (1996).

57. Y. Garreau, M. Sauvage-Simkin, N. Jedrecy, R. Pinchaux, and M.B. Veron, Atomicstructure and faulted boundaries in the GaAs(001) β(2×4) surface as derivedfrom x-ray diffraction and line-shape analysis, Phys. Rev. B, 54, 17638 (1996).

58. Q.-K. Xue, T. Hashizume, and T. Sakarai, Scanning tunneling microscopy ofIII-V compound semiconductor (001) surface, Prog. Surf. Sci., 56, 1 (1997).

59. V.P. LaBella, H. Yang, D.W. Bullock, P.M. Thibado, P. Kratzer, and M. Scheffler,Atomic structure of the GaAs(001)-(2×4) surface resolved using scanning tun-neling microscopy and first-principles theory, Phys. Rev. Lett., 83, 2989 (1999).

60. M. Gothelid, Y. Garreau, M. Sauvage-Simkin, R. Pinchaux, A. Cricenti, and G.Le Lay, Atomic structure of the As-rich InAs(100)β2(2×4) surface, Phys. Rev. B,59, 15285 (1999).

61. C. Ratsch, W. Barvosa-Carter, F. Grosse, J.H.G. Owen, and J.J. Zinck, Surfacereconstructions for InAs(001) studied with density-functional theory and STM,Phys. Rev. B, 62, R7719 (2000).




62. W.G. Schmidt, S. Mirbt, and F. Bechstedt, Surface phase diagram of (2×4) and(4×2) reconstructions of GaAs(001), Phys. Rev. B, 62, 8087 (2000).

63. R.H. Miwa and G.P. Srivastava, Structure and electronic states of InAs(001)-(2×4) surfaces, Phys. Rev. B, 62, 15778 (2000).

64. F. Grosse, W. Barvosa-Carter, J.J. Zinck, and M.F. Gyure, Microscopic mech-anisms of surface phase transitions on InAs(001), Phys. Rev. B, 66, 75321(2002).

65. Yu. G. Galitsyn, D.V. Dmitriev, V.G. Mansurov, S.P. Moshchenko, and A.I.Toropov, Critical phenomena in the β – (2 × 4) → α – (2 × 4) reconstructiontransition on the (001) GaAs surface, JETP Lett., 81, 629 (2005) (translated fromRussian).

66. L. Li, B.-K. Han, C.H. Li, and R.F. Hicks, Example of a compound semiconduc-tor surface that mimics silicon: the InP(001)-(2×1) reconstruction, Phys. Rev.Lett., 82, 1879 (1999).

67. L. Li, Q. Fu, B.-K. Han, C.H. Li, and R.F. Hicks, Determination of InP (001)surface reconstruction by STM and infrared spectroscopy of adsorbed hydro-gen, Phys. Rev. B, 61, 10223 (2000).

68. W.G. Schmidt, P.H. Hahn, F. Bechstedt, N. Esser, P. Vogt, A. Wange, and W.Richter, Atomic structure of InP (001)-(2×4): a dimer reconstruction, Phys. Rev.Lett., 90, 126101 (2003).

69. G. Chen, S.F. Cheng, D.J. Tobin, L. Li, K. Raghavachari, and R.F. Hicks, Indiumphosphide (001)-(2×1): direct evidence for a hydrogen-stabilized surface recon-struction, Phys. Rev. B, 68, 121303 (2003).

70. R.R. Wixon, N.A. Modine, and G.B. Stringfellow, Theory of surfactant (Sb)induced reconstructions on InP (001), Phys. Rev. B, 67, 115309 (2003).

71. X.-Q. Shen, T. Ide, S.-H. Cho, M. Shimizu, S. Hara, H. Okumura, S. Sonoda,and S. Shimizu, Optimization of GaN growth with Ga-polarity by referring tosurface reconstruction reflection high-energy electron diffraction patterns, Jpn.J. Appl. Phys., 40, L23 (2001).

72. W.A. Harrison, E.A. Kraut, J.R. Waldrop, and R.W. Grant, Polar heterojunctioninterfaces, Phys. Rev. B, 18, 4402 (1978).

73. E.A. Kraut, Summary abstract: dipole driven diffusion across polar heterojunc-tion interfaces, J. Vac. Sci. Technol. B, 1, 643 (1983).

74. D.E. Aspnes and J. Ihm, Biatomic steps on (001) silicon surfaces, Phys. Rev. Lett.,57, 3054 (1986).

75. S.F. Fang, K. Adomi, S. Iyer, H. Morkoc, H. Zabel, C. Choi, and N. Otsuka,Gallium arsenide and other compound semiconductors on silicon, J. Appl. Phys.,68, R31 (1990).

76. V. Lebedev, J. Jinschek, U. Kaiser, B. Schroter, and W. Richter, Epitaxial rela-tionship in the AlN/Si(001) heterosystem, Appl. Phys. Lett., 76, 2029 (2000).

77. T. Sakamoto and G. Hashiguchi, Si(001)-2x1 single-domain structure obtainedby high temperature annealing, Jpn. J. Appl. Phys., 25, L78 (1986).

78. M. Kawabe and T. Ueda, Self-annihilation of antiphase boundary in GaAson Si(100) grown by molecular beam epitaxy, Jpn. J. Appl. Phys., 26, L944(1987).

79. D. Saloner, J.A. Martin, M.C. Tringides, D.E. Savage, C.E. Aumann, and M.G.Lagally, Determination of terrace size and edge roughness in vicinal Si{100}surfaces by surface-sensitive diffraction, J. Appl. Phys., 61, 2884 (1987).

80. I. Bhat and W.-S. Wang, Growth of (100) oriented CdTe on Si using Ge as abuffer layer, Appl. Phys. Lett., 64, 566 (1994).




81. M. Grundmann, A. Krost, and D. Bimberg, Observation of the first-order phasetransition from single to double stepped Si (001) in metalorganic chemicalvapor deposition of InP on Si, J. Vac. Sci. Technol. B, 9, 2158 (1991).

82. V. Lebedev, J. Jinschek, J. Krausslich, U. Kaiser, B. Schroter, and W. Richter,Hexagonal AlN films grown on nominal and off-axis Si(001) substrates, J. Cryst.Growth, 230, 426 (2001).

83. H.H. Farrell, M.C. Tamargo, and J.L. de Miguel, Optimal GaAs(100) substrateterminations for heteroepitaxy, Appl. Phys. Lett., 58, 355 (1991).

84. M.C. Tamargo, J.L. de Miguel, D.M. Hwang, and H.H. Farrell, Structural char-acterization of GaAs/Znse interfaces, J. Vac. Sci. Technol. B, 6, 784 (1988).

85. A.A. Chernov, Nucleation and epitaxy, in Modern Crystallography III: CrystalGrowth, Springer-Verlag, New York, 1984, pp. 48–103.

86. F.C. Frank and J.H. van der Merwe, One-dimensional dislocations. II. Misfittingmonolayers and oriented overgrowth, Proc. R. Soc. London A, 198, 216 (1949).

87. M. Volmer and A. Weber, Keimbildung in übersättigten Gebilden, Z. PhysikChem., 119, 277 (1926).

88. J.A. Venables, Rate equation approaches to thin film nucleation kinetics, Phil.Mag., 27, 693 (1973).

89. J.A. Venables, G.D.T. Spiller, and M. Hanbücken, Nucleation and growth ofthin films, Rep. Prog. Phys., 47, 399 (1984).

90. J.A. Venables, Nucleation calculations in a pair-binding model, Phys. Rev. B,36, 4153 (1987).

91. M.J. Stowell and T.E. Hutchinson, Nucleation kinetics in thin film growth. II.Analytical evaluation of nucleation and growth behavior, Thin Solid Films, 8,41 (1971).

92. M.J. Stowell, Thin film nucleation kinetics, Phil. Mag., 26, 361 (1972).93. G.D.T. Spiller, P. Akhter, and J.A. Venables, UHV-SEM study of nucleation and

growth of Ag/W(110), Surf. Sci., 131, 517 (1983).94. M. Hanbücken, M. Futamoto, and J.A. Venables, Nucleation, growth, and the

intermediate layer in Ag/Si(100) and Ag/Si(111), Surf. Sci., 147, 433 (1984).95. J.A. Venables and D.J. Ball, Nucleation and growth of rare-gas crystals, Proc.

R. Soc. London A, 322, 331 (1971).96. J. Tersoff, A.W. Denier van der Gon, and R.M. Tromp, Critical island size for

layer-by-layer growth, Phys. Rev. Lett., 72, 266 (1994).97. M. Schneider, I.K. Schuller, and A. Rahman, Epitaxial growth of silicon: a

molecular-dynamics simulation, Phys. Rev. B, 36, 1340 (1987).98. M.H. Grabow and G.H. Gilmer, Thin film growth modes, wetting and cluster

formation, Surf. Sci., 194, 333 (1988).99. J.D. Weeks and G.H. Gilmer, Dynamics of crystal growth, Adv. Chem. Phys., 40,

157 (1979).100. C. Ratsch and J.A. Venables, Nucleation theory and the early stages of thin film

growth, J. Vac. Sci. Technol. A, 21, S96 (2003).101. E.G. Bauer, B.W. Dodson, D.J. Ehrlich, L.C. Feldman, C.P. Flynn, M.W. Geis,

J.P. Harbison, R.J. Matyi, P.S. Peercy, P.M. Petroff, J.M. Phillips, G.B. Stringfel-low, and A. Zangwill, Fundamental issues in heteroepitaxy: a Department ofEnergy, Council on Materials Science Panel Report, J. Mater. Res., 5, 852 (1990).

102. I. Daruka and A.-L. Barabási, Dislocation-Free island formation in heteroepi-taxial growth: a study at equilibrium, Phys. Rev. Lett., 79, 3708 (1997).

103. I. Daruka, Strained Island Formation in Heteroepitaxy, doctoral dissertation,University of Notre Dame, Indiana (1999).




104. H. Brune, Growth modes, in Encyclopedia of Materials: Science and Technology,Elsevier, Amsterdam, 2001.

105. G. Wulff, Zur frage der geschwindigkeit des wachsturms under auflösung derkristallflächen, Z. Krist., 34, 449 (1901).

106. J. Tersoff, Stress-induced layer-by-layer growth of Ge on Si(100), Phys. Rev. B,43, 9377 (1991).

107. C. Roland and G.H. Gilmer, Growth of germanium films on Si(001) substrates,Phys. Rev. B, 47, 16286 (1993).

108. V.A. Shchukin, N.N. Ledentsov, P.S. Kop’ev, and D. Bimberg, Spontaneousordering of arrays of coherent strained islands, Phys. Rev. Lett., 75, 2968 (1995).

109. V.I. Marchenko, Theory of the equilibrium shape of crystals, Sov. Phys. JETP,54, 605 (1981).

110. W. Ostwald, On the assumed isomerism of red and yellow mercury oxide andthe surface-tension of solid bodies, Z. Phys. Chem., 34, 495 (1900).

111. M. Zinke-Allmang, L.C. Feldman, and M.H. Grabow, Clustering on surfaces,Surf. Sci. Rep., 16, 377 (1992).

112. J. Drucker, Coherent islands and microstructural evolution, Phys. Rev. B, 48,18203 (1993).

113. G. Ehrlich and F.G. Hudda, Atomic view of surface self-diffusion: tungsten ontungsten, J. Chem. Phys., 44, 1039 (1966).

114. R.L. Schwoebel, Step motion on crystal surfaces: II, J. Appl. Phys., 40, 614 (1969).115. R. Kunkel, B. Poelsema, L.K. Verheij, and G. Comsa, Reentrant layer-by-layer

growth during molecular-beam epitaxy of metal-on-metal substrates, Phys. Rev.Lett., 65, 733 (1990).

116. R. Heitz, T.R. Ramachandran, A. Kalburge, Q. Xie, I. Mukhametzhanov, P. Chen,and A. Madhukar, Observation of reentrant 2D to 3D morphology transitionin highly strained epitaxy: InAs on GaAs, Phys. Rev. Lett., 78, 4071 (1997).

117. A. Ohtake and M. Ozeki, In situ observation of surface processes in InAs/GaAs(001) heteroepitaxy: the role of As on the growth mode, Appl. Phys. Lett., 78,431 (2001).


119. R.A. Oliver, M.J. Kappers, C.J. Humphreys, and G.A. Briggs, Growth modesin heteroepitaxy of InGaN on GaN, J. Appl. Phys., 97, 13707 (2005).

120. L. Mandreoli, J. Neugeberger, R. Kunert, and E. Schöll, Adatom density kineticMonte Carlo: a hybrid approach to perform epitaxial growth simulations, Phys.Rev. B, 68, 155429 (2003).

121. S. Yoshida, S. Misawa, and S. Gonda, Epitaxial growth of GaN/AlN hetero-structures, J. Vac. Sci. Technol. B, 1, 250 (1983).

122. S. Yoshida, S. Misawa, and S. Gonda, Improvements on the electrical andluminescent properties of reactive molecular beam epitaxially grown GaN filmsby using AlN-coated sapphire substrates, Appl. Phys. Lett., 42, 427 (1983).


124. H. Amano, I. Akasaki, K. Hiramatsu, and N. Sawaki, Effects of the buffer layerin metalorganic vapour phase epitaxy of GaN on sapphire substrates, Thin SolidFilms, 163, 415 (1988).




125. Y. Koide, N. Itoh, X. Itoh, N. Sawaki, and I. Akasaki, Effect of AlN buffer layeron AlGaN/α-Al2O3 heteroepitaxial growth by metalorganic vapor phase epi-taxy, Jpn. J. Appl. Phys., 27, 1156 (1988).


127. N. Kuznia, M.A. Khan, D.T. Olsen, R. Kaplan, and J. Freitas, Influence of bufferlayers on the deposition of high quality single crystal GaN over sapphiresubstrates, J. Appl. Phys., 73, 4700 (1993).

128. M. Copel, M.C. Reuter, E. Kaxiras, and R.M. Tromp, Surfactants in epitaxialgrowth, Phys. Rev. Lett., 63, 632 (1989).

129. C.M. Fetzer, R.T. Lee, J.K. Shurtleff, G.B. Stringfellow, S.M. Lee, and T.Y. Seong,The use of a surfactant (Sb) to induce triple period ordering in GaInP, Appl.Phys. Lett., 76, 1440 (2000).

130. G.B. Stringfellow, C.M. Fetzer, R.T. Lee, S.W. Jun, and J.K. Shurtleff, Surfactanteffects on ordering in GaInP grown by OMVPE, Proc. Mater. Res. Soc. Symp.,583, 261 (2000).

131. S.N. Filimonov, V. Cherepanov, N. Paul, H. Atsaoka, J. Brona, and B. Voigt-länder, Dislocation networks in conventional and surfactant-mediated Ge/Si(111) epitaxy, Surf. Sci., 599, 76 (2005).

132. G.B. Stringfellow, J.K. Shurtleff, R.T. Lee, C.M. Fetzer, and S.W. Jun, Surfaceprocesses in OMVPE: the frontiers, J. Cryst. Growth, 221, 1 (2000).

133. B. Voigtländer and A. Zinner, Structure of the Stranski-Krastanov layer insurfactant-mediated Sb/Ge/Si(111) epitaxy, Surf. Sci., 292, L775 (1993).

134. H. Hibino, N. Shimizu, K. Sumitomo, Y. Shinoda, T. Nishioka, and T. Ogino,Pb preabsorption facilitates island formation during Ge growth on Si (111), J.Vac. Sci. Technol. A, 12, 23 (1994).

135. S. Tixier, M. Adamcyk, E.C. Young, J.H. Schmid, and T. Tiedje, Surfactantenhanced growth of GaNAs and InGaNAs using bismuth, J. Cryst. Growth, 251,439 (2003).

136. D. Wu, Z. Niu, S. Zhang, H. Ni, Z. He, Z. Sun, Q. Han, and R. Wu, The role ofSb in the molecular beam epitaxy growth of 1.30-1.55 μm wavelength GaInNAs/GaAs quantum well with high indium content, J. Cryst. Growth, 290, 494 (2006).

137. J.C. Harmand, L.H. Li, G. Patriarche, and L. Travers, GaInAs/GaAs quantum-well growth assisted by Sb surfactant: toward 1.3 μm emission, Appl. Phys.Lett., 84, 3981 (2004).

138. S. Gupta, H. Kang, M. Strassburg, A. Asghar, M. Kane, W.E. Fenwick, N. Dietz,and I.T. Ferguson, A nucleation study of group III-nitride multifunctional nano-structures, J. Cryst. Growth, 287, 596 (2006).

139. F. Widmann, B. Daudin, G. Feuillet, N. Pelekanos, and J.L. Rouvière, Improvedquality GaN grown by molecular beam epitaxy using In as surfactant, Appl.Phys. Lett., 73, 2642 (1998).

140. W.K. Fong, C.F. Zhu, B.H. Leung, C. Surya, B. Sundaravel, E.Z. Luo, J.B. Xu,and I.H. Wilson, Characteristics of GaN films grown with indium surfactantby RF-plasma assisted molecular beam epitaxy, Microelectron Reliability, 42, 1179(2002).

141. T.A. Witten, Jr., and L.M. Sander, Diffusion-limited aggregation, a kinetic crit-ical phenomenon, Phys. Rev. Lett., 47, 1400 (1981).

142. H. Brune, Microscopic view of epitaxial metal growth: nucleation and aggre-gation, Surf. Sci. Rep., 31, 121 (1998).




143. I.S. Hwang, T.-C. Chang, and T.T. Tsong, Exchange-barrier effects on nucleationand growth of surfactant-mediated epitaxy, Phys. Rev. Lett., 80, 4229 (1998).

144. T.-C. Chang, I.S. Hwang, and T.T. Tsong, Direct observation of reaction-limitedaggregation on semiconductor surfaces, Phys. Rev. Lett., 83, 1191 (1999).

145. G.B. Stringfellow, Organometallic Vapor Phase Epitaxy: Theory and Practice, 2nded., Academic Press, Boston, 1999, pp. 261–298.

146. L.C. Su, I.H. Ho, N. Kobayashi, and G.B. Stringfellow, Order/disorder hetero-structure in Ga0.5In0.5P with ΔEg = 160 meV, J. Cryst. Growth, 145, 140 (1994).

147. H. Ishikuro and T. Hiramoto, Quantum mechanical effects in the silicon quan-tum dot in a single-electron transistor, Appl. Phys. Lett., 71, 3691 (1997).

148. M. Saitoh, T. Saito, T. Inukai, and T. Hiramoto, Transport spectroscopy of theultrasmall silicon quantum dot in a single-electron transistor, Appl. Phys. Lett.,76, 1440 (2000).

149. L. Harris, D.J. Mowbray, M.S. Skolnick, M. Hopkinson, and G. Hill, Emissionspectra and mode structure of InAs/GaAs self-organized quantum dot lasers,Appl. Phys. Lett., 73, 969 (1998).

150. A. Patanè, A. Polimeni, M. Henini, L. Eaves, P.C. Eaves, P.C. Main, and G. Hill,Thermal effects in quantum dot lasers, J. Appl. Phys., 85, 625 (1999).

151. O.B. Shchekin, G. Park, D.L. Huffaker, and D.G. Deppe, Discrete energy levelseparation and the threshold temperature dependence of quantum dot lasers,Appl. Phys. Lett., 77, 466 (2000).

152. D. Pan, E. Towe, and S. Kennerly, Normal-incidence intersubband (In,Ga)As/GaAs quantum dot infrared photodetectors, Appl. Phys. Lett., 73, 1937 (1998).

153. D. Pan, E. Towe, and S. Kennerly, A five-period normal incidence (In,Ga)As/GaAs quantum-dot infrared photodetector, Appl. Phys. Lett., 75, 2719 (1999).

154. Z. Chen, O. Baklenov, E.T. Kim, I. Mukhametzhanov, J. Tie, A. Madhukar, Z.Ye, and J.C. Campbell, Normal incidence InAs/AlxGa1–xAs quantum dot infra-red photodetectors with undoped active region, J. Appl. Phys., 89, 4558 (2001).

155. G. Bernstein, C. Bazan, M. Chen, C.S. Lent, J.L. Merz, A.O. Orlov, W. Porod,G.L. Snider, and P.D. Tougaw, Practical issues in the realization of quantum-dot cellular automata, Superlattices Microstruct., 20, 447 (1996).

156. T.I. Kamins and R.S. Williams, Lithographic positioning of self-assembled Geislands on Si(001), Appl. Phys. Lett., 71, 1201 (1997).

157. S. Yu Shiryaev, F. Jensen, J.L. Hansen, J.W. Petersen, and A.N. Larsen, Nanoscalestructuring by misfit dislocations in Si1–xGex/Si epitaxial systems, Phys. Rev.Lett., 78, 503 (1997).

158. Y.H. Xie, S.B. Samavedam, M. Bulsara, T.A. Langdo, and E.A. Fitzgerald, Re-laxed template for fabricating regularly distributed quantum dot arrays, Appl.Phys. Lett., 71, 3567 (1997).

159. Z.M. Zhao, T.S. Yoon, W. Feng, B.Y. Li, J.H. Kim, J. Liu, O. Hulko, Y.H. Xie,H.M. Kim, K.B. Kim, H.J. Kim, K.L. Wang, C. Ratsch, R. Caflisch, D.Y. Ryu,and T.P. Russell, The challenges in guided self-assembly of Ge and InAs quan-tum dots on Si, Thin Solid Films, 508, 195 (2006).

160. Q. Xie, A. Madhukar, P. Chen, and N.P. Kobayashi, Vertically self-organizedInAs quantum box islands on GaAs(100), Phys. Rev. Lett., 75, 2542 (1995).

161. I. Mukhametzhanov, R. Heitz, J. Zeng, P. Chen, and A. Madhukar, Independentmanipulation of density and size of stress-driven self-assembled quantum dots,Appl. Phys. Lett., 73, 1841 (1998).




162. C. Teichert, M.G. Lagally, L.J. Peticolas, J.C. Bean, and J. Tersoff, Stress-inducedself-organization of nanoscale structures in SiGe/Si multilayer films, Phys. Rev.B., 53, 16334 (1996).

163. E. Mateeva, P. Sutter, J.C. Bean, and M.G. Lagally, Mechanism of organizationof three-dimensional islands in SiGe/Si multilayers, Appl. Phys. Lett., 71, 3233(1997).

164. R. Hull, J.L. Gray, M. Kammler, T. Vandervelde, T. Kobayashi, P. Kumar, T.Pernell, J.C. Bean, J.A. Floro, and F.M. Ross, Precision placement of heteroepi-taxial semiconductor quantum dots, Mater. Sci. Eng. B, 101, 1 (2003).

165. M. Kammler, R. Hull, M.C. Reuter, and F.M. Ross, Lateral control of self-assembled island nucleation by focused-ion-beam micropatterning, Appl. Phys.Lett., 82, 1093 (2003).

166. Y.-W. Mo, D.E. Savage, B.S. Swartzentruber, and M.G. Lagally, Kinetic pathwayin Stranski-Krastanor Growth of Ge on Si(001), Phys. Rev. Lett., 65, 1020 (1990).



161

5

Mismatched Heteroepitaxial Growth and

Strain Relaxation

5.1 Introduction

Rarely is heteroepitaxial growth lattice-matched. In almost all cases of inter-est, the epitaxial layer has a relaxed lattice constant that is different fromthat of the substrate. The lattice mismatch strain* can be defined as

(5.1)

where is the relaxed lattice constant of the substrate and is the relaxedlattice constant of the epitaxial layer. The absolute value of the lattice mis-match may exceed 10%, but is much smaller in many heteroepitaxial systemsof practical interest. The mismatch may take on either sign, with someinteresting differences observed between tensile (

f

> 0) and compressive(

f

< 0) systems. This chapter is concerned with several important aspects ofmismatched heteroepitaxial growth: the critical layer thickness, lattice relax-ation and the introduction of dislocation defects, and the dynamics of dis-location reactions and removal from thick, mismatched layers.

In heteroepitaxial systems with low mismatch (|

f

| < 1%), the initial growthtends to be coherent, or pseudomorphic. In other words, a thin epitaxiallayer takes on the relaxed lattice constant of the substrate within the growthplane. Therefore, a pseudomorphic layer exhibits an in-plane strain equal tothe lattice mismatch:

(pseudomorphic) (5.2)

* Two other definitions for lattice mismatch are often used in the literature:. All three definitions yield approximately the same abso-

lute value, but there is a difference in sign that must be accounted for:′ ≡ − ′′ ≡ −f a a a f a a ae s e e s s( )/ ( )/and

′′ ≈ ′ = −f f f .

fa a

as e

e

≡ −

as ae

ε|| = f



162


As the epitaxial layer thickness increases, so does the strain energy storedin the pseudomorphic layer. At some thickness, called the

critical layer thicknessh

c

, it becomes energetically favorable for the introduction of

misfit dislocations

in the interface that relax some of the mismatch strain. Beyond the criticallayer thickness, therefore, part of the mismatch is accommodated by misfitdislocations (plastic strain) and the balance by elastic strain. In this case,

(partially relaxed) (5.3)

The residual strain in a heteroepitaxial layer is generally a function of themismatch and layer thickness. It can be calculated based on a thermody-namic model, as long as the growth occurs near thermal equilibrium. Insome cases, however, there are kinetic barriers to the lattice relaxation. Theseare associated with the generation and movement of dislocations. Kineticmodels have been devised to explain and predict the lattice relaxation behav-ior in these situations. These predict that the residual strain in the layer willdepend on the growth conditions and postgrowth thermal cycling, as wellas the mismatch and layer thickness.

In thick, lattice-mismatched heteroepitaxial layers,

most

of the mismatchmay be accommodated by misfit dislocations during growth, even if kineticfactors are important. Therefore, the grown layer will be nearly relaxed at thegrowth temperature. However, the strain measured at room temperature maybe quite different if the epitaxial layer and substrate have different thermalexpansion coefficients. Then a thermal strain will be introduced during thecool-down to room temperature. Moreover, thermal cycling during deviceoperation will result in a temperature dependence of the built-in strain.

The introduction of crystal dislocations and other defects is an importantaspect of lattice-mismatched heteroepitaxy. The misfit dislocations locatedat the heterointerface will degrade the performance of any device whoseoperation depends on it. On the other hand, any device fabricated in theheteroepitaxial layer will tend to be compromised by the presence of thread-ing dislocations in this layer. The threading dislocations are associated withthe misfit dislocations and are introduced during the relaxation process.Whereas the misfit dislocations are expected to be present in partially relaxedlayers under the condition of thermal equilibrium, threading dislocationsare

nonequilibrium defects

. It is possible, at least in principle, to engineerprocessing approaches to remove them entirely from the grown layer.

There are important differences between low-mismatch and high-mis-match heteroepitaxial systems, which are not simply a matter of degree. Theactual mechanisms of strain relaxation and defect introduction have beenfound to be different. This is due, at least in part, to the three-dimensionalnucleation mode of highly mismatched heteroepitaxial layers.

It is often expected that a heteroepitaxial layer will take on the samecrystal orientation as its substrate. In practice, both pseudomorphic andpartly relaxed layers often exhibit small misorientations with respect to their

ε δ|| = −f



Mismatched Heteroepitaxial Growth and Strain Relaxation

163

substrates. In highly mismatched material systems, gross misorientationsare sometimes observed. These come about due to a close match in theatomic spacings for the substrate and the epitaxial layer in different crys-tallographic directions.

The purpose of this chapter is to explore all of these issues in more detail,from both the theoretical and experimental perspectives. This body of knowl-edge forms the basis for the defect engineering approaches described inChapter 7.

5.2 Pseudomorphic Growth and the Critical Layer Thickness

If there is a small lattice mismatch between the epitaxial layer and substrate,and if the growth mode is two-dimensional (layer-by-layer growth), theinitial growth will be coherently strained to match the atomic spacings ofthe substrate in the plane of the interface. This situation is depicted sche-matically in Figure 5.1a, where the epitaxial layer has a larger lattice constantthan the substrate (

a

e

>

a

s

and

f

< 0). The substrate is assumed to be sufficientlythick so that it is unstrained by the growth of the epitaxial layer. Theunstrained substrate crystal is cubic with a lattice constant

a

s

. The pseudo-morphic layer matches the substrate lattice constant in the plane of theinterface (

a

=

a

s

) and therefore experiences in-plane biaxial compression.Using the definition for the mismatch adopted here, the in-plane strain is

(5.4)

where is the lattice relaxation. In the pseudomorphic layer, for which nolattice relaxation has occurred, and so . The epitaxial layer isunconstrained in the direction perpendicular to the interface (the stress inthis direction is zero). Therefore, the out-of-plane strain will have theopposite sign compared to and is given by

(5.5)

where is the biaxial relaxation constant of the growing epitaxial layer.The pseudomorphic epitaxial layer is

tetragonally distorted

with an out-of-plane lattice constant

c

, which is greater than the relaxed lattice constant ofthe epitaxial layer (

c

>

a

e

).As the thickness of the growing layer increases, so does the strain energy

in the layer. At some thickness, it becomes energetically favorable for theintroduction of misfit dislocations to relax some of the strain. The thickness

ε δ|| = −f

δδ = 0 ε|| = f

ε⊥ε||

ε ε ε⊥ = − = −RCCB || ||

2 12

11

RB



164


at which this happens is called the critical layer thickness. For the partiallyrelaxed layer of Figure 5.1b, the in-plane lattice constant of the epitaxial layerhas not relaxed to its unstrained value, but it is greater than the substratelattice constant (

a

e

>

a

>

a

e

). So some of the mismatch is still accommodatedby elastic strain. But a portion of the mismatch has been accommodated bymisfit dislocations (plastic strain). One such misfit dislocation exists at theinterface in Figure 5.1b. Because

a

e

>

a

s

, this misfit dislocation is associatedwith an

extra half-plane

of atoms in the

substrate

.As the description above suggests, it is possible to determine the critical

layer thickness by the minimization of energy. The total energy is the sumof the strain energy and the energy of the misfit dislocations. We can differ-entiate the total energy with respect to the strain and determine the minimum

FIGURE 5.1

Growth of a heteroepitaxial layer on a mismatched substrate: (a) pseudomorphic layer; (b)partially relaxed layer.

as

as

c a

Partially relaxed layer

c as

as

as

Pseudomorphic layer

Substrate

Substrate

(a)

(b)




165

(equilibrium) value. The corresponding strain will equal the mismatch at thecritical thickness.

Energy calculations for a mismatched epitaxial layer on a substrate weremade by Frank and van der Merwe, van der Merwe, and Matthews. Noneof these models, however, considered the mechanism by which misfit dislo-cations would be introduced.

The most widely used theoretical model for the critical layer thickness isthe force balance model of Matthews and Blakeslee,

1

which will be describedfirst. Next, the energy derivation of Frank and van der Merwe and Matthewswill be outlined, and it will be shown that this derivation gives the sameresult as the force balance approach, as long as consistent assumptions aremade. Finally, the energy derivation of People and Bean will be outlined.

5.2.1 Matthews and Blakeslee Force Balance Model

The Matthews and Blakeslee

1

model is used most often to calculate thecritical layer thickness for heteroepitaxy. Here it is considered that a preex-isting threading dislocation in the substrate replicates in the growing epilayerand can bend over to create a length of misfit dislocation in the interfaceonce the critical layer thickness is reached. This process is shown schemat-ically in Figure 5.2. For the threading dislocation shown, the resolved shearstress acting in the direction of slip is

2

(5.6)

where is the biaxial stress, is the angle between the Burgers vector andthe line in the interface plane that is perpendicular to the intersection of theglide plane with the interface, and is the angle between the interface andthe normal to the slip plane. The glide force acting on the dislocation is

(5.7)

where

b

is the length of the Burgers vector for the threading dislocation and

h

is the film thickness. Assuming biaxial stress in an isotropic semiconductor,

FIGURE 5.2

The bending of a grown-in threading dislocation to create a length of misfit dislocation at theinterface between an epitaxial layer and its lattice-mismatched substrate.

FGFL

Substrate

Epitaxial layer

σ σ λ φres = ||cos cos

σ|| λ

φ

F bh bhG res= =σ φ σ λ/ cos cos||



166


(5.8)

so that

(5.9)

where

G

is the shear modulus and is the Poisson ratio. The line tension ofthe misfit segment of the dislocation is given by

(5.10)

where

G

has been assumed to be equal for the epitaxial layer and the sub-strate, is the angle between the Burgers vector and the line vector for thedislocations, and

h

is the layer thickness.To find the critical layer thickness, we equate the glide force to the line

tension for the misfit segment of the dislocation and solve for the thickness.As a result of this procedure, the critical layer thickness

h

c

is found to be

(5.11)

For layers with , the glide force is unable to overcome the line tension,and grown-in dislocations are stable with respect to the proposed mechanismof lattice relaxation. On the other hand, for layers thicker than the criticallayer thickness , threading dislocations will glide to create misfit dis-locations at the interface and relieve the mismatch strain. In the applicationof Equation 5.11 to (001) zinc blende semiconductors, it is assumedthat and , corresponding to 60° dislocationson slip systems. A typical value for the Poisson ratio is .For GaAs, for example, and . Figure 5.3 shows theMatthews and Blakeslee critical layer thickness vs. the lattice mismatchstrain, calculated assuming and .

5.2.2 Matthews Energy Calculation

To determine the critical layer thickness based on the consideration of energy,we can differentiate the total energy with respect to the strain and determinethe minimum (equilibrium) value. The corresponding strain will equal themismatch strain at the critical thickness.

σ νν

ε νν|| ||

( )( )

( )( )

= +−

= +−

2 11

2 11

G Gf

FGbfh

G = +−

2 11( )cos( )

ν λν

ν

FGb

h bL = −−

+( cos )( )

[ln( / ) ]14 1

12ν α

π ν

α

hb h b

fcc= − +

+( cos )[ln( / ) ]

( )cos1 1

8 1

2ν απ ν λ

h hc<

( )h hc>

cos cos /α λ= = 1 2 b a= / 2a2

110 111{ } ν ≈ 1 3/b = 4 0. Å ν( ) .001 0 312=

b = 4 0. Å ν = 1 3/




167

Matthews

3

derived the critical layer thickness in this manner starting withthe areal strain energy in a pseudomorphic mismatched layer of thicknesswith in-plane strain given by

(5.12)

where

G

is the shear modulus and is the Poisson ratio. The energy per unitarea of a square array of misfit dislocations with average separation

S

is

(5.13)

where is the angle between the Burgers vector and the line vector for thedislocations,

b

is the length of the Burgers vector, and

R

is the cutoff radiusfor the determination of the dislocation line energy. This cutoff radius shouldbe taken as the film thickness, or the spacing of the misfit dislocations,whichever is smaller:

(5.14)

FIGURE 5.3

Matthews and Blakeslee critical layer thickness vs. the lattice mismatch strain, calculated as-suming ,

b

= 4.0 Å, and

ν

= 1/3.

1

10

100

1000

0.01 0.1 1 10|f| (%)

h c (n

m)

cos cos /α λ= = 1 2

hε||

E G he = +−

⎛⎝⎜

⎞⎠⎟

211

2νν

ε||

ν

ES

Gb R bd = − +

−1 1 1

2 1

2 2( cos )[ln( / ) ]( )

ν απ ν

α

R S h= min( , )



168


Here it will be assumed that . If the extent of the lattice relaxationis , then the average spacing of misfit dislocations is

(5.15)

where is the angle between the interface and the normal to the slip plane.The total energy of the system is . The condition for energy minimi-zation is

(5.16)

Solving, we find the in-plane strain for minimum energy, or the equilibriumstrain:

(5.17)

Here, the factor accounts for the sign of the strain. The criticallayer thickness is the thickness for which . Solving,

(5.18)

which is exactly the same as the Matthews and Blakeslee critical layer thick-ness as determined by force balance for a threading dislocation.

5.2.3 van der Merwe Model

van der Merwe

4

developed an alternative expression for the critical layerthickness by equating the strain energy in a pseudomorphic film to the inter-facial energy of a network of misfit dislocations. In the same fashion asMatthews, the strain energy in the pseudomorphic layer with thickness

h

wasassumed to be

(5.19)

where

G

is the shear modulus and is the Poisson ratio. The areal energydensity of a misfit dislocation network was estimated to be

R h=δ ε= −f ||

Sb

f=

−cos cos

||

α φε

φE Ee d+

∂ +∂

=( )

||

E Ee d

ε0

ε ν απ ν||( ) ( cos )[ln( / ) ]

( )coseq f

fb h b

h= − +

+1 1

8 1

2

λλ

f f sign f/ ( )=ε||( )eq f=

h b h bf

c = − ++

( cos )[ln( / ) ]( )cos

1 18 1

2ν απ ν λ

E G hfe = +−

⎛⎝⎜

⎞⎠⎟

211

2νν

ν



Mismatched Heteroepitaxial Growth and Strain Relaxation 169

(5.20)

By equating these, van der Merwe found the critical layer thickness to be

(5.21)

van der Merwe’s predictions are quite similar to those of Matthews andBlakeslee, but the absence of the logarithmic term changes the mismatchdependence somewhat.

5.2.4 People and Bean Model

People and Bean5 developed an alternative expression for the critical layerthickness by equating the strain energy in a pseudomorphic film to theenergy of a dense network of misfit dislocations at the interface. FollowingMatthews, the strain energy in the pseudomorphic layer with thicknesswas assumed to be

(5.22)

where G is the shear modulus and is the Poisson ratio. People and Beanconsidered a dense network of misfit dislocations, assumed to have screwcharacter, and with a spacing of . With these assumptions, theycalculated the areal energy density of the misfit dislocation array to be

(5.23)

Equating this result with the strain energy and solving for the thickness,they estimated the critical layer thickness to be

(5.24)

where a is the lattice constant for the epitaxial layer. By assuming a ≈ 0.554nm and b ≈ 0.4 nm, they obtained

E fGb

d ≈⎛⎝⎜

⎞⎠⎟

9 54 2

.π

haf

c =⎛⎝⎜

⎞⎠⎟

−+

⎛⎝⎜

⎞⎠⎟

18

112

0

πνν

h

E G hfe = +−

⎛⎝⎜

⎞⎠⎟

211

2νν

ν

S a= 2 2

EGb

a

hbd ≈

⎛⎝⎜

⎞⎠⎟

2

8 2πln

hba f

c = +−

⎛⎝⎜

⎞⎠⎟

⎛

⎝⎜⎞

⎠⎟⎛

⎝⎜⎞

⎠⎟⎛⎝⎜

⎞⎠

11

1

16 2

12

2

νν π ⎟⎟

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

lnhb

c




(5.25)

People and Bean used this expression to calculate the critical layer thick-ness as a function of composition in Si1–xGex/Si (001), for which the latticemismatch strain is . These results are shown in Figure 5.4, alongwith the calculations by van der Merwe and by Matthews and Blakeslee.Also shown for comparison are experimental data for several heteroepitaxialmaterial systems. Data for the Si1–xGex/Si (001) heteroepitaxial system mea-sured by Bean et al.6 and Bevk et al.7 appear to be in agreement with calcu-lations of the People and Bean model. However, the Matthews and Blakesleemodel appears to agree with many of the available experimental results. Itis known that the combined effects of finite experimental resolution withinitially sluggish lattice relaxation can cause experimental results to overes-timate the critical layer thickness. This could explain why the People andBean model is in fair agreement with some experimental results.

The People and Bean model is attractive because its predictions are in fairagreement with some of the experimental results for SixGe1–x/Si (001) and

FIGURE 5.4Critical layer thickness vs. the lattice mismatch strain. The Matthews and Blakeslee critical layerthickness was calculated assuming , b = 4.0 Å, and ν = 1/3. The People andBean critical layer thickness was calculated using Equation 5.26. The van der Merwe criticallayer thickness was calculated using Equation 5.22.

hnm

fhnmcc= ×⎛

⎝⎜⎞

⎠⎟⎛⎝⎜

⎞⎠⎟

−1 9 100 4

3

2

.ln

.

f x= −0 04.

1

10

100

1000

0.01 0.1 1 10 |f| (%)

h c (n

m)

Matthews and Blakeslee People and Beanvan der MerweHoughton et al. (GeSi/Si)Elman et al. (In GaAs/GaAs)Bean et al. (GeSi/Si)

cos cos /α λ= = 1 2




also InxGa1–xAs/GaAs (001). However, it was developed with the assumptionof a dense net of misfit dislocations having a fixed spacing of , and soit is not physical. Since this close spacing of misfit dislocations correspondsto a fully relaxed layer with , the People and Bean model shouldoverestimate the critical layer thickness for heteroepitaxial systems with lessthan 6.2% mismatch. Moreover, experimental studies of mismatched het-eroepitaxial layers have shown that the lattice mismatch occurs graduallywith the increase of thickness, and not abruptly.

Although the experimental results shown in Figure 5.4 exhibit considerablescatter, the smallest value for a given mismatch will generally be the mostreliable. This is because the combined effects of sluggish lattice relaxationand finite experimental resolution will increase the apparent critical layerthickness obtained by experimentation. The Matthews and Blakeslee modelis in good agreement with the most reliable experimental results and is themost widely accepted model for the critical layer thickness.

5.2.5 Effect of the Sign of Mismatch

The models developed by van der Merwe and Matthews and Blakeslee onlyconsider the absolute value of the lattice mismatch strain, and not its sign.However, it is of technological importance to determine whether the criticallayer thickness is different in the tensile and compressive cases. Petruzzelloand Leys8 considered differences in the lattice relaxation mechanisms forcompressive and tensile layers arising from the nucleation of Shockley partialdislocations in diamond and zinc blende semiconductors. (This topic is dis-cussed in Section 5.5.4.) However, these differences do not impact the criticallayer thickness for the bending over of threading dislocations as consideredby Matthews and Blakeslee. On the other hand, Cammarata and Sieradzki9

modeled the effect of surface tension on the critical layer thickness andshowed that, in principle at least, this should make the critical layer thicknesssmaller for the tensile case and larger for the compressive case. Physically,this asymmetry arises because the surface tension is always compressive.This theoretical treatment will be summarized in what follows.

The elastic strain energy per unit area associated with a uniform elasticstrain in an elastically isotropic layer of thickness h with in-plane strainis given by3

(5.26)

where Y is the biaxial modulus, and for an isotropic crystal, Y = 2G(1 + ν)(1– ν). The misfit dislocation energy per unit area, for a square array of misfitdislocations along the two directions, and with a spacing such thatthey relieve an amount of mismatch strain , is given by

2 2a

f ≈ 0 062.

Ue

ε||

E Y he = ε||2

110δ ε= −f ||




(5.27)

where G is the shear modulus, b is the length of the Burgers vector for themisfit dislocations, is the Poisson ratio, is the angle between the Burgersvector and the line vector for the dislocations, and is the angle betweenthe Burgers vector and the line in the interface plane that is perpendicularto the intersection of the glide plane with the interface. The terms andare essentially those used by van der Merwe and Matthews for the calcula-tion of the critical layer thickness by energy balance. Cammarata andSieradzki9 introduced another term, , due to the surface energy of thestrained heteroepitaxial layer, given by

(5.28)

where is the surface energy. It was assumed that is isotropic and inde-pendent of the strain in the layer, and the critical layer thickness was deter-mined by

(5.29)

yielding

(5.30)

where the + and – apply to the compressive and tensile cases, respectively.Apart from the influence of the logarithmic factor, this amounts to the Mat-thews and Blakeslee critical layer thickness plus or minus a factor propor-tional to the surface energy.

The variation of the critical thickness with the lattice mismatch strain isplotted in Figure 5.5 for the (a) tensile and (c) compressive cases, assuminga surface energy of γ = 2 Jm–2 and G = 3 × 1010

Pa, cos α = cos λ = 1/2, b =0.4 nm, and ν = 1/3. Using these values,

(5.31)

EGb f h b

d =− − +

−( cos ) [ln( / ) ]

( )cos||1 1

4 1

2ν α επ ν λ

ν αλ

Ee Ed

Es

E ds = ∫2 γ ε

γ γ

∂ + +∂

=( )U U Ue d s

ε0

hb h b

fcc= − +

+± −( cos )[ln( / ) ]

( )cos(1 1

8 112ν α

π ν λγ νν

ν)

( )f G2 1+

hhf fcc= + ±0 022 0 4 1 0 0167. [ln( / . ) ] .nm nm nm




Also shown is the Matthews and Blakeslee critical layer thickness (b),calculated by neglecting the surface energy. (The second term in Equation5.31 was neglected.) These results show that, in principle, the surface energycan modify the critical layer thickness and also create an asymmetry betweencompressive and tensile films. This work has been extended by Cammarataet al.10 to include interfacial stresses, for both single heteroepitaxial layersand strained layer superlattices. However, the uncertainties inherent in crit-ical layer thickness measurements have hindered experimental verificationof this effect.

5.2.6 Critical Layer Thickness in Islands

The theoretical models presented thus far assume that the heterointerface isof infinite extent in the lateral directions. Luryi and Suhir11 showed that inislands with finite lateral size the critical thickness depends on the islanddiameter. In their work, Luryi and Suhir calculated the critical layer thicknessfor mismatched heteroepitaxial islands that make rigid contact with thesubstrate only on round seed pads having a diameter of 2l. They showedthat in a pseudomorphic structure of this sort, the strain in the heteroepitaxiallayer decays with distance from the interface. Further, the characteristiclength for this decay is on the order of the seed pad dimension. Because

FIGURE 5.5Critical layer thickness hc vs. the absolute value of the lattice mismatch strain for (a) tensilefilms with γ = 2 Jm–2, (b) tensile or compressive films with γ = 0 (Matthews and Blakeslee model),and (c) compressive films with γ = 2 Jm – 2 . The fol lowing values were as-sumed: , , b = 0.4 nm, and ν = 1/3.

1

10

100

1000

0.01 1 10|f| (%)

h c (n

m)

(a) (b)(c)

0.1

f

G = ×3 1010 Pa cos cos /α λ= = 1 2

he




of this behavior, the critical layer thickness increases as the seed pads arescaled down in size. For a particular value of the lattice mismatch, there isan island diameter for which the critical layer thickness diverges to infinity,so that structures entirely free from misfit dislocations may be produced.

The analysis of island growth by Luryi and Suhir11 started with theassumption that the lattice-mismatched heteroepitaxial material makes rigidcontact with a noncompliant substrate only at round seed pads having adiameter of 2l. Here, the y-axis lies in the plane of the interface, along amajor cord of a seed pad. The z-axis is perpendicular to the substrate andpasses through the center of this seed pad. The thickness of the island growthmaterial is h. If the substrate is unstrained, then the in-plane stress in theepitaxial deposit is given by

(5.32)

where f is the lattice mismatch strain, E is the Young’s modulus, ν is thePoisson ratio, and

(5.33)

where is the effective range for the stress in the z direction, to be deter-mined below, and the interfacial compliance parameter k is given by

(5.34)

The strain energy density per unit volume is

(5.35)

and is maximum at y = 0. The strain energy per unit area may be found byintegrating over the thickness of the epitaxial deposit and takes on a maxi-mum value at y = 0, which is

(5.36)

σν

χ π|| ( , )exp( / )=−

−fE

y z z l1

2

χ( , )cosh( )cosh( )y z

kykl

z h

z h

e

e

=− ≤

≥

⎧⎨⎪

⎩⎪

1

1

he

kh he e

= −+

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ ≡3

211

11 2

νν

ζ/

ω ν σ( , ) ||y zE

= −1 2

E zE

f hs

h

e= ≡−∫ ω

ν( , )0

10

2 2




In this calculation, there is little contribution from , so that it is a goodapproximation to use the form of for . The right-hand side ofEquation 5.36 defines the characteristic thickness , which is then givenimplicitly by

(5.37)

The right-hand side of this equation defines the reduction factor .For , , and for ,

(5.38)

The strain energy per unit area from Equation 5.36 may be used in con-junction with an energy calculation for the critical layer thickness to find thecritical layer thickness for an island of radius l. The result is

In their work, Luryi and Suhir used the People and Bean model for thedetermination of the critical layer thickness. However, the Matthews energycalculation of the critical layer thickness may also be used, with

(5.39)

The critical thickness is shown in Figure 5.6 as a function of the latticemismatch, with the island diameter 2l as a parameter. The Matthews andBlakeslee curve corresponds to . For nanometer-scale islands, the crit-ical layer thickness can be increased significantly. Also, at a given mismatch,there is a critical island diameter for which the critical thickness diverges toinfinity. The critical island size is plotted as a function of the lattice mismatchstrain in Figure 5.7.

5.3 Dislocation Sources

It is well established that lattice relaxation commences after the critical layerthickness is exceeded in a mismatched heteroepitaxial layer. The strain relax-

z he>χ( , )y z z he≤

he

h h hl

hh l

le

e

= −⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ − −1 1

2

sec [ exp( / )]ζ π

πhhh

lh

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪=

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥φ

2

φ( / )l hl h>> h he ≈ l h<<

hlh

he ≈ −[ sec ( )]1 2ζπ

hcl

h h l h fcl

c cl= [ ( / ) ]φ

hb h b

l h fcl c

l

cl

= − ++

( cos )[ln( / ) ]

( / ) (

1 1

8 1

2ν απ φ ν))cos λ

2l → ∞




FIGURE 5.6Critical layer thickness as a function of lattice mismatch strain, with island diameter 2l as aparameter. The case of 2l = ∞ corresponds to planar growth (the Matthews and Blakeslee limit).

FIGURE 5.7Critical island diameter as a function of lattice mismatch strain. For islands equal to or lessthan this diameter, the critical layer thickness diverges to infinity and pseudomorphic structuresmay be grown with arbitrary thickness.

1

10

100

1000

0.01 0.1 1 10|f| (%)

h c (n

m)

20 nm200 nm∞

1

10

100

1000

10000

0 0.2 0.4 0.6 0.8 1|f| (%)

Criti

cal i

sland

dia

met

er (n

m)




ation occurs by the introduction of misfit dislocations at the interface, butwhere do these dislocations come from? Matthews and Blakeslee derived thecritical layer thickness based on the assumption that substrate dislocationsbend over to produce misfit dislocations in the interface. However, heteroepi-taxial layers may contain 105 times the threading dislocation density of theirsubstrates; thus, substrate dislocations are usually not the sole source. Instead,dislocation nucleation (homogeneous or heterogeneous) or dislocation mul-tiplication must take place during mismatched heteroepitaxy. The followingsections will review these sources of dislocations, and it will be shown thatheterogeneous nucleation and multiplication can both be important.

5.3.1 Homogeneous Nucleation of Dislocations

A possible mechanism for the introduction of misfit dislocations is the homo-geneous nucleation of half-loops at the surface. The glide of such a half-loopto the interface results in a misfit dislocation segment with two associatedthreading dislocations. The nucleation of such half-loops has been consid-ered in detail by Matthews3,12 and Matthews et al.13

Consider a dislocation half-loop that has nucleated at the surface of a het-eroepitaxial layer, as shown in Figure 5.8. This situation is analogous to thenucleation of a deposit on a substrate as described in Section 4.3.2. Thus, thereis a critical half-loop radius above which the loop will continue to growuntil it reaches the interface, creating a length of misfit dislocation. Subcriticalloops will shrink and disappear. The homogeneous nucleation of dislocationhalf-loops will occur only if the thermal energy is sufficient for the spontane-ous formation of loops having a radius equal to the critical radius.

The critical radius and the associated half-loop energy were calculated byMatthews as follows. The formation of the dislocation half-loop with a radiusR involves the line energy*

FIGURE 5.8A dislocation half-loop of radius R that has nucleated at the surface of a heteroepitaxial layer,lying on its glide plane.

* Matthews used a slightly different expression for the self-energy of the dislocation half-loop.However, the end result is not changed significantly by this difference.

Epitaxial layer

SubstrateGlide plane

Half loop

Rc




(5.40)

where G is the shear modulus, b is the length of the Burgers vector, and ν isthe Poisson ratio. This is offset by the strain energy released by the formationof the half-loop, which is

(5.41)

where λ is the angle between the Burgers vector and the line in the interfaceplane that is perpendicular to the intersection of the glide plane with theinterface, and φ is the angle between the interface and the normal to the slipplane. If the loop is imperfect, there is a stacking fault on the inside of itwith energy

(5.42)

where σ is the stacking fault energy per unit area. The energy of the surfacestep created by the introduction of the loop is

(5.43)

where λ is the surface energy per unit area and α is the angle between theBurgers vector and the line vector for the misfit portion (bottom) of the loop.The total energy of the loop is found by summing these terms:

(5.44)

This energy is zero for , increases to a maximum value of at thecritical half-loop size , and then decreases for larger values of R. Thecritical half-loop size can then be found by

(5.45)

If a complete dislocation is assumed (no stacking fault is involved), thecritical radius is given by

EGb R R

bl = −−

⎡

⎣⎢

⎤

⎦⎥ +

⎛⎝⎜

⎞⎠⎟

2

821

1( )( )

lnνν

ERGb

επ ν ε

νλ φ= +

−( )

( )cos cos

11

ER

sf = π σ2

2

E R bs = 2 γ αsin

E E E E El sf s= − + +ε

R = 0 Ecrit

Rcrit

∂∂

=ER

0




(5.46)

The activation energy for the nucleation of half-loops may be determinedfrom

(5.47)

If it is assumed that the available thermal energy does not exceed 50 kT,then the homogeneous nucleation of half-loops is not expected to occurunless the mismatch strain is greater than 1.5% (at room temperature).

Due to the large amount of energy involved in this process, it is believedthat the homogeneous nucleation of half-loops will be insignificant in mostheteroepitaxial materials. Instead, it is likely that other processes producemisfit dislocations and start to relieve the misfit strain before this homoge-neous nucleation process can become active.

5.3.2 Heterogeneous Nucleation of Dislocations

The calculations of the previous section show that the homogeneous nucle-ation of dislocation half-loops at the surface should be negligible at typicalgrowth temperatures. On the other hand, the heterogeneous nucleation ofhalf-loops is much more likely. Here, heterogeneous nucleation refers to thenucleation of a half-loop at an existing crystal defect, such as a dislocation,void, precipitate, or scratch. The local strain field associated with such adefect could greatly reduce the activation energy for the creation of a dislo-cation half-loop, thereby allowing this process to occur at an appreciable rate.

Direct evidence of dislocation half-loop nucleation (for example, in theform of transmission electron microscopy (TEM) micrographs) is lacking inthe literature. This does not necessarily mean that this process is inactive.Instead, it may be an indication that super-critical half-loops expand rapidlyonce nucleated. However, Zou et al.14 presented experimental evidence fromSiGe islands on Si (001) that suggests a mechanism of strain relaxation bythe nucleation of partial dislocation half-loops at the surface.

5.3.3 Dislocation Multiplication

In many heteroepitaxial semiconductors, the observed lattice relaxation canonly be explained by invoking either the nucleation of new dislocations ordislocation multiplication. This has been shown by Beanland,15 based on thework of Matthews et al.16 Consider a heteroepitaxial layer on a square sub-strate with sides L, parallel to the misfit dislocation lines. If the threadingdislocation density in the substrate is D, then the total number of dislocation

RGb R b b

crit = − + + −( / )( )[ln( / ) ] ( ) sin2 8 2 2 2 12

ν σ ν αππ νε λ φG b( cos cos )1+

E E Rcrit crit= ( )




sources is . If, in the process of lattice relaxation, misfit dislocations areproduced along the two possible directions with equal numbers, then therewill be misfit dislocations in each direction. If these misfit disloca-tions run to the edge of the sample (an optimistic assumption), and theirsources (the threading dislocations) are uniformly distributed across thesample, their average length will be . The linear density of misfit dislo-cations in the interface will be

(5.48)

The amount of strain that can be relaxed by this density of misfit disloca-tions is

(5.49)

where b is the length of the Burgers vector, α is the angle between the Burgersvector and line vector, and φ is the angle between the interface and the normalto the slip plane. Therefore, the amount of strain that can be relieved by auniform density of sources without multiplication is proportional to thelinear size of the sample. This conclusion holds true for any substrate shape,although geometrical factors must be included in the analysis. For example,in the (001) heteroepitaxy of zinc blende semiconductors, a maximum of 1%mismatch strain may be relieved with a substrate dislocation density of 105

cm–2, in the absence of dislocation multiplication. Figure 5.9 shows theamount of misfit strain that may be relieved as a function of the substratesize, with the substrate threading dislocation as a parameter, for the (001)heteroepitaxy of a zinc blende semiconductor with 60° dislocations on {111}glide planes. A square wafer is assumed, and the values of threading dislo-cation density considered are 102, 104, and 105 cm–2, which encompass thetypical range for practical substrates for heteroepitaxy. This figure showsthat in order to account for the observed lattice relaxation in heteroepitaxialsemiconductors, we must invoke either dislocation multiplication or anextremely high density of sources for the heterogeneous nucleation of dis-locations. Dislocation nucleation sources other than substrate threading dis-locations, if present, are unlikely to have a density much greater than D incarefully prepared, high-quality substrates. We therefore conclude that dis-location multiplication will be important in the lattice relaxation of nearlyall heteroepitaxial semiconductors.

5.3.3.1 Frank–Read Source

One possible mechanism for the multiplication of dislocations is theFrank–Read source,17 illustrated in Figure 5.10. Here, the preexisting dislo-cation is anchored at points D and D′. It is important to note that the

L D2

L D2 2/

L / 2

ρ = =L DL

LD2 2

2/

/

δ ρ α φ α φ= =b LDbcos cos cos cos




FIGURE 5.9The maximum misfit that can be relieved by the bending over of existing threading dislocations,without dislocation multiplication, as a function of the substrate linear size. The substratethreading dislocation was assumed to be 102, 103, and 104 cm–2, as indicated.

FIGURE 5.10The Frank–Read source.

0.0001

0.0010

0.0100

0.1000

1.0000

0.1 1 10 100Wafer size L (cm)

δ (%

)D = 104 cm−2

103 cm−2

102 cm−2

τb

D D′ D D′

D D′

(a) (b) (c)

D D′

(d)

D D′

(e)




dislocation may not end within a perfect crystal, so the points D and D′represent bends in the crystal dislocation, rather than actual terminations.There are many possible reasons why the dislocation could be immobile atpoints D and D′. Whereas the plane of the paper is assumed to be a glideplane, the bent-over portions of the dislocation may not lie on easy glideplanes, rendering them essentially sessile. Another possibility is that defectscould pin the dislocation at points D and D′. These pinning defects couldbe inclusions, voids, or even other dislocations, but the important feature isthat they immobilize the dislocation at these two points.

Regardless of these details, an applied stress as shown in Figure 5.10a willcause the dislocation to bow as shown in Figure 5.10b. Eventually, the dis-location can start to bend back upon itself, as shown in Figure 5.10c. It shouldbe recognized that whereas the Burgers vector is conserved along the lengthof the dislocation, the line vector is reversed on the trailing edge of the boweddislocation relative to the leading edge. Therefore, the leading and trailingedges experience forces of opposite sign, as shown, tending to further expandthe bowing dislocation, as in Figure 5.10d. Eventually, the bowing dislocationcloses upon itself, as in Figure 5.10e. The dislocation loop so created cancontinue to expand under the applied stress. The dislocation segmentbetween the pinning defects can now snap back into its original configura-tion, and the multiplication process can repeat. Hence, such a Frank–Readsource can continue to eject dislocation loops as long as the necessary stressis applied.

Frank–Read dislocation sources have been observed experimentally byDash and by Meieran. Dash decorated dislocations in a Si crystal using aCu precipitation technique, which rendered them visible by infrared trans-mission microscopy.18 The roughly concentric hexagonal dislocation loopsobserved in the sample were attributed to a Frank–Read type source. Meie-ran observed a Frank–Read source in a Si crystal using x-ray topography(Figure 5.11).

Beanland15 considered the operation of Frank–Read sources in mismatchedheteroepitaxial layers. A possible configuration for a Frank–Read source inthis situation is shown in Figure 5.12. In Figure 5.12a, the threading dislo-cation is anchored at points A and B. With an applied stress, the dislocationbows out between A and B, as shown in Figure 5.12b. Upon reaching thesurface, the bowing loop breaks into two dislocations, as in Figure 5.12c.Finally, interaction of the two dislocations results in the formation of a half-loop (the right side of which has glided out of the picture), as shown inFigure 5.12d. This process leaves the original dislocation intact, and it canparticipate in further multiplication.

The critical thickness for the operation of such a Frank–Read source hasbeen calculated by Beanland.15 In this treatment, a force balance relationshipwas applied for the bowing of the pinned dislocation. The critical thicknessso determined depends on the positions of the pinning points and the ori-entation of the pinned segment. However, assuming the pinned segment AB




lies along the [112] direction, the minimum thickness for which theFrank–Read source may operate is given by

(5.50)

where is the Matthews and Blakeslee critical layer thickness and isgiven by

FIGURE 5.11X-ray topograph ( reflection) of a sawed and chemically polished Si wafer, showing abowing Frank–Read dislocation source. The magnification is ×15. (Reprinted from Meieran, E.S.,J. Appl. Phys., 36, 1497, 1965. With permission. Copyright 1965, American Institute of Physics.)

FIGURE 5.12A possible configuration for a Frank–Read source in a heteroepitaxial layer. (a) The threadingdislocation is anchored at points A and B. (b) With an applied stress, the dislocation bows outbetween A and B. (c) Upon reaching the surface, the bowing loop breaks into two dislocations.(d) Interaction of the two dislocations results in the formation of a half-loop (the right side ofwhich has glided out of the picture), leaving a dislocation similar to the original source defect.(Reprinted from Beanland, R., J. Appl. Phys., 72, 4031, 1992. With permission. Copyright 1992,American Institute of Physics.)

[ ]220

(a)

Epitaxial layer

Substrate

A

B

(b)

τb

(c) (d)

hf

h h hf c p= + 2

hc hp




(5.51)

where is the lattice mismatch strain, ν is the Poisson ratio,and b is the length of the Burgers vector. Figure 5.13 shows the criticalthickness for Frank–Read multiplication as a function of the lattice mismatchstrain. Also shown are the Matthews and Blakeslee critical layer thicknessfor lattice relaxation and the critical thickness for multiplication by spiralsources (considered in the next subsection). Typically, is four to seventimes the Matthews and Blakeslee critical layer thickness. In many cases, alarge fraction of the observed relaxation occurs after the thickness is severaltimes the critical layer thickness. It is therefore likely, based on Beanland’sestimates, for Frank–Read multiplication to be active in mismatched het-eroepitaxial layers.

Frank–Read type sources have been observed in heteroepitaxial layers bya number of workers. Lefevbre et al.19 observed such a source in InGaAs/GaAs (001). LeGoues et al.20 reported the observation of Frank–Read type

FIGURE 5.13The critical thicknesses for dislocation multiplication by Frank–Read and spiral sources, asfunctions of the lattice mismatch strain . Also shown is the Matthews andBlakeslee critical layer thickness for lattice relaxation, for comparison. (Reprinted from Beanland,R., J. Appl. Phys., 72, 4031, 1992. With permission. Copyright 1992, American Institute of Physics.)

hb

f

h

bpp= +

−

⎛

⎝⎜⎜

⎞

⎠⎟⎟

+ −+

( )( )

ln( )( )

24 1

4 6 22

νπ ν

νν

⎡⎡

⎣⎢⎢

⎤

⎦⎥⎥

f a a as e e= −( )/

hf

0

10

20

30

40

50

0 1 2 3 4 5 |f| (%)

h (n

m)

Frank-Readmultiplication

Spiral multiplication

Matthews and Blakeslee

f a a as e e= −( )/




sources in SiGe/Si (001). Capano et al.21 observed regular cross-slip in SiGe/Si (001) by x-ray topography and explained this as a result of the operationof Frank–Read type sources.

5.3.3.2 Spiral Source

Another type of dislocation source is the spiral source, also proposed byFrank and Read.17 Such a spiral source is shown in Figure 5.14. Here, thedislocation line ABC is bent out of the horizontal glide plane at point B.Suppose the segment BC lies on a glide plane but segment AB is sessile. Ifa shear stress is applied on the glide plane and in the slip direction, thesegment BC will sweep around the axis BC like the hand of a clock, produc-ing one unit of slip for each revolution. Such a source is expected to sweepout a spiral if glide is inhibited at the outer edge relative to the inner section.Although this mechanism does not produce new dislocations, it can increasethe length of dislocation line arbitrarily.

The spiral source has also been observed experimentally in Si by Dash22

and Authier and Lang.23 Figure 5.15 shows such a spiral source in a Sispecimen. The image is an x-ray projection topograph obtained usingthe reflection. The Si specimen was a rectangular bar, which wasstressed by twisting about its long axis, which was [111], at 900°C.

A possible configuration for the spiral source in a heteroepitaxial layerwas described by Beanland15 and is shown in Figure 5.16. It is assumed thata threading dislocation is anchored at a single point A, as shown in Figure5.16a. With an applied stress, the dislocation may bow out above the pinningpoint, as in Figure 5.16b. The bowed section will continue to expand andmay glide to the interface to relieve mismatch strain, as in Figure 5.16c.Further expansion of the bowed portion may lead to production of a half-loop if the bow reaches the surface and splits in two, as in Figure 5.16d. The

FIGURE 5.14A bent dislocation line ABC, which can act as a spiral source. (Reprinted from Frank, F.C. andRead, W.T., Phys. Rev., 79, 722, 1950. With permission. Copyright 1950, American Physical Society.)

A

B

C

Slip direction

Slippedarea

1 1 1




FIGURE 5.15A spiral dislocation source in a Si crystal (x-ray projection topograph). The approximatesize of the spiral is 1.6 mm. (Reprinted from Authier, A. and Lang, A.R., J. Appl. Phys., 35, 1956,1964. With permission. Copyright 1964, American Institute of Physics.)

FIGURE 5.16A possible configuration for a spiral source in a heteroepitaxial layer. (a) The threading dislo-cation is anchored at point A, which is located a distance hp from the interface. (b) With anapplied stress, the dislocation bows out above A. (c) The bowed section will continue to expandand may glide to the interface to relieve mismatch strain. (d) Upon reaching the surface, thebowing loop breaks into two dislocations. This results in the formation of a half-loop (the rightside of which has glided out of the picture), leaving a dislocation similar to that of the originalsource defect. (Reprinted from Beanland, R., J. Appl. Phys., 72, 4031, 1992. With permission.Copyright 1992, American Institute of Physics.)

1 1 1

(a) (b)

(c) (d)

τbEpitaxial layer

Substrate

Ahp




original dislocation is then available to produce more dislocations by thesame process.

The critical thickness for the operation of such a spiral source has beenalso calculated by Beanland.15 This value depends on the position of thepinning point, but the minimum thickness for which the spiral source mayoperate is given by

(5.52)

where is the Matthews and Blakeslee critical layer thickness and isthe height of the pinning point above the interface.

In Figure 5.13, the critical thicknesses for multiplication by spiral andFrank–Read sources are compared. The spiral source can become active attwo to four times the Matthews and Blakeslee critical thickness for latticerelaxation. It is therefore likely that both the spiral and Frank–Read mecha-nisms are active in relaxing heteroepitaxial layers.

Spiral sources have been seen by Mader and Blakeslee24 in GaAsP/GaAs(113) and by Wasburn and Kvam25 in GeSi/Si (001).

5.3.3.3 Hagen–Strunk Multiplication

The previously considered mechanisms for multiplication involve the pin-ning of a dislocation at one (spiral source) or two (Frank–Read source) pointsin the presence of an applied stress. However, other multiplication mecha-nisms are possible that do not result from pinning of a dislocation, butinstead involve the intersection of two gliding dislocations. One such mech-anism has been proposed by Hagen and Strunk26 and is illustrated in Figure5.17. Here, it is assumed that two dislocations AB and CD have the sameBurgers vector but are on different glide planes. If these dislocations reactat the cross-point, they may create two angular dislocations, as shown inFigure 5.17b. The repulsion of these dislocations with like Burgers vectors,in conjunction with the image force in a thin layer, can push the bent tip ofone of the dislocations toward the surface on its inclined glide plane (forexample, a {111} plane for (001) zinc blende heteroepitaxy). Upon reachingthe surface, the dislocation can split into two, as shown in Figure 5.17c. Aftera process involving the combined cross-slip and glide of the broken dislo-cation segments, there will be three misfit dislocations, AE, FD, and COB,all of which may participate in further multiplication by the same process.The three misfit dislocations will all have the same Burgers vector and mayappear in a configuration like that of Figure 5.17e.

Hagen et al. observed dislocations with a configuration similar to thatshown in Figure 5.17e in heteroepitaxial Ge/GaAs (001) by TEM.26,27 Theyinterpreted these results as evidence for the operation of the Hagen–Strunkmechanism in heteroepitaxial layers. However, simple dislocation reac-tions at the interface could result in similar configurations,28 and so it has

hf

h h hs c p= +

hc hp




been argued that the TEM results do not provide evidence for Hagen–Strunk multiplication.15

Obayashi and Shintani29 made a theoretical investigation of the Hagen–Strunk mechanism in Ge/GaAs (001) and SiGe/Si (001) systems. They con-sidered the forces acting on a dislocation segment created by the reactionbetween two crossing misfit dislocations. They found that, because of theinvolvement of the image forces in this multiplication scheme, there is acritical thickness above which the mechanism cannot operate. They calcu-lated the Hagen–Strunk critical thickness to be smaller than the Matthewsand Blakeslee critical layer thickness for lattice relaxation. Based on thisfinding, they concluded that Hagen–Strunk multiplication is unlikely tooccur in heteroepitaxial layers.

In summary, multiplication of dislocations in heteroepitaxial semiconduc-tors is rather complex. Several mechanisms for dislocation multiplicationhave been proposed, including the Frank–Read source and spiral source.Frank–Read sources have been observed experimentally in both bulk andheteroepitaxial semiconductors. Similarly, there have been reports ofobserved spiral sources in both bulk semiconductors and mismatched layers.

FIGURE 5.17The Hagen–Strunk dislocation multiplication mechanism. (Reprinted from Beanland, R., J. Appl.Phys., 72, 4031, 1992. With permission. Copyright 1992, American Institute of Physics.)

(a) (b)

A B

C

D

O

X

(c)

C

D

O

A

A A

B

B B

C

DO

bb

(d)

C

D

O

A B

(e)

C

D

O

E

F




Theoretical calculations by Beanland point to the likelihood of these mecha-nisms being active in mismatched heteroepitaxy. Despite this, clear experi-mental observations of them remain extremely rare. Often the dislocationsin partially relaxed heteroepitaxial layers take on complex configurations thatare difficult to interpret. One contributing factor might be that these multi-plication sources are only able to operate a few times in highly dislocatedcrystals, whereas they might go unnoticed unless they operate many times.Still, the body of experimental and theoretical work indicates that sources ofthese types must be active during heteroepitaxial growth. That is, any con-figuration giving rise to the pinning of a dislocation at one or more points,in the presence of applied stress, should lead to dislocation multiplication.

It should be noted that other multiplication mechanisms are possible thatdo not require the anchoring of existing dislocations, but instead involve theinteraction of two or more dislocations. One such process that has beenproposed is the Hagen–Strunk mechanism. There is very limited experimen-tal evidence for this mechanism, and theoretical calculations show that it isunlikely in heteroepitaxial semiconductors. Still, there might be two-dislo-cation multiplication processes that are important but remain undiscovered.

Further experimental investigations may lead to a better understandingof the complex processes involved in dislocation multiplication. Wurtzitesemiconductors and SiC, barely studied until now, may reveal yet other typesof mechanisms for dislocation multiplication.

5.4 Interactions between Misfit Dislocations

In diamond and zinc blende (001) heteroepitaxial layers, the misfit disloca-tions usually have 60° or edge (90°) character and lie along the orthogo-nal directions. There are three basic types of interactions that can occurat the intersections of these misfit dislocations,30–32 shown in Figure 5.18. Ifthe Burgers vectors are (a) parallel or (b) antiparallel, the intersecting dislo-cations will form two L-shaped dislocations. (c) If the Burgers vectors makean angle of 60°, a linking dislocation will form. (d) If the Burgersvectors are perpendicular, no reaction is expected.

If only 60° misfit dislocations are present at the interface, for (001) dia-mond or zinc blende heteroepitaxy, then there are four possible Burgersvectors. This results in 16 possible interactions, 4/16 (25%) of which shouldproduce L-shaped dislocations; 8/16 (50%), linked dislocations; and 4/16(25%), no reaction.

If edge dislocations are present as well as 60° dislocations, then there arefive possible Burgers vectors. This results in 25 possible interactions, 4/25(16%) of which should produce L-shaped dislocations; 16/25 (64%), linkeddislocations; and 5/25 (20%), no reaction. And if the possible Burgers vectors

110

a / 2 110




are present in equal number in each type of dislocation, the observed dislo-cation interactions should closely follow the percentages above.

In TEM observations of interacting misfit dislocations, it may be difficultto distinguish between linked and unreacted dislocations. In the former case,the links are expected to be very short. This is because the slip planes of thelinked dislocations lie out of the plane of the interface, and so they areunlikely to move apart significantly. Still, approximately 16% of the interac-tions should produce L-shaped dislocations if edge dislocations are excluded,

FIGURE 5.18Three possible interactions between misfit dislocations along directions in a diamond orzinc blende (001) heteroepitaxial layer. If the Burgers vectors are parallel (a) or antiparallel (b),the intersecting dislocations will form two L-shaped dislocations. (c) If the Burgers vectors makean angle of 60°, a linking dislocation will form. (d) If the Burgers vectors are per-pendicular, no reaction is expected.

b2

b2

b2

b2

Parallel burgers vectors L reaction

Antiparallel burgers vectors L reaction

Link reaction

No reaction

Burgers vectors at 60°

Perpendicular burgers vectors

(a)

(b)

(c)

(d)

b1

b1

b1

b1

110

a/2 110




but the inclusion of edge dislocations should result in L-shaped dislocationsat a greater fraction of the intersections.

Dixon and Goodhew28 examined about 1000 intersections between misfitdislocations in 20-nm-thick In0.2Ga0.8As/GaAs (001) grown by molecularbeam epitaxy (MBE). They found that L-shaped dislocations were present at18% of the intersections and interpreted this result as indicating the presenceof some edge dislocations.

5.5 Lattice Relaxation Mechanisms

Some simple lattice relaxation mechanisms have already been consideredbriefly, in the sections on the critical layer thickness, dislocation nucleation,and dislocation multiplication. For example, Matthews and Blakeslee con-sidered the bending over of substrate dislocations in deriving the criticallayer thickness. However, as has been shown in Section 5.3.3, this mechanismalone cannot account for the measured extent of lattice relaxation in mostheteroepitaxial systems. This leads us to invoke mechanisms involving thenucleation of new dislocations, or dislocation multiplication, as consideredin the previous sections. Further, in the growth of heteroepitaxial islands(Volmer–Weber growth mode) new relaxation mechanisms are possible, suchas the injection of misfit dislocations at the island boundaries. The purposeof this section is to describe the lattice relaxation processes in more detailand the resulting defect structures that are to be expected.

5.5.1 Bending of Substrate Dislocations

Practical substrates for heteroepitaxy typically contain threading dislocationswith a density of 10 to 105 cm–2. These dislocations are replicated in theepitaxial layer and can glide to create misfit dislocations at the interface. Thisis the mechanism considered by Matthews and Blakeslee in their model forthe critical layer thickness.

This lattice relaxation mechanism is shown schematically in Figure 5.19.In Figure 5.19a, a substrate threading dislocation AO has replicated in theepitaxial layer. If the layer is sufficiently thick (h > hc), the threading segmentwill glide under the influence of the misfit stress, creating a misfit segmentOC, as shown in Figure 5.19b. The dislocation will continue to glide as shownin Figure 5.19c, increasing the total length of the misfit segment and reducingthe average strain, unless it is impeded by a pinning defect or other dislo-cation. In rare circumstances, the threading segment may glide all the wayto the wafer edge, annihilating the threading segment in the epitaxial layer,as shown in Figure 5.19d. Usually, however, the original threading segmentBC will remain in the epitaxial layer. It is important to note that if only this




FIGURE 5.19Relaxation mechanism involving the bending over of an existing substrate dislocation. (a) Thesubstrate threading dislocation AO is replicated in the epitaxial layer. (b) If the layer thicknessexceeds hc, the threading segment will glide under the influence of the misfit stress, creating amisfit segment OC. (c) Unless impeded by a pinning defect or other dislocation, the threadingsegment will continue to glide to the right, increasing the length of the misfit dislocation andrelaxing the mismatch strain in the epitaxial layer. (d) In rare circumstances, the threadingsegment may glide all the way to the wafer edge, annihilating the threading segment in theepitaxial layer. Otherwise, the threading segment BC will remain in the epitaxial layer.

O

B

A

Epitaxial layer

Substrate

O

A

B

O

B

A

O

A

(a)

(b)

(c)

(d)

C

C




mechanism is active, the epitaxial layer must have a threading dislocation densityequal to or less than that of the starting substrate.

The amount of lattice mismatch that may be relieved by this mechanismdepends on the substrate dislocation density and the average length for themisfit segments of the dislocations.

First, suppose that the substrate is square with sides of length L, parallelto the misfit dislocation lines. If the threading dislocation density in thesubstrate is D, then the total number of dislocations to be bent over is .If, in the process of lattice relaxation, misfit dislocations are produced alongthe two possible directions with equal numbers, then there will bemisfit dislocations in each direction. If these misfit dislocations run to theedge of the sample (an optimistic assumption), and their sources (the thread-ing dislocations) are uniformly distributed across the sample, their averagelength will be . The linear density of misfit dislocations in the interfacewill be

(5.53)

The amount of strain that can be relaxed by this density of misfit disloca-tions is

(5.54)

where b is the length of the Burgers vector, α is the angle between the Burgersvector and line vector, and φ is the angle between the interface and the normalto the slip plane.

Typically, impediments to dislocation glide will limit the lengths of themisfit segments to be much less than the size of the substrate. If the averagelength for a misfit segment is , then the amount of lattice mismatch strainthat can be relieved by bending over all of the threading dislocations is

(5.55)

For the (001) heteroepitaxy of a zinc blende semiconductor,

(5.56)

So, with an average misfit segment length of 100 μm and a substrate thread-ing dislocation density of 105 cm–2, only 0.0033% mismatch strain may berelieved by this mechanism.

L D2

L D2 2/

L / 2

ρ = =L DL

LD2 2

2/

/

δ ρ α φ α φ= =b LDbcos cos cos cos

Lave

δ α φ= 2DL bave cos cos

δ = × −( . )3 3 10 8 cm DLave




5.5.2 Glide of Half-Loops

In most cases, the observed extent of lattice relaxation cannot be explainedsolely on the basis of bending of substrate dislocations, even though this maybe the first mechanism to become active. Instead, it is necessary to invokemechanisms involving the nucleation of new dislocations or dislocation mul-tiplication. Even though homogeneous nucleation of dislocation half-loops isnot expected, their heterogeneous nucleation at a surface defect or concentra-tion of stress is likely to occur. Such a half-loop can expand to create a lengthof misfit dislocation, as illustrated in Figure 5.20. Suppose that a dislocationhalf-loop ABCD is nucleated at a defect or region of concentrated stress at thesurface, as shown in Figure 5.20a. This half-loop can expand by the glide ofits segments AB, BC, and CD, as shown in Figure 5.20b. By continued expan-sion, the half-loop may reach the interface, as shown in Figure 5.20c, resultingin a misfit segment BC as well as two threading segments AB and CD. Latticerelaxation can continue by the expansion of the half-loop, as shown in Figure5.20d. In rare circumstances, one of the threading segments may glide all theway to the wafer edge and annihilate. In this case, only one threading segmentwill remain in the epitaxial layer; otherwise, there will be two threadingsegments associated with each misfit segment. In contrast to the bending overof substrate dislocations, the half-loop mechanism causes an increase in the thread-ing dislocation density compared to that in the starting substrate.

If a mismatched heteroepitaxial layer is completely relaxed by the glide ofhalf-loops, and the average size of the half-loops (i.e., the average length oftheir misfit segments) is , and each misfit segment has two threadingsegments associated with it, then the threading dislocation density will be

(5.57)

where b is the length of the Burgers vector, α is the angle between the Burgersvector and line vector, and φ is the angle between the interface and the normalto the slip plane. Assuming 60° misfit dislocation segments in a (001) zincblende semiconductor, with an average half-loop width of 100 μm, the relax-ation of 1% lattice mismatch will result in a threading dislocation density ofabout 6 × 107 cm–2.

5.5.3 Injection of Edge Dislocations at Island Boundaries

Many highly mismatched heteroepitaxial layers grow in a Volmer–Weber(three-dimensional) mode. In such a case, pure edge misfit dislocations canbe injected at the boundaries of the growing islands, prior to island coales-cence, and then glide on the interfacial plane. This phenomenon has beenobserved in a number of highly mismatched zinc blende (001) heterointer-faces by high-resolution TEM. The presence of these edge dislocations cannot

Lave

Df

L bave

= | |cos cosα φ




FIGURE 5.20Relaxation mechanism involving the nucleation and glide of a half-loop. (a) A dislocation half-loop ABCD is nucleated at a defect or region of concentrated stress at the surface. (b) In responseto the mismatch stress, the loop can expand on its glide plane by the glide of the segments AB,BC, and CD. (c) The half-loop may reach the interface by continued expansion, resulting in amisfit segment BC as well as two threading segments AB and CD. (d) The half-loop can continueto expand, lengthening its misfit segment and relaxing mismatch strain in the process. (e) Inrare circumstances, one of the threading segments may glide all the way to the wafer edge andannihilate. In this case, only one threading segment will remain in the epitaxial layer; otherwise,there will be two threading segments associated with each misfit segment.

(a)

Epitaxial layer

Substrate

A

B C

D

(b)

A

B C

D

(c)

A

B C

D

(d)

A

B C

D

(e)

A

B




be explained by nucleation and glide from the sample surface, because theslip direction for these dislocations lacks a component perpendicular to theinterface. Another possible mechanism is the reaction of two 60° dislocationsat the intersection of their glide planes. This Lomer–Cottrell mechanism islikely to be active, but it cannot account for the high numbers of edgedislocations observed in some heteroepitaxial systems. Climb is anotherpossible explanation for the introduction of edge dislocations, but requireslong-range diffusion and is expected to proceed too slowly at typical growthtemperatures to explain the experimental observations.

It might be expected that layers relaxing by this mechanism would exhibitlow threading dislocation densities. If the strain is relaxed by edge disloca-tions in this manner, there is no need for the nucleation of dislocations atthe surface of the growing layer. At the same time, it is not possible for theedge misfit dislocations to glide upward toward the film surface. However,if the misfit strain is not fully relaxed at the time of island coalescence, thenfurther relaxation may proceed by the glide of dislocation half-loops fromthe surface, accompanied by the introduction of threading dislocations.Additionally, the matching of atomic bonds at the region of coalescencebetween neighboring islands may introduce geometrically necessary dislo-cations during the process of coalescence, and these can thread to the filmsurface. For these reasons, heteroepitaxial layers growing by a Volmer–Webermechanism typically have large threading dislocations.

A material system exhibiting this mechanism of lattice relaxation is GaSb/GaAs (001), which was studied by Qian et al.33 They examined the interfacialmisfit dislocations in MBE-grown structures using high-resolution TEM.They found an array of pure edge dislocations with having aspacing of 57 ± 2 Å along each direction. Within the experimentalerror, this is equal to the spacing of 55 Å, at which the edge dislocationswould completely relieve all of the mismatch strain (f = –8.2%). Figure 5.21shows a high-resolution TEM lattice image of the GaSb/GaAs (001) interfacealong the [110] direction. Each edge dislocation is associated with two extra{111} half-planes in the GaAs substrate, as marked.

In a separate experiment Qian et al.34 studied the initial stages of relaxationin GaSb/GaAs (001) grown by MBE. They found that the edge dislocationsexisted in the growing islands, prior to coalescence. The misfit dislocationsin the interior part of each island had a uniform spacing, but the spacing ofthe outermost dislocation was typically larger. The suggested interpretationof this observation was that the misfit dislocations nucleate at the leadingedges of the {111} planes of the islands and then glide inward on the (001)plane, i.e., the 90° misfit dislocations are injected at the advancing boundariesof the islands.

5.5.4 Nucleation of Shockley Partial Dislocations

Petruzzello and Leys8 found differences in the misfit dislocation structurebetween tensile and compressive interfaces in GaP/GaAsP and GaAsP/GaP

b a= ± / [ ]2 110110




interfaces, which they attributed to lattice relaxation by the nucleation ofpartial dislocations. Moreover, they showed that this mechanism results indifferences in the lattice relaxation behavior in tensile vs. compressive layers.

In this work, Petruzzello and Leys investigated the misfit dislocation struc-ture at interfaces having both signs of mismatch strain in a GaP/GaAs0.3P0.7/GaP 001 heterostructure grown by metalorganic vapor phase epitaxy(MOVPE). In this structure, the GaAs0.3P0.7 layer was 2600 Å thick and theGaP cap was 900 Å thick. The room temperature lattice mismatch strains atthese interfaces are ±1.1%, corresponding to a Matthews and Blakeslee crit-ical layer thickness of ~80 Å. (The mismatch strain is compressive for theGaAs0.3P0.7/GaP interface and tensile for the GaP/GaAs0.3P0.7 interface.)

At the tensile interface (positive mismatch strain), Petruzzello and Leysfound a square grid network of perfect and partial dislocations aligned withthe directions. These observations are consistent with the nucleationof partial dislocations in the mismatched layer with tensile strain. At thecompressive interface (negative mismatch strain), however, the network ofmisfit dislocations involved only perfect dislocations, some of which werecurved. This might indicate the involvement of a cross-slip mechanism thatcan only occur with perfect dislocations.

Petruzzello and Leys explained these differences between the compressiveand tensile layers using the model of Marée et al.35 for relaxation by thenucleation of Shockley partial dislocations. In a zinc blende heteroepitaxiallayer, a 30° partial and a 90° partial can nucleate and then react at the interfaceto produce a 60° misfit dislocation. For example, in a layer with (001) orien-

FIGURE 5.21High-resolution TEM lattice image of the GaSb/GaAs (001) interface along the [110] direction.Each edge dislocation is associated with two extra {111} half-planes in the GaAs substrate asmarked. (From Qian, W. et al., J. Electrochem. Soc., 144, 1430, 1997. Reproduced by permissionof ECS–The Electro-Chemical Society.)

GaSb

GaAs

5 nm

110




tation, if the lines of the dislocations are parallel to the direction, theBurgers vectors for the 90 and 30° partials could be a/6[112] and a/6[211],respectively. These Shockley partials can react to form a single 60° misfitdislocation by the reaction

(5.58)

In a tensile layer, the 90° partial will nucleate first, followed by the 30°partial. In the compressive layer, the partials are nucleated in the reverse order.This can be shown by consideration of the atomic arrangement on the {111}-type planes, shown schematically in Figure 5.22. The solid circles representatoms in a layer of the {111}-type plane, and the dashed circles represent atomsin the underlying layer. The Burgers vectors b1, b2, and b are for the 90° partial,30° partial, and 60° perfect dislocation, respectively. u represents the line ofthe dislocations. The direction of the resolved shear stress ττττ corresponds tothe tensile case. For the situation shown, the slip of atoms in the layer by b1

will bring them to low-energy positions over the voids in the underlying layer,but the same is not true for slip by the partial Burgers vector b2. Therefore,in the tensile case, the 90° partial will nucleate first. Following this, the 30°Shockley partial will nucleate, with a stacking fault existing between the twopartials. The 30° partial will glide toward the 90° partial, and they will even-tually react to annihilate the stacking fault and form a perfect 60° dislocation.

Following the same arguments, we expect the 30° partial to nucleate firstin the layer with compressive stress, in which the sign of ττττ is reversed.However, negligible dissociation is expected in this case because of thegreater force on the 90° partial (whose Burgers vector is parallel to ττττ) com-

FIGURE 5.22Atomic arrangement of the {111} planes of diamond and zinc blende semiconductors. TheBurgers vectors b1, b2, and b are for the 90° partial, 30° partial, and 60° perfect dislocation,respectively. u represents the line of the dislocations. The resolved shear stress on the plane isττττ, and the direction shown corresponds to the case of tensile stress. (Reprinted from Petruzzello,J. and Leys, M.R., Appl. Phys. Lett., 53, 2414, 1988. With permission. Copyright 1988, AmericanInstitute of Physics.)

uτb1

b2

b

[ ]110

a a a6

1126

2112

101[ ] [ ] [ ]+ →




pared to that on the 30° partial (whose Burgers vector is at a 60° angle to ττττ).Therefore, in the compressive case we expect that the lattice relaxation willoccur predominantly by the glide of perfect 60° dislocations. The cross-slipof such a perfect dislocation from one {111} plane to another as it glides tothe interface can result in the curved dislocation lines that Petruzzello andLeys observed at the compressive interface.

It should be noted that this mechanism of relaxation by Shockley partialdislocations does not alter the Matthews and Blakeslee critical layer thicknessfor the bending over of threading dislocations from the substrate. However,it could affect the critical layer thickness for the nucleation or multiplicationof dislocations, and therefore the observable critical layer thickness. Sincethe more stressed 90° partial is nucleated first in the tensile case, this differ-ence would cause the measurable onset of relaxation to occur in tensile layerswith a smaller thickness than in compressive layers.

5.5.5 Cracking

Another lattice relaxation mechanism is cracking, which has been observedin the case of wurtzite III-nitride semiconductors grown with tensile mis-match strain. Ito et al.36 studied the lattice relaxation of AlxGa1–xN/GaN(0001). For this heteroepitaxial system, the lattice mismatch strain is positive(tensile strain) and given by f ≈ x (3.5%) at room temperature. Ito et al. foundthat the tensile AlGaN layers exhibited cracking if the critical layer thicknesswas exceeded. They showed that this cracking resulted from the latticerelaxation mechanism, rather than the thermal strain introduced during cool-down. Cracking cannot relieve mismatch strain in compressive films, how-ever. Therefore, the lattice relaxation by cracking in the tensile layers isindicative of a fundamental difference in lattice relaxation mechanismsbetween the tensile and compressive cases.

5.6 Quantitative Models for Lattice Relaxation

Heteroepitaxial layers with moderate mismatch strain will growcoherently strained to match the lattice spacings of the substrate inthe plane of the interface, up to the critical layer thickness hc. Beyond thecritical layer thickness, it becomes energetically favorable for the introduc-tion of misfit dislocations to relieve some of the mismatch strain.

A number of models have been developed to describe the variation of theresidual strain with film thickness in partially relaxed layers, which aregreater than the critical layer thickness. Matthews and Blakeslee developedan equilibrium model that adequately describes the strain relaxation in het-eroepitaxial layers for which there exist no significant kinetic barriers to the

( %)f < 1( )||ε = f




nucleation or glide of dislocations. Experimentally, however, it has beenfound that it is possible to grow metastable layers, with residual strainsgreatly exceeding those predicted by the equilibrium model. This has moti-vated the development of kinetic models for strain relaxation in mismatchedheteroepitaxial layers.

The first such kinetic model appears to be that of Matthews, Mader, andLight, who modified the equilibrium theory with a term to account for thePeierls (lattice friction) force on moving dislocations. They made use of themodel for dislocation motion developed by Haasen. However, it has beenfound that the Matthews, Mader, and Light model cannot accurately predictboth the initial and later stages of the lattice relaxation. This is becausedislocation multiplication was not included in their model.

Dodson and Tsao38 developed a kinetic model that included a phenome-nological model for dislocation multiplication as well as an empirical modelfor dislocation glide under the influence of stress. This model has been usedto successfully fit the relaxation characteristics of a number of heteroepitaxiallayers from different material systems. Though the model involves twoadjustable parameters, it appears to provide a satisfactory description of thedislocation dynamics and lattice relaxation.

This section will outline the equilibrium and kinetic models describedabove. In each case, the starting assumptions and underlying equations willbe given, along with the resulting model equations. The practical applicationof these models and their limitations will also be summarized.

5.6.1 Matthews and Blakeslee Equilibrium Model

The Matthews and Blakeslee equilibrium model is based on force balancefor an existing threading dislocation, with the same physical basis as theMatthews and Blakeslee critical layer thickness. The resulting equilibriumstrain in a heteroepitaxial layer of thickness h, with , is given by

(5.59)

where b is the length of the Burgers Vector, ν is the Poisson ratio, α is theangle between the Burgers vector and the line vector for the dislocations,and λ is the angle between the Burgers vector and the line in the interfaceplane that is perpendicular to the intersection of the glide plane with theinterface. The term takes on a value of ±1 to account for the sign ofthe strain.

It is important to note that the equilibrium strain is inversely proportionalto the layer thickness. Therefore, heteroepitaxial layers of finite thicknesswill not relax completely even in equilibrium.

h hc>

ε ν απ ν||( ) ( cos )[ln( / ) ]

( )coseq f

fb h b

h= − +

+1 1

8 1

2

λλ

f f/




For application to the heteroepitaxy of zinc blende or diamond semicon-ductors with (001) orientation, we assume that the gliding dislocations are

of the 60° type, with Burgers vectors of the type and line vectors of

the type . The glide planes for these dislocations are {111}-type

planes. Thus, , , and . Figure 5.23 shows theequilibrium strain vs. the thickness for the heteroepitaxy of a diamond orzinc blende semiconductor.

5.6.2 Matthews, Mader, and Light Kinetic Model

The first kinetic model for lattice relaxation was developed by Matthews etal.16 As in the equilibrium model, they considered the forces acting on agrown-in threading dislocation. The glide force exerted on the dislocation,which tends to make it glide in a sense, so as to produce a length of misfitdislocation in the interface, is

(5.60)

where f is the lattice mismatch, Y is the biaxial modulus, b is the length ofthe Burgers vector, λ is the angle between the Burgers vector and the line inthe interface plane that is perpendicular to the intersection of the glide planewith the interface, and φ is the angle between the interface and the normal

FIGURE 5.23Equilibrium strain vs. thickness for a heteroepitaxial zinc blende layer with (001) orientation,calculated using the Matthews and Blakeslee model assuming cos α = cos λ = 1/2, b = 4.0 Å,and ν = 1/3.

0.000

0.002

0.004

0.006

0.008

0.010

h (nm)

Equl

ibriu

m in

-pla

ne st

rain

0 100 200 300 400 500

a2

011

1

2110

b a= / 2 cos /α = 1 2 cos /λ = 1 2

F fYbG = cos cosλ φ




to the slip plane. In the isotropic case considered by Matthews, Mader, andLight, the line tension of the misfit segment of the dislocation is given by

(5.61)

where ν is the Poisson ratio, α is the angle between the Burgers vector andthe line vector for the dislocations, and G is the shear modulus (assumed tobe the same for the epitaxial layer and the substrate).

For the gliding dislocation, there is also a Peierls force (lattice friction force)that opposes the motion. Following the work of Haasen,37 Matthews, Mader,and Light assumed that the Peierls force acting on the bowing dislocationwas given by

(5.62)

where h is the layer thickness, v is the dislocation glide velocity, is thediffusion constant, U is the activation energy for the diffusion of the dislocationcore, k is the Boltzmann constant, and T is the absolute temperature.

The linear density of misfit dislocations was considered to be constant withtime (dislocation multiplication processes were not included). Within thisassumption, the time rate of change of the lattice relaxation is

(5.63)

where D is the threading dislocation density in the substrate. Solving, in theanisotropic case, the time-dependent lattice relaxation is given by

(5.64)

where*

(5.65)

and β is the limiting (equilibrium) value of the lattice relaxation for the layer,

* The equation given here differs by a factor of four from that given by Matthews, Mader, andLight, due to the correction of Fitzgerald (Fitzgerald, E.A., Mater. Sci. Rep., 7, 87, 1991).

FGb

h bL = −−

+( cos )( )

[ln( / ) ]1

11

2ν απ ν

Fh vkT

bDU kTF =

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟cos

exp( / )φ 0

D0

ddt

vDbδ φ= cos

δ β α= − −[ ]1 e t

α ν φ λν

= + −−

Gb D D U kTkT

3 201

2 1( )cos cos exp( / )

( )




(5.66)

Figure 5.24 shows the predicted behavior for heteroepitaxial Ge on GaAs.In the figure, the ×’s represent data for a sample with . The data forthis sample fall on the dashed curve calculated for (the equilibriumcurve, marked A in the figure). The data point shown by the open circle wasmeasured for a sample with and closely matches the curve calculatedfor (labeled C in the figure). Therefore, depending on the value of ,it is possible to grow samples with equilibrium values of strain, or valuesthat greatly exceed the predictions of equilibrium theory.

5.6.3 Dodson and Tsao Kinetic Model

Dodson and Tsao38 built upon the Matthews, Mader, and Light model byincluding a dislocation multiplication term. In the Dodson and Tsaomodel, it was assumed that the glide velocity for a dislocation follows theempirical relationship

(5.67)

where is the effective stress and B is a constant. (In most materials, theexponent m is found to be between 1 and 1.2; Dodson and Tsao assumed

FIGURE 5.24Elastic strain vs. thickness for heteroepitaxial Ge/GaAs (011). The filled circles represent datafor a sample with αt >> 1, and the open circle was measured for a sample with αt ≈ 1. Thedashed curves were calculated from the kinetic model. For curve A, it was assumed that αt >>1, and for curve C, it was assumed that αt = 1. (Reprinted from Matthews, J.W. et al., J. Appl.Phys., 41, 3800, 1970. With permission. Copyright 1970, American Institute of Physics.)

0

1

2

3

4

5

6

7

0.0 0.5 1.0 1.5 2.0h (μm)

ε ×

104

A

C

B

ε = f

hc

× ××

×

β ε ν απ ν

= − = − − +−

f eq fb h b

h||( )( cos )[ln( / ) ]

(1 1

8 1

2

))cos λ

αt >> 1αt >> 1

αt ≈ 1αt = 1 αt

v B U kTeffm= −τ exp( / )

τeff




the value to be unity.) Dislocation multiplication was modeled by theassumption

(5.68)

where is the density of mobile dislocations and K is a phenomenologicalparameter. The strain relief is proportional to the linear density of misfitdislocations, so that

(5.69)

Combining Equations 5.68 and 5.69, with the assumption that all disloca-tions are mobile so that , we obtain

(5.70)

where κ is a constant. The absolute value of the effective stress in the het-eroepitaxial layer is

(5.71)

Combining the above equations, we obtain a single differential equationfor the time-dependent relaxation:

(5.72)

The Dodson and Tsao model has been applied to a number of heteroepi-taxial systems, in an attempt to better understand their lattice relaxationprocesses. This involves the adjustment of the parameters C and to pro-duce a good fit with the measured results. For example, Dodson and Tsao38

applied this model to the case of SiGe/Si (001) grown by Bean et al.,6 at atemperature of 823K, or roughly 70% of the growth temperature. They foundthat the experimentally measured strains could be fit usingand . More recently, Yarlagadda et al.39 applied the Dodson andTsao model to ZnSe1–xTex/InGaAs/InP (001), grown at 653K, or roughly 40%of the melting temperature. In that work, it was necessary to useand to reproduce the experimental results. The calculations areinsensitive to the value of . It is significant, however, that Yarlagadda et

ddt

K vm effρ ρ τ=

ρm

ddt

bddt

γ λ φ ρ= cos cos

ρ ρm =

ddt

v teffγ κ τ γ= ( )

τ νν

γ εeffG

f t eq= +−

− −2 11( )

( )[ ( ) ( )]||

d tdt

CG f t eq tγ γ ε γ( )

[ ( ) ( )] ( )||= − −2

γ 0

CG2 146= −sγ 0

53 10= × −

CG2 180= −sγ 0

910= −

γ 0




al. needed to use a value of that was roughly twice the value utilizedby Dodson and Tsao. This indicates that dislocations glide more readily inZnSe1–xTex at 40% of the growth temperature than in SiGe at 70% of thegrowth temperature. In summary, even though the Dodson and Tsao modelis not predictive, its application can be helpful in understanding the relax-ation process and comparing different materials.

5.7 Lattice Relaxation on Vicinal Substrates: Crystallographic Tilting of Heteroepitaxial Layers

Heteroepitaxial semiconductors grown on vicinal substrates generallyexhibit a crystallographic tilt with respect to the underlying substrate. Thus,for nominally (001) heteroepitaxy of a zinc blende semiconductor, the [001]axes for the deposit and substrate are not parallel if the substrate [001] axisis inclined from the normal. This effect has been observed in many materialsystems, including GaN/Al2O3 (0001),40 GaN/6H-SiC (0001),41 AlGaAs/GaAs (001),42 InGaAs/GaAs (001),43,44 InGaAs/GaP (001),51 InGaP/GaP(001),44 ZnSe/GaAs (001),45 ZnSe/Ge (001),46,47 CdTe/InSb (001),44 CdZnTe/GaAs (001),49 CdTe/ZnTe/Si (112),50 GaAs/Si (001),51–55 wurtzite ZnS/Si(111),56 diamond/Si (001),57–59 and Si3N4/Si (111).60

Typically, if the substrate inclination is about an axis of symmetry, the tiltis about the same axis as the substrate inclination. Therefore, the surfacenormal, the low-index axis of the epitaxial layer, and the low-index axis ofthe substrate are coplanar. For this situation, the tilt is either away from(positive) or toward (negative) the surface normal. For pseudomorphic lay-ers, the magnitude of the tilt increases with both the substrate inclinationand the lattice mismatch. The tilt is positive (away from the surface normal)if , but negative if . In partially relaxed layers, the sign of the tiltis usually the opposite. However, the dependence of the tilt on the substrateinclination and mismatch is rather complex and incompletely understood atthe present time.

5.7.1 Nagai Model

In the case of pseudomorphic growth, with no misfit dislocations at theinterface, the tilt can be predicted by the Nagai model,43 which can beunderstood with the aid of Figure 5.25. The vicinal substrate is assumed tocomprise terraces of uniform length L separated by steps of height h. If thesubstrate inclination is Φ, then

(5.73)

CG2

a ae s> a ae s<

Φ =⎛⎝⎜

⎞⎠⎟

−tan 1 hL




This equation applies for single, double, or other step heights, as long asthe step height is uniform across the wafer. For specificity, the steps wereassumed to have a height of for the creation of Figure 5.25. Further,the epitaxial layer was assumed to be a cubic crystal, but tetragonally dis-torted (by the applied biaxial stress) with unit cell dimensions . Thesubstrate was assumed to be cubic and unstrained, with a lattice constant .If coherency is maintained at the steps, so that the lattice constant of theepitaxial layer relaxes from to c over the length of the terrace, then theepitaxial layer will be tilted with respect to the substrate by an amountgiven by*

(5.74)

This model predicts that the direction of tilt will be away from the surfacenormal (positive tilt) in the case of (or ), but toward the surfacenormal if (or ). The magnitude of tilt is predicted to increasewith the substrate misorientation and lattice mismatch, as has been observedfor pseudomorphic layers. Although this model was developed to explaintilting in zinc blende crystals, it should apply to hexagonal semiconductorsas long as the correct biaxial relaxation constant is used to calculate the out-of-plane lattice constant.

In general, the introduction of dislocations at the interface will modify thetilt from the value predicted by the Nagai model. This will be true if thedislocations have Burgers vectors that are inclined to the interface. Here, theedge component of the Burgers vector that is normal to the interface can be

FIGURE 5.25Nagai’s model for tilting in a pseudomorphic heteroepitaxial layer deposited on a vicinalsubstrate. The vicinal substrate has uniform steps of height as/2 and separation L. For the caseshown here, c > as, resulting in tilt away from the substrate normal (positive tilt). (Reprintedfrom Ayers, J.E. et al., J. Cryst. Growth, 113, 430, 1991. With permission. Copyright 1991, Elsevier.)

* The sign conventions used here differ from those sometimes used in the literature. In Equations5.73 and 5.74, the substrate inclination is considered to always be positive; thus, the value of Φcontains no information about the direction of this inclination. Further, the tilt of the epitaxiallayer is considered positive if it adds to the substrate inclination but negative if it subtracts fromit. Using these conventions, Equation 5.74 will correctly predict the sign of ΔΦ.

L

as/2 c/2

as / 2

a a c× ×as

as

ΔΦ

ΔΦ Φ= −⎛⎝⎜

⎞⎠⎟

−tan tan1 c aa

s

s

c as> a ae s>c as< a ae s<




considered to be a tilt component. (The edge component of the Burgers vectorthat is in the plane of the interface is a misfit-relieving component.)

5.7.2 Olsen and Smith Model

Olsen and Smith44 proposed a model to explain the tilting of a heteroepitaxialzinc blende semiconductor due to the introduction of misfit dislocations withBurgers vectors inclined to the growth interface. Suppose one type of dislo-cation is involved, with a tilt component (edge component perpendicular tothe interface) and a misfit component (edge component parallel to theinterface) . Then, if the linear density of dislocations is just sufficient torelax the strain in the mismatched layer, the absolute value of the tilt willbe approximately

(5.75)

where f is the lattice mismatch. This expression is approximate because itdoes not consider the component necessary to relieve the lattice mismatchat the steps.

There are two important limitations to the Olsen and Smith model. First,misfit dislocations exist in a two-dimensional array in the interface. There-fore, it is not possible to predict the direction of the tilt. Second, the Olsenand Smith model only predicts an upper bound for the tilt. This is becausedislocations on different slip systems will have different components. Ifsome are negative while some are positive, there will be partial cancellation,which will reduce the magnitude of the tilt. A more complete model for thecrystallographic tilting of partially relaxed heteroepitaxial layers should takeinto consideration all of the active slip systems.

5.7.3 Ayers, Ghandhi, and Schowalter Model

Ayers, Ghandhi, and Schowalter61 presented one such model for (001) het-eroepitaxy of zinc blende semiconductors. Here, it was assumed that therelaxation was by 60° dislocations on {111}-type glide planes for layersgreater than the critical layer thickness. Dislocation glide was modeled usingthe kinetic relaxation model of Matthews et al.16 It was shown that the tiltingof the substrate would create an asymmetry in the resolved shear stresseson the various slip systems. Because of this, preferential glide of dislocationson certain slip systems would lead to the crystallographic tilting of a het-eroepitaxial layer on a vicinal substrate. This model is summarized below.

The eight active slip systems for (001) heteroepitaxy of zinc blende semi-conductors are summarized in Table 5.1. In the case of an exact (001) sub-strate, the {111} glide planes all meet the interface at an angle of 54.7° along

b1

b2

ΔΦ ≈ fbb

1

2

b1

b1




<110> directions. For each type of 60° dislocation, the Burgers vector containsa tilt component and a misfit component . Due to the symmetry, theeight slip systems are all identically stressed and will contribute equal num-bers of dislocations as the heteroepitaxial layer relaxes. Therefore, the tiltcomponents of their Burgers vectors will cancel and there will be zero nettilt of the epitaxial layer. In the case of a vicinal substrate, for which the [001]axis is inclined from the normal by an angle of Φ, this is no longer true.

The effect of various Burgers vectors components may be understood withthe aid of Figure 5.26. The four dislocations shown all have pure edgecharacter. The line vectors are each into the plane of the paper. ClockwiseBurgers circuits have been drawn in each case for the determination of theBurgers vector. It can be seen that the pure misfit dislocation of Figure 5.26awith its Burgers vector to the right will relieve mismatch strain in a layerwith (tensile strain), whereas that of Figure 5.26b will relieve com-pressive strain. The tilt dislocation of Figure 5.26c with its Burgers vector upintroduces clockwise tilt, but the dislocation of Figure 5.26d with its Burgersvector down causes counterclockwise tilt in the overlying crystal. (A screwcomponent will neither relieve misfit nor introduce a macroscopic tilt in theepitaxial layer.)

The 60° dislocations in a heteroepitaxial zinc blende layer contain misfit,tilt, and screw components; however, we can still use the same principlesoutlined above to understand their behavior. In the case of a layer withtensile strain, the dislocations will be introduced with misfit components tothe right. Then, with a counterclockwise substrate inclination as shown inFigure 5.27a, dislocations with Burgers vector will be more stressed than

TABLE 5.1

Eight Active Slip Systems for the (001) Heteroepitaxy of Zinc Blende Semiconductors

System Line Vector l Glide Plane Burgers Vector b

S1 (111)

S2 (111)

S3

S4

S5 [110]

S6 [110]

S7 [110]

S8 [110]

[ ]11012

101a[ ]

[ ]11012

011a[ ]

[ ]110 ( )1 1 112

101a[ ]

[ ]110 ( )1 1 1 12

011a[ ]

( )11112

101a[ ]

( )11112

011a[ ]

( )1 1 112

101a[ ]

( )1 1 1 12

011a[ ]

b1 b2

a ae s<

b2




those with Burgers vector . The preferential introduction of dislocationswill introduce positive tilt, which adds to the substrate inclination. Figure5.27b shows the situation for a counterclockwise tilt but compressive mis-match. Here, the strain must be relieved by dislocations with their misfitcomponents to the left. The preferential introduction of dislocations,which are more stressed in this case, will introduce a negative tilt, whichsubtracts from the substrate inclination.

The quantitative determination of the tilt in the heteroepitaxial layerrequires (1) the determination of the densities of dislocations on the eightslip systems and (2) the summing of their contributions to the epitaxial layertilt. This was done for two limiting cases. In the case of type I relaxation, itwas assumed that all eight slip systems would become active, most of therelaxation would occur with h >> hc, and the more stressed systems wouldcontribute more misfit dislocations. In the case of type II relaxation, it wasassumed that the relaxation would be affected only by the most stressed slipsystems for the two <110> directions. In other words, the least stressed slipsystems are excluded as a consequence of relaxation by the others. This couldbe caused by differences in critical thickness, glide, multiplication, or nucle-

FIGURE 5.26Pure misfit dislocations (a, b) and pure tilt dislocations (c, d) with edge character. In each case,the dislocation line vector is into the page, and a clockwise Burgers circuit is drawn from s tof. The Burgers vector is fs. The dislocation shown will (a) relieve strain in an epitaxial layerwith ae < as (tensile strain), (b) relieve strain in an epitaxial layer with ae > as (compressive strain),(c) introduce clockwise tilt, and (d) introduce counterclockwise tilt. (Reprinted from Ayers, J.E.et al., J. Cryst. Growth, 113, 430, 1991. With permission. Copyright 1991, Elsevier.)

(a) (c)

(b) (d)

s

f

s

f

f s

s f

b1 b2

b1




ation of the dislocations on the different slip systems. In the limiting case oftype II relaxation, the tilt would be equal to that predicted by the Olsen andSmith model.

For example, the imbalance of the dislocation populations could be theresult of differences in the critical layer thicknesses (which arise from thesubstrate inclination) for the dislocations in the different slip systems.62 Themost stressed slip systems (MSSSs), which have a lower critical layer thick-ness, will initiate relaxation by glide before the least stressed slip systems(LSSSs). After the MSSSs become active, they can continually reduce thestrain in the growing layer, thus keeping the LSSSs inactive. In such a situ-ation, the LSSSs may be completely excluded from the relaxation process.

Tsao and Dodson63 have shown that a slip system will become active(introduce misfit dislocations to relax strain) only when its excess stressbecomes positive, where

(5.76)

and where is the angle between the slip direction and that direction inthe plane of the interface that is perpendicular to the intersection of the glideplane and the interface, G is the shear modulus, is the average strain inthe epitaxial layer, is the Poisson ratio, α is the angle describing thedislocation character (60°), h is the layer thickness, and b is the length of theBurgers vector. In the case of a vicinal substrate, the different slip systemswill have different values of due to the different values of λ. Type IIrelaxation is affected entirely by the MSSSs in one of two scenarios. In thefirst, the relaxation takes place near equilibrium, so that the MSSSs main-tain . Then is negative for the LSSSs, and they will not participate

FIGURE 5.27Burgers vectors of 60° dislocations in a heteroepitaxial zinc blende semiconductor on a vicinal(001) substrate. (a) Tensile mismatch, with a counterclockwise substrate inclination. (b) Com-pressive mismatch, with a counterclockwise substrate inclination. (Reprinted from Ayers, J.E.et al., J. Cryst. Growth, 113, 430, 1991. With permission. Copyright 1991, Elsevier.)

(a)

Growth plane

b1

b2

45° + φ

45° – φ

(b)

45° + φ 45° – φ

Growth plane

b1

b2

σ exc

σ λ ε νν π

ν ανexc

G G= +−

− −−

⎡

⎣⎢

⎤

⎦⎥4 1

1 21

1

2cos ( ) cos ln(( / )/

4h bh b

⎡

⎣⎢

⎤

⎦⎥

λ

εν

σ exc

σ exc = 0 σ exc




in relaxation. In the second (more likely) situation, the relaxation is limitedby kinetics and does not take place near equilibrium. The LSSSs will expe-rience less negative, or perhaps even positive, values of . Nonetheless,all slip systems are subject to essentially the same kinetic limitations. Thismeans the MSSSs can still maintain values of the excess stress that are muchgreater than those for the LSSSs, and the MSSSs can relieve the strain whileessentially excluding the LSSSs.

For the case of type I relaxation, the dislocation populations were estimatedusing the Matthews, Mader, and Light kinetic model for lattice relaxationdescribed in Section 5.3.2. In the type I case, all slip systems are active, andso the calculation of the tilt requires the determination of their relativecontributions. In contrast to the type II case, it is necessary to assume alimiting mechanism for lattice relaxation by the individual slip systems. Forthis model, it is assumed that lattice relaxation is limited by the glide ofdislocations. Dislocations on slip systems S1 through S2 relieve strain inthe direction, while S5 through S8 are associated with strain relief inthe [110] direction. If is the strain relaxation by dislocations on the ith slipsystem, then the strains in the two <110> directions are

(5.77)

and

(5.78)

The time rate of change of the lattice relaxation by the ith slip system isgiven by

(5.79)

where G is the shear modulus, b is the length of the Burgers vector, is thePoisson ratio, is the strain in the appropriate <110> direction, is thelinear density of misfit dislocations, is the misfit component of the Burgersvector, is the angle between the slip direction and that direction in theplane of the film that is perpendicular to the intersection of the glide planeand the plane of the film, is the angle between the film surface and thenormal to the slip plane, and U is the activation energy for dislocation glide.

The geometric factors for the eight slip systems can be found as follows.If the vicinal (001) substrate is inclined degrees toward the [100] and βdegrees toward the [010], then the substrate unit normal is

(5.80)

σ exc

[ ]110δ i

ε δ δ δ δ[ ] ( )110 1 2 3 4= − + + + +f

ε δ δ δ δ[ ] ( )110 5 6 7 8= − + + + +f

ddt

Gb bkT

i i i i iδ ν ερ λν

≈ +−

2 11

22

2( ) cos cos( )exp(

ΨUU kT/ )

νε ρi

bi2

λ i

Ψ i

α

ˆ [sin , sin ,( sin sin ) ]/n = − −α β α β1 2 2 1 2




If is the unit vector in the direction of the Burgers vector and is theunit normal to the glide plane, then

(5.81)

If the substrate inclination is small, then the direction cosines for the eightslip systems are given approximately by

(5.82)

Similarly,

so that

bi gi

cosˆ [ ˆ ( ˆ ˆ )]ˆ [ ˆ ( ˆ ˆ )]

λii i

i i

b n n g

b n n g= ⋅ × ×

⋅ × ×

cos λ βα β1

12

≈ +− −

cos λ αα β2

12

≈ +− −

cos λ βα β3

12

≈ −+ +

cos λ αα β4

12

≈ −+ +

cos λ βα β5

12

≈ ++ −

cos λ αα β6

12

≈ −+ −

cos λ βα β7

12

≈ −− +

cos λ αα β8

12

≈ +− +

Ψi ig n= ⋅−sin { ˆ ˆ }1

Ψ Ψ1 21

2 2 1 21

3= = + + − −⎧−sin

[sin sin ( sin sin ) ]/α β α β⎨⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

Ψ Ψ3 41

2 2 1 21

3= = − − + − −−sin

[ sin sin ( sin sin ) ]/α β α β⎧⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

Ψ Ψ5 61

2 2 1 21

3= = − + + − −−sin

[ sin sin ( sin sin ) ]/α β α β⎧⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪




(5.83)

The magnitude and direction of the tilt can be calculated if the misfit strainrelaxed by each slip system (the ) is known. A slip system with line vector

will produce tilt about the axis but relieve strain in the direction.Also, a slip system with line vector will produce tilt about the axis butrelieve strain in the direction. The strain relaxed by each slip system wasestimated as follows.

The values of are assumed to be approximately equal, so the relaxationrates for two different slip systems will be in the ratio

(5.84)

Then, if nearly complete relaxation has occurred, the lattice relaxation by theith slip system can be found from

, for i = 1, 2, 3, or 4 (5.85)

, for i = 5, 6, 7, or 8 (5.86)

The resulting tilt can be calculated by combining the contributions of theeight slip systems as follows. If and are the tilts about the and[110] axes, respectively, then in the case of complete relaxation,

(5.87)

and

(5.88)

Ψ Ψ7 81

2 2 1 21

3= = − + − −⎧−sin

[sin sin ( sin sin ) ]/α β α β⎨⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

δ i

l1 l1 l2

l2 l2

l1

ρi

d dtd dt

i

j

i i

j j

δδ

λλ

//

cos coscos cos

≈2

2

ΨΨ

δ λ

λi

i i

j j

j

f≈

=∑

cos cos

cos cos,

2

2

1 4

Ψ

Ψ

δ λ

λi

i i

j j

j

f≈

=∑

cos cos

cos cos,

2

2

5 8

Ψ

Ψ

γ η [ ]110

γ α β δ= + −⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭

−

=∑tan tan[ ( )]

,

1 1

21 4

2fb

bi i

ii ⎪⎪

η α β δ= − −⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭

−

=∑tan tan[ ( )]

,

1 1

25 8

2fb

bi i

ii ⎪⎪




The intrinsic tilt due to the surface steps has been included here, and itwill cancel the dislocation contributions at least in part. Finally, the overalltilt can be found from

(5.89)

As a result, the tilt is predicted to be (approximately) proportional to thesubstrate inclination and the lattice mismatch, as has been observed. Thepredictions of this model for type I and type II relaxation are shown in Figure5.28, for the case of 2° substrate inclination, along with experimental datafrom several epitaxial systems.

This model successfully predicts the direction of the tilt for tensile andcompressive layers, both pseudomorphic and relaxed. It also intro-duced a framework for the quantitative prediction of the absolute tilt. As aresult, the tilt was predicted for the two limiting cases of type I relaxation(all eight slip systems active) and type II relaxation (only the most stressedsystems active). The key limitation of the model is that it did not accountfor either dislocation nucleation or multiplication in the type I case, althoughboth are known to be important in determining the dynamics of lattice

FIGURE 5.28Tilt ΔΦ vs. the absolute value of the lattice mismatch f for (001) heteroepitaxy of zinc blendesemiconductors. The two lines were calculated for the type I and type II limiting cases, asindicated, for a substrate inclination of 2°. The experimental data shown by open circles werefor substrates having a 2° inclination. For the filled circles, the substrate inclination was asindicated. (Reprinted from Ayers, J.E. et al., J. Cryst. Growth, 113, 430, 1991. With permission.Copyright 1991, Elsevier.)

0.000 0.2 0.4 0.6 0.8 1

0.25

0.50

0.75

|f| (%)

Δφ (d

egre

es)

Type II

Type I

φ = 4°

φ = 0.5°

φ = 2°

ΔΦ = + + −cos[( tan tan ) ]/1 2 2 1 2γ η

( )δi = 0




relaxation. However, the effect of these phenomena, acting either alone orin conjunction, will be to exaggerate the imbalance in lattice relaxationamong the eight slip systems. It will therefore produce a tilt that is greaterthan the type I case and closer to the type II limit. And, as pointed outpreviously, the type II case is mechanism independent, as long as the moststressed slip systems relax the strain at the exclusion of the others, and nonew slip systems become active. Therefore, the type I and type II cases canstill be considered to give the minimum and maximum tilt that should beexpected for (001) heteroepitaxy of zinc blende semiconductors. Nonetheless,a far more detailed model is needed to make accurate quantitative predic-tions of the tilt or even to use the observed tilts to extract informationregarding the dislocation dynamics.

5.7.4 Riesz Model

Riesz64 extended the model of Ayers, Ghandhi, and Schowalter to includedislocation multiplication, using the approach of Dodson and Tsao,38 for the(001) heteroepitaxy of zinc blende semiconductors. Here, it was assumed thattwo types of slip systems are active: the most stressed slip systems (MSSSs),called set A, and the least stressed slip systems (LSSSs), called set B.* Foreach set, the dislocation multiplication was assumed to be described by

(5.90)

where is the excess stress, G is the shear modulus, is the linear densityof misfit dislocations, and C is a thermally activated factor given by

(5.91)

where U is the activation energy for dislocation glide. The dislocation mul-tiplication processes for the A and B dislocations are assumed to be inde-pendent, because dislocation multiplication sources usually emitdislocations with the same Burgers vector.

The strain relaxation in either <110> direction is due to the combined effectof dislocations from sets A and B; hence,

(5.92)

* Riesz considered the case of substrate inclination toward a <100> direction α = 0 (or β = 0), forwhich there are only two distinct values of λ among the eight slip systems. In the general case(both α and β nonzero), there will be four distinct values λ among the eight slip systems. Thiswill complicate the model considerably, but is not expected to change the qualitative results.

∂∂

=⎛⎝⎜

⎞⎠⎟

+ρ σ ρ ρt

CG

texc

2

0[ ( ) ]

σ exc ρ

C C U kT= −0 exp( / )

δ ρ λ ψ ρ λ ψ= +b A A A B B B( sin sin sin sin )




The resulting tilt of the epitaxial layer may be calculated from

(5.93)

where is the inclination of the vicinal substrate. Here, the first term is theNagai contribution due to steps at the interface and the second term is dueto the asymmetric lattice relaxation by the dislocations from sets A and B.

Using this model, the epitaxial layer tilt was calculated as a functionof the growth temperature, with the initial dislocation density as a param-eter. For these calculations, it was assumed that and U =1.7 eV. The results of these calculations are shown in Figure 5.29 for substratemiscut angles of 0.2, 2, and 4°. Also shown in the figure are the type I andtype II limits predicted by the Ayers, Schowalter, and Ghandhi model.

The tilt behavior is predicted to be intermediate between the type I andtype II limits, as expected. In all cases, the predicted tilt increases with thegrowth temperature. For substrate inclinations of 2 or 4°, type II behavior isapproached for high growth temperatures. This shows that if the MSSSshave sufficiently fast slide and multiplication processes, they may largelyexclude the other slip systems.

Type I behavior was approximated over much of the temperature rangeif the substrate inclination was small. This is to be expected; in the limit ofzero substrate inclination, the lattice relaxation is symmetric, so all slipsystems will participate. With larger substrate inclinations, however, type Ibehavior was predicted only if the initial dislocation densities were excessive.

The tendency toward type I or type II behavior therefore appears to becontrolled by the starting dislocation density, as well as the substrate incli-nation and the growth temperature. This is further illustrated in Figure 5.30.Here, the epitaxial layer tilt is plotted as a function of the substrate inclina-tion, with the initial defect density as a parameter. The behavior is approx-imately type I over the range of miscut angles only if the initial dislocationdensity is very high. Otherwise, there is a tendency toward type II behavioras the substrate inclination, and therefore the asymmetry between the slipsystems, is increased.

The characteristics of Figure 5.29 and Figure 5.30 were calculated with theassumption of complete lattice relaxation. In partially relaxed layers, the tiltvaries monotonically as the extent of lattice relaxation increases. This isshown in Figure 5.31, for the case of a substrate inclination ,with . In the pseudomorphic case (δ = 0), a small positivetilt is observed due to the interfacial steps (Nagai contribution). For thepartially relaxed layers, the tilt varies monotonically with the extent of therelaxation.

In summary, the tilts observed in heteroepitaxial layers are caused byasymmetric relaxation by the different slip systems. The asymmetric relax-ation arises from differences in both dislocation glide and multiplication

ΔΦ

ΔΦ Φ= +−

⎛⎝⎜

⎞⎠⎟

−−

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ −f b

11

21

νν

δ νν

λsin coos ( )ψ ρ ρA B−

Φ

ΔΦρ0

C011 15 10= × −s

Φ = °4ρ ρ0 0

3 110A B= = −m



Mism

atched Heteroepitaxial G

rowth and Strain R

elaxation217

FIGURE 5.29Predicted tilt ΔΦ vs. growth temperature, with the initial dislocation density ρ0 as a parameter. It was assumed that C0 = 5 × 1011 s–1 and EA = 1.7 eV. Thesubstrate inclination was assumed to be 0.2, 2, and 4° for the first, second, and third graphs, respectively. Also shown are the type I and type II limitspredicted by the Ayers, Schowalter, and Ghandhi model. (Reprinted from Riesz, F., J. Appl. Phys., 79, 4111, 1996. With permission. Copyright 1996, AmericanInstitute of Physics.)

Growth temperature, °C Growth temperature, °C Growth temperature, °C300 400 500 300 400 500600 600700 700800 800 300 400 500 600 700 800

Parameter:P0A = P0B

P0A = P0B

P0A = P0B

Tilt

angl

e, de

gree

s0.0

–0.1

–0.2

–0.3

–0.4

β = 0.2°

β = 2°

β = 4°

10–410–310–210–1 100 101 102 103 104 10–410–310–210–1 100 101 102 103 104 10–410–310–210–1 100 101 102 103 104–0.4

–0.3

–0.2

–0.1

0.0

C, 1/s C, 1/s C, 1/s

Parameter:

Parameter:

Type-II tiltType-II tilt

Type-II tilt

Type-I tiltType-I tilt

105/m

105/m105/m 106/m106/m

106/m 107/m107/m

104/m

104/m104/m

103/m

103/m103/m

102/m

102/m

102/m

10/m

10/m

10/m

1/m

1/m

1/m

7195_book.fm Page 217 T

hursday, Decem

ber 21, 2006 8:59 AM



with an inclined substrate. The overall behavior is intermediate between thetype I and type II limits. (Type I relaxation involves relaxation by all of theslip systems, with asymmetries introduced by weak geometric factors. TypeII relaxation involves relaxation by only the most stressed slip systems, atthe exclusion of the others.) Type I behavior is favored only for very highinitial dislocation densities or small substrate inclinations. Type II behavioris approached as the substrate inclination or temperature is increased.

5.7.5 Vicinal Epitaxy of III-Nitride Semiconductors

The III-nitrides such as AlN and GaN have been grown on vicinal SiC orsapphire substrates. It has been found that vicinal surface epitaxy (VSE)results in heteroepitaxial layers of improved crystal quality with either typeof substrate.65–69

In significantly relaxed (nonpseudomorphic) nitride layers grown on vic-inal surfaces, the tilts are as predicted by the Nagai model. This has beenfound to be the case for AlN/6H-SiC (0001) and also for GaN/Al2O3 (0001)with small offcut angles. The same is true for heteroepitaxial GaN on AlN,when the AlN is a buffer layer grown on vicinal SiC (0001).

FIGURE 5.30Predicted tilt ΔΦ as a function of the substrate inclination, with the initial dislocation densityas a parameter. It was assumed that C0 = 5 × 1011 s–1, Ea = 1.7 eV, and T = 300°C. (Reprintedfrom Riesz, F., J. Appl. Phys., 79, 4111, 1996. With permission. Copyright 1996, American Instituteof Physics.)

Tilt

angl

e, de

gree

s

–0.4

–0.3

–0.2

–0.1

0.0

Substrate miscut angle, degrees0 2 4 6 8

Type-II tilt

Type-I tilt

Parameter:P0A = P0B

Tgrowth = 300°C

107/m

106/m

105/m

104/m103/m

1/m

10/m 102/m




For diamond and zinc blende semiconductors, the Nagai model onlyapplies to the pseudomorphic case. This is because the 60° misfit dislocationson slip systems have Burgers vectors with tilt components,which in general do not cancel. However, for the (0001) heteroepitaxy of III-nitrides, the misfit dislocations have in-plane Burgers vectors. Therefore,they do not affect the crystallographic tilting of the heteroepitaxial layer, andNagai’s model should apply to pseudomorphic, partially relaxed, or fullyrelaxed layers.

Huang et al.41 studied the crystallographic tilting in MOVPE-grown GaN/AlN/6H-SiC (0001) heterostructures using TEM and x-ray diffraction. Theycompared layers grown on two types of substrate: exact (0001) and

. For the case of the vicinal substrate, they found that theAlN was tilted with respect to the substrate by 142 arc sec, consistent withthe Nagai model and the measured change in the out-of-plane lattice con-stant, . Also, the tilting of the GaN overlayer with respect tothe AlN buffer was –370 arc sec, also consistent with the Nagai model andthe measured change in the out-of-plane lattice constant for that interface,which was .

The agreement between the Nagai model and the experimental tilts indi-cates that the misfit dislocations at both the GaN/AlN (0001) and AlN/6H-

FIGURE 5.31Predicted tilt ΔΦ as a function of the extent of lattice relaxation. On the abscissa, zero representsthe pseudomorphic case and 1.0 represents complete lattice relaxation. It was assumed that C0

= 5 × 1011 s–1, EA = 1.7 eV, and . (Reprinted from Riesz, F., J. Appl. Phys., 79,4111, 1996. With permission. Copyright 1996, American Institute of Physics.)

Percentage relaxation0.0 0.2 0.4 0.6 0.8 1.0

Parameter:growth temperature

Tilt

angl

e, de

gree

s

–0.3

–0.2

–0.1

0.0

300 °C

400 °C500 °C

600 °C

700 °C

900 °C

800 °C

β = 4°P0A = P0B = 103/m

ρ ρ0 03 110A B= = −m

a/ { }2 011 111

( ) . [ ]0001 3 5 1120° →

Δc c/ . %= −1 05

Δc c/ . %= 3 94




SiC (0001) interfaces have in-plane Burgers vectors. This was confirmed forthe AlN/SiC interface using cross-sectional HRTEM images. The pro-jection HRTEM image was filtered using a masked fast Fourier transform.The -filtered image revealed misfit dislocations with extra half-planesin the SiC substrate. However, the 0004-filtered image showed no verticaldisplacement around the misfit dislocations, indicating that their Burgersvectors were indeed within the plane of the interface.

GaN on sapphire (0001) shows a similar behavior for small values ofsubstrate inclination. Huang et al.40 studied the crystallographic tilting inMOVPE-grown GaN/Al2O3 (0001). The substrate inclination and the epitax-ial layer tilt with respect to the substrate were determined using back-reflection synchrotron Laue x-ray diffraction patterns. For the case of a

substrate, the direction and magnitude of the tilt werein agreement with the Nagai model. This result is taken to mean that themisfit dislocations have zero tilt components (they are in the plane of theinterface). For larger substrate inclinations, the tilting behavior was quitedifferent. However, this is believed to be affected by step heights greaterthan two atomic layers on these substrates. (This is described in greater detailin Section 5.7.6.)

The tilting behavior of the hexagonal III-nitrides on 6H-SiC and sapphiresubstrates indicates that the misfit dislocations (MDs) have zero tilt compo-nents. In other words, these MDs have in-plane Burgers vectors. Unless theMDs come about by the reaction of dislocations having out-of-plane Burgersvectors, they must be introduced by basal plane slip (slip on the (0001) plane).The former possibility can probably be ruled out; otherwise, some unreacteddislocations with out-of-plane Burgers vectors should have been observed.If basal plane slip is the dominant means for the introduction of MDs in III-V nitrides on c-face substrates, this is in sharp contrast with the diamondand zinc blende semiconductors for which MDs glide to the interface on{111}-type planes.

5.7.6 Vicinal Heteroepitaxy with a Change in Stacking Sequence

An interesting feature of AlN/6H-SiC (0001) heteroepitaxy is that these twocrystals have different stacking sequences in the growth direction. In the[0001] direction, the stacking sequence for the 6H-SiC substrate isABCACBA, but for the wurtzite AlN it is ABA. (The wurtzite structure canbe considered to have a 2H stacking sequence.) If a vicinal substrate is used,defects must be introduced at the interfacial steps to accommodate thechange in stacking sequence.

Huang et al.41 studied the misfit dislocation structure in MOVPE-grownGaN/AlN/6H-SiC (0001) heterostructures using TEM. The vicinal 6H-SiCsubstrates were (0001) and . From the analysis of TEMresults, they concluded that most of the misfit dislocations were 60° Shockleypartial dislocations, which they called “geometrical partial misfit disloca-

[ ]1100

1120

( ) . [ ]0001 1 85 1120° →

( ) . [ ]0001 3 5 1120° →




tions (GPMDs).” Because the GPMDs were generally on different terraces,they appeared to be unpaired partial dislocations, rather than the pairedpartials that would be expected if they arose from the dissociation of perfectdislocations on a basal glide plane.

Huang et al. showed that the high density of unpaired partial dislocationsin their samples could serve to accommodate the difference in stackingsequence between the AlN epitaxial layer and its 6H-SiC substrate. Thestructural model they proposed is shown in Figure 5.32. Figure 5.32a showsa side view of the interface with the steps on the (0001) surface. (The planeof the paper is the face.) Above the terraces, the AlN layer may takeon the ABA or ACA stacking sequence, either of which results in the wurtzitestructure. A transition between these two stacking sequences at a step canbe accommodated by the introduction of a 60° Shockley partial dislocation,as shown in Figure 5.32b. This transition is gradual and preserves the hex-agonal structure of the AlN, albeit with some distortion. Moreover, thegradual change in the stacking sequence, as shown in Figure 5.32c, does notcreate vertical boundaries in the epitaxial layer.

These results for AlN/6H-SiC (0001) have interesting implications for vic-inal substrate epitaxy (VSE) of III-nitrides. If the geometric partial disloca-tions are introduced to accommodate the difference in stacking sequencesbetween the epitaxial layer and the substrate, then their introduction iscontrolled in part by the substrate inclination. This offcut angle might beused to affect the lattice relaxation, the introduction of dislocations, and thethreading dislocation density in the heteroepitaxial material.

5.7.7 Vicinal Heteroepitaxy with Multilayer Steps

Up to now, we have only considered vicinal heteroepitaxy with monatomicor bilayer steps between the substrate terraces. However, step bunching canoccur on some substrates with large miscut angles. For example, vicinal Al2O3

(0001) (c-face sapphire) substrates annealed at temperatures above 1200°Cexhibit steps of n-bilayer height, with 1 < n < 6.70

Huang et al.40 investigated the tilting of GaN/Al2O3 (0001). They foundthat the tilt was in agreement with the Nagai model only for small substrateinclinations. For offcuts of 6.29 or 10.6° toward the , the measured tiltswere very different from those predicted by the Nagai model, and for thecase of 10.6° inclination, the direction of the tilt was opposite that predicted.

Huang et al. explained their measured results using the schematics ofFigure 5.33. Assuming that the misfit dislocations have in-plane Burgersvectors, the tilt of the epitaxial layer should be the same as predicted by theNagai model for n = 1 or n = 2, as shown in Figure 5.33a and b, respectively:

(5.94)

( )1100

[ ]1100

ΔΦ Δ Φ=⎛⎝⎜

⎞⎠⎟

−tan tan1 cc




For a three-bilayer step as shown in Figure 5.33c, however, the local tiltwill have a magnitude given by . Becausethis tilt can take on either sign with equal probability, the average tilt isexpected to be zero. For the four-bilayer step of Figure 5.33d, the tilt would be

(5.95)

FIGURE 5.32A possible arrangement for the accommodation of the stacking sequence difference at the AlN/6H-SiC (0001) interface by geometric partial misfit dislocations (GPMDs). (a) Partial dislocationsare introduced at the steps. (b) The gradual transition preserves the hexagonal structure of thecells, with some distortion. (c) The gradual transition does not result in vertical boundarieswithin the epitaxial AlN. (Reprinted from Huang, X.R. et al., Phys. Rev. Lett., 95, 86101, 2005.With permission. Copyright 2005, American Physical Society.)

(c)

(b)

(a)

C

B

b

AB

Dislocation line

60°(0001)

B

B

B B B

B BBB

B B

B

B B B B B B B

BB

B

B

B

B

B

B

B

B

B B B B B B B B B AA

A

A

A

A

A

A

A

A

A

AAA

A

A

AA

A A

AA

A

A

AAA

AAAA

A A A A

A

A

A

A

A

A

AAAAAAA C

CCCCCCCCC

C C C C C C C C

C

CC

CC

CCC

C

C

C

6H-SiC(11–20)

6H-SiC(1–1–20)

~~~

~~~~~~

~~~~~~~~~

C

C

C C C C C CA

A

B B B B B B

B

BA A A A A

A

A

2H~

ΔΦ Φ= −−tan {[( )/ ]tan }1 3 2 3c c ce

ΔΦ Φ= − −⎡

⎣⎢

⎤

⎦⎥

−tan( )

tan1 4 34

c cc

e




This tilt will have a sign opposite to that predicted by the Nagai model forfour bilayer steps. For the five-bilayer step,

(5.96)

and for the six-bilayer step,

(5.97)

FIGURE 5.33Schematic diagrams of GaN grown heteroepitaxially on vicinal sapphire (0001) having variousstep heights of nc, where c is the lattice constant of the substrate: (a) n = 1 (step height = c); (b)n = 2; (c) n = 3; (d) n = 4; (e) n = 5; (f) n = 6. c is the substrate lattice constant and ce is theepitaxial layer lattice constant. ns is the substrate surface normal and nc is the offcut direction.(Reprinted from Huang, X.R. et al., Appl. Phys. Lett., 86, 211916, 2005. With permission. Copyright2005, American Institute of Physics.)

(a) (d)

(b) (e)

(c) (f)

3c

6c 5ce

5c 4ce

2.5ce

2ce

2ce2c

3ce

(0001) (0001)

GaN

α

c ce

α

Al2O3

nc ns ns

4c l

l2

ϕ

ϕ

ϕ

ΔΦ Φ= − −⎡

⎣⎢

⎤

⎦⎥

−tan( )

tan1 5 45

c cc

e

ΔΦ Φ= − −⎡

⎣⎢

⎤

⎦⎥

−tan( )

tan1 6 56

c cc

e




In this case, the approximate matching between and results in neg-ligible tilt.

Thus, the Nagai model may be used for pseudomorphic layers or relaxedlayers for which the misfit dislocations have in-plane Burgers vectors. Butif the steps on the vicinal substrate are higher than two bilayers, it is neces-sary to consider the matching between different integral multiples of thesubstrate and epitaxial layer lattice constants.

5.7.8 Tilting in Graded Layers: LeGoues, Mooney, and Chu Model

In the heteroepitaxial system SiGe/Si (001), the observed tilts can be muchgreater in graded layers than in single heterostructures for the same finalcomposition. The measured tilts fall within the limits predicted for type Iand type II relaxation in both cases. However, the graded layers exhibit tiltsmuch closer to the type II limit. This has been attributed to an anomalousstrain relaxation mechanism,71 which is unique to graded layers having highpurity and low densities of surface defects.

With type II relaxation, the misfit dislocations along each [110] directionare introduced only by the most stressed slip systems (MSSSs). Therefore,in the graded layers exhibiting large tilts, there is a lattice relaxation mech-anism that essentially excludes all but these MSSSs.

LeGoues, Mooney, and Chu72 developed a model for the epitaxial layer tiltin graded layers exhibiting this anomalous strain relaxation mechanism,which they called a modified Frank–Read (MFR) mechanism.71 The under-lying assumptions relating the tilt to the lattice relaxation by the eight slipsystems are the same as in the Ayers, Schowalter, and Ghandhi model;however, the individual values of are assumed to be limited by dislocationnucleation rather than glide.

In their model, LeGoues, Mooney, and Chu grouped the eight active 60°slip systems for (001) heteroepitaxy in pairs, each of which is an MFR system.By this lattice relaxation mechanism, corner dislocations are associated withthe simultaneous glide of two orthogonal dislocation segments on different{111} planes. Hence, the MFR1 system involves two dislocation segments,one from the 60° glide system S3 and another from the glide system S5.Similarly, MFR2 involves corner dislocations made of segments from S1 andS7. The MFR systems as defined by LeGoues, Mooney, and Chu are relatedto the slip systems tabulated by Ayers, Ghandhi, and Schowalter in Table 5.2.

By the MFR mechanism, the nucleation of a new dislocation produces onesegment along each of the two orthogonal <110> directions. If the orthogonalsegments always remain equal in length, then the MFR mechanism will resultin equivalent strain relaxation in the two <110> directions.72 In developingtheir model, LeGoues, Mooney, and Chu assumed this to be true, and thatthe miscut of the substrate introduced a change in the activation energy Δfor the nucleation of dislocations on the most stressed slip system.

6c 5ce

δ i




As a specific example, consider a (001) substrate inclined toward the [100].(The axis of rotation associated with the substrate miscut is [010].) Thisshould result in an epitaxial layer tilt about the [010] axis, requiring animbalance between the MFR1 and MFR2 systems, but not between the MFR3and MFR4 systems. Therefore, the numbers of dislocations in the four MFRsystems are assumed to be such that

(5.98)

where Δ is the change in nucleation energy arising from the miscut. The totalnumber of dislocations is the sum

(5.99)

The imbalance in lattice relaxation by MFR1 and MFR2 results in the tiltso that

(5.100)

where is the tilt component of the Burgers vector for MFR1 and MFR2.Finally, the ratio of the total dislocation density to the number producing tiltis expected to be

(5.101)

TABLE 5.2

Relationship between the Slip Systems Used by LeGoues, Mooney, and Chu and Those Defined by Ayers, Ghandhi, and Schowalter

MFR System(LeGoues, Mooney, and Chu)

Slip Systems(Ayers, Ghandhi, and Schowalter)

MFR1 S3, S5

MFR2 S1, S7

MFR3 S4, S8

MFR4 S2, S6

N N3 4=

N N kT1 3= −exp( / )Δ

N N kT2 3= exp( / )Δ

N N N N NT = + + +1 2 3 4

ΔΦ = = −− −tan [ ] tan [ ( )]1 11 2b N b N Ntilt tilt tilt

btilt

NN

kTkT

T

tilt

= + −−

1 cosh( / )sinh( / )

ΔΔ




Here, due to the exponential dependence, any appreciable change in thenucleation energy (e.g., ) will tend to drive the above ratio to 1,which is the type II limit.

This type of behavior is expected for any graded layer exhibiting themodified Frank–Read (MFR) mechanism of lattice relaxation. The MFRmechanism is believed to be active in graded SiGe/Si (001) and also gradedInGaAs/GaAs (001), when the layers of high purity and the substrate sur-faces are relatively free from defects. In both material systems, dislocationloops have been observed to propagate deep into the substrate, and thesesubstrate dislocations have been identified as a signature of the MFR mech-anism. It is possible that the MFR mechanism is active in other heteroepi-taxial material systems involving diamond or zinc blende semiconductors.However, this mechanism can only operate with low dislocation densitiesand does not appear to be active in abrupt heterostructures.

The dependence of the nucleation energy on the substrate inclination ispoorly understood at the present time, and this hinders the theoretical esti-mation of Δ. Therefore, it is not possible to know if the lattice relaxation, andtherefore the crystallographic tilting, will be dominated by glide or nucle-ation of dislocations a priori. On the other hand, if it is assumed that the tiltis governed by nucleation, then the measured tilt ΔΦ and dislocation den-sity can be used to estimate the change in activation energy Δ using theabove equations. Such calculations have been made for graded GeSi grownon Si (001).73

In summary, it is now well established that tilting of heteroepitaxial layersis affected by substrate surface steps in strained heteroepitaxial layers.43 Inrelaxed (or partly relaxed) heteroepitaxial layers, both the steps at the inter-face and the misfit dislocations44 may contribute to the crystallographictilting of the heteroepitaxial layer. It is generally accepted that net tilt resultsfrom an imbalance in the dislocation populations on the various slip sys-tems.61 The underlying cause for this imbalance is not entirely clear, butmay relate to imbalances in the glide, multiplication, or nucleation of misfitdislocations. It is possible, in fact, that all three phenomena contribute tothe dislocation imbalance (and hence the tilt) under certain conditions,depending on the material system and the growth conditions. It is likelythat glide and multiplication of dislocations dominate the relaxation processand the tilt in most heteroepitaxial systems. However, nucleation may bethe governing phenomenon in some compositionally graded systems thatrelax by a modified Frank–Read mechanism, such as graded layers of SiGe/Si (001). Further work, both theoretical and experimental, is needed toclarify this behavior.

Most of the work, both theoretical and experimental, has been directed atdiamond and zinc blende semiconductors. However, it has been shown inrecent work that the crystallographic tilts in relaxed AlN/6H-SiC (0001) canbe predicted by the Nagai model for pseudomorphic layers.41 This showsthat the misfit dislocations in this material system do not contribute to thetilt. GaN on sapphire (001) behaves similarly for small values of the substrate

Δ ≈ −3 kT

NT




inclination. Preliminary results show that, with larger offcut angles, thepresence of larger steps alters the tilting in this heteroepitaxial system. Moreexperimental results are needed to characterize the tilting behavior of wurtz-ite semiconductors under a variety of conditions with various substrates.This will provide a better understanding of the mechanisms involved in thetilting of the materials, and therefore their relaxation mechanisms.

5.8 Lattice Relaxation in Graded Layers

Graded buffer layers are of commercial importance for the production oflight-emitting diodes (i.e., GaAsP light-emitting diodes (LEDs) on GaAssubstrates) and high-electron-mobility transistors (i.e., InGaAs high-elec-tron-mobility transistors (HEMTs) on GaAs substrates). In a graded buffer,the composition (and therefore the relaxed lattice constant) is varied contin-uously throughout the growth process. The discussion here will be limitedto linearly graded layers, in which the relaxed lattice constant varies linearlywith distance from the interface.

Grading in a mismatched heteroepitaxial layer will change the dislocationdynamics and relaxation process compared to the case of a single abruptheteroepitaxial layer. Both the critical layer thickness and the final threadingdislocation density become functions of the grading constant. Also, the misfitdislocation segments become distributed throughout the thickness of thegraded layer, instead of all being concentrated near the interface.

GaAs1–xPx/GaAs (001)30,74–77 was one of the first graded material systemsto be studied, due to its importance for the production of LEDs. Morerecently, graded layers of Si1–xGex/Si (001), InxGa1–xAs/GaAs (001), andInxGa1–xP/GaP (001) have been studied extensively due to potential appli-cations in electronics and optoelectronics. In all cases, the use of a gradedbuffer layer is intended to reduce the dislocation density or strain in thedevice layer. The following sections will outline some simple models andexperimental results that bear on these applications.

5.8.1 Critical Thickness in a Linearly Graded Layer

Fitzgerald et al.78 have calculated the critical layer thickness for the onset oflattice relaxation in a linearly graded layer, using an approach similar to theMatthews energy derivation for an abrupt heterostructure. Suppose the dis-tance from the interface is y and the lattice mismatch varies linearly withthis distance so that , where is the grading constant in cm–1. Atany distance from the interface, , where is the in-plane strainand δ is the lattice relaxation. The dislocation dynamics and strain relaxationin a graded layer are rather complex, because the dislocations have distrib-

f C yf= Cf

f = +ε δ|| ε||




uted misfit components rather than well-defined misfit segments lying at ornear the interface. However, the analysis can be greatly simplified by theassumption that and δ vary linearly with distance from the interface (assupported by experimental evidence), so that and . Theelastic strain energy per unit area of total thickness h will therefore be

(5.102)

where G is the shear modulus and ν is the Poisson ratio. The energy of misfitdislocations per unit area, assuming (001) heteroepitaxy of a diamond orzinc blende semiconductor with 60° misfit segments, will be

(5.103)

where b is the length of the Burgers vector for the misfit dislocations. Thecritical layer thickness can be determined by , yielding

(5.104)

Therefore, the critical thickness decreases with increasing grading coefficient.

5.8.2 Equilibrium Strain Gradient in a Graded Layer

Fitzgerald et al.78 found the equilibrium strain gradient in a linearly gradedlayer by extending the analysis of the previous section. Here, it is assumedthat the graded layer is thicker than its critical layer thickness as given above.Then the equilibrium strain gradient is

(5.105)

It has been assumed that the dislocation density is low enough so the cutoffradius for the integration of the dislocation strain field is equal to the layerthickness h.

5.8.3 Threading Dislocation Density in a Graded Layer

Abrahams et al.30 developed the first model for the threading dislocationdensity in a linearly graded layer. More recently, Fitzgerald et al.79 derived

ε||

ε ε|| = C y δ δ= C y

E G C y dye

h

= +−

⎛⎝⎜

⎞⎠⎟ ∫2

11

2 2

0

νν ε

EGbhC h

bd = −−

⎛⎝⎜

⎞⎠⎟

+⎡

⎣⎢

⎤

⎦⎥δ ν

π ν( / )

( )ln

1 41

1

∂ +( ) ∂ =E E he d / 0

hb

Chbc

f

c2 3 1 44 1

1= −+

⎛⎝⎜

⎞⎠⎟

+⎡

⎣⎢

⎤

⎦⎥

( / )( )

lnν

π ν

Cb

hhbε

νπ ν

= −+

⎛⎝⎜

⎞⎠⎟

+⎡

⎣⎢

⎤

⎦⎥

3 1 44 1

12

( / )( )

ln




a model for dislocation dynamics in a graded layer that can be used to predictthe threading dislocation density. Both models predict that the threadingdislocation density will scale with the grading coefficient, as has been exper-imentally observed. The value of the Fitzgerald et al. model is that it predictsthe dependence of the dislocation density on the growth rate.

5.8.3.1 Abrahams et al. Model

The structures of dislocations in graded layers are quite complex, but Abra-hams et al. made the simplifying assumption that the misfit dislocationcontent comprises many small segments. Moreover, it was assumed that thismisfit dislocation content would be distributed uniformly throughout thethickness of the graded layer and that the lattice mismatch would be com-pletely relaxed by the misfit dislocation segments. Assuming that the gradingcoefficient is , and (001) heteroepitaxy of a zinc blende or dia-mond semiconductor, the areal density of misfit dislocation segments inter-secting the {110} planes of the epitaxial layer was estimated to be

(5.106)

where is the mismatch-relieving component of the Burgers vector forthe misfit dislocation segments (the projection of the edge component intothe plane of the interface). Now, if it is assumed that the threading dislocationdensity increases to a constant value at a thickness equal to , and thatall dislocations are bent-over substrate dislocations, the (constant) threadingdislocation density in the top part of the graded layer will be

(5.107)

where l is the average length of the misfit segments. This length is assumedto be proportional to the separation of the misfit dislocations, with a constantof proportionality m, because of mutual repulsion. Then and

(5.108)

Therefore, the threading dislocation density at the top of the graded layerwill be proportional to the grading coefficient. This prediction was verifiedby Abrahams et al.30 in experimental measurements of dislocation densitiesin GaAsxP1–x graded layers on GaAs (001) substrates. They found that thedislocation density increased in approximately linear fashion with the grad-ing coefficient, from D = 8 × 105 cm–2 with Cf = 0.8 cm–1, to D = 4 × 107 cm–2

for Cf = 20 cm–1.

C f yf = Δ Δ/

nC

bAf=

cos λ

b cos λ

nA−1 2/

Dn

lA=−2 1 2/

l m nA= −1 2/

DC

mbf=

2cos λ




The limitation of the Abrahams et al. model is that it does not considerkinetic factors and cannot predict the dependence of the threading disloca-tion density on the growth rate or temperature. To address this, Fitzgeraldet al. developed a model for dislocation flow in a linearly graded heteroepi-taxial layer.

5.8.3.2 Fitzgerald et al. Model

Fitzgerald et al. have presented a model for dislocation flow in a gradedlayer based on a Rowan-type equation.78,79 If the graded layer has a threadingdislocation density D, and each dislocation glides to create a length ofmisfit dislocation, then the amount of strain relaxed will be approximately

(5.109)

The dislocation glide velocity is assumed to be given by the empiricalrelationship

(5.110)

where B is a constant (cm/s), is the effective stress, is a constanthaving units of stress, and U is the activation energy for dislocation glide.If the dislocation density is assumed to be constant, the time rate of strainrelaxation is

(5.111)

If the dislocations are all half-loops, then any particular misfit segmentwill grow by the glide of its associated threading segments in oppositedirections at a velocity v. Therefore,

(5.112)

where Y is the biaxial modulus and is the effective strain, assumed to beconstant throughout the thickness of the graded layer. Substituting this result intoEquation 5.111, we obtain

(5.113)

l

δ ≈ Dbl4

v BUkT

eff

m

=⎛

⎝⎜⎞

⎠⎟−

⎛⎝⎜

⎞⎠⎟

σσ0

exp

σ eff σ0

� �δ = Dbl

4

�l v BYUkT

meffm= = −

⎛⎝⎜

⎞⎠⎟

2 2 ε exp

εeff

�δ ε= −⎛⎝⎜

⎞⎠⎟

DbBY

UkT

meffm

2exp




If it is assumed that the graded layer is much thicker than its critical layerthickness, and that the effective strain is constant with thickness, so that thestrain relief is a linear function of the thickness, then the threading disloca-tion density is found to be

(5.114)

where g is the growth rate. Therefore, the threading dislocation density atthe top of the graded layer will be proportional to the growth rate as wellas the grading coefficient.

5.9 Lattice Relaxation in Superlattices and Multilayer Structures

Superlattices and multilayer structures are useful for the fabrication ofdiverse electronic and optoelectronic devices. Some of these utilize the elec-tronic properties of heterointerfaces and require pseudomorphic structuresfree from misfit dislocations. It is therefore of interest to determine theconditions under which a multilayer structure will begin to relax by theintroduction of misfit dislocations.

There are two requirements for the realization of a stable, coherentlystrained (pseudomorphic) multilayer structure. First, the entire multilayerstack must be stable against lattice relaxation by the glide of a threadingdislocation through the entire stack. Second, each of the individual layers inthe stack must be stable against the glide of threading dislocations to createmisfit dislocations at either interface.

Consider the first condition, that the stack must be stable against latticerelaxation. This condition can be stated simply using the Matthews andBlakeslee critical layer thickness: , where is the total thicknessof the multilayer stack comprising n layers:

(5.115)

and is the thickness of the ith layer. If the lattice mismatch strain in theith layer is , then the effective mismatch strain for the multilayer structure is

(5.116)

DgC

bBYUkT

fm

effm

=⎛⎝⎜

⎞⎠⎟

2

εexp

h htot c eff< , htot

h h itot

i

n

==∑ ( )

0

h i( )f i( )

fh

h i f iefftot i

n

==∑1

1

( ) ( )




The critical layer thickness for the multilayer stack is approximately

(5.117)

where λ is the angle between the Burgers vector and the line in the interfaceplane that is perpendicular to the intersection of the glide plane with theinterface, α is the angle between the Burgers vector and the line vector forthe dislocations, is the average Poisson ratio for the stack, and is theaverage length of the Burgers vector for the stack.

Now consider an individual layer in the stack. Assuming the entire stackis pseudomorphic, every layer in the stack has the same in-plane latticeconstant as the substrate. Because of this, the force balance condition for adislocation threading the ith layer depends only on the lattice mismatch of the ith

layer with respect to the substrate. The glide of a grown-in threading dislocationin the ith layer will create two misfit segments, one at each interface, as shownin Figure 5.34. The ith layer will be stable against lattice relaxation by glideof the threading dislocation if

(5.118)

The critical thickness for the ith layer of the stack is thus*

FIGURE 5.34Force balance for a threading dislocation in the ith layer of an n-layer multilayer stack.

* In the original equation derived by Matthews and Blakeslee,1 there was a factor of two, ratherthan four, in the denominator. This is because they considered the mismatch with respect to theadjacent layer, rather than the substrate, and they assumed the multilayer was a superlatticewith the same average lattice constant as the substrate. In the general case, it is more convenientto define the mismatch of the layers with respect to the substrate.

FG

FL

Substrate

ith layer

(i + 1)th layer

(i − 1)th layer

FL

nth layer

hb h b

c effeff eff c eff eff

,,( cos )[ln( / ) ]

=− +1 12ν α

88 1π ν λfeff eff( )cos+

νeff beff

2F FL G>




(5.119)

Both conditions for stability must be checked in the design of a multilayerdevice structure. It is possible to design a structure that appears stableagainst relaxation based on one of the two conditions but is unstable becauseof the other.

5.10 Dislocation Coalescence, Annihilation, and Removal in Relaxed Heteroepitaxial Layers

In most thick, (nearly) relaxed heteroepitaxial layers, it is found that (1) thethreading dislocation density greatly exceeds that of the substrate and (2)this dislocation density (measured at the surface or averaged over the thick-ness) decreases approximately with the inverse of the thickness, as noted bySheldon et al.80 for a number of heteroepitaxial material systems. Figure 5.35

FIGURE 5.35Threading dislocation density vs. epitaxial layer thickness for several mismatched heteroepi-taxial material systems. The data are from Sheldon et al.,80 Ayers et al.,89 Akram et al.,90 andKalisetty et al.91 as indicated in the legend.

hb h b

f ic ii i c i i

,,( cos )[ln( / ) ]

( ) (=

− ++

1 14 1

2ν απ νii)cos λ

0.1 1 10Epitaxial layer thickness (μm)

Thre

adin

g di

sloca

tion

dens

ity (c

m–2

)

InAs/GaAs Sheldon et al.GaAs/Ge/Si Sheldon et al.GaAs/InP Sheldon et al.InAs/InP Sheldon et al.GaAs/Si Ayers et al.ZnSe/GaAs Akram et al.ZnSe/GaAs Kalisetty et al.

1010

109

108




shows the observed threading dislocation densities as a function of layerthickness, for several heteroepitaxial systems.

To better understand this behavior, Ayers et al.89 measured the threadingdislocation densities in GaAs/Si (001) heterostructures grown by MOVPE,both as grown and after postgrowth annealing treatments. Uncracked layerswith thicknesses up to about 4 μm were studied. For as-grown samples, thedislocation density was found to be inversely proportional to the layer thick-ness as expected. The dislocation density could be reduced by postgrowthannealing. However, for all annealing temperatures investigated, the dislo-cation density saturated at a minimum value and could not be furtherreduced by additional annealing time. It is significant that the minimum valueof the dislocation density, which presumably represented some stable configurationof dislocations, was found to be inversely proportional to the thickness, but with asmaller constant of proportionality than for the as-grown samples.

Several models have been proposed to explain these experimental results;all of them involve dislocation–dislocation reactions that can reduce thethreading dislocation density. During the early stages of relaxation, newdislocations are created by heterogeneous nucleation and multiplication pro-cesses. But once most of the lattice mismatch has been relieved by misfitdislocations, the threading dislocations (which are nonequilibrium defects)can react with other threading dislocations, leading to coalescence or anni-hilation. In some cases, dislocations may glide to the edge of the sample andbe removed in that way. However, this is only expected to be important insmall (patterned) regions of heteroepitaxial material. Therefore, in a planar(unpatterned) layer, coalescence and annihilation are the important processesfor dislocation removal. Here, coalescence refers to a reaction between twothreading dislocations having different Burgers vectors; the end product isa single threading dislocation, and so one threading dislocation is removed.Annihilation refers to the reaction of two dislocations having antiparallelBurgers vectors, which leads to the removal of both. These processes, involv-ing thermally activated glide of dislocations, will only occur during thegrowth itself or subsequent thermal processing.

A semiempirical model for dislocation coalescence and annihilation wasdeveloped by Tachikawa and Yamaguchi.92 The equation governing thereduction of the dislocation density D with the thickness h was assumed toinclude first-order and second-order dislocation interactions, so that

(5.120)

where and are constants. The physics underlying the term linear inD (some process involving single dislocations) is not clear, since both anni-hilation and coalescence processes are expected to be two-dislocation reac-tions. However, the solution of this equation provides a model for thethreading dislocation density as a function of thickness, given by

dDdh

C D C D= − −1 22

C1 C2




(5.121)

where is a constant. Tachikawa and Yamaguchi used this model to fittheir experimental results for GaAs on Si, with , ,and . However, they noted that the dislocation density wasinversely proportional to the thickness for all uncracked samples. Thecracked samples (thicker than about 10 μm) appeared to exhibit a differentthickness dependence, which could be fit by Equation 5.121, but not by theinverse law. However, this behavior has not been observed in other het-eroepitaxial material systems for uncracked samples. It therefore remainsunclear whether the departure from the inverse law in those samples wasrelated to cracking.

Romanov et al.81 extended the annihilation and coalescence model ofTachikawa and Yamaguchi to selective area growth and provided a physicalanalysis of the constants. Here, the starting equation was the same as thatgiven by Tachikawa and Yamaguchi:

(5.122)

However, it was assumed that the first-order reaction was due to the lossof threading dislocations to sidewalls in the case of selective area epitaxy.The first-order constant was calculated from , where λ is a lengthcharacterizing the travel necessary to reach a mesa sidewall and G is ageometric factor associated with the inclination of threading dislocationsand . The second-order constant was calculated from ,where is a characteristic length for the second-order reaction. Romanovet al. wrote the solution in the form

(5.123)

and discussed two limiting cases. First, for planar (unpatterned) layers, theglide of dislocations to sidewalls will be negligible, so and

(5.124)

For large-area growth, the dislocation density will exhibit an (approxi-mately) inverse relationship with the thickness. In the second limiting caseof selective epitaxy with small mesas, the first-order reaction dominates,leading to

DD C C C h C C

=+( ) ( ) −

11 0 2 1 1 2 1/ / exp /

D0

D012 210= −cm C1

1200= −cmC2

51 8 10= × −. cm

dDdh

C D C D= − −1 22

C G1 = / λ

G ≈ 1 C Gr2 12=r1

DD

C D C C h h D C C=

+ − −0

2 0 1 1 0 0 2 11( / )exp[ ( )] /

C h h1 0 1( )− <<

DD

D C h h=

+ −0

0 2 01 ( )




(5.125)

This might explain the different thickness dependence found by Tachikawaand Yamaguchi for thick, cracked films of GaAs/Si (001), which would havecontained many sidewalls.

A limitation of the annihilation/coalescence models is that they do notexplicitly address the influence of the lattice mismatch.* However, it isknown that the dislocation densities in relaxed heteroepitaxial layers exhibita weak dependence on the lattice mismatch. The glide model82 was devel-oped in an attempt to include the lattice mismatch dependence. Here, it wasassumed that reaction (annihilation or coalescence) between two dislocationsis limited by their ability to overcome the line tensions of their misfit seg-ments so they can glide toward one another. Figure 5.36 shows the forcesacting on dislocations in a relaxed heteroepitaxial layer.† Here, two neigh-boring threading dislocations have opposite Burgers vectors, resulting in anattractive glide force , which acts on each. At the same time, each dislo-cation experiences a line tension associated with its misfit dislocationsegment. If the sample is held at elevated temperature for a sufficiently longtime, during either growth or other thermal processing, the dislocations willcome to a stable configuration in which the glide and line tension forcesbalance. Therefore, the dislocation density can be estimated by consideringthe balance of these forces.

The attractive glide force between dislocations with a separation r is given by

(5.126)

where G is the shear modulus, b is the length of the Burgers vector, and φ isthe angle between the threading segments and the interface. The line tensionof the misfit segment is given by

FIGURE 5.36Forces acting on dislocations in a heteroepitaxial structure. (Reprinted from Ayers, J.E., J. Appl.Phys., 78, 3724, 1995. With permission. Copyright 1995, American Institute of Physics.)

* The mismatch dependence is embodied in the parameter D0.† The interaction shown in the figure involves one bent-over substrate dislocation and one half-loop. Any combination of these types of dislocations can interact, but the stability condition willbe the same.

FGFL

Substrate

Epitaxial layerFG

FL

D D C h h= − −0 1 0exp[ ( )]

FG

FL

FGb hrG =

2

π φcos




(5.127)

where α is the angle between the Burgers vector and line vector, ν is thePoisson ratio, and R is the average spacing between dislocations (perpendic-ular to the intersection of the glide plane and the heterointerface) or the filmthickness, whichever is smaller. The average spacing for misfit segments is

(5.128)

where λ is the angle between the interface and the normal to the slip plane.Thus,

(5.129)

We can find the minimum stable separation of threading dislocations byequating the glide and line tension forces, yielding

(5.130)

In the development of the glide model, the average value of r was consid-ered to be twice the minimum. (001) heteroepitaxy of a zinc blende semi-conductor was assumed, with threading dislocations separated by Rave

between glide planes and rave within glide planes. Using these assumptions,the threading dislocation density was calculated to be

(5.131)

Therefore, this model predicts the 1/h dependence and also that thethreading dislocation density will increase in sublinear fashion with thelattice mismatch. This model correctly predicts the threading dislocationdensities in a number of heteroepitaxial systems, with a factor of two accu-racy and without adjustable parameters. However, more experimentalresults are needed to test this model adequately with respect to the latticemismatch dependence.

FGb R

bL = +−

⎡

⎣⎢

⎤

⎦⎥

⎛⎝⎜

⎞⎠⎟

22

2

4 4 1 4πα α

νcos

sin( )

ln

Rb

fave = sin cosα λ

FGb

fL = +−

⎡

⎣⎢

⎤

⎦⎥

22

2

4 4 1 4πα α

να λ

cossin( )

lnsin cos⎛⎛

⎝⎜

⎞

⎠⎟

14 4 1

22

r hmin

coscos

sin( )

lnsin co= +

−⎡

⎣⎢

⎤

⎦⎥

φ θ αν

α ss λ4 f

⎛

⎝⎜

⎞

⎠⎟

DR r

f

bh fave ave

= =−

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎛

⎝⎜

⎞216 1

14

cos

( )ln

φν ⎠⎠

⎟




5.11 Thermal Strain

Often heteroepitaxial layers have thermal coefficients of expansion verydifferent from their substrates. This causes the introduction of a thermalstrain during the cool-down process. This is a severe problem in cases forwhich the epitaxial layer has a larger thermal coefficient of expansion,because this leads to tensile strain and the possibility of cracking or grossfailure of the heteroepitaxial layer. Whereas lattice mismatch strain is oftennearly relaxed at the growth temperature, thermal strain is applied duringthe cool-down and cannot usually be relaxed by dislocation motion. Thereason is that dislocation glide velocities are thermally activated and mayreduce by a decade for every 25°C reduction in temperature.

Consider a heteroepitaxial layer grown at a temperature and cooleddown to room temperature . If the linear coefficients of thermal expan-sion are and for the epitaxial layer and substrate, respectively, andif no lattice relaxation occurs during cool-down, then the thermal strainwill be

(5.132)

If the coefficients of thermal expansion are considered to be independentof temperature, then

(5.133)

The thermal strain is therefore tensile (positive) if the epitaxial layer hasa larger coefficient of thermal expansion than its substrate. For a heteroepi-taxial layer with a growth temperature in-plane strain of , the roomtemperature strain will be

(5.134)

An interesting practical application of the thermal strain arises in someheteroepitaxial material systems, including certain II-VI semiconductorsdeposited on GaAs (001) substrates. If the lattice mismatch strain is com-pressive but the thermal strain is tensile, it is possible to have perfect straincompensation at room temperature.

Tg

Tr

αe αs

ε α αTh s eT

T

T T dTg

r

= −∫ [ ( ) ( )]

ε α αTh e s g rT T≈ − −( )( )

ε||( )Tg

ε ε α α|| ||( ) ( ) ( )( )T T T Tr g e s g r≈ + − −




5.12 Cracking in Thick Films

Heteroepitaxial films under tensile thermal stress, such as GaN/sapphire(0001), GaN/Si (111), GaAs/Si (001), or InP/Si(001), often develop macro-scopic cracks if grown too thick. Figure 5.37 shows an example of cracks inheteroepitaxial AlGaN/sapphire (0001). The simple model of Griffith83–85

predicts the thickness at which cracks may propagate for a given tensilestrain. This calculation is based on a balance between the strain energyrelieved by the crack and the surface energy of the crack walls; a crack maypropagate if the film thickness h is equal to or greater than the thickness atwhich these two energy contributions balance. Films under compressivestress will not crack because cracking would increase their strain energy.They may separate from the substrate, but this is expected to occur at greaterthicknesses than cracking, because the separation process would create morenew surface area than cracking.

The Griffith criterion for crack propagation in heteroepitaxial films maybe derived as follows.83–85 Consider a semiconductor layer of thicknessunder tensile strain, with a crack passing through the entire thickness andwith a length L. The change in surface energy associated with the two wallsof the crack will be , where γ is the surface energy per unit area for thesemiconductor. At the same time, the crack will relieve strain energy equalto , where is the biaxial stress and Y is the biaxial modulus.The total change in free energy associated with the crack is

FIGURE 5.37SEM image of cracks in heteroepitaxial Al0.2Ga0.8N grown on sapphire (0001) by MOVPE, grownwith an AlN buffer layer. (Reprinted from Zhang, J.P. et al., Appl. Phys. Lett., 80, 3542, 2002.With permission. Copyright 2002, American Institute of Physics.)

10 μm

h

2γ hL

π σLh Y2 2|| / σ||




(5.135)

The condition for propagation of the crack is

(5.136)

or

(5.137)

In terms of the strain, the Griffith criterion for crack propagation in a layerwith a tensile strain is

(crack propagation) (5.138)

For (001) heteroepitaxy of a zinc blende semiconductor, the surface energyfor the (110) cleavage planes may be approximated by

(5.139)

where a is the lattice constant and is the Poisson ratio, so that the Griffithcriterion becomes

(crack propagation, zinc blende (001)) (5.140)

Figure 5.38 shows the Griffith thickness as a function of the in-plane tensilestrain for (001) heteroepitaxy of a zinc blende or diamond semiconductor.These calculations show that cracking will be unimportant if the tensilestrain is less than about 10–4. On the other hand, a tensile strain of greaterthan 1% will result in a Griffith crack thickness less than 0.2 μm and severelylimit device design.

The Griffith criterion describes the condition under which existing cracksmay propagate, but it is not in itself a sufficient condition for cracking.

ΔW hLLh

Y= −2

2 2

γπ σ||

0 22 2

= ∂∂

= −( ) ||ΔWL

hh

Yγ

π σ

hY= 2

2

γπσ||

hY

≥ 22

γπ ε||

γ νπ

( )( )

1101

2 2 2= −Y a

ν

ha≥ −( )

||

1

2

2

3 2

νπ ε




However, defects or regions of concentrated stress near the edges of a samplemay act as nucleation sites for cracks, and we can expect cracks to appearat a thickness close to the Griffith thickness .

The Griffith model as developed above cannot account for cracks withdifferent orientations or irregular geometries. Nonetheless, it predicts thethickness for the onset of cracking within about a factor or two or three. Forexample, in the GaAs/Si (001) system, cracking is observed at thicknessesgreater than about 4 μm, whereas the Griffith thickness is about 1.5 μm fortypical values of the thermal tensile strain.

The thermal strain increases linearly with the growth temperature. There-fore, due to the dependence, the Griffith thickness decreases stronglywith increasing growth temperature. In GaN/Si (111), for example, GaN canbe grown crack-free on Si substrates by MBE at 800°C up to a thickness of3 μm.86 But for MOVPE growth, typically carried out at 1090°C, the maxi-mum thickness for crack-free growth is 1.4 μm.87

For nitride semiconductors grown on Si (111) or sapphire (0001), it hasbeen found that cracking can be suppressed by the insertion of a strainedlayer superlattice (SLS). For example, Feltin et al.88 found that they couldgrow crack-free GaN on Si (111) up to a thickness of 2.5 μm by MOVPE whenan AlN/GaN superlattice was inserted (without such a superlattice, themaximum crack-free thickness is about 1.4 μm). The benefit of the SLS was

FIGURE 5.38Griffith thickness for cracking in a heteroepitaxial layer as a function of in-plane tensile strain.(001) heteroepitaxy of a zinc blende or diamond semiconductor was assumed, with a = 0.565 nmand ν = 1/3.

0.1

1.0

10.0

100.0

0.1 1 10In-plane tensile strain (10–3)

G

riffith

thic

knes

s for

crac

king

( μm

)

hG

1 2/ ||ε




found to come from the introduction of a compressive component of strainin the overlying GaN. For a 0.9-μm-thick layer of GaN on Si (111) the roomtemperature strain without an SLS buffer is about 0.15%, but can be madenegative (compressive) by the insertion of four SLSs. Unfortunately, thiseffect is diminished with the thickness of the GaN overlayer, and this limitsthe maximum thickness for crack-free growth.

A similar enhancement in the Griffith thickness was found for MOVPEAl0.2Ga0.8N grown on sapphire (0001) by Zhang et al.,93 with the insertion ofAlN/Al0.2Ga0.8N strained layer superlattices. For Al0.2Ga0.8N top layers grownwithout the SLS, the maximum crack-free thickness was 1.2 μm, but with anSLS layer could be grown up to 3.0 μm thick without cracks. As in thepreviously mentioned study, Zhang et al. found that the insertion of the SLScould compensate the tensile thermal strain and even result in a net com-pressive strain at room temperature.

Problems

1. Calculate the critical layer thickness for In0.15Ga0.85As/GaAs (001)assuming (a) 60° misfit dislocations and (b) pure edge misfit dis-locations.

2. For InxGa1–xAs grown on InP (001) with , (a) determine thecomposition x; (b) calculate the Matthews and Blakeslee critical layerthickness; (c) estimate the critical layer thickness assuming the sur-face energy is 2000 erg/cm2; (d) repeat (a) for ; and (e)repeat (c) for .

3. For a zinc blende epitaxial layer with (001) orientation, the biaxialmodulus is

Show that in the general case, the biaxial modulus for a cubic crystal is

where l, m, and n are the direction cosines that relate the unit normalto the cube axes.

4. Suppose that mismatched heteroepitaxy is to be used to bend overall of the dislocations in a substrate having a threading dislocation

f = 0 4. %

f = −0 4. %f = −0 4. %

Y C C C C= + −11 12 122

112 /

YC C C C

C C C C= + − +

+ − +( )

( )(11 12 11 12

11 44 11 12

22

32

2 2 ll m m n n l2 2 2 2 2 2+ +⎡

⎣⎢

⎤

⎦⎥)




density of 104 cm–2. Find the necessary lattice mismatch. What is therequirement on the thickness of the heteroepitaxial layer?

5. Suppose a pseudomorphic layer of AlAs is grown on a vicinal (001)GaAs substrate for which the surface normal is inclined by 2° towardthe [110]. Find the magnitude and direction for the expected tiltbetween the heteroepitaxial layer and its substrate.

6. GaN is grown on a vicinal sapphire (0001) substrate that is inclinedby 4° toward the . Find the magnitude and direction of thetilt between the GaN and the sapphire, for the case of substratesurface steps with a height of (a) two bilayers, (b) three bilayers, (c)four bilayers, and (d) five bilayers.

7. Find the critical layer thickness for a linearly graded Si1–xGex/Si (001)layer, if the germanium composition x is graded by 4%/μm. Com-pare this to the critical layer thickness for a uniform alloy layerhaving the same germanium surface concentration, if both layers are0.5 μm thick.

8. (a) Estimate the room temperature thermal strain in InP on Si (001)grown at 650°C. (b) Estimate the thickness beyond which the thermalstrain will cause cracking.

References

1. J.W. Matthews and A.E. Blakeslee, Defects in epitaxial multilayers. I. Misfitdislocations, J. Cryst. Growth, 27, 118 (1974).

2. W.A. Jesser and J.W. Matthews, Evidence for pseudomorphic growth of ironon copper, Phil. Mag., 15, 1097 (1967).

3. J.W. Matthews, Epitaxial Growth, Part B, Academic Press, New York, 1975.4. J.H. van der Merwe, Crystal interfaces. II. Finite overgrowths, J. Appl. Phys.,

34, 123 (1962).5. R. People and J.C. Bean, Calculation of critical layer thickness versus lattice

mismatch for GexSi1–x/Si strained-layer heterostructures, Appl. Phys. Lett., 47,322 (1985); Appl. Phys. Lett., 49, 229 (1986).


7. J. Bevk, J.P. Mannaerts, L.C. Feldman, B.A. Davidson, and A. Qurmazd, Ge-Silayered structures: Artificial crystals and complex cell ordered superlattices,Appl. Phys. Lett., 49, 286 (1986).

8. J. Petruzzello and M.R. Leys, Effect of the sign of misfit strain on the dislocationstructure at interfaces of heteroepitaxial GaAsxP1–x films, Appl. Phys. Lett., 53,2414 (1988).

9. R.C. Cammarata and K. Sieradzki, Surface stress effects on the critical filmthickness for epitaxy, Appl. Phys. Lett., 55, 1197 (1989).

[ ]1100




10. R.C. Cammarata, K. Sieradzki, and F. Spaepen, Simple model for interfacestresses with application to misfit dislocation generation in epitaxial thin films,J. Appl. Phys., 87, 1227 (2000).

11. S. Luryi and E. Suhir, New approach to the high quality epitaxial growth oflattice-mismatched materials, Appl. Phys. Lett., 49, 140 (1986).

12. J.W. Matthews, Defects associated with the accommodation of misfit betweencrystals, J. Vac. Sci. Technol., 12, 126 (1975).

13. J.W. Matthews, A.E. Blakeslee, and S. Mader, Use of misfit strain to removedislocations from epitaxial thin films, Thin Solid Films, 33, 253 (1976).

14. J. Zou, X.Z. Liao, D.J.H. Cockayne, and Z.M. Jiang, Alternative mechanism formisfit dislocation generation during high-temperature Ge(Si)/Si (001) islandgrowth, Appl. Phys. Lett., 81, 1996 (2002).

15. R. Beanland, Multiplication of misfit dislocations in epitaxial layers, J. Appl.Phys., 72, 4031 (1992).

16. J.W. Matthews, S. Mader, and T.B. Light, Accommodation of misfit across theinterface between crystals of semiconducting elements or compounds, J. Appl.Phys., 41, 3800 (1970).

17. F.C. Frank and W.T. Read, Multiplication processes for slow moving disloca-tions, Phys. Rev., 79, 722 (1950).

18. W.C. Dash, Copper precipitation on dislocations in silicon, J. Appl. Phys., 27,1193 (1956).

19. A.T. Lefevbre, C. Herbeaux, C. Bouillet, and J. Di Persio, A new type of misfitdislocation multiplication process in InxGa1–xAs/GaAs strained-layer superlat-tices, Phil. Mag. Lett., 63, 23 (1991).

20. F.K. LeGoues, B.S. Meyerson, and J.M. Morar, Anomalous strain relaxation inSiGe thin films and superlattices, Phys. Rev. Lett., 66, 2903 (1991).

21. M.A. Capano, L. Hart, D.K. Bowen, D. Gordon-Smith, C.R. Thomas, C.J. Gib-bings, M.A.G. Halliwell, and L.W. Hobbs, Strain relaxation in Si1–xGex layerson Si(001), J. Cryst. Growth, 116, 260 (1992).

22. W.C. Dash, Dislocations and Mechanical Properties of Crystals, J. Fisher, Ed., JohnWiley, New York, 1957, p. 57.

23. A. Authier and A.R. Lang, Three-dimensional x-ray topographic studies ofinternal dislocation sources in silicon, J. Appl. Phys., 35, 1956 (1964).

24. A.T. Mader and A.E. Blakeslee, On dislocations in GaAs1–xPx, IBM J. Res. Dev.,19, 151 (1985).

25. A.T. Washburn and E.P. Kvam, Possible dislocation multiplication source in(001) semiconductor epitaxy, Appl. Phys. Lett., 57, 1637 (1990).

26. W. Hagen and H. Strunk, A new type of source generating misfit dislocations,Appl. Phys., 17, 85 (1978).

27. H. Strunk, W. Hagen, and E. Bauser, Low-density dislocation arrays at het-eroepitaxial Ge/GaAs interfaces investigated by high voltage electron micros-copy, Appl. Phys. A, 18, 67 (1979).

28. R.H. Dixon and P.J. Goodhew, On the origin of misfit dislocations in InGaAs/GaAs strained layers, J. Appl. Phys., 68, 3163 (1990).

29. Y. Obayashi and K. Shintani, Is the Hagen-Strunk multiplication mechanismof misfit dislocations in heteroepitaxial layers probable? Phil. Mag. Lett., 76, 1(1997).

30. M.S. Abrahams, L.R. Weisberg, C.J. Buiocchi, and J. Blanc, Dislocation mor-phology in graded heterojunctions: GaAs1–xPx, J. Mater. Sci., 4, 223 (1969).




31. G.R. Booker, J.M. Titchmarsh, J. Fletcher, D.B. Darby, M. Hockly, and M. Al-Jassim, Nature, origin and effect of dislocations in epitaxial semiconductorlayers, J. Cryst. Growth, 45, 407 (1978).

32. V.I. Vdovin, L.A. Matveeva, G.N. Semenova, M. Ya. Skorohod, Yu. A. Tkhorik,and L.S. Khazan, Misfit dislocations in epitaxial heterostructures: mechanismsof generation and multiplication, Phys. Status Solidi A, 92, 379 (1985).

33. W. Qian, M. Skowronski, and R. Kaspi, Dislocation density reduction in GaSbfilms grown on GaAs substrates by molecular beam epitaxy, J. Electrochem. Soc.,144, 1430 (1997).

34. W. Qian, M. Skowronski, R. Kaspi, and M. De Graef, Nucleation of misfit andthreading dislocations during epitaxial growth, J. Appl. Phys., 81, 7268 (1997).

35. P.M.J. Marée, R.I.J. Olthof, J.W.M. Frenken, J.F. van der Veen, C.W.T. Bulle-Lieuwma, M.P.A. Viegers, and P.C. Zalm, Silicon strained layers grown onGaP(001) by molecular beam epitaxy, J. Appl. Phys., 58, 3097 (1985).

36. K. Ito, K. Hiramatsu, H. Amano, and I. Akasaki, Preparation of AlxGa1–xN/GaNheterostructure by MOVPE, J. Cryst. Growth, 104, 533 (1990).

37. P. Haasen, On the plasticity of Germanium and Indium Antimonide, Acta. Met.,5, 598 (1957).

38. B.W. Dodson and J.Y. Tsao, Relaxation of strained-layer semiconductor struc-tures by plastic flow, Appl. Phys. Lett., 51, 1325 (1987); Appl. Phys. Lett., 52, 852(1988).

39. B. Yarlagadda, A. Rodriguez, P. Li, B.I. Miller, F.C. Jain, and J.E. Ayers, Elasticstrains in heteroepitaxial ZnSe1–xTex on InGaAs/InP (001), J. Electron. Mater., 351327 (2006).

40. X.R. Huang, J. Bai, M. Dudley, R.D. Dupuis, and U. Chowdhury, Epitaxial tiltingof GaN grown on vicinal surfaces of sapphire, Appl. Phys. Lett., 86, 211916 (2005).

41. X.R. Huang, J. Bai, M. Dudley, B. Wagner, R.F. Davis, and Y. Zhu, Step-con-trolled strain relaxation in the vicinal surface epitaxy of nitrides, Phys. Rev.Lett., 95, 86101 (2005).

42. A. Leiberich and J. Levkoff, A double crystal x-ray diffraction characterizationof AlxGa1–xAs grown on an offcut GaAs (100) substrate, J. Vac. Sci. Technol. B,8, 422 (1990).

43. H. Nagai, Structure of vapor-deposited GaxIn1–xAs crystals, J. Appl. Phys., 45,3789 (1974).

44. G.H. Olsen and R.T. Smith, Misorientation and tetragonal distortion in het-eroepitaxial vapor-grown III-V structures, Phys. Status Solidi A, 31, 739 (1975).

45. A. Ohki, N. Shibata, and S. Zembutsu, Lattice relaxation mechanism of ZnSelayer grown on a (100) GaAs substrate tilted toward <011>, J. Appl. Phys., 64,694 (1988).

46. J. Kleiman, R.M. Park, and H.A. Mar, On epilayer tilt in ZnSe/Ge heterostruc-tures prepared by molecular-beam epitaxy, J. Appl. Phys., 64, 1201 (1988).

47. E. Yamaguchi, I. Takayasu, T. Minato, and M. Kawashima, Growth of ZnSe onGe (100) substrates by molecular-beam epitaxy, J. Appl. Phys., 62, 885 (1987).

48. I.B. Bhat, K. Patel, N.R. Taskar, J.E. Ayers, and S.K. Ghandhi, X-ray diffractionstudies of CdTe grown on InSb, J. Cryst. Growth, 88, 23 (1988).

49. W.L. Ahlgren, S.M. Johnson, E.J. Smith, R.P. Ruth, B.C. Johnston, M.H. Kalisher,C.A. Cokrum, T.W. James, D.L. Arney, C.K. Ziegler, and W. Lick, Metalorganicchemical vapor deposition growth of Cd1–yZnyTe epitaxial layers on GaAs andGaAs/Si, J. Vac. Sci. Technol. A, 7, 331 (1989).




50. T.J. de Lyon, D. Rajavel, S.M. Johnson, and C.A. Cockrum, Molecular-beamepitaxial growth of CdTe(112) on Si(112) substrates, Appl. Phys. Lett., 66, 2119(1995).

51. S.J. Rosner, J. Amano, J.W. Lee, and J.C.C. Fan, Initial growth of gallium ars-enide on silicon by organometallic vapor phase epitaxy, Appl. Phys. Lett., 53,110 (1988).

52. L.J. Schowalter, E.L. Hall, N. Lewis, and S. Hashimoto, Strain relief of largelattice mismatch heteroepitaxial films on silicon by tilting, Thin Solid Films, 184,437 (1990).

53. S.K. Ghandhi and J.E. Ayers, Strain and misorientation in GaAs grown onSi(001) by organometallic epitaxy, Appl. Phys. Lett., 53, 1204 (1988).

54. F. Riesz, J. Varrio, A. Pesek, and K. Lischka, Correlation between initial growthplanarity and epilayer tilting in the vicinal GaAs/Si system, Appl. Surface Sci.,75, 248 (1994).

55. M. Calamiotou, N. Chrysanthakopoulos, Ch. Lioutas, K. Tsagaraki, and A.Georgakilas, Microstructural differences of the two possible orientations ofGaAs on vicinal (001) Si substrates, J. Cryst. Growth, 227/228, 98 (2001).

56. Y.-Z. Yoo, T. Chikyow, M. Kawasaki, T. Onuma, S. Chichibu, and H. Koinuma,Heteroepitaxy of hexagonal ZnS thin films directly on Si (111), Jpn. J. Appl.Phys., Part 1, 42, 7029 (2003).

57. X. Jiang, R.Q. Zhang, G. Yu, and S.T. Lee, Local strain in interface: origin ofgrain tilting in diamond (001)/silicon (001) heteroepitaxy, Phys. Rev. B, 58, 15351(1998).

58. X. Jiang, K. Schiffmann, C.-P. Klages, D. Wittorf, C.L. Jia, K. Urban, and W.Jager, Coalescence and overgrowth of diamond grains for improved heteroepi-taxy on silicon (001), J. Appl. Phys., 83, 2511 (1998).

59. E. Maillard-Schaller, O.M. Kuttel, P. Groning, O. Groning, R.G. Agostino, P.Aebi, and L. Schlapbach, Local heteroepitaxy of diamond on silicon (100): astudy of the interface structure, Phys. Rev. B, 55, 15895 (1997).

60. B.W. Dodson, D.R. Meyers, A.K. Datye, V.S. Kaushik, D.L. Kendall, and B.Martinez-Tovar, Asymmetric tilt boundaries and generalized heteroepitaxy,Phys. Rev. Lett., 61, 2681 (1988).

61. J.E. Ayers, S.K. Ghandhi, and L.J. Schowalter, Crystallographic tilting of het-eroepitaxial layers, J. Cryst. Growth, 113, 430 (1991).

62. J.E. Ayers and L.J. Schowalter, Comment on “measurement of the activationbarrier to nucleation of dislocations in thin films,” Phys. Rev. Lett., 72, 4055 (1994).

63. J.Y. Tsao and B.W. Dodson, Excess stress and the stability of strained hetero-structures, Appl. Phys. Lett., 53, 848 (1988).

64. F. Riesz, Crystallographic tilting in lattice-mismatched heteroepitaxy: a Dod-son-Tsao relaxation approach, J. Appl. Phys., 79, 4111 (1996).

65. T.W. Weeks, Jr., M.D. Bremser, K.S. Ailey, E. Carlson, W.G. Perry, and R.F. Davis,GaN thin films deposited via organometallic vapor phase epitaxy on α(6H)-SiC(0001) using high-temperature monocrystalline AlN buffer layers, Appl.Phys. Lett., 67, 401 (1995).

66. M.H. Xie, L.X. Zheng, S.H. Cheung, Y.F. Ng, H. Wu, S.Y. Tong, and N. Ohtani,Reduction of threading defects in GaN grown on vicinal SiC (0001) by molec-ular-beam epitaxy, Appl. Phys. Lett., 77, 1105 (2000).

67. J.-I. Kato, S. Tanaka, S. Yamada, and I. Suemune, Structural anisotropy in GaNfilms grown on vicinal 4H-SiC surfaces by metalorganic molecular-beam epi-taxy, Appl. Phys. Lett., 83, 1569 (2003).




68. L. Lu, H. Yan, C.L. Yang, M. Xie, Z. Wang, J. Wang, and W. Ge, Study of GaNthin films grown on vicinal SiC(0001) substrates by molecular beam epitaxy,Semicond. Sci. Technol., 17, 957 (2002).

69. X.-Q. Shen, M. Shimizu, and H. Okumura, Impact of vicinal sapphire (0001)substrates on the high-quality AlN films by plasma-assisted molecular beamepitaxy, Jpn. J. Appl. Phys., 42, L1293 (2003).

70. L.P. Van, O. Kurnosikov, and J. Cousty, Evolution of steps on vicinal (0001)surfaces of α-alumina, Surf. Sci., 411, 263 (1998).

71. F.K. LeGoues, B.S. Meyerson, J.F. Morar, and P.D. Kirchner, Mechanism andconditions for anomalous strain relaxation in graded thin films and superlat-tices, J. Appl. Phys., 71, 4230 (1992).

72. F.K. LeGoues, P.M. Mooney, and J.O. Chu, Crystallographic tilting resultingfrom nucleation limited relaxation, Appl. Phys. Lett., 62, 140 (1993).

73. F.K. LeGoues, P.M. Mooney, and J. Tersoff, Measurement of the activationbarrier to nucleation of dislocations in thin films, Phys. Rev. Lett., 71, 396 (1993);Phys. Rev. Lett., 71, 3234 (1993).

74. J.J. Tietjen, J.I. Pankove, I.J. Hegyi, and H. Nelson, Vapor-phase growth ofGaAs1–xPx, room-temperature injection lasers, Trans. AIME, 239, 385 (1967).

75. C.J. Nuese, J.J. Tietjen, J.J. Gannon, and H.F. Gossenberger, Electroluminescenceof vapor-grown GaAs and Gas1–xPx diodes, Trans. AIME, 242, 400 (1968).

76. D. Richman and J.J. Tietjen, Rapid vapor phase growth of high-resistivity GaPfor electro-optic modulators, Trans. AIME, 239, 418 (1967).

77. R.H. Saul, Effect of a GaAsxP1–x transition zone on the perfection of GaP crystalsgrown by deposition onto GaAs substrates, J. Appl. Phys., 40, 3273 (1969).

78. E.A. Fitzgerald, Y.-H. Xie, D. Monroe, P.J. Silverman, J.M. Kuo, A.R. Kortan,F.A. Thiel, and B.E. Weir, Relaxed GexSi1–x structures for III-V integration withSi and high mobility two-dimensional electron gases in Si, J. Vac. Sci. Technol.B, 10, 1807 (1992).

79. E.A. Fitzgerald, A.Y. Kim, M.T. Currie, T.A. Langdo, G. Taraschi, and M.T.Bulsara, Dislocation dynamics in relaxed graded composition semiconductors,Mater. Sci. Eng. B, 67, 53 (1999).

80. P. Sheldon, K.M. Jones, M.M. Al-Jassim, and B.G. Yacobi, Dislocation densityreduction through annihilation in lattice-mismatched semiconductors grownby molecular beam epitaxy, J. Appl. Phys., 63, 5609 (1988).

81. A.E. Romanov, W. Pompe, G.E. Beltz, and J.S. Speck, An approach to threadingdislocation “reaction kinetics,” Appl. Phys. Lett., 69, 3342 (1996).

82. J.E. Ayers, New model for the thickness and mismatch dependencies of thread-ing dislocation densities in mismatched heteroepitaxial layers, J. Appl. Phys.,78, 3724 (1995).

83. A.A. Griffith, The phenomenon of rupture and flow in solids, Phil. Trans. R.Soc., 221, 163 (1921).

84. A. Kelly, Strong Solids, Clarendon Press, Oxford, 1966, pp. 45–47.85. J.W. Matthews and E. Klockholm, Fracture of brittle films under the influence

of misfit stress, Mater. Res. Bull., 7, 213 (1972).86. F. Semond, P. Lorenzini, N. Grandjean, and J. Massies, High-electron-mobility

AlGaN/GaN heterostructures grown on Si (111) by molecular-beam-epitaxy,Appl. Phys. Lett., 78, 335 (2000).

87. H.M. Liaw, R. Venugopal, J. Wan, R. Doyle, P. Fejes, M.J. Loboda, and M.R.Melloch, Crack-free, single-crystal GaN grown on 100 mm diameter silicon,Mater. Sci. Forum, 338–342, 1463 (2000).




88. E. Feltin, B. Beaumont, M. Laügt, P. de Mierry, P. Vennéguès, H. Lahrèche, M.Leroux, and P. Gibart, Stress control in GaN grown on silicon (111) by metal-organic vapor phase epitaxy, Appl. Phys. Lett., 79, 3230 (2001).

89. J.E. Ayers, L.J. Schowalter, and S.K. Ghandhi, Post-growth thermal annealingof GaAs on Si(001) grown by organometallic vapor phase epitaxy, J. Cryst.Growth, 125, 329 (1992).

90. S. Akram, H. Ehsani, and I.B. Bhat, The effect of GaAs surface stabilization onthe properties of ZnSe grown by organometallic vapor phase epitaxy, J. Cryst.Growth, 124, 628 (1992).

91. S. Kalisetty, M. Gokhale, K. Bao, J.E. Ayers, and F.C. Jain, The influence ofimpurities on the dislocation behavior in heteroepitaxial ZnSe on GaAs, Appl.Phys. Lett., 68, 1693 (1996).

92. M. Tachikawa, and M. Yamaguchi, Film thickness dependence of dislocationdensity reduction in GaAs-on-Si substrates, Appl. Phys. Lett., 56, 484 (1990).

93. J.P. Zhang, H.M. Wang, M.E. Gaerski, C.Q. Chen, Q. Fareed, J.W. Yang, G. Simin,and M.A. Khan, Crack-free thick AlGaN grown on sapphire using AlN/AlGaNsuper lattices for strain management, Appl. Phys. Lett., 80, 3542 (2002).



249

6

Characterization of Heteroepitaxial Layers

6.1 Introduction

Numerous and diverse characterization techniques have been used for theevaluation of heteroepitaxial semiconductors, and have enabled the advance-ment of the field to its present state. It would be impossible to describe allof them here. Instead, this chapter will emphasize some of the most com-monly used techniques, such as x-ray diffraction, electron diffraction, elec-tron microscopy, crystallographic etching, and photoluminescence. Thesebasic methods have contributed greatly to our current understanding ofheteroepitaxy. Some have also been adapted as routine characterizationmethods for commercial production of heteroepitaxial materials and devices.

High-resolution x-ray diffraction (HRXRD) is the most widely used tech-nique for the

ex situ

characterization of heteroepitaxial layers. HRXRD isnondestructive and yields a wealth of structural information, including thelattice constants and strains, crystallographic orientation, and defect densi-ties. Reciprocal space mapping and rocking curve measurements at differentazimuths can also be used to study the asymmetries in the defect densitieson different slip systems. In multilayer device structures, dynamical rockingcurve simulations can be used to extract the thicknesses, lattice constants,and compositions in the individual layers. The versatility and nondestructivenature of HRXRD have led to its common use in production environmentsas well as in basic studies.

Electron diffraction methods have also been used to a great extent for thestudy of surface structures, surface reconstructions, nucleation, and growthmodes. The commonly used methods include reflection high-energy electrondiffraction (RHEED) and low-energy electron diffraction (LEED) and theirvariants. The high-vacuum environment used for molecular beam epitaxy(MBE) growth allows the

in situ

use of electron diffraction techniques, whichprovide valuable feedback during the growth process.

Microscopic methods include optical microscopy (OM), scanning electronmicroscopy (SEM), transmission electron microscopy (TEM), atomic forcemicroscopy (AFM), and scanning tunneling electron microscopy (STEM).Most of these are used to image the surface, and thus characterize the surface



250


morphology and growth mode. They may also be used in conjunction withcrystallographic etches to determine defect densities. AFM has sufficientresolution for the study of the surface atomic structure. TEM enables theimaging of dislocations and other crystal defects within the volume of thecrystal; it is therefore important in the study of dislocations and latticerelaxation in mismatched heteroepitaxial layers.

Crystallographic etching techniques are used routinely to determinethreading dislocation densities in single-crystal substrates as well as het-eroepitaxial layers. Crystallographic etches, due to their surface-sensitiveetch rates, reveal pits at the points of emergence for defects. Subsequentmicroscopic inspection can be used to determine the dislocation density. Insome materials, different types of dislocation defects can be distinguishedby their characteristic pit shapes. Moreover, crystal orientation may be deter-mined by the alignment of oval pits.

Photoluminescence (PL) is commonly employed

ex situ

to assess the suit-ability of heteroepitaxial structures for optoelectronic devices such as light-emitting diodes (LEDs) and laser diodes. A wealth of information may beobtained from PL spectra, especially by taking measurements at differenttemperatures or with different excitation wavelengths or intensities. Muchof this information is particularly useful in studies of doping, which arebeyond the scope of this book and will not be elaborated here. It is alsopossible to use PL for the determination of structural information, such ascompositions and strains in heteroepitaxial layers. This is not commonlydone, though, because the analysis involves many nonstructural factors andis far more complex than structural characterization by HRXRD. An excep-tion is the case of quaternary layers, for which PL is commonly used inconjunction with HRXRD for determination of the composition and strain.Relative intensities measured by PL are useful in assessing the influence ofcrystal defects on the minority carrier lifetime. An important extension ofthis is photoluminescence microscopy (PLM), which can be used to imageindividual dislocations for the study of lattice relaxation in heteroepitaxialsemiconductors. Related luminescence imaging techniques can be used sim-ilarly; these include cathodoluminescence and electroluminescence.

This chapter describes the principles of these commonly used characteriza-tion techniques and the application of these methods to heteroepitaxial layers.

6.2 X-Ray Diffraction

High-resolution x-ray diffraction (HRXRD) is important in the structuralcharacterization of heteroepitaxial layers, revealing lattice constants, strains,crystallographic orientation, and defect densities. It can also be used inconjunction with dynamic simulations for the extraction of the compositions,strains, and thicknesses of individual layers in multilayer device structures.




251

A typical high-resolution x-ray diffractometer is illustrated in Figure 6.1.The source of x-rays is usually an x-ray tube, producing a divergent beamwith a broad spectrum. The beam is conditioned (limited in both angulardivergence and wavelength spread) by four diffracting surfaces arranged ina

Bartels monochromator

. The conditioned beam is then diffracted by the spec-imen crystal and measured using a scintillation detector. In a rocking curvemeasurement, the specimen is rotated about the

ω

-axis (which is perpendic-ular to the plane of the page and passes through the point where the beamstrikes the sample). The diffracted intensity is measured as a function of thespecimen angle

ω

. The positions, intensities, and widths of the intensity peaksin this diffraction profile (or rocking curve) are used to characterize the struc-tural properties of the specimen crystal. Application of this method thereforerequires an understanding of how the rocking curve relates to the specimencrystal structure. This section will describe the basic principles needed for theapplication of this technique to the characterization of heteroepitaxial layers,as well as some practical aspects of diffractometer instruments.

6.2.1 Positions of Diffracted Beams

Diffraction from a three-dimensional crystal is the constructive interferenceof waves scattered by the atoms in the lattice. A necessary condition fordiffraction is that the path length difference for beams scattered from differ-ent atoms be an integral multiple of the x-ray wavelength. This conditionmay be stated in two equivalent ways: the Bragg equation, which is a geo-metric equation in real space, and the Laue equations, which are the equiv-alent condition expressed in reciprocal space. The positions of the atomswithin the unit cell impose further conditions on diffraction.

6.2.1.1 The Bragg Equation

The Bragg equation for diffraction may be understood with the aid of Figure6.2. Here an x-ray beam is incident on a set of crystal planes with separation

d

. If the angles of incidence and reflection are equal to

θ

(specular reflection),

FIGURE 6.1

High-resolution x-ray diffractometer.

X-ray source

Bartelsmonochromator

Specimen

Detector

ω



252


then the path difference

Δ

between the beams a and b is . The con-dition for constructive interference is , where

n

is an integer and

λ

isthe x-ray wavelength. Thus, the condition for diffraction is

(6.1)

where

n

is the order of the reflection. This is the Bragg equation,

1

and isthe Bragg angle.

For a cubic crystal with lattice constant

a

, the spacing of the (hkl) planes is

(6.2)

The hkl Bragg angle is then

(6.3)

For a hexagonal crystal with lattice constants

a

and

c

, the spacing of the(

hkil

) planes is

(6.4)

The hkil Bragg angle is

(6.5)

FIGURE 6.2

The Bragg condition for diffraction.

dd sin θd sin θ

θ

a b

2d sin θΔ = nλ

2d nBsin θ λ=

θB

d hkl a h k l( ) ( ) /= + + −2 2 2 1 2

θ λB hkl h k l a( ) sin [ ( ) /( )]/= + +−1 2 2 2 1 2 2

d hkilh hk k

alc

( )/

/

= + + +⎛

⎝⎜⎞

⎠⎟

−2 2

2

2

2

1 2

3 4

θ λB hkilh hk k

alc

( ) sin/

= + + +⎛

⎝⎜⎞

⎠⎟⎡

⎣

−12 2

2

2

2

1 2

3 4⎢⎢⎢

⎤

⎦

⎥⎥




253

6.2.1.2 The Reciprocal Lattice and the von Laue Formulation for Diffraction

For a crystal lattice in real space, there is a corresponding lattice in recip-rocal space. This reciprocal lattice finds application in the analysis of waveinteractions with the crystal, as in diffraction. The reciprocal lattice pointslie at the tips of all wave vectors

K

that yield plane waves with the peri-odicity of the real lattice.

2

Thus, the reciprocal lattice vectors

K

are the setof vectors satisfying

(6.6)

for all real lattice vectors

R

, which are generated by

(6.7)

where

a

,

b

, and

c

are the primitive translation vectors of the real lattice and

m

,

n

, and

o

are integers. The reciprocal lattice vectors are generated by

(6.8)

where

h

,

k

, and

l

are integers. The primitive translation vectors of the recip-rocal lattice are given by

(6.9)

(6.10)

and

(6.11)

Any arbitrary set of primitive vectors for the crystal lattice will reproducethe unique reciprocal lattice. Whereas the crystal lattice vectors have unitsof length in real space, the reciprocal lattice vectors have units of length

–1

inthe associated reciprocal space. The reciprocal of the face-centered cubiclattice with lattice constant

a

is body-centered cubic with cube side .The von Laue condition for diffraction may be understood using Figure

6.3. Incident radiation with wave vector

k

is scattered at two points of thelattice that are displaced by a vector

d

. For constructive interference of thescattered waves, the path difference must be an integral number of wave-lengths, or

exp{ ( }i K R⋅ =) 1

R = + +m n oa b c

K = + +h k lA B C

A b c a b c= × ⋅ ×2π( /() )

B c a a b= × ⋅ ×2π( )/( )c

C a b a b c= × ⋅ ×2π( )/( )

4π / a



254


(6.12)

where

n

and

n

′

are the unit vectors parallel to

k

and

k

′

, respectively, and

m

is an integer. Multiplying both sides by , which is the magnitude ofthe wave vectors, we obtain

(6.13)

The condition for diffraction from the crystal is that Equation 6.13 holdfor all pairs of scatterers, or

(6.14)

for all lattice vectors

R

. This may be written in the equivalent form

(6.15)

Combining this equation with the defining equations for the reciprocal lat-tice, we arrive at the von Laue condition for diffraction. That is, constructiveinterference will occur if the scattering vector is equal to areciprocal lattice vector, or

(6.16)

where the integers

h

,

k

, and

l

are the indices for the reflection.

FIGURE 6.3

The von Laue condition for diffraction.

φ

k

k

k′

φ'

d

d cos φ = d.n

d cos φ' = d.n′

k′

d n n'⋅ − =( ) mλ

2π λ/

d k k'⋅ − =( ) 2π m

R k k⋅ − ′ =( ) 2π m

exp{ ( ) }i k k' R− ⋅ = 1

Δk k k'= −( )

( )k k A B C− ′ = + +h k l




255

6.2.1.3 The Ewald Sphere

The diffraction condition in reciprocal space may be represented by thegeometric construction of Ewald, as shown in Figure 6.4. The vector

k

isdrawn in the direction of the incident x-ray beam and terminating on theorigin. A sphere of radius is drawn with its center at the other end ofthe vector

k

and passing through the origin. This is the Ewald sphere. Areflection will be excited if any reciprocal lattice point

G

lies on the surfaceof the sphere.

6.2.2 Intensities of Diffracted Beams

Whereas the positions of the diffracted beams depend only on the unit celldimensions, the intensities of Bragg reflections depend on how the x-rays arescattered within each unit cell, by the electrons surrounding individual atoms.

In this section, the relative intensities of the Bragg reflections will bedetermined, starting with a description of the scattering of x-rays by a singleisolated electron. The individual atoms will be treated as ensembles of scat-tering electrons. Then it will be shown that the diffracted intensity is deter-mined by the interference of scattered x-rays from all of the atoms in theunit cell.

These considerations lead to many important features of x-ray diffraction,which pertain to its use as a characterization tool. These include the differentintensities observed for different reflections from a single crystal, the finiteangular widths of diffraction profiles from perfect crystals, and the forbid-den reflections.

FIGURE 6.4

The Ewald sphere construction.

C

OG

kk′

Δk = G

90° – θB 90° – θB

2θB

k = k



256


6.2.2.1 Scattering of X-Rays by a Single Electron

X-rays are scattered in all directions by a single electron, with the scatteredintensity strongly dependent on the scattering angle,

α

. This dependencewas derived by J.J. Thompson and is

(6.17)

where is the scattered intensity at a distance

r

and an angle

α

,,

q

is the electronic charge, , and

m

is theelectron rest mass, .

α

is the angle between the scattering direc-tion and the direction of acceleration for the electron, and therefore dependson the polarization of the x-ray beam.

X-ray beams obtained from x-ray tubes are unpolarized so that the electricvector

E

has a random orientation in the plane perpendicular to the beam.Referring to Figure 6.5, an unpolarized x-ray beam issuing from point Nencounters an electron at point O and the scattered beam is observed at pointP.

E

may be resolved into two orthogonal components,

E

σ

and , whereis perpendicular to both the line NO and the scattering plane NOP andis the component parallel to this plane. Because of the random nature of thedirection of

E, the mean square values are equal: . The intensityis therefore evenly divided between the two polarizations:

(6.18)

The intensity scattered to point P is the sum of the intensities for the twopolarizations. For σ polarization, , but for π polarization, π = 90° –2θ, where θ is the scattering angle. Therefore, the intensity scattered to thepoint of observation P is

(6.19)

FIGURE 6.5Scattering of a randomly polarized x-ray beam by an electron.

O

NP

Eσ

Eπ

I Iq

m r=

⎛⎝⎜

⎞⎠⎟

⎛

⎝⎜⎞

⎠⎟0

0

2 4

2 22

4μ

παsin

Iμ π0

7 14 10= × − −Hm 1 602 10 19. × − C9 11 10 31. × − kg

Eπ EσEπ

E E Eσ π2 2 2= =

I I I0 0 0 2σ π= = /

α = °90

II q

m rP =⎛⎝⎜

⎞⎠⎟

⎛

⎝⎜⎞

⎠⎟+0 0

2 4

2 22

2 41 2

μπ

θ[ cos ( )]



Characterization of Heteroepitaxial Layers 257

This is the Thompson equation for the scattering of an unpolarized x-raybeam by a single electron. In an x-ray diffraction experiment, all of the termsin this equation are constant except for , which is known as thepolarization factor.

6.2.2.2 Scattering of X-Rays by an Atom

Each electron in an atom scatters an x-ray beam according to the Thompsonequation. The net effect of the electrons is described by the atomic scatteringfactor f, defined as the ratio of the amplitude of the wave scattered by theatom to the amplitude that would be scattered by one electron. The atomicscattering factor depends on the atomic number, scattering angle, and x-raywavelength. The wavelength dependence arises from the scattering angledependence and the anomalous dispersion corrections. Anomalous disper-sion corrections are significant when the incident x-ray wavelength is com-parable to the absorption edge of the scatterer. The imaginary component ofthe anomalous dispersion correction accounts for x-ray absorption.

Numerical values for the atomic scattering factors may be obtained usinganalytic expressions available in the International Tables for X-Ray Crystallog-raphy.3 These expressions are best fits to experimentally determined atomicscattering factors and have the form

(6.20)

where . The nine coefficients a(atom,i), i = 1, 2, 3, 4, b(atom,i),i = 1, 2, 3, 4, and c(atom) are tabulated in the International Tables for X-RayCrystallography for many elements. Figure 6.6 shows the atomic scatteringfactors for the atoms N, C, Si, Ga, In, As, and P. In the case of forwardscattering (θ = 0), the atomic scattering factor is equal to the atomicnumber. There is a monotonic decrease in the atomic scattering factor withscattering angle that diminishes the intensities of reflections with largeBragg angles.

Anomalous dispersion corrections to the atomic scattering factors are alsogiven in the International Tables for X-Ray Crystallography. These correctionsaccount for the fact that bound electrons in the atom scatter differently fromfree electrons, if the frequency of the incident radiation is comparable to anabsorption frequency of the atom. The anomalous dispersion corrections arecomplex, to account for corrections to the magnitude and phase of the scat-tered radiation. For a particular atom scattering a wavelength λ, the atomicscattering factor may be adjusted for anomalous dispersion by

(6.21)

[ cos ( )]1 22+ θ

f x atom a atom i b atom i xi

02

1

4

( , ) { ( , )exp[ ( , ) ]}= −=∑∑ + c atom( )

x B= sin /θ λ

f x atom f x atom f atomcorr( , , ) ( , ) ( , )λ λ= +0




where fcorr(atom,λ) is the anomalous dispersion correction, which has beentabulated for a number of commonly used x-ray wavelengths.

6.2.2.3 Scattering of X-Rays by a Unit Cell

The x-ray amplitude scattered by a unit cell is the vector sum of the ampli-tudes scattered by the individual atoms, taking into account phase differ-ences. It is described by the structure factor, the magnitude of which isnormalized to the scattering amplitude for a single electron. If the unit cellcontains N atoms with atomic scattering factors , then the structure factorfor the hkl reflection is given by

(6.22)

where is the position of the nth atom normalized to the primitiveunit cell vectors a, b, and c.

For the zinc blende crystal structure made of atoms A and B, the structurefactor may be simplified to

(6.23)

FIGURE 6.6Atomic scattering factors for the atoms N, C, Si, Ga, In, As, and P.

00 0.5 1 1.5

10

20

30

40

50 InAsGaPSiNC

Atom

ic sc

atte

ring

fact

or, f

Sin θB/λ (Å–1)

fn

F f i hu kv lwhkl n n n n

n

N

= + +=

∑ exp{ ( )}21

π

( , , )u v wn n n

F f f i h k lhkl A B= + + +4 2{ exp[( / )( )]}π




The intensity of a reflection is proportional to , which is found bymultiplying the structure factor by its complex conjugate. For zinc blendesemiconductors, if the anomalous dispersion is neglected, there are fourcases of :

When h, k, and l are mixed even

and odd (forbidden reflection)

When is an odd multiple

of two (weak in zinc blende,forbidden in diamond)

When is odd (strong)

When is an even

multiple of two (very strong) (6.24)

If anomalous dispersion is included, Equation 6.21 must be applied, usingthe values of determined in the previous section. An immediateconsequence is that .

An important result from Equation 6.24 is that those hkl reflections withmixed reflections are forbidden for the zinc blende and diamond crystals.The 112, 001, and 003 are examples of these forbidden reflections. The 002,006, 222, and 024 are examples of reflections that are forbidden in the dia-mond structure and weak in zinc blende crystals. The 113, 115, and 333 arestrong, and the 004, 026, and 044 are very strong in both types of crystals.

For (001) heteroepitaxy of zinc blende semiconductors, the 002, 004, and006 reflections may be excited from planes parallel to the interface (sym-metric reflections). Of these, the 004 is usually preferred because of itsgreater intensity.

For (0001) heteroepitaxy of III-nitrides, the 0002 is a strong reflection thatis commonly used.

6.2.2.4 Intensities of Diffraction Profiles

The absolute intensities of diffraction peaks depend on many factors and aredifficult to predict. However, the relative intensities for hkl reflections fromsingle perfect crystals using an ideal diffractometer with zero divergenceand monochromatic radiation can be estimated by4

F2

F2

F2

0=

F f fA B

2 216= −( ) ( )h k l+ +

F f fA B

2 2 216= +( ) ( )h k l+ +

F f fA B

2 216= +( ) ( )h k l+ +

f x atom( , , )λF hkl F h k l( ) ( )≠




(6.25)

where is the magnitude of the structure factor, (1 + cos2 θB)/(sin θB cos θB)is the Lorentz polarization factor, and is the temperature factor. As anexample, for a perfect GaAs (001) crystal, this equation predicts that the 004reflection will have about 250 times the peak intensity of the 002 reflection,for the case of Cu kα radiation. This is approximately the intensity ratioobserved using a double-crystal diffractometer or Bartels diffractometer.

It should be noted that for the case of a real diffractometer, the intensitiesof broader diffraction lines are enhanced; this is because a broader Braggpeak can reflect a greater portion of the beam that is both divergent andcontains a spread of wavelengths.

Relative intensities of lines are of more fundamental importance than theabsolute intensities, which depend strongly on the instrument. Therefore,the temperature factor, which scales all reflections in equal fashion, is of littleimportance unless measurements are taken at different temperatures. Thetemperature factor has been treated in detail by James5 and Warren.6

6.2.3 Dynamical Diffraction Theory

X-ray diffraction profiles (or rocking curves) from heteroepitaxial structuresoften exhibit interesting shapes and multiple peaks, which are difficult tointerpret directly. Instead, the depth profiles of strain and composition areguessed based on the growth process. Based on this guess, the x-ray diffrac-tion profile is simulated and then compared with the experimental results.Subsequent refinement of the model structure continues until there is rea-sonable agreement between the simulated and experimental profiles. Thenthe simulation structure is assumed to closely represent the physical sample.

Early work was often based on the kinematical theory, which has beendescribed by Speriosu and coworkers.7–12 This theory, appropriate for thinheteroepitaxial films as well as powders and polycrystalline samples, treatsvolume elements of the crystal independently except for the inclusion ofincoherent power losses to the diffracted beam. Important applications ofthe kinematical theory include the simulation of x-ray diffraction profilesfrom semiconductor multilayers and superlattices, thin films, and ion-implanted regions of crystals. The advantage of the kinematical treatmentis that it reduces computational complexity and time compared to the moregenerally applicable dynamical theory. However, this advantage has becomeless important due to the advances in computer speed.

The dynamical theory has been described in detail by Darwin,13 Ewald,14–16

von Laue,17 Batterman and Cole,18 Zachariasen,19 Klar and Rustichelli,20 Tak-agi,21,22 and Taupin.23 In this theoretical treatment, all wave interactionswithin the crystal are included. The dynamical theory must be applied for

IF eB

M

B B

∝+ −2 2 21( cos )

sin cos

θθ θ

Fe M−2




the accurate determination of diffraction profiles from thick or nearly perfectcrystals, but is generally applicable to heteroepitaxial structures as well.

The foundation of the dynamical theory is a solution of Maxwell’s equa-tions in the periodic electron density of the crystal. It has enabled the calcu-lation of the intensities and shapes of diffraction profiles from infinitely thick,perfect crystals, which serves as a starting point for the study of real crystals.Dynamical theory has also been extended to the case of laminar (layered)structures,20,21,24,25 and crystals with any arbitrary distortion,22,23 for the sim-ulation of diffraction profiles from heteroepitaxial multilayered structures;this aspect is described in Section 6.12.

6.2.3.1 Intrinsic Diffraction Profiles for Perfect Crystals

Usually, the substrate for a heteroepitaxial structure may be treated approx-imately as a perfect crystal. Also, the diffraction profile for a perfect crystalserves as a starting point for the analysis of an imperfect heteroepitaxial layer.

Solution of Maxwell’s equations in the crystal yields the Takagi–Taupinequations,21–23 which describe the change in scattering amplitude with depthin the diffracting crystal. The complex scattering amplitude is the ratio ofthe diffracted and incident waves, which exchange energy through multiplescattering. Taupin23 combined this set of equations into a single differentialequation for the centrosymmetric Bragg case, which was subsequently gen-eralized to the case of polar crystals by Bartels.26 The resulting equation is

(6.26)

where T is the thickness parameter, given by

(6.27)

where t is the depth measured from the diffracting surface, is the deviationparameter, and X is the scattering amplitude, given by

(6.28)

and are the amplitudes of the diffracted and incident waves, respec-tively. and are the structure factors for the and reflection,respectively. and are the direction cosines of the diffracted and inci-dent waves with respect to the inward surface normal, and their ratio is theasymmetry factor b,

− = − +idXdT

X X2 2 1η

T tF FH H

H

=π

λ γ γ

Γ

0

η

XF

FDD

H

H H

H= γγ

0

0

DH D0

FH FH hkl h k lγ H γ 0




(6.29)

which accounts for any differences in the angles of incidence and exit forthe x-rays. For the Bragg case, the diffracted beam passes back through thesame surface through which the incident beam enters, and . For thesymmetric Bragg case, i.e., equal angles of incidence and exit, .

The deviation parameter describes the departure from the Bragg condition,

(6.30)

where is the Bragg angle, is the actual angle of incidence on the dif-fracting planes, C is the polarization factor, and is given by

(6.31)

where is the classical electron radius, , is the x-ray wave-length, and V is the crystal volume for which we have calculated the struc-ture factor. (For a cubic crystal, .) When the polarization of the incidentbeam is in the plane of incidence (π polarization), , and whenthe x-rays are polarized perpendicular to the plane of incidence (σ polariza-tion), .

For an infinitely thick, perfect crystal, the solution of the Takagi–Taupinequations yields the Darwin–Prins formula:27

(6.32)

Here, the sign must be chosen to be opposite of that of the real componentof ; in other words,

(6.33)

This equation allows the calculation of the diffracted intensity as afunction of angle for a perfect crystal. Figure 6.7 shows the calculated 004rocking curve for an infinitely thick, perfect GaAs (001) crystal.

6.2.3.2 Intrinsic Widths of Diffraction Profiles

The Darwin–Prins equation predicts that intrinsic diffraction profiles forperfect crystals should have finite width. Physically, this arises because the

b H= γ γ0 /

b < 0b = −1

η

η θ θ θ= − − − −b b F

b C F FB B

H H

( )sin( ) . ( )2 0 5 1 0Γ

Γ

θB θΓ

Γ = rV

eλπ

2

re 2 818 10 5. × − Å λ

V a= 03

C Bπ θ= ( )cos 2

Cσ = 1

X = ± −η η2 1

η

X Sign= − −η η η( ) 2 1

I X=2

θ




incident x-ray intensity reduces with each successive plane due to extinctionand absorption, and the destructive interference with is not perfect.We thus expect stronger reflections to have broader intrinsic profiles.

In the absence of absorption, the intrinsic diffraction profile has a flat topcorresponding to total Bragg reflection over a finite angular range . Thiswidth approximates the full width at half maximum (FWHM) for the intrin-sic profile and can be calculated as

(6.34)

where is the natural width of the diffraction profile for the symmetriccase (equal angles of incidence and exit), is the classical electron radius,

, is the x-ray wavelength, V is the crystal volume for whichwe have calculated the structure factor, for a cubic crystal, C is thepolarization factor, which is usually assumed to be 1, is the magnitudeof the structure factor for the reflection, and is the Bragg angle. TheFWHMs for symmetric reflections from absorbing crystals are closelyapproximated by Equation 6.34, except for the very weak reflections.

For the case of an asymmetric reflection (Bragg planes inclined to thecrystal surface), the natural width of the diffraction profile depends on theangle of incidence for the exploring beam. If the (hkl) planes are inclined tothe (mno) surface by an angle , then incidence will give a narrowerrocking curve than incidence. If is the symmetric profile widthfor , then

FIGURE 6.7004 rocking curve for a perfect GaAs (001) crystal, calculated using the Darwin–Prins formula.

0–5–10–15

0.5

1.0GaAs 004

Diff

ract

ed in

tens

ity (a

.u.)

ω – θB(Arc sec)5 10 15

θ θ≠ B

WS

Wr C F

VSe H

B

=2

2

2λπ θsin( )

WS

re

2 818 10 5. × − Å λV a= 0

3

FH

hkl θB

Φ ( )θB + Φ( )θB − Φ WS

Φ = °0




(6.35)

This effect is significant for highly asymmetric reflections. For example, inthe case of the 353 Bragg reflection from (001) GaAs using Cu kα radia-tion, and . The tables in Appendix F providethe natural widths of rocking curves for selected semiconductor crystals forthe case of Cu kα radiation.

6.2.3.3 Extinction Depth and Absorption Depth

The intensity of an exploring x-ray beam diminishes with depth in a diffract-ing crystal, as intensity is transferred to the diffracted beam; this effect iscalled extinction. On top of this, the exploring beam loses energy to photo-electric absorption within the crystal. Because of these two effects, the x-raybeam probes a finite depth of a crystal specimen. In the absence of absorption,most of the integrated intensity for a reflection originates within a distancefrom the surface, which is called the extinction depth. It is given by19,28

(6.36)

Here , which is the structure factor for the reflection, is due to thereal components of the atomic scattering factors. Typical extinction depthsfor reflections used in the characterization of heteroepitaxial layers are ofthe order of 10 μm. Weak reflections may have extinction depths of hundredsof microns; an example is the 006 reflection from GaAs (001), forwhich using Cu kα radiation. Strong reflections have smallerextinction depths; an example is the 224 reflection from HgTe (001), forwhich using Cu kα radiation.

Experimentally, diffraction profiles from layers of thickness less than are obtained with great effort due to their weak intensity. Glancing

angle geometry29 is therefore sometimes used for thin layers to reduce theextinction depth. For typical double-axis x-ray diffraction experiments withheteroepitaxial layers, 0.1 μm is the approximate minimum thicknessrequired to obtain usable intensity.

If the extinction is weak, then the thickness of crystal that contributes mostof the diffracted intensity is equal to the absorption depth , given by19,28

(6.37)

W WsB

BB+

−

= ++

⎛

⎝⎜

⎞

⎠⎟ +sin( )

sin( ); ( )

/θθ

θΦΦ

Φ1 2

inccidence

W WsB

B−

−

= ++

⎛

⎝⎜

⎞

⎠⎟

sin( )sin( )

; (/

θθ

ΦΦ

1 2

θθB − Φ) incidence

θB = ° = °63 3 62 8. , .Φ W W− + =/ 78

text

tF

ext

H

H

=πλ γ γ0

Γ

FH h k l

text = 531 μm

text = 1 6. μm

text / 100

tabs

tFabs

H

H

=′ +

λπ

γ γγ γ2 0

0

0Γ ( )




where is the magnitude of the 000 structure factor that is due to theimaginary components of the atomic scattering factors. Typical absorptiondepths in semiconductors are of the order of 10 μm, but depend strongly onthe crystal density. Absorption depths with the 004 reflection and Cu kαradiation are 8.0 μm in GaAs (density = 5.32 g/cm3) and 1.3 μm in CdTe(density = 8.17 g/cm3).

A practical implication of the absorption depth is the following: is(approximately) the maximum thickness of a mismatched heteroepitaxiallayer through which strong substrate diffraction may be observed.

In general, both extinction and absorption may be important, and thepenetration depth for the x-ray beam is then

(6.38)

Values of the extinction, absorption, and penetration depth are tabulatedin Appendix F.

6.2.4 X-Ray Diffractometers

The high-resolution x-ray diffractometers used for the characterization ofheteroepitaxial semiconductors are usually of the double-axis or triple-axistype. Therefore, there will be two or three axes perpendicular to the plane ofthe diffractometer.*

The source of x-rays is usually a sealed x-ray tube, the output of whichconsists of strong characteristic lines superimposed on a broad spectrum (thebraking radiation). Typically, the kα lines of Co, Cr, Cu, Fe, or Mo are used.The kα spectra of these elements contain one very strong peak, the kα1, anda strong peak, the kα2. The wavelengths for these lines are given in Table6.1 for the commonly used anodes. The kα1 typically has twice the integratedintensity of the kα2, and the two wavelengths usually differ by 0.2 to 0.6%in wavelength, with the kα2 at the longer wavelength.

Figure 6.8 shows the Cu kα spectrum.33 The Cu kα1 has peak intensity atλ1 = 1.540594 Å31 and a full width at half maximum of W1 = 4.61(9) × 10–4

Å.34 The Cu kα2 has peak intensity at λ2 = 1.544423 Å31 and a full width athalf maximum of W2 = 6.1(4) × 10–4 Å.34 Both peaks are asymmetric Lorent-zian distributions.

The output intensity for the x-ray tube increases linearly with the emissioncurrent and sublinearly with the accelerating voltage. Sometimes rotatinganode tubes are used because they permit higher operating currents andtherefore intensity. Occasionally synchrotron radiation is used if intensity isneeded (i.e., for the characterization of very thin layers).

* This is the plane that includes the beams incident on and diffracted by the specimen.

′F0

tabs

tp

tt tp

ext abs

= +⎛⎝⎜

⎞⎠⎟

−1 1

1




The raw x-ray beam produced by an x-ray tube is divergent and containsa spread of wavelengths. To reduce these effects, the first axis of the diffrac-tometer is fitted with one or more diffracting crystals, in order to conditionthe x-ray beam.

The specimen is mounted on the second axis, which is the most critical inthe instrument design. For high-resolution measurements, the axis 2 goni-ometer must have a step size of less than 1 arc sec, with 0.1 arc sec typical.Usually, only peak separations need to be measured, and these can be foundfrom the step size and the number of steps that have been taken. Sometimes,absolute angle encoders are affixed to the second axis of the diffractometer.

Other axes are provided that allow for tilt adjustment of the monochro-mator and the specimen. These make it possible to bring the diffractionvectors of the monochromator crystals and specimen into the plane of thediffractometer. Usually, another axis is provided for the rotation of the spec-imen about its azimuth (about the surface normal). Axis 2 is driven by a

TABLE 6.1

Commonly Used kα Wavelengths

Elementkαααα1

(Å)kαααα2

(Å)kαααα (Weighted Average)

(Å)

Co 1.78896530 1.7928530 1.790260a

Cr 2.2897030 2.29360630 2.29100a

Cu 1.54059431 1.54442331 1.54194232

Fe 1.93604230 1.93998030 1.937355a

Mo 0.7093030 0.71359030 0.710730a

a The kα1 was assigned twice the weight of the kα2.

FIGURE 6.8The Cu kα spectrum.

0.0

0.2

0.4

0.6

0.8

1.0

1.538Wavelength (Å)

kα1

Inte

nsity

(a.u

.)

kα2

1.5461.5441.5421.540




stepper motor to facilitate computer control. Often, the other axes are alsocomputer controlled so that series of experiments may be fully automated.

Usually a scintillation detector is used. This device employs a NaI crystalfollowed by a photomultiplier. Each x-ray photon produces a pulse of currentfrom the scintillation detector, and these pulses are counted for the measure-ment of the intensity. However, the detector will saturate when the currentpulses begin to overlap, typically at ~106 counts per second.

The following sections give some specifics of the three most importantinstruments for characterizing heteroepitaxial layers: the double-crystal dif-fractometer, the Bartels diffractometer, and the triple-axis diffractometer.

6.2.4.1 Double-Crystal Diffractometer

The double-crystal diffractometer is a double-axis instrument in whichmatched crystals are placed on the first and second axes. As shown in Figure6.9, there are two possible configurations for the instrument. In the (+, –)

FIGURE 6.9Double-crystal diffractometer.

(a)

(b)

X-ray source

First crystal

Specimen

Detector

(+, –) Configuration

ω

X-ray source

First crystalSpecimen

Detector

(+, +) Configurationω




configuration, the first crystal and the specimen bend the beam in oppositedirections (counterclockwise and clockwise, respectively), but in the (+, +)configuration both crystals bend the beam in the same direction. Only the(+, –) configuration is useful because the (+, +) setup is dispersive. This isbecause the first crystal does not act as a true monochromator, but insteaddisperses the various wavelengths according to the Bragg law.

This can be understood with the aid of the Dumond diagram35 shown inFigure 6.10. This dispersion relationship for the (fixed) first crystal is given by*

FIGURE 6.10Dumond diagram for double crystal diffractometer with matched crystals. Each crystal has adispersion relation given by the Bragg equation: . The finite angular width foreach crystal (exaggerated here) is due to the finite rocking curve width for the crystal. (a) Inthe parallel (+, –) arrangement, the rocking curve is narrow, because the two dispersion rela-tionships overlap only for a narrow range of ω. (b) In the antiparallel (+, +) setting, a broadrocking curve is obtained because of the wide overlap of the two dispersion characteristics.

* A first-order reflection was assumed (n = 1).

Angle θ(a)

Parallel (+, –)setting

First crystalSecond crystal(specimen)

ω

Wav

elen

gth

λ

Angle θ(b)

Anti-parallel(+, +) setting

Second crystal(specimen)

Wav

elen

gth

λ

ω

First crystal

λ θ= 2d nsin /




(6.39)

where is the spacing of the diffracting planes in the first crystal. With the(+, –) parallel configuration, the dispersion relationship for the specimencrystal is

(6.40)

where is the spacing of the diffracting planes in the specimen and isthe rocking angle (the rotation of the specimen crystal from the positionwith peak intensity). Each crystal has a finite rocking curve width, as shownin the diagram. The diffracted intensity vs. angle may be found by inte-grating the overlap of the two dispersion functions for each angle. In the(+, –) parallel arrangement, significant overlap of the two dispersion char-acteristics occurs only with . But for the (+, +) antiparallel configura-tion shown in Figure 6.10b, the dispersion relationship for the specimen isgiven by

(6.41)

and it can be seen that the two dispersion characteristics will overlap fora significant range of if the x-ray source produces a range of wave-lengths.

Mathematically, the dispersion of the double-crystal diffractometer (thebroadening due to the source wavelength spread) is given by

(6.42)

where is the Bragg angle for the first crystal and is the Bragg anglefor the specimen. The minus (plus) sign applies to the parallel (antipar-allel) arrangement.

The double-crystal diffractometer is nondispersive only if the first crystaland specimen are matched ( ) and the (+, –) parallel configuration isused. A practical difficulty in the use of the double-crystal instrument witha heteroepitaxial sample is that the first crystal cannot be matched to boththe substrate and the heteroepitaxial layer. A second limitation is that mea-surements cannot be taken in the (+, +) configuration with high resolution;this precludes the use of the double-crystal diffractometer for absolute mea-surements of lattice constants by the Bond method.

λ θ= 2 1d sin

d1

λ θ ω= +2 2d sin( )

d2 ω

ω ≈ 0

λ θ ω= − +2 2d sin( )

ω

ΔΔ

ωλ

θ θλ

= ±tan tan1 2

θ1 θ2

θ θ1 2=




6.2.4.2 Bartels Double-Axis Diffractometer

The Bartels diffractometer uses an arrangement of two channel-cut crystalswith four diffracting surfaces as a monochromator,* as shown in Figure 6.11.Typically, the monochromator uses four symmetric 220 or 440 reflectionsfrom two channel-cut Ge crystals with (110) faces. Each of the channel-cutcrystals acts as a double-crystal diffractometer in the (+, –) configuration. Inthe first channel-cut crystal, the first reflection passes a wide range of wave-lengths, but each wavelength is diffracted at a particular angle. The secondreflection accepts this entire wavelength spread, but bends the beam backinto line with the source beam. The third reflection (from the first surface ofthe second channel-cut crystal) can accept a narrow piece of this spectrum,because this crystal is antiparallel with the second and its acceptance anglefor a particular wavelength is approximately the intrinsic rocking curvewidth for this reflection. The fourth reflection brings the beam back into theline of the source beam. Therefore, the Bartels monochromator produces a

FIGURE 6.11Bartels diffractometer: (a) (+, –, +, –, +) configuration; (b) (+, –, –, +, +) configuration.

* Sometimes this arrangement is called a monochrocollimator because it reduces the beam diver-gence as well as the wavelength spread.

(a)

X-ray source


Specimen

Detector

(+, –, –, +, –) Configuration

ω

(b)

X-ray source


Specimen

Detector

(+, –, –, +, +) Configuration

ω




conditioned beam with a divergence and wavelength spread that are bothdetermined by the intrinsic rocking curve width of the monochromatorreflections. Using Ge 440 reflections, the conditioned beam exiting the mono-chromator has a divergence of 5 arc sec and a wavelength spreadof (23 parts per million). Because of this, diffraction pro-files may be measured in either geometry, (+, –, –, +, –) or (+, –, –, +, +), withvery little dispersion. Also, the monochromator need not be matched to thespecimen. Therefore, a single monochromator can be used to measure near-ideal rocking curves for a wide range of specimen crystals, even specimensthat contain layers with different lattice constants.

6.2.4.3 Triple-Axis Diffractometer

The triple-axis diffractometer proposed by Fewster36,37 has additional versa-tility due to the use of an analyzer placed between the specimen and thedetector. This instrument is illustrated in Figure 6.12. The analyzer is a crystaloriented to diffract the beam from the specimen, with an angle of acceptanceequal to its intrinsic rocking curve width. Typically, a single analyzer reflec-tion is used; however, if the analyzer is a channel-cut crystal, then it may bearranged to diffract the beam two or three times. The essential behavior ofthe instrument is similar in all three cases, however.

There are two modes of operation for the triple-axis diffractometer: thediffraction profile mode and the mapping mode. In the diffraction profilemode, the computer control is set to couple the rotations of the specimen, ,and the analyzer crystal, , such that . The final profile is given by

(6.43)

where , , and are the reflectivity profiles for the monochro-mator, specimen, and analyzer, respectively. This scan yields a near-ideal rocking curve for the specimen. It is similar to the scan measuredby the Bartels diffractometer, except that in the triple-axis case, the rockingcurve is relatively unaffected by sample curvature and distortions.

FIGURE 6.12Triple-axis diffractometer.

X-ray source


Specimen

DetectorAnalyzercrystal

ω

ω'

Δλ λ/ .= × −2 3 10 5

ω′ω ′ =ω ω2

R R R R d dm s a( ) ( ) ( ) ( )'

ω α ω α ω ω α ωαω

= − − ′ ′∫∫ 2

Rm( )ϕ Rs( )ϕ Ra( )ϕω ω− 2

ω




In the mapping mode of operation, the specimen and analyzer axes areuncoupled. In this case, the specimen axis is scanned for each particularsetting of . This results in a two-dimensional map of the reflectivity vs.and . For any particular scan of with fixed , the measured profile isgiven by

(6.44)

Also, for a scan of with fixed , the measured profile is given by

(6.45)

The two-dimensional map so obtained contains the scan as onecross section, and in practice the mapping mode is often used to obtain thisprofile, in order to eliminate difficulties associated with the critical alignmentof the analyzer crystal in the diffraction profile mode.

It is common to translate a triple-axis diffraction map from the angularcoordinates and to the reciprocal space coordinates and , with unitsof nm–1. The resulting map is called a reciprocal space map (RCS).

Reciprocal space mapping is a useful tool for the characterization ofheteroepitaxial layers, because it allows the separation of strain broadeningand angular broadening of defects. By the matching of simulated and mea-sured reciprocal space maps it should be possible to determine defect typesand distributions.

6.3 Electron Diffraction

Electron diffraction techniques, especially reflection high-energy electrondiffraction (RHEED) and low-energy electron diffraction (LEED), are impor-tant for the characterization of semiconductor surfaces. RHEED, in fact, is acritical in situ diagnostic tool for MBE growth. It allows the verification of asmooth, contaminant-free surface prior to growth, as is necessary for theepitaxy of high-quality material. It can also be used to determine the growthrate, composition, and growth mode in situ. Surface structure can be studiedby RHEED as well, although LEED is used almost exclusively for this pur-pose. Electron diffraction techniques require a high vacuum, and thereforecannot be used for in situ diagnostics during vapor phase epitaxy.

ω′ω ω

′ω ω ′ω

R R R R dm s a′ = − − ′∫ω

α

ω α ω α ω ω α( ) ( ) ( ) ( )2

′ω ω

R R R R dm s aω

ω

ω α ω ω ω ω( ) ( ) ( ) ( )′ = − ′∫ 2

ω ω− 2

′ω ω q|| q⊥




6.3.1 Reflection High-Energy Electron Diffraction (RHEED)

In a typical RHEED experiment, a high-energy (10- to 100-keV) beam ofelectrons is incident on the sample surface at a shallow angle of 1 to 2°.Diffraction of the electrons is governed by the Bragg law, as with x-raydiffraction. However, there are two important differences between RHEEDand the x-ray case. First, the electrons do not penetrate significantly into thesample, so diffraction is essentially from the two-dimensional lattice on thesurface. Second, for the high-energy electrons used in RHEED, the Ewaldsphere is large in diameter, so many reflections are excited at once.

Because the diffraction occurs from a two-dimensional net of atoms on thesurface, the reciprocal lattice comprises a set of rods perpendicular to thesurface in real space. These rods can be indexed using the two Miller indices hk.

The electrons in a RHEED experiment behave as waves, with a de Brogliewavelength given by

(6.46)

where h is the Planck constant, c is the speed of light, and E is the electronenergy. For example, an electron energy of 100 keV corresponds to a deBroglie wavelength of 3.7 pm. The radius of the Ewald sphere is = 1700nm–1, whereas the separation of the rods in the reciprocal lattice mightbe about 20 nm–1. The Ewald sphere is so large compared to the separationof the reciprocal lattice rods that it will intersect several rods, exciting severalBragg reflections for any given geometry. The diffraction pattern thereforecomprises a set of streaks, as shown in Figure 6.13.

FIGURE 6.13Reflection high-energy electron diffraction experiment.

λ = hcE

k0

2π / a

Shadow

Specimen

00

01

0–1

t

L

φ

ψ

Electron beam

Fluorescent screen




If the distance from the specimen to the screen is L and the separationbetween the 00 and hk streaks on the screen is t, then

(6.47)

If the surface structure is a square lattice with lattice constant a, then bythe Bragg law,

(6.48)

In a RHEED experiment, so that

(6.49)

Therefore, the lattice constant for the surface may be determined. This anal-ysis can be extended to other surface lattices, and by performing RHEEDexperiments at different azimuths ψ, it is possible to determine the dimen-sions of the surface unit mesh. In principle, it is also possible to determinethe surface structure (the positions of the atoms in the unit mesh), althoughthis is not usually done.

RHEED is commonly used in situ during MBE growth to discern thegrowth rate and growth mode. The growth rate may be determined fromRHEED intensity oscillations, for which the period corresponds to one mono-layer of growth. The surface roughness, and therefore the growth mode, maybe discerned from the nature of the RHEED pattern. As noted previously, astreaky pattern is an indication of an atomically flat surface. In the case ofa rough surface, the electron beam will penetrate islands or other structureson the surface, giving rise to diffraction from a three-dimensional lattice.Therefore, the RHEED pattern becomes spotty in this case.

6.3.2 Low-Energy Electron Diffraction (LEED)

LEED is an electron diffraction method that utilizes a beam of low-energy(<1 keV) electrons at normal incidence, as shown in Figure 6.14. Here, too,it can be assumed that only the top layer of atoms gives rise to the diffraction.Due to the normal incidence, however, the reciprocal lattice rods are nearlyperpendicular to the (approximately flat) Ewald sphere surface. This givesrise to a pattern of spots on the pattern corresponding to the intersection ofthe reciprocal lattice rods with this Ewald sphere. The positions of thesespots can be used to determine the unit cell of the surface mesh, and thesimulation of the spot intensities makes it possible to determine the positionsof the atoms within the unit cell. LEED is therefore used to study surfacestructure and surface reconstructions.

t L hk= tan( )2θ

λ θ θ= =+

22

2 2 1 2da

h khkhksin

sin( ) /

λ << a

a h k L t≈ +( ) //2 2 1 2 λ




Figure 6.15 shows example LEED patterns for a S(111) surface that hasbeen exposed to In, inducing a Si (111) 4 × 1-In structure.38 The left patternwas obtained using an electron energy of 70 eV, and the 4 × 1 unit cell hasbeen highlighted. The pattern on the right was obtained using an electronenergy of 120 eV. With the higher electron energy, the diffraction spots aremore closely spaced.

6.4 Microscopy

Microscopic methods include optical microscopy (OM), scanning electronmicroscopy (SEM), transmission electron microscopy (TEM), atomic force

FIGURE 6.14LEED experiment.

FIGURE 6.15LEED patterns for the Si(111) 4 × 1-In surface. The left pattern was obtained using an electronenergy of 70 eV, and the 4 × 1 unit cell has been highlighted. The pattern on the right wasobtained using an electron energy of 120 eV. With the higher electron energy, the diffractionspots are more closely spaced. (Reprinted from Wang, J. et al., Phys. Rev. B, 72, 245324, 2005.With permission. Copyright 2005, American Physical Society.)

Grids

Filament–Vp

Fluorescentscreen

WindowSpecimen

–Vp + V




microscopy (AFM), and scanning tunneling electron microscopy (STEM).Most of these are used to image the surface, and thus characterize the surfacemorphology and growth mode. They may also be used in conjunction withcrystallographic etches to determine defect densities. AFM has sufficientresolution for the study of the surface atomic structure. TEM enables theimaging of dislocations and other crystal defects within the volume of thecrystal; it is therefore important in the study of dislocations and latticerelaxation in mismatched heteroepitaxial layers.

6.4.1 Optical Microscopy

Optical microscopy (OM) is routinely used to characterize the surface mor-phology of heteroepitaxial layers, because it is rapid and nondestructive.However, the method offers only modest resolution and depth of field. Thelateral resolution may be estimated by the Rayleigh criterion to be

(6.50)

where is the optical wavelength and NA is the numerical aperture of theobjective lens. It can be improved by increasing the numerical aperture.However, this involves a trade-off with the depth of field, or axial resolu-tion , which can be estimated as

(6.51)

where n is the index of refraction. Typical objectives have numerical aper-tures of 0.1 to 1, so both the lateral resolution and depth of field are measuredin microns.

Nomarski interference contrast microscopy39 is often used for the micro-scopic inspection of heteroepitaxial layers, for the evaluation of the surfacemorphology, or for the counting of etch pits after the use of a crystallo-graphic etch. This method produces an image from the gradient of therefractive index, and therefore acts as a high-pass filter that accentuatesedges and boundaries.

6.4.2 Transmission Electron Microscopy (TEM)

Transmission electron microscopy (TEM)40 is a valuable technique for theobservation of dislocations, stacking faults, twin boundaries, and other crys-tal defects in heteroepitaxial layers. TEM characterization is applicable tomost heteroepitaxial semiconductor samples, provided that they can be

rlateral

rNAlateral = 0 6. λ

λ

raxial

rn

NAaxial = 1 4

2

. λ




thinned to transmit electrons and that they are stable when exposed to ahigh-energy electron beam in an ultrahigh vacuum. Conventional TEMs useelectron energies of ~100 keV, whereas this number can be ~1 MeV in a high-voltage TEM. For observation in a conventional TEM, typical heteroepitaxialsemiconductor samples must be thinned to less than about 100 nm. Thisrequirement may be relaxed somewhat if the sample is made up of lightatoms with low atomic number (such as Si, SiC, or sapphire) or if high-voltage electrons are used. The sample preparation is destructive, and insome cases, it can alter the defects that are to be observed.

The electrons in a TEM behave as waves, with a de Broglie wavelengthgiven by

(6.52)

where h is the Planck constant, c is the speed of light, and E is the electronenergy. For example, an electron energy of 100 keV corresponds to a deBroglie wavelength of 3.7 pm. The TEM uses lenses to produce an image,but the lenses are electromagnetic in nature. Lens aberrations, along withmechanical and electrical instabilities, usually limit the resolution of the TEMto 2 Å.

The operation of a TEM instrument is shown schematically in Figure 6.16.Collimated high-energy electrons from a condenser lens impinge on thesemiconductor specimen and are transmitted through it. The electrons arescattered into particular directions by the crystalline sample according to theBragg law for diffraction. These diffracted beams are brought into focus atthe focal plane for the objective lens.

In the diffraction mode, the first intermediate lens is focused on the backfocal plane of the objective lens, thus capturing the diffraction pattern. Thisdiffraction pattern is magnified and projected by the combination of theintermediate and projection lenses. The diffraction pattern displayed on thescreen comprises an array of spots, each corresponding to a particular dif-fraction vector g. The diffraction mode is used to index the diffraction beamsand to facilitate the selection of the diffraction spots to be used in ultimatelyforming an image.

In the imaging mode, the intermediate lens is focused on the inverted imageof the sample formed by the objective lens. This image is magnified andprojected onto the screen with an overall magnification of up to 106. An aper-ture at the back focal plane of the objective lens is used to select only onediffracted beam to form the image. If the beam transmitted directly throughthe image is chosen, a bright-field image results. If one of the dif-fracted beams is chosen to form the image, then a dark-field image is produced.

The variation of the image intensity leaving the specimen may be under-stood using two simplifying approximations. First, the specimen is assumedto behave as if made up of narrow columns (the column approximation)

λ = hcE

g = [ ]000




with axes parallel to the incident beam. The image therefore represents anintensity bit map for the array of columns. Second, it is assumed that theimage is formed by the directly transmitted beam plus only one diffractedbeam (the two-beam approximation).

A uniform, perfect crystal, with uniform thickness, will produce an imagewith uniform electron intensity. Image contrast results from crystal nonuni-formities, including variations in thickness, changes in composition, inclu-sions, and voids. Dislocations may also produce image contrast, if theydisplace the diffracting planes such that their separation or orientationchanges. Based on the column approximation, we can state that the conditionfor image contrast by a crystal defect is , where u is the vector bywhich atoms are displaced from their normal sites within a particular col-umn. Put another way, the condition for invisibility is .

For an edge or screw dislocation, image contrast will result if the Burgersvector has a component in the direction of the diffraction vector. In other

FIGURE 6.16Transmission electron microscope.

Electron gun

First condenser lens

Second condenser lens

Objective lens

First intermediate lens

Second intermediate lens

Projector lens

Specimen

Condenser aperture

Objective aperture

Selected area aperture

Phosphor screen

g u⋅ ≠ 0

g u⋅ = 0




words, the condition for invisibility (zero contrast) for an edge or screwdislocation is

(invisibility condition) (6.53)

For dislocations of mixed character, there is no condition for which .Instead, the invisibility criterion is approximately satisfied for andweak contrast is observed even if .

The invisibility condition can be applied to the determination of the Burg-ers vector direction for a dislocation. If a dislocation is invisible (or nearlyinvisible, in the case of a mixed dislocation) in two images produced usingthe diffraction vectors g1 and g2, then its Burgers vector must be perpendicularto both diffraction vectors. This means that its Burgers vector is in the direc-tion . In this way, the Burgers vector and the character (edge, screw,or mixed) may be determined for dislocations in a heteroepitaxial layer.

A critical step in any TEM experiment is the sample preparation. It isnecessary to prepare a thin foil that includes the region to be examined (forexample, the heterointerface) and that is thin enough to transmit the elec-trons. Often this can be achieved by a combination of wet etching and ionmilling. In some cases, etch stop layers can be used for the preparation ofthin foils. For example, plan view samples of heteroepitaxial zinc blendelayers on GaAs can be prepared by the use of a thin etch release layer suchas AlAs. Selective etching of the AlAs layer allows release of the heteroepi-taxial layers above, which can be floated on water and picked up by a carbon-coated TEM grid for microscopic investigation.41

If the results from TEM examination are to be meaningful, the samplestudied must be stable under irradiation by a high-energy electron beam.This condition is not always met in heteroepitaxial samples. For example,in the case of InxGa1–xAs, it has been found that electron irradiation of thesample can excite motion of glissile dislocations, and that the glide could bestarted or stopped by condensing or expanding the electron beam.41

6.4.3 Scanning Tunneling Microscopy (STM)

The scanning tunneling microscope (STM) can measure surfaces withatomic-scale resolution. It was invented by Binnig and Rohrer in 1981, forwhich they received the Nobel Prize in 1986. The basic principle of operationcan be understood using Figure 6.17. Here, Px and Py are piezoelectric ele-ments that allow a metal tip to be scanned over a surface with Angstrompositioning accuracy. A feedback control unit (CU) biases the third piezo-electric element such that a constant tunneling current flows between the tipand the surface. Because the tunneling current varies exponentially with thetip-to-surface separation, the tip will follow the contour of the surface duringthe scan. The deflections of the three piezoelectric elements are proportional

g b⋅ = 0

g u⋅ = 0g b⋅ = 0

g b⋅ = 0

g g1 2×




to the three bias voltages Vz, Vx, and Vy. Therefore, a map of Vz as a functionof Vx and Vy gives a topographic map of the surface.

The tunnel current density flowing from the tip to the surface dependsstrongly on the work function of the surface and the tip-to-surface sepa-ration s:

(6.54)

where . Therefore, for a typical surface work function of~1 eV, a single atomic step on the surface would change the tunneling currentby three orders of magnitude, in the absence of vertical adjustment by Pz.This leads to a vertical resolution of 0.2 Å.

As shown in Figure 6.17, false surface features can emerge as a consequenceof surface contaminants that modify the work function (C in Figure 6.17).However, the work-function-mimicked features can be separated from truesurface structures by modulating the tip distance during the scan.

Even though practical probe tips will tend to be blunt and irregular inshape (see Figure 6.18), the exponential dependence of the tunneling cur-rent on tip separation causes a localized fine tip, or even a single atom onit, to be active in the tunneling. Therefore, the lateral resolution of 10 Å canbe achieved.

The STM provides atomic-scale images of surfaces and can be used tostudy surface steps and kinks, the growth mode, and the earliest stages ofislanding. Also, because there are surface trenches associated with subsurfacemisfit dislocations in very thin layers, STM can be used to study the initiationof lattice relaxation in highly mismatched heteroepitaxial layers. A drawbackof this instrument, however, is the necessity for a conducting specimen. Thesurfaces of insulators may not be investigated due to charging effects. More-

FIGURE 6.17Scanning tunneling microscope (STM). (Reprinted from Binnig, G. et al., Phys. Rev. Lett., 49, 57,1982. With permission. Copyright 1982, American Physical Society.)

CU s

VT

JT

Py

Pz

PxVP

A

B

C

δ

Δs

ψ

J A sT ∝ −exp( )/ψ1 2

A = − −1 025 1 1 2. /Å eV




over, semiconductors with high resistivity or native oxide films cannot beexamined by STM.

6.4.4 Atomic Force Microscopy (AFM)

The atomic force microscope is an instrument with similar resolution andapplications as the STM, but it can be used with insulating surfaces. TheAFM, first proposed by Binnig et al.42 in 1986, combines the principles of theSTM and the stylus profilometer, resulting in an atomic-scale profilometer.

The operation of the AFM can be understood with the aid of Figure 6.19.A cantilever with a sharp tip (which need not be conducting) is placedbetween the STM tunneling tip and the sample to be examined by atomicforce microscopy. While scanning, a very small and constant force is main-tained on the AFM tip. This can be done by different means, in one of severaloperating modes. In one mode of operation, the force exerted on the AFMstylus by its piezoelectric element is adjusted for constant tunneling currentin the STM. Regardless of the details of how the feedback system isemployed, the z displacement of the STM corresponds to the z displacementof the surface examined by the AFM.

Some of the capabilities of this characterization technique are apparent inthe AFM micrographs of Figure 6.20. The images show surfaces ofIn0.65Ga0.35As layers of different thicknesses, grown on InP (001) substratesby MOVPE and reported by Jasik et al.43 The layer of Figure 6.20a is 23.4 Åthick. Monolayer surface steps can be seen clearly, with an average stepspacing of 200 nm, corresponding to an off-cut angle of about 0.25°. The layer

FIGURE 6.18STM tip over a surface. The tip may be blunt and irregular; however, the exponential depen-dence of the tunneling current causes one microtip to be active. In this picture, the uppermicrotip is expected to give rise to a tunneling current 1/1000th of the tunneling currentassociated with the lower tip. (Reprinted from Binnig, G. and Rohrer, H., Rev. Mod. Phys., 59,615, 1987. With permission. Copyright 1987, American Physical Society.)

Tunnel tip




of Figure 6.20b is 103.4 Å thick and displays a more irregular arrangementof surface steps. The new features that have appeared on the 125-Å-thicklayer of Figure 6.20c are associated with misfit dislocations at the interface.

6.5 Crystallographic Etching Techniques

Crystallographic etches can be used to reveal crystal defects, such as dislo-cations or stacking faults. They are also useful for the delineation of interfacesand also p-n junctions. Moreover, crystallographic etches produce pits withorientation-dependent characteristics, so they can be used to identify crystaldirections and detect inversion domains. In all of these applications, theusefulness of the etch comes about because of its sensitivity to the surface;defects, interfaces, or junctions are revealed due to the modified etch rate intheir vicinity.

Quite generally, an etching process may be polishing or crystallographic.Either type of process involves the transport of reactants through a diffusionboundary layer, a reaction or reactions at the surface, and transport of thereaction products away from the etching surface. If the etch rate is limitedby either the diffusion of reactants to the surface or the products away from

FIGURE 6.19Atomic force microscope (AFM). (a) System schematic. (The tip is not shown to scale.) (b)Cantilever with diamond tip, showing the dimensions. (Reprinted from Binnig, G. et al., Phys.Rev. Lett., 56, 930, 1986. With permission. Copyright 1986, American Physical Society.)

Scanners.feedback

AFMFeedback

STMA

B

C

DF

E

(a)

Block (aluminum)

x

zy

F 1 cm

(b)

A: AFM sampleB: AFM diamond tip

E: Modulating piezoF: Viton

C: STM tip (Au)D: Cantilever,

STM sample

25 μm

0.25 mm

Lever(Au-foil)

Diamondtip

0.8 mm




the surface, it is termed diffusion limited or mass transport controlled. A diffu-sion-limited etch is insensitive to the crystallographic orientation and localvariations in the surface structure; it is therefore polishing. On the otherhand, a reaction-rate-limited etch is sensitive to the surface and will be acrystallographic etch. The etch rate in such a case will vary across the surfaceand can be used to reveal defects such as dislocations.

Wet chemistry is used almost exclusively for crystallographic etching.However, gaseous etching processes may also be reaction rate limited underthe proper conditions of temperature and flow rates. For example, Tachikawaand Mori44 demonstrated the use of HCl-GaCl for crystallographic etchingof GaAs/Si (001) and GaP/Si (001) in the growth chamber, at the growth

FIGURE 6.20Surface steps as observed by AFM on a 23.4-Å-thick layer of In0.65Ga0.35As on InP (001).

(Reprinted from Jasik, A. et al., Thin Solid Films, 412, 50, 2002. With permission. Copyright2002, Elsevier.)

(a) (b)

(c)

0 1.0 2.00

1.0

2.0

μm0 1.00 2.00

μm

0

1.0

2.0

μm

μm

0

1.0

2.0

0 1.0 2.0




temperature. As another example, Ohno et al.45 found that a hydrogenplasma revealed dislocation etch pits on the surface of ZnSe.

Wet chemical etches usually comprise an oxidizer, a complexing agent,and a diluent. The most commonly used oxidizers are H2O2 and HNO3. Thecomplexer reacts with the oxidized surface to create a water-soluble complex.While HF is the most common complexer, nitric, sulfuric, phosphoric, andcitric acids are also sometimes used. Br2, which is used in some etch formu-lations, serves the dual role of oxidizer and complexer. The diluent, usuallywater or CH3COOH, is sometimes omitted.

The most common application for crystallographic etching is the evaluationof dislocation densities in bulk or heteroepitaxial semiconductors. Here, thepoints where dislocations emerge at the surface are marked by the appear-ance of hillocks or, more commonly, pits. These features occur due to thereduced or enhanced etch rate in the strained region around the dislocation.

Etch pits on the surface of a semiconductor crystal usually reveal thecrystal symmetry and can be used to determine the orientation. For example,Si (111) treated by Sirtl etch shows triangular pits, while molten KOH etchingof 6H-SiC (0001) reveals hexagonal pits. On the other hand, molten KOHetching of GaAs (001) produces approximately rectangular pits, which areelongated along the direction. This makes it possible to distinguishthe [110] and directions in the surface, and therefore find whetherinversion domains are present.

Crystallographic etching has also been used extensively to delineate p-njunctions or other interfaces in multilayered heteroepitaxial structures. Forexample, dilute (15%) A-B etch in water etches p-type material much fasterthan n-type material and has been used to delineate p-type regions in cleavedAlGaAs/GaAs laser structures.46 A modified A-B etch has also been used todelineate compositional steps on the cleaved edge of a GaAsP/GaAs/Ge(001) heterostructure.47

The application of crystallographic etching to the determination of thread-ing dislocation densities is detailed in Section 6.11.2. The compositions ofsome commonly used crystallographic etches are tabulated in Appendix E.

6.6 Photoluminescence

Photoluminescence (PL) is commonly employed ex situ to assess the suit-ability of heteroepitaxial structures for optoelectronic devices such as LEDsand laser diodes. A wealth of information may be obtained from PL spectra,especially by taking measurements at different temperatures or with differ-ent excitation wavelengths or intensities. Much of this information is partic-ularly useful in studies of doping, which are beyond the scope of this bookand will not be elaborated here. It is also possible to use PL for the deter-mination of structural information, such as compositions and strains in het-

[ ]110[ ]110




eroepitaxial layers. The application to quaternary layers is of special interest;here, PL is commonly used in conjunction with HRXRD for determinationof the composition and strain. Photoluminescence microscopy (PLM) is animaging technique that creates a map of the PL intensity. This can be usedto study electronically active defects such as dislocations, which show up asdark regions on PLM images.

A typical photoluminescence setup is shown schematically in Figure 6.21.A laser with above-bandgap photons excites electron–hole pairs in the sam-ple. Recombination of the excess carriers gives rise to the emission of char-acteristic wavelengths associated with the electronic transitions in thesample. The emitted radiation (which may span the range from ultravioletto the infrared, depending on the specimen) is collimated by a collectinglens, L1, and then focused on the entrance slit of the monochromator by asecond lens, L2. The monochromator is scanned in wavelength during theexperiment, so an intensity vs. wavelength spectrum is obtained. At eachwavelength, the intensity passing through the monochromator is measuredusing a photomultiplier tube. Usually, the exciting laser beam is chopped tofacilitate phase-sensitive detection with a lock-in amplifier; this greatlyimproves the signal-to-noise ratio.

Luminescence imaging techniques are important for the study of defectsin heteroepitaxial layers. This is because enhanced nonradiative recombina-tion of carriers occurs in the vicinity of electronically active defects such asdislocations. They therefore show up as darkened regions on a map ofluminescence intensity.

Photoluminescence microscopy (PLM) is an imaging technique that canbe performed in a micro-PL system or a scanning near-field optical micro-

FIGURE 6.21Photoluminescence setup.

Specimen

Laser

Gratingmonochromator

S2

Chopper

Collectinglens

Focussinglens

Photomultiplier

Slit

Slit

S1

L1L2




scope (SNOM). As with standard PL, the PLM technique involves the exci-tation of electron–hole pairs in the sample by laser illumination, therecombination of the resulting free carriers, and the detection of emittedlight. However, PLM requires spatial resolution of the signal so that an imagecan be obtained. Typically, a microimaging system with a resolution of 1 μmis used.48

In a micro-PL system, the laser excitation is spot focused (typical spot size,~50 μm) to allow illumination of a small area of the sample. In the case ofmultilayer structures, the incident wavelength can be chosen to stimulatephotoluminescence from only the layers with the smallest bandgaps. Theresulting images are captured using a digital camera. Typically, cryogenictemperatures (e.g., 77K) are used with the advantage of one to three ordersof magnitude increase in the photoluminescence intensity.

PLM images may also be obtained using a scanning near-field opticalmicroscope (SNOM).49 In this case, both the laser illumination and collectedlight pass through optical fibers. Using specially prepared metal-coatedfibers with tapered tips,50 images with submicron resolution can be obtained.A typical SNOM-based PLM arrangement is shown in Figure 6.22. Theexcitation comes from a laser diode coupled to an optical fiber. This can bescanned over the surface of the specimen. The luminescence is collected byan ellipsoidal mirror and a lens, and then fed to the monochromator througha second optical fiber. Scanning of the excitation source fiber aperture in xand y allows the creation of a PL intensity map. The monochromator is fixedupon a particular wavelength for the measurement of the PLM image. Often,this is the wavelength corresponding to peak intensity in the PL spectrum.Sometimes, other wavelengths are chosen to study subtle aspects of theelectronic behavior of defects.

The depth sensitivity of PLM may be tailored by adjustment of the exci-tation wavelength.51 Generally, a shorter wavelength will have a largerabsorption coefficient and will be absorbed close to the sample surface.Therefore, the resulting PLM image will be associated with photons emittedfrom the material near the surface. On the other hand, a longer wavelengthwill penetrate more deeply into the sample.

Figure 6.23 shows the PLM image for a GaAs/In0.15Ga0.85As multiquantumwell structure (seven periods, 50 nm of GaAs, and 16 nm of In0.15Ga0.85As)grown on a GaAs (001) substrate by MBE.52 Dark lines corresponding tomisfit dislocations are seen to run parallel to the [110] and directions.This image was obtained using the peak emission wavelength from the PLspectrum and a bandwidth of 60 meV. The areas of the dark lines exhibit a25% reduction in PL intensity compared to the surrounding regions.

Cathodoluminescence is another imaging technique that can be used tostudy defects in heteroepitaxial structures. This method bears many similar-ities to PLM, except that it is conducted in a scanning electron microscopeusing an electron beam for the excitation.

[ ]110




FIGURE 6.22Photoluminescence microscopy setup. (Reprinted from Ohizumi, Y. et al., J. Appl. Phys., 92, 2385,2002. With permission. Copyright 2002, American Institute of Physics.)

FIGURE 6.23PLM image for a GaAs/In0.15Ga0.85As multiquantum well structure (seven periods, 50 nm ofGaAs, and 16 nm of In0.15Ga0.85As) grown on a GaAs (001) substrate by MBE. (Reprinted fromOhizumi, Y. et al., J. Appl. Phys., 92, 2385, 2002. With permission. Copyright 2002, AmericanInstitute of Physics.)

Optical fiber

Monochromator

LensPhotomultiplier

Ellipsoidal mirror

Laser diodeλ = 635 nm

Optical fiber probe

Filter

Bimorph

Sample



[110][110]


6.7 Growth Rate and Layer Thickness

Layer thicknesses and growth rates have been determined by many means;some are simple (gravimetric measurements), whereas others are quitesophisticated (RHEED oscillations, Fourier transform infrared spectroscopy,x-ray Pendellosung).

RHEED provides a means for the in situ determination of growth rates andlayer thicknesses in an MBE growth chamber. Because the electron beam isincident on the sample at a very shallow angle (typically 1 to 2°), RHEEDanalysis can be performed while the heteroepitaxial layer is growing. In thecase of Frank–van der Merwe (layer-by-layer) growth, a streaky diffractionpattern is obtained and the intensity of a particular streak oscillates with time.Much information is contained in the RHEED oscillation characteristics, andsophisticated models have been developed for their analysis. To determine thegrowth rate and layer thickness, it suffices to recognize that one period of theRHEED oscillations corresponds to the growth of 1 ml. The intensity is max-imum for a smooth surface. The nucleation of a new layer on this surfacecauses its roughening until the new layer completes, and then the processrepeats. If the period of the RHEED oscillations is T, then the growth rate is

(6.55)

Figure 6.24 shows representative RHEED oscillations measured during theMBE growth of GaAs (001) with incident angles of 1.33° and 0.93°, as indicated.Here the period of oscillations is 2.2 s, corresponding to a growth rate of 0.45ml/s, or 230 nm/h. Usually, the RHEED oscillations decay after only a fewmonolayers; in Figure 6.24 only about eight periods of oscillation are observed.

A convenient method for rapid, nondestructive layer thickness measure-ment is based on the reflectance characteristic, measured with a Fouriertransform infrared (FTIR) spectrometer. The reflectance (or transmittance)curve will contain an interference pattern, due to the interference of thewaves reflected at the epitaxial layer surface and the layer–substrate inter-face, as long as the epitaxial layer and substrate have different indices ofrefraction. In practice, this condition is satisfied even in the case of homoepi-taxy due to the change in doping at the interface. If the measured interferencefringe pattern contains m periods in the range of wavenumbers from to ,the layer thickness is given by

(6.56)

gMLT

= 1

ν1 ν2

hm

n=

− −

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

2

11 12 2

1 2

sin θν ν




where n is the index of refraction for the heteroepitaxial layer and is theangle of incidence. Figure 6.25 shows example FTIR spectra for silicon epi-taxial layers having different thicknesses.

For layers having high crystal perfection and smooth surfaces, the thick-ness can be determined from the x-ray rocking curve using the Pendellosunginterference fringes. The spacing of the Pendellosung is given by

(6.57)

where is the x-ray wavelength, is the Bragg angle, is the anglebetween the diffracting planes and the surface (is the angle of inci-dence), and h is the layer thickness. For example, for a symmetric 004 reflec-tion from a (001) layer of GaAs, 1 μm thick, the Pendellosung spacing isexpected to be 28 arc sec. Pendellosung can only be observed with a small-area x-ray beam. Otherwise, thickness nonuniformities or bending will sup-

FIGURE 6.24RHEED oscillations observed during the homoepitaxial growth of GaAs (001) on a 2 × 4reconstructed surface. The primary beam energy was 20 keV. (a) Original curves; (b) normalizedcurves. Here, an energy filter was applied in front of the RHEED screen; the filter settings wereas shown. (Reprinted from Braun, W. et al., J. Vac. Sci. Technol. B, 16, 2404, 1998. With permission.Copyright 1998, American Institute of Physics.)

Time (s)151050

0.91°

1.33°

0

0

< 30 eV< 15 eV

< 2 eV

< 30 eV< 15 eV

< 2 eV

< 30 eV

< 30 eV

< 15 eV

< 15 eV

< 2 eV

< 2 eV

20 25

RHEE

D in

tens

ity (A

rb.u

nits

)

(b)

(a)

θ

Sh

B

B

= ±λ θ φθ

sin( )sin( )2

λ θB φθ φB ±




press the fringing. Usually, Pendellosung are observed only for thin layers,whereas in thicker layers the nonuniformities, surface roughening, or defectswill extinguish them. Figure 6.26 shows a simulated rocking curve for 0.2μm of AlAs on GaAs (001), in which the Pendellosung are clearly seen.

Layer thicknesses can be determined directly using TEM cross-sectionalmicrographs. In a multilayer structure, there is usually sufficient contrast sothat all individual layers can be distinguished. In partially relaxed structures,defects enhance the visibility of interfaces. Marker layers (e.g., AlAs in GaAs,Ge in SiGe) can be inserted to delineate a particular point in the growthprocess, which is a unique feature associated with TEM.

6.8 Composition and Strain

The strain in a single heteroepitaxial layer is most easily determined usingdouble-axis x-ray diffraction (double-crystal or Bartels diffractometer). It isusually appropriate to assume the strain is constant with depth. Then, for abinary layer, the relaxed lattice constant is known, so there is only one

FIGURE 6.25FTIR spectra measured for three Si epitaxial layers of different thicknesses, as indicated. (Re-printed with permission from Thermo Electron Corporation, Madison, WI.)

0.360.350.350.340.340.330.330.320.320.310.310.300.300.290.290.280.28

Tran

smitt

ance

1200.0 1100.0 1000.0 900.0 800.0

50 Micron film

15 Micron film

10 Micron film

Wavenumber (cm–1)




independent unknown. (Once the in-plane or out-of-plane lattice constantis known, the other may be calculated.) In a strained ternary alloy (or thealloy Si1–xGex), there are two independent unknowns. One possible set is therelaxed lattice constant and the in-plane lattice constant. Thus, two measure-ments are necessary to characterize the layer. The number of required mea-surements increases with the number of degrees of freedom in the layer.Also, if there is a crystallographic tilt between the epitaxial layer and thesubstrate, this introduces two more unknown variables (the magnitude anddirection of tilt), thus necessitating additional measurements. This sectionwill detail the application of x-ray measurements to characterize the com-position and strain in these heteroepitaxial layers.

6.8.1 Binary Heteroepitaxial Layer

In a nonalloyed heteroepitaxial layer such as Ge or GaAs (referred to as abinary layer here), the relaxed lattice constant is known, and this greatlysimplifies the analysis. If the substrate is much thicker than the epitaxiallayer, it is assumed to be unstrained (thick substrate approximation), withits normal Bragg angle, . It is most convenient to use symmetric rocking

FIGURE 6.26Simulated x-ray rocking curve for 0.2 μm of AlAs on GaAs (001) using Cu kα1 radiation. Thespacing of the Pendellosung interference fringes is inversely proportional to the epitaxial layerthickness. (Reprinted from Kim, I. et al., J. Appl. Phys., 83, 3932, 1998. With permission. Copyright1998, American Institute of Physics.)

100

10–1

10–2

10–3

10–4

10–5

10–6

X-ra

y refl

ecta

nce

0.2 μm AlAs/GaAsGaAs substrateInteger n

n = 0

–1000 –500 0 500Rocking angle (Arc sec)

θBS




curves (i.e., 004 for zinc blende (001) semiconductors or 0002 for wurtzite(0001)) for the analysis.

The angular separation between the epitaxial layer and substrate peaks inthe rocking curve (symmetric 00m reflection) at an azimuth is

(6.58)

where is the difference in Bragg angles between the epitaxiallayer and the substrate, , is the crystal-lographic tilt between the [001] axes of the epitaxial layer and the substrate,and specifies the direction of this tilt. In order to find the strain in theepitaxial layer, it is necessary to measure rocking curves at two or moreazimuths. If rocking curves are measured at opposing azimuths*and , then

(6.59)

The out-of-plane lattice constant is then determined from the Bragg anglefor the epitaxial layer.

For a diamond or zinc blende heteroepitaxial layer, using the reflection,

(6.60)

The out-of-plane strain is

(6.61)

and assuming biaxial stress and tetragonal distortion, the in-plane strain is

(6.62)

Similarly, for a wurtzite or hexagonal SiC crystal, using the reflection,

(6.63)

* The reference for the azimuth is arbitrary and can be set for convenience.

ψ

Δ Δ ΔΦθ ψ θ ψ ψ00 00 0m B m( ) cos( )= + −

ΔθB m00 00mΔθ θ θB m B m epitaxial B m substrate00 00 00= −, , ΔΦ

ψ0

ψ = °0ψ = °180

Δ Δ Δθ θ ψ θ ψB m00

0 1802

= = ° + = °( ) ( )

00m

cm

B m substrate B m

=+

λθ θ2 00 00sin( ), Δ

ε⊥ = −c aa

0

0

ε ε|| = − ⊥2 12

11

CC

000m

cm

B m substrate B m

=+

λθ θ2 00 00sin( ), Δ




The out-of-plane strain is

(6.64)

and assuming biaxial stress, the in-plane strain is

(6.65)

If it is necessary to determine the crystallographic tilt, then the rocking curvemust be measured at least one more azimuth.

6.8.2 Ternary Heteroepitaxial Layer

For a ternary alloy layer such as InxGa1–xAs, or an alloy such as GexSi1–x, theindependent determination of the relaxed lattice constant (and therefore thecomposition) and the state of strain requires measurements of two differenthkl rocking curves. Sometimes the analysis is simplified with the assumptionthat the heteroepitaxial layer has grown coherently on the substrate.53–55 Withthis pseudomorphic assumption, the in-plane lattice constant is assumed tobe equal to that of the substrate. Then a single rocking curve measurement,using a symmetric reflection, is used for the estimation of the compositionand state of strain in a ternary layer. This simplified approach has beenextended to quaternary semiconductors, for which a single x-ray rockingcurve measurement is combined with a photoluminescence measurement todetermine the bandgap (and therefore the composition and relaxed latticeconstant) for the material. Such a simplified approach is suitable for a het-eroepitaxial system such as AlGaAs/GaAs, for which the lattice mismatchstrain is small over the entire range of composition. Usually, however, it isnot possible to start with the pseudomorphic assumption.

Typically, for heteroepitaxy of a (001) zinc blende substrate, rocking curvesare obtained for one symmetric reflection such as the 004 and one asymmetricreflection such as 115 or 044. Then, with the assumption that the strainedalloy is tetragonally distorted, the in-plane and out-of-plane lattice constants(a and c, respectively) may be determined. However, it is necessary toaccount for both the crystallographic tilting of the epitaxial layer with respectto the substrate and the additional tilting of the asymmetric planes due tothe tetragonal distortion. The standard procedure for analysis of a zincblende epitaxial layer using a symmetric 00m reflection and an asymmetrichkl reflection will be described below. However, the adaptation of this pro-cedure to hexagonal epitaxial layers is a straightforward extension.

As in the case of the nonalloyed (binary) semiconductor layer, the analysisstarts with a symmetric reflection, which is measured at two opposing azi-

ε⊥ = −c cc

0

0

ε ε|| = − ⊥2 13

33

CC




muths. If the 004 reflection is used, then the rocking curve peak separationat an azimuth will be

(6.66)

where is the difference in 004 Bragg angles between the epitaxial layerand the substrate, is the crystallographic tilt between the epitaxial layerand the substrate, and specifies the direction of this tilt. The differencein Bragg angles for the 004 reflection is found by averaging the peak sepa-ration for two opposing azimuths:

(6.67)

The out-of-plane lattice constant can be determined as before, with theassumption that the substrate is unstrained (thick substrate approximation):

(6.68)

An additional complication arises if one attempts to use the aboveapproach with an asymmetric reflection such as 044. In such cases there isan additional tilt component, , if the heteroepitaxial layer is tetragonallydistorted:

(6.69)

Like before, the measurement of the asymmetric rocking curves at oppos-ing azimuths, for the same set of planes, allows elimination of the tilt compo-nent, .56–58 However, the disadvantage of that approach is that it requiresmeasuring the rocking curve for one azimuth using incidence, asshown in Figure 6.27. This leads to a rocking curve peak that is broadenedand weakened in intensity. Specifically, the intensity ratio for the two anglesof incidence can be estimated as59

(6.70)

where and are the intensities for and inci-dence, respectively. For example, in the case of the 044 reflection from GaAs(001) with Cu kα radiation, the intensity ratio is 112. This means that thereflected intensity will be insufficient for the purpose of an accurate mea-

ψ

Δ Δ ΔΦθ ψ θ ψ ψB B004 004 0( ) cos( )= + −

ΔθB004

ΔΦψ0

Δ Δ Δθ θ ψ θ ψB004

0 1802

= = ° + = °( ) ( )

cm

B m substrate B m

=+

λθ θ2 00 00sin( ), Δ

ΔΦtet

Δ Δ ΔΦ ΔΦθ ω θ ψ ψB B tet044 044 0( ) cos( )= + − +

ΔΦtet

θB − Φ

II

B

B

B

B

( )( )

sin ( )sin ( )

θθ

θθ

+−

= +−

ΦΦ

ΦΦ

2

2

I B( )θ + Φ I B( )θ + Φ θB + Φ θB − Φ




surement in that case. Instead, it is necessary to measure the rocking curvesat both azimuths with the incidence, as shown in Figure 6.28. Thenthe two rocking curves are measured from two different sets of {011} planes.The tilt, , does not cancel out when we average the two peak separations,but must be extracted in the analysis.

FIGURE 6.27Asymmetric 044 reflections from a (001) zinc blende crystal, at opposing azimuths using thesame set of diffracting planes. (a) with incidence; (b) with in-cidence. (Reprinted from Zhang, X.G. et al., J. Vac. Sci. Technol. B, 18, 1375, 2000. With permission.Copyright 2000, American Institute of Physics.)

FIGURE 6.28Asymmetric 044 reflections from a (001) zinc blende crystal, at opposing azimuths using twodifferent sets of diffracting planes. (a) ψ = 90°; (b) ψ = 180°. Both rocking curves are measuredat incidence. (Reprinted from Zhang, X.G. et al., J. Vac. Sci. Technol. B, 18, 1375, 2000.With permission. Copyright 2000, American Institute of Physics.)

Detector Detector

Incidentbeam

Incidentbeam

φ = 45

°

φ = 45°

θB = 50.4° θB = 50.4°

(001)(010)

(010)(001)

Diffractedbeam

Diffractedbeam

(a)

(011) (011)

(b)

ψ = °0 θB + Φ ψ = °180 θB − Φ

θB + Φ

ΔΦtet

(a) (b)

Detector

Incidentbeam

(001)

(010)

Diffractedbeam

(011) (011)φ =

45°

θB = 50.4°

Detector

Incidentbeam

(001)

(010)

Diffractedbeam

φ = 45°

θB = 50.4° −

θB + Φ




Using an asymmetric hkl reflection, the peak separation between the epi-taxial layer and the substrate is averaged for the two opposing azimuths,with both rocking curves measured at incidence. For example,if corresponds to the projection of the incident beam aligned withthe [011] direction, then

(6.71)

(Note that is not the same as the Bragg angle difference.) The spac-ing of the hkl planes can be found from

(6.72)

The in-plane lattice constant in the strained heteroepitaxial layer is foundfrom

(6.73)

The tilting of the hkl planes due to the tetragonal distortion can be calcu-lated as

(6.74)

Equations 6.72 to 6.74 must be solved iteratively, starting with an assumedvalue of . Typically, the values of , a, and converge after six orfewer iterations.

Once the in-plane and out-of-plane lattice constants have been determined,the relaxed lattice constant may be calculated from

(6.75)

where is the Poisson ratio. The relaxed lattice constant is used to deter-mine the composition, either by using the known lattice constant vs. com-

θB + Φψ = °0

Δ Δ Δθ θ ψ θ ψAVE hkl,

( ) ( )= = ° + = °45 2252

ΔθAVE hkl,

dhklBhkl substrate AVE hkl tet

=+ −λ

θ θ2 sin( ), ,Δ ΔΦ

ah k

l c l dhkl

= +−

⎛

⎝⎜⎞

⎠⎟

−2 2

2 2 2

1 2

/ /

/

ΔΦtetl c

h a k a l c=

( ) + ( ) + ( )

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟

−cos/

/ / /

1

2 2 2 ⎟⎟−

+ +

⎛

⎝⎜

⎞

⎠⎟

−cos 1

2 2 2

1

h k l

ΔΦtet dhkl ΔΦtet

ac a

0

21

12

1

=+

−⎛⎝⎜

⎞⎠⎟

+−

⎛⎝⎜

⎞⎠⎟

νν

νν

ν




position characteristic or by using a linear interpolation with the latticeconstants of the binaries (Vegard’s law). If the Poisson ratio varies stronglywith composition, iteration can be used to find a consistent solution forand . Finally, the in-plane and out-of-plane strains in the heteroepitaxiallayer are found using

(6.76)

and

(6.77)

6.8.3 Quaternary Heteroepitaxial Layer

In the case of a quaternary alloy such as AlxInyGa1–x–yP, the compositionis not uniquely determined once the relaxed lattice constant is known.For this reason, the analysis of the composition and strain cannot be doneusing x-ray rocking curves alone. If, however, the composition (and there-fore the relaxed lattice constant) is determined by some other technique,such as auger electron spectroscopy (AES) or secondary ion mass spec-troscopy (SIMS), the in-plane and out-of-plane strains may be determinedby following the procedure above. Another approach involves the esti-mation of the bandgap from photoluminescence (PL) measurements. Thecomposition is not uniquely determined by the bandgap. Nonetheless,the knowledge of the bandgap (from PL) and the relaxed lattice constant(from XRD) together allows the determination of the composition. Thein-plane and out-of-plane strains are then determined as before from theXRD values of a, c, and . An important source of error in such a proce-dure is the strain-induced shift in the PL emission peak. Therefore, theXRD strain results must be used to estimate this shift if reliable resultsare to be obtained.

6.9 Determination of Critical Layer Thickness

Experimental methods for the determination of the critical layer thicknessinclude transmission electron microscopy (TEM),60,61 scanning tunnelingmicroscopy (STM),62 photoluminescence (PL),60,63,64 photoluminescencemicroscopy (PLM),52,65 electrical measurements on modulation-doped struc-tures,66,67 reflection high-energy electron diffraction (RHEED),63,68 x-ray dif-

a0

ν

ε|| = −a aa

0

0

ε⊥ = −c aa

0

0

a0




fraction (XRD),60,61,68,69 x-ray topography (XRT),71 electron beam-inducedcurrent (EBIC),72 and ion channeling.73 The traditional strain-based methodsfor measuring involve the determination of in-plane strain by PL or x-ray diffraction. On the other hand, with imaging techniques such as TEM,photoluminescence microscopy, or x-ray topography, individual dislocations(or the material surrounding them) can be imaged after the onset of latticerelaxation. There are also several methods for the indirect observation oflattice relaxation. For example, lattice relaxation introduces surface steps thatbroaden specular spots measured by RHEED. Also, the introduction ofthreading dislocations during lattice relaxation causes several observablechanges in the x-ray diffraction profiles. These include the broadening of themain diffraction peak, broadening and extinguishing of the Pendellosungfringes,61,70 and reduction in the ratio of the epitaxial layer peak intensity tothe substrate peak intensity. All of these phenomena have been used for theexperimental determination of the critical thickness in mismatched het-eroepitaxial layers.

In nearly all of the methods above, samples of various thicknesses areexamined for evidence of strain relaxation. Therefore, thickness resolution(stemming from the use of a finite number of samples) is an important sourceof error unless many samples are grown and characterized.

Strain resolution is another source of error, which could be important ifcoupled with sluggish strain relaxation. Strain-based methods such as PLand XRD typically have strain resolutions of 10–5 to 10–4. The resolution ofan imaging technique such as TEM or PLM, though harder to quantify, canbe better than 10–5 if a sufficiently large area is examined. All of these tech-niques appear to have sufficient resolution for the determination of thecritical layer thickness. Despite this, anomalously large critical layer thick-nesses have sometimes been reported in studies based on XRD. New modelsfor the critical layer thickness have been proposed to explain these results,for example, by People and Bean73 and Fischer et al.74 It has also been shownby Fritz75 that the anomalously large critical layer thicknesses could beexplained by initially sluggish lattice relaxation combined with finite exper-imental resolution. On the other hand, the anomalously large critical layerthicknesses reported in XRD studies could be due to errors introduced bythe crystallographic tilting of the epitaxial layers.76 However, this cause forerror can be readily eliminated.77,78

The choice of experimental technique will therefore be dictated in largepart by the application. In the case of strained quantum wells for optoelec-tronic devices, PLM may be the most appropriate technique. If, however,strained layers are to be used in high-electron-mobility transistors (HEMTs),then measurements of the carrier mobility in modulation-doped structuresare indicated. The study of strain relaxation kinetics can best be done usingXRD. Finally, the types of dislocations and the mechanisms for their intro-duction are best studied using TEM.

hc




6.9.1 Effect of Finite Resolution

Fritz75 has shown that limited strain resolution, coupled with initially slug-gish lattice relaxation, can lead to the experimental determination of appar-ent critical layer thicknesses (CLTs) that are much greater than the actualvalues. Essentially, any experimental method for CLT determination allowsthe measurement of the in-plane strain (directly or indirectly) for samplesof various thickness. According to the Matthews and Blakeslee model, theequilibrium strain in the layer is given by

(6.78)

The critical layer thickness is considered to correspond to the smallestthickness in which the measured strain departs measurably from the mis-match strain f.

Now suppose the experimental method has finite resolution and can detecta change in strain no smaller than R. The critical layer thickness deter-mined with this finite experimental resolution will satisfy

(6.79)

By way of example, the resolution of XRD techniques falls typically in therange of 10–5 to 10–4. Figure 6.29 shows the Matthews and Blakeslee criticallayer thickness vs. the lattice mismatch strain f, and also the experimentalcritical layer thickness , assuming a worst case of . So, althoughexperimentally determined critical layer thicknesses sometimes exceed thepredictions of the Matthews and Blakeslee model significantly, this cannotbe explained as a consequence of finite experimental resolution acting alone.

Partially relaxed layers with are usually found to contain fewermisfit dislocations than expected according to the equilibrium theory. If theratio of the actual misfit dislocation density to the expected density for a layerin equilibrium is Q, then the apparent critical layer thickness will satisfy

(6.80)

Fritz simplified this expression for (001) heteroepitaxy of diamond or zincblende semiconductors to

ε ν απ

||( )

;

( cos )[ln( / ) ](

eq

f h h

ff

b h bh

c

=<

− +1 18 1

2

−−>

⎧

⎨⎪

⎩⎪ ν λ)cos

; h hc

hc1

f Rb h b

hc

c

− = − +−

( cos )[ln( / ) ]( )cos

1 18 1

21

1

ν απ ν λ

hc

hc1 R = −10 4

h hc>

hc1

fRQ

b h bh

c

c

− = − +−

( cos )[ln( / ) ]( )cos

1 18 1

21

1

ν απ ν λ




(6.81)

where the apparent critical layer thickness is in angstroms.Anomalously large critical layer thickness values have been determined

using strain-based methods for InGaAs/GaAs (001) by Orders and Usher69

and Anderson et al.79 and for SiGe/Si (001) by People and Bean.73 Fritz showedthat these results can be (approximately) reconciled with the Matthews andBlakeslee critical layer thickness if it is assumed that . Figure6.30 shows the previously mentioned experimental results, along with theapparent critical layer thickness curve calculated assuming R/Q = 7.5 × 10–3.

On the other hand, the anomalously large critical layer thicknessesreported in XRD studies could be due to errors introduced by the crystallo-graphic tilting of the epitaxial layers.76 This potential source of error in x-raymeasurements, now understood, can easily be removed by averaging peakseparations from rocking curves taken at opposing azimuths.77,78

Therefore, the apparent critical layer thickness determined by any exper-imental method will generally be larger than the actual value in the presenceof initially sluggish strain relaxation. The extent of this effect is determinedby the ratio R/Q, where R is the experimental resolution and Q is the ratioof the misfit dislocation density to the density expected in equilibrium. The

FIGURE 6.29Critical layer thickness vs. the lattice mismatch strain f. The solid line shows the Matthews andBlakeslee critical layer thickness hc; the dashed line shows the apparent critical layer thicknesshc1, which would be measured using an experimental technique with a resolution R = 10–4.

1

10

100

1000

0.01 0.1 1 10 |f| (%)

h c (n

m)

Matthews and BlakesleeFinite resolution (R = 10−4)

fRQ h

h

c

c− =⎛⎝⎜

⎞⎠⎟

+⎡

⎣⎢

⎤

⎦⎥

0 224

11

1.ln

hc1

R Q/ .= × −7 5 10 3




physics of the parameter Q are poorly understood, however, and whereasFritz assumed Q to be constant, it is actually a function of time and layerthickness. Considerable work remains before these issues can be satisfacto-rily resolved.

6.9.2 X-Ray Diffraction

The critical thickness has been determined using x-ray rocking curves basedon strain69,80,81 and also the rocking curve full width at half maximum.81 Ineither case, a series of heteroepitaxial samples is produced with a range oflayer thicknesses. A rocking curve is measured for each sample, and thecritical layer thickness is deduced from the strain vs. thickness or rockingcurve width vs. thickness characteristic.

6.9.2.1 Strain Method

In the strain method, values of the in-plane elastic strain are determinedfrom the separation of the substrate and epitaxial layer diffraction peaks. Itis most convenient to use a symmetric x-ray reflection for this purpose. Forexample, the 004 reflection is typically used for (001) heteroepitaxy of dia-

FIGURE 6.30Apparent critical layer thickness vs. mismatch curve. The curve was calculated using the modelof Fritz, with R/Q = 7.5 × 10–3. The squares are data from Orders and Usher,69 the triangles aredata from Anderson et al.,79 and the circles are data from People and Bean.73 (Reprinted fromFritz, I.J., Appl. Phys. Lett., 51, 1080, 1987. With permission. Copyright 1987, American Instituteof Physics.)

1

10

100

1000

0 1 2 3 4 |f| (%)

h c (n

m)

Fritz model InGaAs/GaAs Orders and UsherInGaAs/GaAs Anderson et al.SiGe/Si People and Bean




mond or zinc blende semiconductors, and the 0002 reflection is often usedfor (0001) wurtzite semiconductors.

The resolution of the XRD strain method is typically 10–5 to 10–4, butdepends on a number of factors, as will be shown in detail below. This levelof resolution allows accurate determination of the critical layer thickness, solong as the difference in Bragg angles between the epitaxial layer and sub-strate is determined correctly. Large errors can be introduced by the crystal-lographic tilting between the epitaxial layer and substrate, if this is notproperly accounted for. This is because the rocking curve peak separationis given by

(6.82)

where is the difference in Bragg angles between the epitaxiallayer and substrate, is the azimuthal angle for the incident x-raybeam, is the crystallographic tilting between the epitaxial layer and thesubstrate, and specifies the direction of the epitaxial layer tilt. In earlywork, x-ray rocking curve results were analyzed with the assumptionthat . However, this is only correct in the absence of epitaxial layertilting and leads to large errors in most cases. It is therefore necessary to findthe Bragg angle difference by measuring the rocking curve at two opposingazimuths and then taking the average:

(6.83)

It should be noted that the above equation applies only in the case of asymmetric reflection. If an asymmetric reflection is used, an additional tiltingof the diffracting planes is introduced by tetragonal distortion of the epitaxiallayer. This term does not zero-out when the rocking curve peak separationsare averaged from opposing azimuths, as is detailed in Section 6.8.3.

The out-of-plane lattice constant c for the epitaxial layer can be determinedusing the Bragg law,

(6.84)

where d is the interplanar spacing, is the Bragg angle, n is the order ofthe reflection, and is the x-ray wavelength. Often, Cu kα1 radiation isused, with . For the 004 reflection from a zinc blende semicon-ductor, and . For the 0002 reflection from a wurtzite semiconduc-tor, and . If it is assumed that the substrate is unstrained, and itslattice constant is known, then the out-of-plane lattice constant for the epi-taxial layer may be determined by applying the Bragg law to each.

Δθ

Δ Δ ΔΦθ θ ψ ψ= + −B cos( )0

Δθ θ θB Be Bs= −ψ

ΔΦψ0

Δ Δθ θ≈ B

Δ Δ Δθ θ ψ θ ψB = = ° + = °( ) ( )0 180

2

2d nBsin θ λ=

θB

λλ = 1 540594. Å

c d= n = 4c d= n = 2




The out-of-plane strain in the epitaxial layer is then

(6.85)

where is the relaxed lattice constant of the epitaxial layer. Assumingisotropic elasticity and tetragonal distortion of the epitaxial layer, the in-plane strain can be found by

(6.86)

where is the Poisson ratio.The sensitivity of the x-ray strain method is limited by the uncertainty in

the peak separation between the diffraction peaks of the epitaxial layerand the substrate. In other words, the minimum detectable peak shift of theepitaxial layer rocking curve leads to an uncertainty, , in the calculatedout-of-plane strain, , and a corresponding uncertainty, , in the in-planestrain, .

The resolution of the strain method may be analyzed as follows. By dif-ferentiating the Bragg law with respect to , we obtain

(6.87)

Then, if the strained layer is tetragonally distorted, with a symmetric reflec-tion we obtain

(6.88)

Zhang et al.81 found that the uncertainty in the peak position for theepitaxial layer was directly proportional to the epitaxial layer rocking curveFWHM, β. This is expected to hold in the general case, to the extent that therocking curves have approximately the same shape, because a narrowerrocking curve has a more rapidly changing first derivative. For a pseudo-morphic heteroepitaxial layer, in which there are no significant sources ofrocking curve broadening other than the finite layer thickness, the uncer-tainty in the Bragg angle is , where is the rocking curve for a perfectcrystal and r is a unitless constant of proportionality. Zhang et al. determinedthe constant of proportionality to be in their experiments. How-

ε⊥ = −c aa

e

e

ae

ε νν

ε|| = −−

⎛⎝⎜

⎞⎠⎟ ⊥

21

ν

ΔθB

Δε⊥ε⊥ Δε||

ε||

θB

Δ Δdd B B= − θ θcot

Δ Δ Δ Δ Δε θ θ⊥ = ≈ = = −ca

cc

dde

B Bcot

Δθ

rβ0 β0

r = 1 25/




ever, this value may vary depending on the experimental conditions anddiffractometer used, as well as the profile shape.

For a pseudomorphic heteroepitaxial layer having a thickness , theuncertainty in the epitaxial layer Bragg angle is

(6.89)

where is given by the Scherrer formula.82,83 For this development, itis necessary to use the Scherrer equation to obtain an analytical result. Forits application, however, either the Scherrer equation or the results of dynam-ical simulations may be used.

According to the Scherrer formula, the FWHM for a perfect pseudomor-phic epitaxial thin layer is given by

(6.90)

so that

(6.91)

Substituting Equation 6.91 into Equation 6.88, we obtain the uncertaintyin the calculated out-of-plane strain:

(6.92)

The minimum lattice relaxation that may be detected using the x-ray strainmethod is

(6.93)

In the case of (001) heteroepitaxy of a diamond or zinc blende semicon-ductor using the 004 reflection and Cu kα1 radiation ( ), assum-ing r = 1/25, ν = 1/3, hc = 200 nm, and , the minimum detectablelattice relaxation is .

Orders and Usher69 applied the XRD strain method to find critical layerthicknesses in MBE-grown InxGa1–xAs/GaAs (001) layers with various com-positions. (In this material system, the room temperature mismatch strain isapproximately .) Using 004 double-crystal rocking curves mea-

hc

Δθ βB cr h≈ ± 0( )

β0( )hc

β λθ0

0 9≈ .cosh B

Δθ β λθB c

c B

r h rh

≈ ± = ±00 9( ) .cos

Δ Δε θ θ λθ⊥ = − =B B

c B

rh

cot .sin

∓ 0 9

R strain( ).

||= =−

⎛⎝⎜

⎞⎠⎟

=−

⎛⎝⎜

⎞⎠⎟⊥Δ Δε ν

νε ν

ν2

12

10 99r

hc B

λθsin

⎛⎝⎜

⎞⎠⎟

λ = 1 540594. ÅθB = °33

R ≈ × −5 10 5

f x≈ 7 2. %




sured at a single azimuth, they estimated the strains in layers having variousthicknesses for several particular compositions. Some representative rockingcurves are shown in Figure 6.31. In this way, they determined the criticallayer thicknesses to be 200, 100, and 30 nm for x = 0.07, 0.14, and 0.25,respectively. In their study, the composition x for each series of samples wasdetermined by the rocking curve peak separation for thin layers. The strainswere also calculated from the rocking curve peak separations. Both deter-minations would be subject to errors introduced by the crystallographictilting of the epitaxial layers, however.

FIGURE 6.31004 x-ray rocking curves from InxGa1–xAs/GaAs (001) heterostructures. (Reprinted from Orders,P.J. and Usher, B.F., Appl. Phys. Lett., 50, 980, 1987. With permission. Copyright 1987, AmericanInstitute of Physics.)

Bragg angle (deg)32.0 32.5 33.0

X-ra

y int

ensit

y (co

unts

/sec

)

X-ray rocking curves

InxGa1−xAs/GaAsInxGa1−xAs

x = 0.07

x = 0.14

x = 0.25

GaAs

h = 0.20 μmstrained



h = 0.50 μmrelaxed

h = 1.00 μmrelaxed

h = 1.00 μmrelaxed

10.000

1000

100

10

100

10

100

10

100

10

100

10

100

10




Petruzzello et al.60 applied the XRD strain method to determine the critical

thickness in MBE-grown ZnSe/GaAs (001). (For this heteroepitaxial system,the lattice mismatch strain is at room temperature and the Mat-thews and Blakeslee critical layer thickness is hc = 47 nm.) For this purpose,they used a precision biaxial diffractometer85 with absolute angle encoders,and thereby avoided problems of crystallographic tilting between the ZnSeand GaAs. The strain vs. thickness characteristic they measured in this way(Figure 6.32) was corroborated by TEM, PL, and powder diffractometermeasurements. They found that the 87-nm layer was pseudomorphic, butthe 180-nm layer was partially relaxed. From this observation, it is clear thatthe critical layer thickness is between 87 and 180 nm. The value of the criticallayer thickness was estimated to be 150 nm, but no samples close to thisthickness were grown in order to refine this estimate.

Zhang et al.81 also used the XRD strain method to determine the criticallayer thickness in MOVPE-grown ZnSe/GaAs (001). They used a Bartelshigh-resolution diffractometer and corrected for the epitaxial layer tilt. Fig-ure 6.33 shows the in-plane strain determined at room temperature as afunction of the layer thickness for ZnSe/GaAs (001). The solid lines areguides to the eye. Here, the apparent critical layer thickness is approximately210 nm.

Kim et al.85 applied the XRD strain method to determine the critical layerthickness for GaN on AlN in GaN/AlN/α-Al2O3 (0001) structures. (The roomtemperature in-plane lattice mismatch strain for this combination is f =–2.6%.) In this study, MBE was used to produce GaN layers of varyingthickness (50 Å to 1 μm) on 3.2-nm-thick AlN buffer layers on c-face sapphiresubstrates. The thickest GaN layer was measured using selective etching and

FIGURE 6.32Room temperature in-plane elastic strain in ZnSe/GaAs (001) measured by XRD, TEM, and PL.(Reprinted from Petruzzello, J. et al., J. Appl. Phys., 63, 2299, 1988. With permission. Copyright1988, American Institute of Physics.)

0 1 2 3 4 5h (μm)

ε ||(1

0−3)

TEMXRDPL

0

1

2

3

f = −0 27. %




a mechanical profilometer, with 5% precision. The thicknesses of the remain-ing layers were estimated assuming the thickness was linear in the growthtime. XRD measurements were made using synchrotron radiation at theNational Synchrotron Light Source (NSLS). Multiple reflections were usedand least squares fitting was employed to improve the reliability of theresults. They found that the in-plane lattice constant fit the theoretical expres-sion for equilibrium films,

(6.94)

with a critical layer thickness of . This value is in agreementwith the prediction of the Matthews and Blakeslee model ( , assum-ing ν = 0.38, b = 3.084 Å, λ = 30°, and α = 90°).

Their data are shown in Figure 6.34, along with the curve calculated usingEquation 6.94 and the measured in-plane lattice constant for the AlN bufferlayer ( ).

6.9.2.2 FWHM Method

With the FWHM method, the determination of is based on the observationof the rocking curve broadening by dislocations that are introduced duringthe relaxation process. It involves a comparison of the experimental rocking

FIGURE 6.33X-ray strain method for the determination of the critical layer thickness applied to ZnSe/GaAs(001). Here the room temperature in-plane strain has been plotted as a function of the layerthickness for ZnSe/GaAs (001). The solid lines are guides to the eye. (From Zhang, X.G. et al.,J. Electron. Mater., 28, 553, 1999. With permission.)

−0.3

−0.2

−0.1

0

10 100 1000Layer thickness (nm)

In-p

lane

stra

in (%

)

a ahh

a asc

s e= + −( )

hc = ±29Å 4Åhc = 28Å

aS = 3 084. Å

hc




curve widths with the expected widths for perfect crystal layers. The perfectcrystal rocking curve widths should usually be determined using dynamicalsimulations.25,86,87 The Scherrer equation may also be used as long as the layerthickness is much less than the extinction depth for the x-rays.

For a dislocation-free pseudomorphic epitaxial layer, the width of the x-ray rocking curve decreases monotonically with the layer thickness. At theonset of lattice relaxation, misfit and threading dislocations are typicallyintroduced together by the heterogeneous nucleation of dislocation half-loops or dislocation multiplication processes. The dislocations cause signif-icant broadening of the x-ray rocking curve when the epitaxial layer thick-ness is beyond the critical layer thickness for dislocation multiplication. Thisphenomenon is well known and is the basis for the x-ray determination ofthreading dislocation densities in heteroepitaxial layers.88–90

In the following analysis, it was assumed that the dislocation broadeningis due to half-loops and bent-over substrate dislocations that participate inthe strain relaxation process. In either case, the dislocations have threadingand misfit segments, both of which can broaden the rocking curves. Kaganeret al.91 presented an analysis of the rocking curve broadening by the misfitsegments for some particular configurations. However, it is expected thatthe broadening will be dominated by the threading segments in most cases.

Commonly, the threading dislocation densities in heteroepitaxial diamondand zinc blende layers are determined from the broadening of the 004 rock-ing curve. This involves fitting the 004 rocking curve with a Gaussian peak,

FIGURE 6.34Room temperature in-plane lattice constant a vs. thickness for GaN on AlN in GaN/AlN/α-Al2O3 (0001) structures grown by MBE. (Reprinted from Kim, C. et al., Appl. Phys. Lett., 69, 2358,1996. With permission. Copyright 1996, American Institute of Physics.)

3.06

3.08

3.10

3.12

3.14

3.16

3.18

3.20

3.22

1 10 100 1000GaN layer thickness (nm)

Latti

ce co

nsta

nt a

(Å)

hc = 2.9 + 0.4 nm−




to determine the FHWM, and then deconvoluting the contribution due tothe intrinsic rocking curve width for a perfect crystal.90 Hence, the broaden-ing due to the dislocations is given by

(6.95)

where is the experimentally measured FWHM and is the intrinsicrocking curve FHWM. By differentiating this equation with respect to , weobtain the uncertainty in the dislocation broadening, which is

(6.96)

The uncertainty in the rocking curve width, , was found by Zhang etal. to be proportional to the measured rocking curve width, with the sameconstant of proportionality as in Equation 6.96. For a layer at the criticallayer thickness,

(6.97)

It was assumed that the minimum detectable threading dislocation broad-ening, , corresponds to the uncertainty in the broadening, . Then

(6.98)

Solving,

(6.99)

Assuming , the minimum detectable threading dislocation broadeningfor a layer with thickness is

(6.100)

If we apply the Scherrer equation, we have

β β βd = −202

β β0

βd

Δ Δβ ββ

βdd

=⎛⎝⎜

⎞⎠⎟

Δβ

Δβ β≈ ±r hc0( )

βdmin Δβd

β ββ β

ββd d

d

dmin

min

min

= =+⎡

⎣⎢⎢

⎤

⎦⎥⎥

Δ Δ02 2

ββ β β β

d

rmin

/

( / ) ( / )=

+ +⎡⎣⎢

⎤⎦⎥ =

+ +Δ Δ Δ1 1 4

2

1 1 202

1 222

1 2

2

⎡⎣⎢

⎤⎦⎥

/

r << 1hc

β β βd cr

h rmin ( )≈ =Δ0




(6.101)

From the minimum detectable dislocation broadening, , the corre-sponding dislocation density, , may be estimated as follows. If

, then the rocking curve broadening is dominated by the angularmosaic spread rather than the d-spacing mosaic spread. For this situation,is approximately independent of the Bragg angle and

(6.102)

where b is the length of the Burgers vector for the threading dislocations.To compare the FWHM and strain methods in terms of sensitivity, it is

necessary to relate from the former method to obtained from thelatter. The relationship between the misfit dislocation density, , and thein-plane strain, , is

(6.103)

where b is the length of the Burgers vector, is the angle between theBurgers vector and line vector, and φ is the angle between the interface andthe normal to the slip plane. If it is assumed that there are two orthogonalmisfit dislocation arrays with the same average misfit dislocation length L,and that each misfit dislocation has n threading dislocations associated withit, then

(6.104)

where D is the threading dislocation density. The value of n can be 0 (bothends of the misfit dislocation terminate at the sides of the bottom of thesubstrate), 1 (corresponding to the bending over of a substrate threadingdislocation), or 2 (corresponding to a half-loop introduced at the surface ofthe epitaxial layer). The average length of the misfit dislocations L may beestimated if it is assumed that they lie along <110> directions in the (001)interface and all have equal lengths. Then

(6.105)

βθd

c B

rhmin

.cos

≈ 0 92

βdmin

Dmin

tan2 2θB <βd

Db

dmin

min

.= β2

24 36

Dmin Δε||

ρMD

ε||

ε ρ α φ|| cos cos= −f bMD

α

ρMDDL

n=

2

Ln

D= 2




Combining the previous three equations, we can find the minimum detect-able strain relaxation associated with the FWHM method:

(6.106)

In the case of (001) heteroepitaxy of a diamond or zinc blende semicon-ductor using the 004 reflection and Cu kα1 radiation ( ), assum-ing n = 2, r = 1/25, hc = 200 nm, and , the minimum detectable latticerelaxation is about . For this situation, the FWHM method turnsout to be about three times more sensitive than the strain method whenusing the same x-ray diffractometer.

In general, the sensitivities of the two x-ray methods for critical layerthickness determination may be compared using the ratio

(6.107)

If P > 1, then the FWHM will be more sensitive, but if P < 1, the strainmethod will be more sensitive. Typically, P ≈ 3 and the FWHM method ispreferred. Based on this, we can expect critical layer thicknesses determinedby the FWHM method to be smaller than those determined using the strainmethod if the same diffractometer and experimental conditions are used.

Zhang et al.81 compared the x-ray FWHM and strain methods for thedetermination of the critical layer thickness for ZnSe/GaAs (001) grown byphotoassisted MOVPE. (For this heteroepitaxial system, the lattice mismatchstrain is f = –0.27% at room temperature and the Matthews and Blakesleecritical layer thickness is hc = 47 nm.) Figure 6.35 shows the experimentallydetermined 004 rocking curve FWHM vs. the ZnSe thickness (squares witherror bars). Also shown are the calculated FWHM values for perfect crystallayers of ZnSe on GaAs (001), based on the Scherrer formula. It can be seenthat the thin layers exhibit rocking curve widths that are the same as perfectcrystal values, within the experimental errors. Significant dislocation broad-ening was observed for layers thicker than 135 nm. They considered thecritical layer thickness to be the thickness at which the slope of the experi-mental characteristic (FWHM vs. thickness) was zero, giving a critical layerthickness of 140 nm. This result is comparable to those reported by Kamataand Mitsuhashi92 (~150 nm) and Petruzzello et al. (150 nm), and significantlyless than that reported by Reisinger et al.68 (225 ± 5 nm). It was noted thatthis thickness corresponds to the onset of significant dislocation multiplica-tion rather than the critical layer thickness for the initiation of the glide ofsubstrate dislocations. On the other hand, Zhang et al.’s result is significantlylarger than the value of 97 nm reported by O’Donnell and coworkers.93

R FWHMr

h nc B

( ).

cos||= =Δε λ

θ0 15

λ = 1 540594. ÅθB = °33

1 5 10 5. × −

Ρ = =−

R strainR FWHM

rn B( )( )

cot121

ν θν




The layer thickness for significant (observable) dislocation multiplicationvaries somewhat with growth conditions. Nonetheless, for a given heteroepi-taxial system, a smaller value of critical layer thickness suggests better exper-imental sensitivity. Therefore, the x-ray FWHM method appears to havesuperior sensitivity to the x-ray strain method or the PL method. Only theuse of synchrotron x-ray topography appears to be more sensitive than thex-ray FWHM method. However, all of the experimentally measured valuesof the critical layer thickness are significantly greater than the value predictedby Matthews and Blakeslee (hc = 47 nm for ZnSe/GaAs (001)) for the initialonset of lattice relaxation by the bending over of threading dislocations.

Zhang et al. also determined the critical layer thickness from the sameset of samples using the x-ray strain method. The apparent critical layerthickness found in this way was 210 nm. However, they determined theXRD FWHM method ( ) to have better resolution than the XRDstrain method ( ). The value of 140 nm was therefore consideredmore reliable.

6.9.3 X-Ray Topography

X-ray topography (XRT) involves the capture of a photographic image froma diffracting crystal set at the Bragg condition and in reflection mode. Acollimated, monochromatic x-ray beam is incident on the crystal at the Bragg

FIGURE 6.35X-ray FWHM method for the determination of the critical layer thickness applied to ZnSe/GaAs (001). The 004 rocking curve FWHM has been plotted as a function of the layer thickness.The squares with error bars represent experimental data. The solid line was calculated usingthe Scherrer formula for perfect crystals of ZnSe. (From Zhang, X.G. et al., J. Electron. Mater.,28, 553, 1999. With permission.)

0

100

200

300

400

500

600

700

0 100 200 300 400Layer thickness (nm)

004

FWH

M (A

rc se

c)

R = × −1 5 10 5.R = × −5 10 5




angle, diffracted, and collected by a photographic emulsion. A planar viewis normally obtained unless the sample is specially prepared. Strainedregions surrounding a dislocation diffract less intensity, and misfit disloca-tions are revealed in the image as dark lines. XRT can therefore reveal thecreation of a single misfit dislocation in the sample.

The absolute resolution of this technique depends on the area examined.For example, if the heteroepitaxial layer is imaged over an area of ,and a single misfit dislocation can be detected in this area, then the resolu-tion is

(6.108)

where b is the length of the Burgers vector, is the angle between theBurgers vector and line vector, and is the angle between the interface andthe normal to the slip plane. For (001) heteroepitaxy of zinc blende semicon-ductors, an image size of 25 × 25 μm results in a resolution of .

O’Donnell et al.94 used XRT to determine the critical layer thickness forMBE-grown ZnSe/GaAs (001). (For this material system the room temper-ature mismatch strain is –0.27% and the Matthews and Blakeslee criticallayer thickness is 47 nm.) 044 topographs were obtained using synchrotronradiation at a wavelength of 0.148 nm. Samples of various thicknesses wereprepared, and the layer thicknesses were calibrated by comparing the mea-sured x-ray rocking curves to dynamical simulations. In a 95-nm-thickheteroepitaxial layer, no misfit dislocations were imaged. Misfit dislocationswere found to be present in a 100-nm-thick layer with a spacing of 14 μm.Based on these observations, the critical layer thickness was estimated tobe 97 nm.

6.9.4 Transmission Electron Microscopy

Transmission electron microscopy (TEM)60,61 can be used to determine thecritical layer thickness by the examination of samples having various epi-taxial layer thicknesses. The samples may be prepared in either planar orcross-sectional view. Cross-sectional micrographs allow the viewing of indi-vidual misfit dislocations. It is also possible to determine the Burgers vectorsfor individual dislocations using TEM micrographs obtained with variousdiffraction vectors for the electrons, using the criterion for theextinction of dislocation contrast, where b is the Burgers vector, l is thedislocation line vector, and g is the diffraction vector. This capability is aunique advantage of the TEM method. Disadvantages of the TEM methodare the destructive nature of sample preparation and the potential for theintroduction of artifact defects during this preparation.

Edirisinghe et al.94 used TEM to establish the critical layer thickness inMBE-grown InGaAs/GaAs (111)B. They examined samples of different

D D×

Rb

D= cos cosα φ

αφ

R = × −8 10 6

b g l⋅ × =( ) 0




thicknesses and compositions in plan view, bright-field micrographsobtained using the diffraction vector. Figure 6.36 shows such a micro-graph for a 150-nm-thick In0.15Ga0.85As/GaAs (111)B sample. Misfit disloca-tions are clearly visible running along two of the <110> directions in theinterface. However, the lattice relaxation is asymmetric due to the vicinalsubstrate, and so the misfit dislocations do not have equal densities in thetwo <110> directions. Also, misfit dislocations along the direction areinvisible with this diffraction vector.

Edirisinghe et al. found that for In0.25Ga0.75As/GaAs (111)B, misfit disloca-tions were present in a 25-nm-thick layer, but not in a 15-nm-thick layer. Thisindicates that the critical layer thickness for this material system is between15 and 25 nm. (The room temperature lattice mismatch strain is –1.8%.)

Petruzzello et al.60 applied TEM to determine the critical thickness in MBE-grown ZnSe/GaAs (001). Here, cross-sectional and planar samples werestudied, with layer thicknesses varying from 50 to 4900 nm. Samples withepitaxial layer thicknesses of 50 and 87 nm contained stacking faults on {111}planes bounded by Frank partial dislocations having Burgers vectors of thetype . However, only the epitaxial layers with thicknesses of 180nm or greater contained perfect misfit dislocations. From this observation,it is clear that the critical layer thickness is between 87 and 180 nm. Thevalue of the critical layer thickness was estimated to be 150 nm, but nosamples close to this thickness were grown in order to refine this estimate.

Petruzzello et al. also used their TEM results to estimate the residual strainsin the ZnSe/GaAs (001) heteroepitaxial layers as a function of the ZnSe

FIGURE 6.36TEM plan view, bright-field micrograph of a 150-nm-thick layer of In0.15Ga0.85As/GaAs (111)B,with . Misfit dislocations are visible along two of the <110> directions in the interface.Misfit dislocations along the direction are invisible with this diffraction vector. (Reprintedfrom Edirisinghe, S.P. et al., J. Appl. Phys., 82, 4870, 1997. With permission. Copyright 1997,American Institute of Physics.)

250 nm 022

g = 022[ ]011

( )022

[ ]011

a / 3 111




thickness. This was done using the measured densities of 60° and edgedislocations in the samples. Whereas most of the misfit dislocations were ofthe 60° type, the fraction of Lomer-type edge dislocations increased withlayer thickness. It was concluded that the edge dislocations originated fromreactions between 60° misfit dislocations. The calculated residual strainswere in agreement with high-resolution XRD measurements made using ahigh-resolution biaxial diffractometer84 and a conventional powder diffrac-tometer, as well as photoluminescence results.

6.9.5 Electron Beam-Induced Current (EBIC)

Electron beam-induced current (EBIC)71 allows the imaging of individualdislocations located in the vicinity of a p-n junction and has been employedfor critical layer thickness determination. Here, a p-n heterojunction isformed between the substrate and heteroepitaxial layer. Then, in a scanningelectron microscope (SEM), an electron beam is scanned over the surface ofthe sample. The electron beam excites electron–hole pairs in the sample,which are separated by the built-in electric field of the zero-biased p-nheterojunction. The electron beam-induced current flowing in the substrateis measured and recorded at each position of the electron beam. Dislocationsgive rise to nonradiative recombination in the material and a reduction inthe electron beam-induced current. Therefore, if the magnitude of the currentis used to make an image, misfit dislocations and other areas of short minor-ity carrier lifetime show up as dark features on the EBIC image.

Kohama et al.71 applied EBIC for the determination of critical layer thick-nesses in MBE-grown Si1–xGex/Si (001). The layers were grown on n-type Sisubstrates and were unintentionally doped p-type, thus forming a p-n het-erojunction at the interface. EBIC measurements were done in a SEM with20 kV of accelerating voltage and an electron beam current of .

In this study, EBIC and TEM were used to examine Si1–xGex/Si (001) sam-ples having various thicknesses and compositions. The misfit dislocationlines imaged by the two techniques were correlated for a relaxed 300-nm-thick layer of Si0.8Ge0.2/Si (001). Thinner layers were considered to be pseudo-morphic if no misfit dislocations were found in the EBIC image. In this way,the greatest pseudomorphic thickness was plotted as a function of the latticemismatch. For each composition, the smallest thickness in which there weremisfit dislocations was also plotted. The actual critical layer thickness char-acteristic was considered to lie between the pseudomorphic and partiallyrelaxed curves, as shown in Figure 6.37.

6.9.6 Photoluminescence

Photoluminescence (PL) may be used to determine the critical layer thicknessof a variety of structures, including single heteroepitaxial layers, strainedsingle quantum wells (SSQWs), or strained layer superlattices. Several

7 10 7× − A




observable changes in the spectrum accompany lattice relaxation at the crit-ical layer thickness, including a wavelength shift in the near-band-edge peakdue to strain relaxation, the broadening and reduction in intensity of thenear-band-edge peak, and, in some cases, the appearance of deep-level emis-sion. The critical layer thickness can be determined based on any of thesephenomena or some combination. If the material studied is to be used inoptoelectronic devices, then the observed optical characteristics are imme-diately relevant.

If the peak emission energy is assumed to correspond to the bandgap inthe material, a biaxial stress in the layer will induce a shift in the peak, whichfor a (001) zinc blende semiconductor is given by

(6.109)

FIGURE 6.37Critical layer thickness vs. Ge mole fraction x in Si1–xGex/Si (001) as determined using EBIC.The solid dots represent the thinnest samples in which misfit dislocations were found. The opencircles represent the thickest samples at each composition that were pseudomorphic (commen-surate). Also shown are the experimental data of Bean et al. and the predictions of the Peopleand Bean model and the Matthews and Blakeslee model. (Reprinted from Kohama, Y. et al.,Appl. Phys. Lett., 52, 380, 1988. With permission. Copyright 1988, American Institute of Physics.)

1

10

100

1000

0 1 2 3 4|f| (%)

Laye

r thi

ckne

ss (n

m)

Matthews and BlakesleePeople and BeanKohama et al. commensurateKohama et al. incommensurateBean et al.

ΔE a C C C b C C Cg = − − − + +ε||[ ( )/ ( )/ ]2 211 12 11 11 12 11




where is the in-plane strain, and are the elastic constants, and aand b are the hydrostatic and shear deformation potentials, respectively.Usually, PL spectra are measured at cryogenic temperatures. In these cases,care must be taken to account for the temperature variation of the bandgap.Also, the strain at cryogenic temperatures will not be the same as the roomtemperature value due to the thermal strain. This can be accounted for if thethermal expansion characteristics are known for both substrate and deposit,but in some cases, this information is incomplete below 100K. Another com-plication arises in quantum wells, in which the (thickness-dependent) blueshift due to quantum confinement effects must be considered.

Parker et al.95 used PL to determine the critical layer thicknesses inInxGa1–xN on GaN in InxGa1–xN/GaN/α-Al2O3 heterostructures grown byMOVPE. (In this combination, the room temperature lattice mismatch strainis f = –x9.7%.) Film thicknesses were determined by TEM (thin layers) oroptical measurements (thicker layers). Because of the negative (compressive)lattice mismatch strain, the near-band-edge emission was blue shifted inpseudomorphic layers. In partially relaxed layers, the blue shift decreasedand deep-level emission eventually became dominant. Representative PLspectra are shown in Figure 6.38 for In0.15Ga0.85N/GaN layers with variousInGaN thicknesses. Parker et al. considered the critical layer thickness to bethat thickness for which the peak emission wavelength was the same as fora relaxed layer. Using this approach, they determined hc = 100 nm for x =0.08 and hc = 65 nm for x = 0.15. These values are considerably larger thanthe predictions of the Matthews and Blakeslee model, which are 14 and 6.3nm, assuming , , , and . However, a fairly lim-ited number of layer thicknesses were studied. Also, the use of room tem-perature PL spectra resulted in strain resolution below that which can beobtained at cryogenic temperatures.

Reed et al.64 used several techniques, including PL, to determine the criticallayer thicknesses for MOVPE-grown GaN/InxGa1–xN/GaN quantum wellson sapphire (0001) substrates. Here, too, the wavelength shift of the peakintensity was used to deduce the critical layer thickness. The compositionswere determined by XRD measurements on thick layers. They also estimatedthe critical layer thicknesses based on the carrier mobility and quantum wellconductivity, as found from Hall effect measurements. These results areshown in Figure 6.39 for indium compositions up to 16%.

6.9.7 Photoluminescence Microscopy

Photoluminescence microscopy (PLM)48,52,65 is an imaging technique thatallows the detection of individual misfit dislocations. It therefore allowsdetection of lattice relaxation at its very earliest stages, making PLM a highlysensitive method for determination of the critical layer thickness. The abso-lute resolution of this method depends on the area examined, in the samemanner as for XRT.

ε|| C11 C12

ν = 0 38. b = 3 1. Å λ = °30 α = °90




Gourley et al.65 applied PLM to determine the critical layer thickness inIn0.2Ga0.8As/GaAs (001) strained layers and superlattices. In this study,micro-PL images were obtained at 80K for In0.2Ga0.8As layer thicknessesvarying from 50 to 600 Å. Figure 6.40 shows the PLM images for GaAs/In0.2Ga0.8As/GaAs (001) strained single quantum wells (SSQWs) of variousthicknesses. The SSQWs with thicknesses of 150 Å or less are free from darkline defects (DLDs). However, SSQWs of 200 Å or greater show DLDs alignedwith the <110> directions, indicating the presence of misfit dislocations attheir interfaces.

From the PLM images, Gourley et al. determined and plotted the DLDdensity vs. InGaAs thickness for GaAs/In0.2Ga0.8As/GaAs (001) strainedsingle quantum wells (SSQWs) and strained layer superlattices. These resultsare shown in Figure 6.41. For the SSQWs, the DLD density increases dra-matically, and in approximately linear fashion, for thicknesses above 200 Å.By extrapolating the DLD density back to zero, a critical layer thickness of190 Å was inferred. For the same samples, Gourley et al. used the x-ray

FIGURE 6.38Room temperature PL spectra for In0.15Ga0.85N/GaN layers with various InGaN thicknesses inMOVPE-grown InxGa1–xN/GaN/α-Al2O3 heterostructures. (Reprinted from Parker, C.A. et al.,Appl. Phys. Lett., 75, 2776, 1999. With permission. Copyright 1999, American Institute of Physics.)

960 nm

720 nm

360 nm

240 nm

120 nm

80 nm

40 nm

Energy (eV)2.2 2.4 2.6 2.8 3.0 3.2

PL In

tens

ity, A

U




strain method (Section 6.9.2) to determine the critical thickness and found avalue between 400 and 600 Å. This is indicative of the excellent sensitivityafforded by the PLM technique.

6.9.8 Reflection High-Energy Electron Diffraction (RHEED)

Reflection high-energy electron diffraction (RHEED) can be used for the insitu monitoring of lattice relaxation by a growing film in the MBE chamber.This contrasts with the previously described methods, which involve theexamination of many samples having different thicknesses, outside of thegrowth chamber. Also, because RHEED can be used to monitor the layerthickness with atomic layer accuracy, the technique lends itself to high-mismatch heteroepitaxial systems for which the critical layer thickness is ofthis order.

There are several features of RHEED characterization relevant to this pur-pose. First, the intensity of a particular RHEED spot is found to oscillatewith time, and each period of oscillation corresponds to the growth of onemonolayer. Second, a streaky RHEED pattern corresponds to an atomicallysmooth surface, whereas a spotty pattern indicates surface roughening.Third, the rapid damping of the RHEED intensity oscillations is associatedwith a change from two-dimensional-to-three-dimensional growth. Usually,

FIGURE 6.39Critical layer thicknesses determined for GaN/InxGa1–xN/GaN quantum wells on sapphire(0001) substrates as a function of the quantum well indium composition. The critical layerthicknesses were determined from photoluminescence spectra (diamonds), Hall effect quantumwell carrier mobility (circles), and Hall effect quantum well conductivity (triangles). (Reprintedfrom Reed, M.J. et al., Appl. Phys. Lett., 77, 4121, 2000. With permission. Copyright 2000,American Institute of Physics.)

0

20

40

60

80

100

0 5 10 15 20% In N

h c (n

m)




FIGURE 6.40PLM images for GaAs/In0.2Ga0.8As/GaAs (001) strained single quantum wells (SSQWs) ofvarious thicknesses, obtained at 80K with 476.2 nm of excitation and a continuous wave (cw)power density of 100 W/cm2. (Reprinted from Gourley, P.L. et al., Appl. Phys. Lett., 52, 377, 1988.With permission. Copyright 1988, American Institute of Physics.)

InGaAs SSQW T = 80KVR430 50Å VR431 300Å

VR415 90Å VR432 350Å

VR427 150Å VR422 400Å

VR420 200Å VR423 600Å

VR428 250Å

10 μm




the critical layer thickness for the two-dimensional-to-three-dimensionaltransition is assumed to be the same as the critical layer thickness for theonset of lattice relaxation by misfit dislocations, although these two will notnecessarily be the same.

Elman et al.96 used RHEED to investigate the critical layer thickness inInxGa1–xAs/GaAs (001) grown by MBE for various compositions and growthtemperatures. Figure 6.42 shows their measured RHEED intensity oscilla-tions for layers grown at 615°C with the compositions x = 0.24, 0.29, 0.32,0.39, 0.42, and 0.50. From these characteristics, the critical layer thicknesseswere found in terms of monolayers for layers having these compositionsand are shown in Figure 6.43, along with the predictions of the Matthewsand Blakeslee model.

6.9.9 Scanning Tunneling Microscopy (STM)

Scanning tunneling microscopy (STM) is often used to study the morpho-logical aspects of relaxing mismatched layers.97–99 However, under the appro-priate conditions, this technique can also be used for the indirect observationof misfit dislocation formation during growth. STM involves the scanningof a fine metal tip over the surface of the sample, while the vertical position

FIGURE 6.41Dark line defect densities as obtained from PLM images for GaAs/In0.2Ga0.8As/GaAs (001)strained single quantum wells (solid points) and In0.2Ga0.8As/GaAs (001) strained layer super-lattices (open circles) with various InGaAs layer thicknesses. (Reprinted from Gourley, P.L. et al.,Appl. Phys. Lett., 52, 377, 1988. With permission. Copyright 1988, American Institute of Physics.)

0

1000

2000

3000

0 100 200 300 400 500 600 700 800Layer thickness (Å)

Line

def

ect d

ensit

y (cm

−1)




of the tip is adjusted to maintain a nearly constant tunneling current betweenthe tip and the sample. While scanning, the vertical deflection of the tip isrecorded as a function of position on the surface. In this way, a surfacetopographic map is produced for the sample, with atomic-scale resolution.Lattice distortions at the surface having subangstrom scale may be imagedas long as smooth layer-by-layer growth is achieved and the critical layerthickness is on the order of a few atomic layers. Therefore, this method mayonly be applied to large-mismatch heteroepitaxial material systems. Also,the effective use of this method requires a purpose-built STM that is integralto an ultra-high-vacuum MBE chamber, to allow examination of the depositat various thicknesses. These requirements make the method applicable toonly a few heteroepitaxial material systems.

Yamaguchi et al.62 have used STM to determine the critical layer thicknessin MBE-grown InAs/GaAs (111)A. (The room temperature mismatch strain

FIGURE 6.42RHEED intensity oscillations for InxGa1–xAs/GaAs (001) grown by MBE at a temperature of 615°C,with the compositions x = 0.24, 0.29, 0.32, 0.39, 0.42, and 0.50. (Reprinted from Elman, B. et al.,Appl. Phys. Lett., 55, 1659, 1989. With permission. Copyright 1989, American Institute of Physics.)

InxGa1−xAs on GaAsTs = 615°C

x = 0.50tth = 4 ml

x = 0.42tth = 5 ml

x = 0.39tth = 7 ml

x = 0.24tth = 40 ml

x = 0.29tth = 12 ml

x = 0.32tth = 9 ml

Ga ONIn ON

Ga ONIn ON




for this combination is –7.2%.) In this material system, step flow growth canbe maintained for all values of InAs coverage. Also, the InAs (111)A surfaceexhibits a (2 × 2) surface reconstruction with a corrugation of only 0.2 Å.Because of these characteristics, the misfit dislocations at the interface pro-duce measurable distortions at the top surface.

Yamaguchi et al. found that the InAs grew heteroepitaxially in a step flowmode, but with some nucleation of islands. At between 1 and 2 ml thickness,they observed the appearance of ~0.5-Å surface troughs at the coalescedboundaries of islands. Because the (2 × 2) reconstruction pattern was con-tinuous across these troughs, they were interpreted to be the result of misfitdislocations at the heterointerface, rather than gaps between islands. Further,these misfit dislocations were found to form a trigonal network at the (111)Ainterface, with dislocation lines along the <110> directions. From this studyit was concluded that the critical layer thickness is between 1 and 2 ml (1 ml= 3 Å) for this material combination.

6.9.10 Rutherford Backscattering (RBS)

The critical layer thickness may also be determined from Rutherford back-scattering (RBS) experiments. Here, a He ion beam is incident on the sample,which can be adjusted in angle by a goniometer. Along certain low-index

FIGURE 6.43Critical layer thickness (in monolayers) vs. composition x for InxGa1–xAs/GaAs (001). (Reprintedfrom Elman, B. et al., Appl. Phys. Lett., 55, 1659, 1989. With permission. Copyright 1989, AmericanInstitute of Physics.)

0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Indium composition x

Thic

knes

s (m

onol

ayer

s)

Matthews and Blakeslee615°C RHEED550°C RHEED




crystallographic directions such as <100> and <110>, the He ions channeldeeply into the specimen, and only a small percentage of them will bebackscattered to the detector. For other orientations (misaligned crystal), alarge number of ions will be backscattered and detected. Therefore, if thenumber of backscattered ions per unit time is measured as a function of theangle of incidence for the He ion beam, there will be dips in the profilecorresponding to the low-index directions in the crystal.

Figure 6.44 shows that the off-normal crystallographic directions in theheteroepitaxial layer will be shifted by the presence of strain. For a pseudo-morphic, tetragonally distorted layer, as shown in Figure 6.44a, the <110>channeling direction for the heteroepitaxial layer will differ from that of thesubstrate. But for the relaxed layer of Figure 6.44b, the <110> directions willbe parallel. Therefore, if the difference between the substrate and epitaxiallayer channeling directions is measured for samples of different thicknesses,the critical layer thickness can be determined from the onset of observablestrain relaxation.

For (001) heteroepitaxy of zinc blende or diamond crystals, if the <110>channeling directions of the epitaxial layer and substrate differ by , thenthe tetragonal distortion is given by

(6.110)

where c and a are the out-of-plane and in-plane lattice constants for theepitaxial layer, respectively, is the relaxed lattice constant of the epitaxiallayer, and is the angle between the <110> channeling direction and thenormal to the surface. For the <110> channeling direction, .

Bean et al.100 used RBS to determine the critical layer thickness in Si1–xGex/Si (001) and found good agreement with x-ray strain measurements.

6.10 Crystal Orientation

In some cases, the crystal orientation of the heteroepitaxial layer may be verydifferent from that of the substrate, or the two crystals may even havedifferent crystal structures. In these situations, the measurement of three hklx-ray diffraction profiles (with three noncoplanar diffraction vectors) allowsthe determination of the crystal orientation for a nonpolar heteroepitaxiallayer such as GexSi1–x. For a polar semiconductor, a crystallographic etch maybe used in conjunction with the x-ray measurements for the unambiguousassignment of the heteroepitaxial relationship between the deposit and sub-strate. For example, in the case of GaAs, molten KOH etching can be usedto distinguish between the [110] and the directions.

Δφtet

c aae

tet− = − Δφφ φsin cos

ae

φφ = °45

[ ]110




More often, the epitaxial layer and substrate differ only slightly in orien-tation. Then this misorientation may be found by measuring diffractionprofiles at various azimuths. Figure 6.45 illustrates the specimen geometryused in an HRXRD experiment. ω, ξ, and ψ are the rocking, tilt, and azi-muthal angles, respectively. Rocking curves are obtained by measuring thediffracted intensity as a function of the rocking angle ω. Rocking curvesmay be measured at different azimuths; for each setting of the azimuth ψ,the tilt ξ is adjusted so that the diffraction vector is in the plane of thediffractometer. At an azimuthal rotation ψ, the angular separationbetween the epitaxial layer rocking curve intensity peak and the substraterocking curve peak will be

FIGURE 6.44Schematic illustration of backscattering directions in mismatched heteroepitaxial structures. (a)For a pseudomorphic layer, the tetragonal distortion introduces a tilt of the channelingdirections relative to the substrate. (b) For a relaxed layer, this tilt is zero. (Reprinted from Bean,J.C. et al., J. Vac. Sci. Technol. A, 2, 436, 1984. With permission. Copyright 1984, American Instituteof Physics.)

c

Substrate

Substrate

as

ae

as

as

ae

as

as

<100

><1

00>

<110

> Epil

ayer

<110>

<110> Substrate

(a)

(b)

110

Δω




(6.111)

where is the difference in Bragg angles between the epitaxial layer andsubstrate, is the misorientation between the epitaxial layer and the sub-strate, and specifies the direction of the misorientation. Therefore, ifrocking curves are measured at three or more azimuths, a plot of vs. ψallows the determination of the crystallographic misorientation (and )for the epitaxial layer.

To illustrate the technique, Figure 6.46 shows the measured rocking curvepeak separation as a function of azimuth for a 4.7-μm-thick layer of GaAsgrown on Si (001) by MOVPE.77 The filled squares represent the measureddata. The curve is the best sinusoidal fit to the data, given by

(6.112)

In this example, the GaAs layer is tilted by 225 arc sec (0.0625°) toward the[010]. Because the substrate was tilted by 2.25° toward the , the tilt ofthe [001] axis of the epitaxial layer with respect to that of the substrate istoward the surface normal.

6.11 Defect Types and Densities

Defects such as dislocations, stacking faults, twins, and inversion domainboundaries may be investigated using a number of characterization tech-niques. TEM, crystallographic etching, and XRD are employed most often.Of these, TEM is the most powerful, as it can image individual dislocationsor planar defects. It is possible with TEM to determine the Burgers vectors

FIGURE 6.45Geometry used in an HRXRD experiment. A rocking curve (ω scan) involves measuring thediffracted intensity as a function of the rocking angle, ω. Prior to measurement of the rockingcurve, the tilt, ξ, must be adjusted so that the diffraction vector is in the plane of the diffractom-eter. Rocking curves may be measured at different azimuthal positions ψ. However, the tilt mustbe readjusted for each new azimuth.

Specimen

Detector

ω

ψ

X-ray beam

ξ

Δ Δ Δω θ φ ψ ψ= + −B cos( )0

ΔθB

Δφψ0

ΔωΔφ ψ0

Δω ψ= ′′ + ′′ − °5270 225 225cos( )

[ ]010




for dislocations, as well as their line vectors. However, the TEM can imageonly a small volume of the sample. In addition, the sample preparation isdestructive and, in some cases, could alter the configuration of the defectsstudied. Crystallographic etching has been used extensively to determinethe densities of dislocations intersecting a crystal surface. The crystallo-graphic etches reveal pits (or occasionally hillocks) at the points of emergencefor threading dislocations. This technique can be used over a large samplearea, but is not applicable for very high dislocation densities, for which theetch pits overlap. XRD is nondestructive and can be used to estimate theaverage dislocation density in the volume of the sample from the dislocationbroadening of the rocking curve widths. Other techniques, such as photolu-minescence microscopy (PLM), cathodoluminescence (CL), x-ray topogra-phy (XRT), electron beam-induced current (EBIC), or scanning tunnelingmicroscopy (STM), can be used to image individual dislocations. Their usehas been covered in Section 6.9 and will not be repeated here.

6.11.1 Transmission Electron Microscopy

Transmission electron microscopy (TEM) is capable of imaging individualdislocations, similar to PLM, CL, XRT, and EBIC, but it also allows determi-nation of the Burgers vector for a dislocation using the analysisdescribed in Section 6.4.2. This information can be used to understand the

FIGURE 6.46004 rocking curve peak separation, Δω, vs. the azimuthal angle, ψ, for 4.7-μm-thick GaAs/Si (001)grown by MOVPE. The experimental data are shown with filled squares. Also plotted is thecurve given by . (Reprinted from Ghandhi, S.K. and Ayers, J.E.,Appl. Phys. Lett., 53, 1204, 1988. With permission. Copyright 1988, American Institute of Physics.)

5000

5100

5200

5300

5400

5500

0 90 180 270 360Azimuth ψ (degrees)

Peak

sepa

ratio

n Δω

(Arc

sec)

Δω ψ= ′′ + ′′ − °5270 225 225cos( )

g b⋅




geometry of the slip systems in a semiconductor crystal and the characterof the misfit and threading dislocations.

Figure 6.47 illustrates the method of Burgers vector identification usingTEM, from the work of Abrahams et al.102 The sample investigated compriseda graded GaAs1–xPx layer grown on GaAs (001) by vapor phase epitaxy. Thetwo-beam, bright-field TEM micrographs were obtained for the identicalarea of a single sample, but with four different diffraction vectors. The misfitdislocation under the numeral 1 vanishes for the diffraction vectorbut is visible for , , and . Its Burgers vector musttherefore be perpendicular to , but not any of the other diffractionvectors. From this it can be concluded that the Burgers vector is parallel to[011], and so b = (a/2)[011]. The line of this dislocation is parallel to ,

FIGURE 6.47TEM Burgers vector analysis for a graded GaAs1–xPx layer on a GaAs (001) substrate. The fourplan view TEM micrographs were obtained for the identical area of a single sample, but withdifferent diffraction vectors: (a) , (b) , (c) , and (d) . (Re-printed from J. Mat. Sci., 4, 223 (1969), Dislocation morphology in graded heterojunctions:GaAs1–xPx, M.S. Abrahams, L.R. Weisberg, C.J. Buiocchi, and J. Blanc, Figure 4. With kindpermission of Springer Science and Business Media.)

g = [ ]02 2g = [ ]02 2 g = [ ]0 40 g = [ ]00 4

[ ]02 2

[ ]011

1

2

3

1 μm

022g

0221

2

3

1 μm

022

g 022

1

2

3

1 μm

022

g022

1

2

3

1 μm

022 g 004

(a) (b)

(c) (d)

g = [ ]022 g = [ ]022 g = [ ]040 g = [ ]004




so it has edge character. Following similar reasoning, we can conclude thatthe misfit dislocation to the left of the numeral 1 is an edge dislocation, butwith .

In plan view TEM micrographs like those shown in Figure 6.47, the pointswhere misfit dislocations appear to end really correspond to their bendingout of the plane of the image. This implies the existence of a set of threadingdislocations, each of which has a component of its line vector perpendicularto the image plane. The Burgers vector is conserved upon bending of thedislocation, so an indirect Burgers analysis can be performed for the thread-ing dislocations in this way. However, cross-sectional TEM micrographs aremore commonly used for the study of threading dislocations, allowing theirdirect imaging and Burgers analysis.

6.11.2 Crystallographic Etching

Crystallographic etches are often used to evaluate dislocation densities inbulk or heteroepitaxial semiconductors. These etches reveal the points ofemergence of dislocations on the etched surface as hillocks or, more com-monly, pits. These features occur due to the reduced or enhanced etch ratein the strained region around the dislocation.

Etch pits on a specimen surface are counted within a known area usingoptical microscopy, Nomarski phase contrast microscopy, SEM, AFM, orsome other imaging method. The areal density of etch pits (the etch pitdensity (EPD)) can thus be determined and is usually considered to corre-spond to the threading dislocation density.

The one-to-one correspondence between etch pits and dislocations wasfirst established by Vogel et al.102 They etched a bulk Ge crystal with 5:3:3:1/10 HNO3:HF:CH3COOH:Br2 and showed that the spacing of the etch pitsalong a grain boundary was consistent with the angle of the boundary, asmeasured by x-ray diffraction. However, each etchant/crystal system hasunique characteristics, and it is always necessary to verify a one-to-onecorrespondence between etch pits and the threading dislocations usinganother technique, such as TEM.

In some cases, crystallographic etching results in distinctive etch pits thatallow identification of different types of dislocations. An example of this isthe case of offcut 4H- or 6H-SiC (0001) etched by molten KOH, in whichthree types of etch pits allow the identification of threading screw disloca-tions, threading edge dislocations, and basal-plane screw dislocations.104,105

Sometimes the measured EPD is found to be much lower than the dislo-cation density determined by other techniques. In the case of GaAs/Si (001)etched by molten KOH, the measured etch pit densities are sometimes foundto be orders of magnitude lower than the dislocation density measured byTEM.105 This discrepancy has been explained by the existence of a secondset of smaller etch pits that may be observed by TEM.106 However, even thecounting of both sets of pits resulted in underestimation of the dislocation

b a= ( / )[ ]2 011




density. This is because two or more closely spaced dislocations may producea single pit, and because there are no pits associated with some disloca-tions.107 Occasionally, a crystallographic etch will produce pits that do notappear to be associated with dislocations.

EPD measurements are best suited for crystals with low dislocation den-sities, such a semiconductor substrates. Because a large area can be etchedand examined for pits, even Si wafers with dislocation densities of ~10 cm–2

may be characterized in this way. On the other hand, the etch pits fromneighboring dislocations begin to overlap at high defect densities, resultingin undercounting. For this reason, the practical upper limit of the dislocationdensity that may be measured by crystallographic etching is about 106 to 108

cm–2. This upper limit can be maximized by the use of slow etchants, shortetch times, and high-magnification imaging techniques (SEM or TEM).

A number of crystallographic etches have been used with Si. The Dashetch108 was first used to establish the correlation between etch pits anddislocations in Si; however, this mixture is slow and requires hours of etchtime. The Sirtl,109 Secco,110 and Wright111 etches are fast acting and commonlyused with Si. All three utilize chromium salt oxidizers. Plating out of thechromium ion helps to delineate the etch pits under microscopic inspection.The Wright etch uses copper nitrate as a plating agent as well. The Sirtl andWright etches are anisotropic, revealing triangular pits on Si (111) and rec-tangular pits on Si (001). The Secco etch is nearly isotropic and producescircular or elliptical pits on both Si (111) and Si (001).

A wide range of crystallographic etches is available for GaAs as well.Molten KOH is commonly used, and an important application for this etchis the discovery of inversion domain boundaries (IDBs). The two-part A-Betch is frequently used for the determination of dislocation densities, butalso for the delineation of interfaces in cleaved multilayered structures. Inaddition, all of the crystallographic etches designed for use with Si can alsobe used for GaAs. The application of Sirtl etch with GaAs is interesting inthat it produces hillocks and mounds, rather than pits, at the emergencepoints for defects.

Molten KOH etching of GaAs was developed by Grabmaier and Watson112

for use with the (001) orientation. Angilello et al.113 subsequently showed thecorrespondence between etch pits revealed by molten KOH and dislocationsimaged in an x-ray transmission topograph using the 004 reflection for GaAs(001). Typically, molten KOH etching is carried out in a crucible maintainedat a temperature of 400°C or higher* for a time of up to several minutes. Onthe (001) surface, the etch pits revealed by molten KOH elongate in thedirection. They can therefore be used to distinguish between the <110>directions on the surface of a GaAs crystal. Moreover, in the presence ofinversion domain boundaries (IDBs), the pits are rotated by 90° upon cross-ing a domain boundary. The presence or absence of these rotations is oftenused to establish whether inversion domains exist in the material.

* The melting point for KOH is 360°C.

[ ]110




Unfortunately, the crystallographic etches that were developed for GaAsseldom prove useful with other zinc blende III-V semiconductors, and sonew etches have been developed and characterized for nearly every newmaterial. In the case of InP, mixtures of HBr with either HF or CH3COOHhave been successfully applied to the EPD characterization of both (001) and(111) surfaces.114 For GaP, H2SO4:H2O2:HF etches115 are used. For InSb,HNO3:HF:CH3COOH etchants have been used for (001) and (111)Sb surfaces,but it is necessary to add stearic acid, Pb, or Ge to reveal etch pits on the(111)In surfaces.116 Even for other arsenides, such as InGaAs, the A-B etchcommonly employed for GaAs has proven to be unreliable in the measure-ment of dislocation densities.117

For zinc blende II-VI semiconductors as well, unique crystallographicetches have been developed for nearly every material of interest. Mixturesof lactic, nitric, and hydrofluoric acids are utilized with the tellurides CdTe118

and CdZnTe,119 whereas the Chen etch120 is used with HgCdTe. Bro-mine–methanol mixtures have been used successfully with the selenidesZnSe,92,121,122 ZnSSe,92 and ZnMgSSe.123 This etch is ineffective with the cad-mium-bearing quaternary ZnCdMgSe, however, so mixtures of hydrobromicand acetic acid are used instead.124

Several etches have been used for the EPD characterization of III-nitrides,including H3PO4:H2SO4 and molten KOH. Ono et al.125 demonstrated the useof H3PO4:H2SO4 to reveal etch pits on GaN (0001), and this etch was usedby Tsai, Chang, and Chen to evaluate GaN free-standing films. Kozawa etal.126 used molten KOH to characterize the dislocation density in GaN/α-Al2O3 (0001). They found hexagonal pits with a density of .

Rosner et al.127 observed pits on the surface of GaN on sapphire (0001)following MOVPE growth, without the intentional use of a crystallographicetch. These pits were confirmed to be associated with threading dislocations.However, it was not clear whether the pits were inherent in the growth processor if they formed by an etching process in the reactor following growth.

SiC is difficult to etch due to its chemical stability, and molten KOH is theonly reported crystallographic etch for this material. Typically, a temperatureof 600°C is used with an etch time of 30 s to several minutes. Three types ofetch pits are observed on vicinal 4H- or 6H-SiC (0001): large hexagonal pits,small hexagonal pits, and shell-like etch pits. These have been shown to beassociated with threading screw dislocations (TSDs), threading edge dislo-cations (TEDs), and basal-plane dislocations (BPDs), respectively.103,104 TSDsand TEDs are parallel to the [0001]; the BPDs are screw dislocations that liewithin the (0001) basal plane but emerge at the surface if the SiC is offcut.

The compositions of some commonly used crystallographic etches aretabulated in Appendix E.

6.11.3 X-Ray Diffraction

The x-ray rocking curve technique is complementary to the TEM and EPDmethods because it is nondestructive and can be used to determine threading

2 107 2× −cm




dislocation densities in the range from 105 to 109 cm–2 and with a factor oftwo accuracy. Gay et al.88 and Hordon and Averbach89 described the theoryof x-ray rocking curve broadening in dislocated metal crystals, and this workwas extended to zinc blende semiconductor crystals.90

The x-ray rocking curve measured from a semiconductor crystal using aBartels diffractometer or a double-crystal diffractometer in the parallel posi-tion has a characteristic full width at half maximum (FWHM) that dependson the crystal examined and the incident wavelength, but which is insensi-tive to instrumental effects. Dislocations broaden the rocking curve in threeways: (1) the dislocation introduces a rotation of the crystal lattice, thusdirectly broadening the rocking curve; (2) the dislocation is surrounded bya strained volume of crystal, in which the Bragg angle is nonuniform; and(3) in grossly dislocated crystals, arrays of dislocations can form the wallsbetween small polycrystals, giving rise to crystal size broadening. In high-quality heteroepitaxial layers, only the first two (angular broadening andstrain broadening) are important.

In developing a quantitative model for the dislocation broadening, it isassumed that the broadening is due to dislocation half-loops or bent-oversubstrate threading dislocations. These dislocations have both threading andmisfit dislocation segments that can give rise to rocking curve broadening.Kaganer et al.91 have presented an analysis of the profile broadening due tomisfit dislocations for certain defect configurations. However, the local strainvariations introduced by misfit dislocations are expected to cancel at dis-tances greater than one half their mean spacing in the interface. For a relaxedlayer, the misfit dislocation spacing in the interface is

(6.113)

where f is the lattice mismatch strain, b is the length of the Burgers vector,is the angle between the Burgers vector and line vector, and is the anglebetween the interface and the normal to the slip plane. For example, wit-h and , the misfit dislocation spacing is about 0.1 μm. Arelaxed layer will be at least 10 times this thickness. Most of the diffractedintensity will originate in the top 90% of the layer, and the broadening willtherefore be dominated by the threading dislocations there. Moreover, sinceboth the critical layer thickness and the misfit dislocation spacing in therelaxed layer are inversely proportional to , the same should be true forall relaxed heteroepitaxial layers.

Experimental results also suggest that the rocking curve broadening isdominated by threading dislocations. For example, Ayers et al.128 showedthat the misfit dislocations are not dominant in broadening the 004 rockingcurves for GaAs/Si (001). In that study, GaAs on Si samples were preparedwith equal misfit dislocation densities but very different threading disloca-tion densities, by postgrowth annealing. It was found that the rocking curve

1/cos cosρ α φ= b

f

αφ

b = 4Å f = 0 1. %

f




widths changed dramatically with annealing, despite the essentiallyunchanged misfit dislocation densities.

It is further assumed that the resulting diffraction profiles are Gaussian.This assumption is supported by experimental evidence128 in the case ofGaAs/Si (001). Figure 6.48 shows the GaAs 004 rocking curve for a 1.5-μm-thick GaAs/Si (001) sample grown by MOVPE, along with the Gaussian bestfit. It can be seen that the two profiles are indistinguishable, except in thetail regions. This was also found to be true for the 002, 113, 224, 115, 006, 026,444, and 117 rocking curves measured for the same GaAs/Si (001) sample.Kaganer et al.129 found a similar result when they investigated the shapes of0002 and 0004 diffraction profiles from dislocated GaN/6H-SiC (0001) grownby plasma-assisted molecular beam epitaxy (PAMBE). They found that thetails of the profiles obeyed a power law, but that most of the intensity residedin the central part of the peak, which was Gaussian in nature.

Assuming that the measured rocking curve is Gaussian, with a fullwidth at half maximum , and results from the convolution of Gaus-sian intensity distributions,

(6.114)

where is the intrinsic rocking curve width for the crystal being exam-ined, is the width of the instrumental broadening function, is

FIGURE 6.48GaAs 004 rocking curve for a 1.5-μm-thick GaAs/Si (001) sample grown by MOVPE, obtainedby a double-crystal diffractometer with Cu kα radiation, along with the Gaussian best fit.(Reprinted from Ayers, J.E., J. Cryst. Growth, 135, 71, 1994. With permission. Copyright 1994,Elsevier.)

0.0−0.10 −0.05 0.00 0.05 0.10

0.2

0.4

0.6

0.8

1.0

ω (degrees)

Nor

mal

ized

inte

nsity

GaussianExperiment

hklβm hkl( )

β β β β βα εm dhkl hkl hkl hkl hkl202 2 2 2( ) ( ) ( ) ( ) (= + + + )) ( ) ( )+ +β βL rhkl hkl2 2

β0( )hklβd hkl( ) βα( )hkl




the rocking curve broadening caused by angular rotations at disloca-tions, is the rocking curve broadening caused by the inhomogeneousstrain surrounding dislocations, is the rocking curve broadening dueto the crystal size (layer thickness), and is the rocking curve broad-ening due to curvature of the heteroepitaxial specimen.

The intrinsic rocking curve width for incidence can be esti-mated by

(6.115)

where is the classical electron radius, , is the x-ray wave-length, V is the crystal volume for which we have calculated the structurefactor, for a cubic crystal, is the magnitude of the structure factorfor the hkl reflection, and is the Bragg angle. Selected values are tabulatedin Appendix F.

The broadening due to angular rotation at the dislocations has been mod-eled by Gay et al.88 In this treatment, the single crystal is considered tocomprise an arrangement of subsidiary mosaics with mutual inclination.Each subsidiary mosaic is associated with a dislocation, which tilts themosaic with respect to its neighbors. If the orientations of the mosaics havea Gaussian distribution, then the probability that a mosaic is tilted by anangle from the mean orientation is equal to ,where is the standard deviation. The probability for two mosaics to betilted with respect to each other by an angle is equal to

. The mean disorientation between mosaics is

(6.116)

The value of the denominator is , and Dunn and Koch130 have integratedthe numerator explicitly to obtain . Therefore,

(6.117)

For a Gaussian distribution, the relationship between the standard devia-tion and the FWHM is given by , so that the meandisorientation between mosaics can be related to the measured FWHM by

βε( )hklβL hkl( )

βr hkl( )

β0 θB + Φ

βλ θ

π θθ

0

22 1 2

2=

+ −r F

Ve B H

B

B[ cos( )]

sin( )sin( )si

Φnn( )

/

θB +⎡

⎣⎢

⎤

⎦⎥Φ

1 2

re 2 818 10 5. × − Å λ

V a= 03 FH

θB

θ ( ) exp( / )σ π θ σ2 21 2 2− −σ

θ φ−( ) exp[ ( )/ ]σ π θ φ σ2 21 2 2 2− − +

θ φθ φ θ φ σ θ φ

− =− − +

−

−∞

∞

−∞

∞

∫∫ exp[ ( )/( )]

exp[

2 2 22 d d

(( )/( )]θ φ σ θ φ2 2 22+−∞

∞

−∞

∞

∫∫ d d

2 2πσ4 3σ π

θ φ σπ

− = 2

σ β σ β= /( ln )2 2 2




(6.118)

If the threading dislocations are arranged in a random network, with anaverage spacing of , where D is the dislocation density, then the aver-age angle between neighboring mosaics is approximately

(6.119)

where b is the length of the Burgers vector. Comparing Equations 6.118 and6.119, we can relate the measured broadening to the dislocation density by

(6.120)

The angular broadening due to threading dislocations is therefore given by

(6.121)

The strain broadening due to dislocations has been modeled by Warren131

and Hordon and Averbach.89 If it is assumed that the random array ofthreading dislocations gives rise to a Gaussian distribution of local strain,then this strain gives rise to rocking curve broadening given by

(6.122)

where is the mean square strain in the direction of the normal to thediffracting planes. may be estimated as follows. If it is assumed that thePoisson ratio is 1/3, then from the known strain distribution around theedge dislocation it is found that

(6.123)

where is the angle between the dislocation glide plane normal and thenormal to the diffracting planes, is the angle between the dislocationBurgers vector and the normal to the diffracting planes, and r and are theupper and lower limits for integration of the strain field in the radial directionfrom the dislocation core. Similarly, for the pure screw dislocation,

θ φ βπ

α− =2 2ln

1/ D

θ φ− ≈ b D

Db b

≈ =βπ

βα α2

2

2

22 2 4 36( ln ) .

β πα α2 22 2≈ =( ln )b D K

β ε θ θε ε2 2 2 28 2= =( ln ) tan tanN B BK

εN2

εN2

ε ψπNe

br

rr

22 2 2

2 20

5 2 45 0 4564

= + ⎛⎝⎜

( . cos . cos )ln

Δ ⎞⎞⎠⎟

Δψ

r0




(6.124)

For dislocations with mixed character,

(6.125)

where is the angle between the Burgers vector and line vector for thedislocations.

For half-loops in a (001) diamond or zinc blende epitaxial layer, with 60°misfit segments and screw (90°) threading segments, it has been shown thatthe dislocation broadening is given by

(6.126)

For heteroepitaxial layers, the crystal size broadening is usually affectedonly by the layer thickness h. Then the crystal size broadening is givenapproximately by the Scherrer equation:82

(6.127)

For example, for the 004 reflection from GaAs (001), arc sec for a 1-μm-thick layer.

Rocking curve broadening due to specimen curvature has been analyzedby Halliwell et al.24 and Flanagan.132 It is given by

(6.128)

where w is the width of the x-ray beam in the plane of the diffractometerand r is the radius of curvature for the heteroepitaxial structure. Usually, thebroadening due to curvature is negligible if the substrate is much thickerthan the epitaxial layer. However, since the curvature contribution is thesame for both the substrate and the epitaxial layer, the substrate rockingcurve width gives an upper limit for the broadening due to the curvature.

Combining the above equations, we obtain

ε ψπNs

br

rr

22 2

3 204

=⎛⎝⎜

⎞⎠⎟

sinln

ε ε α ε αN Ne Ns2 2 2 2 2= +sin cos

α

β θ

θ

ε

ε

2 2 7 2

2

0 09 2 10= ×

=

−. ln( ) tan

tan

b D cm D

K

B

B

βπ

λθL

Bh2

2

2

2

4 2≈⎡

⎣⎢

⎤

⎦⎥

⎛

⎝⎜⎞

⎠⎟ln

cos

βL = 35

βθr

B

wr

22

2 2=

sin




(6.129)

If the effects of the crystal size broadening and curvature are negligible,then the broadening contribution due to dislocations can be found bythe deconvolution of the natural width of the reflection and the instrumentalbroadening:

(6.130)

If the rocking curve width is measured for a number of hkl reflec-tion, and the extracted value is plotted as a function of , thenthe values of and correspond to the intercept and slope of the char-acteristic, respectively. The dislocation density can be found from eithervalue, using angular broadening:

(6.131)

or strain broadening:

(6.132)

Figure 6.49 shows the application of this method for the case of a 3.0-μm-thick layer of GaAs/Si (001) grown by MOVPE. The filled circles representmeasurements and the line is the least squares fit, given by

(6.133)

Therefore, from the angular broadening of dislocations we obtain D = 1.4 ×108 cm–2 and from the strain broadening of dislocations we obtain D = 1.5 ×108 cm–2.

Because the two approaches give nearly the same dislocation density, it isusually adequate to estimate D from the angular broadening alone. Then itsuffices to measure one rocking curve at a small value of . In the case

β β β θα εm d Bhkl hkl hkl K K202 2 2

4 2

( ) ( ) ( ) tan

ln

= + + +

+ππ

λθ θh

K

B

r

B2

2

2 2

⎡

⎣⎢

⎤

⎦⎥⎛

⎝⎜⎞

⎠⎟+

cos sin

βdisl

β β β

θα ε

disl m d

B

hkl hkl

K K

2 2 2

2

= −

= +

( ) ( )

tan

βm hkl( )βdisl hkl( ) tan2 θB

Kα Kε

DK

b= α

4 36 2.

DK

b cm D=

× −ε

0 09 2 102 7. ln( )

β θdisl2 2 2 241 600 5850= +, ( ) ( ) tanarc sec arc sec BB

tan2 θB




of GaAs (001), the 004 reflection can be used because for this reflec-tion, , so the strain broadening may be neglected.

6.12 Multilayered Structures and Superlattices

For multilayered heteroepitaxial structures, a wealth of information can beobtained from the x-ray rocking curve, including the depth profiles of com-position and strain. However, the analysis is not straightforward as in thecase of a single, uniform heteroepitaxial layer. Tanner and Hill137 have shownthat there is no one-to-one correspondence between the layers in the struc-ture and the intensity peaks in the rocking curve. Moreover, because thephase information is lost in the x-ray rocking curve, it is not possible todirectly calculate the structure from it. Instead, this is done indirectly withdynamical x-ray simulations. The starting point for this analysis is a virtualstructure, which represents an educated guess based on the growth condi-tions and times. The virtual structure is then refined by adjusting the thick-nesses, compositions, or lattice constants in the individual layers until thereis an acceptable match between the simulated rocking curve and the exper-imental profile. The refined virtual structure is then assumed to correspond

FIGURE 6.49 vs. for a 3.0-mm-thick layer of GaAs/Si (001) grown by MOVPE. is the square

of the dislocation broadening, extracted from measured rocking curve widths for various hklreflections. θB is the Bragg angle. The filled circles represent the data extracted from measure-ments, and the line is the least squares fit, given by = 41,600 (arc sec)2 + 5850 (arc sec)2tanθB. (Reprinted from Ayers, J.E., J. Cryst. Growth, 135, 71, 1994. With permission. Copyright1994, Elsevier.)

0

100000

200000

0 5 10 15 20Tan2 θB

β dis1

2 (A

rc se

c)2

βdisl2 tan2 θB βdisl

2

βdisl2

tan .2 0 422θB =




closely to the actual one. This analysis requires the ability to calculate the x-ray rocking curve for an arbitrary layered structure, which will be describedbriefly here.

The calculation of the rocking curve for an arbitrary structure is based onrecursion formulae derived from the Tagaki–Taupin equation,24,25,133 whichwas introduced in Section 6.2.3 and is repeated here:

(6.134)

The structure is assumed to comprise N uniform layers, as shown inFigure 6.50. This approach is generally applicable, and continuously gradedlayers may be approximated by a series of steps. The starting point is thecalculation of the scattering amplitude for the substrate, which is assumedto be an infinitely thick, perfect crystal. It is calculated using the Dar-win–Prins formula:27

(6.135)

where the deviation parameter for the substrate is given by

(6.136)

where is the Bragg angle for the substrate, is the actual angle ofincidence on the diffracting planes, , , and are the substrate struc-ture factors for the 000, hkl, and reflections, respectively, C is the polar-ization factor, and is given by

FIGURE 6.50Lamellar (layered) structure assumed for the dynamical simulation of the x-ray rocking curve.The substrate is assumed to be an infinitely thick, perfect crystal. The N layers are assumed tobe uniform both laterally and in the growth direction.

Substrate

layer 1layer 2layer 3

layer N

h1

h2

h3

hN

XS

X1

X2

X3

XN

− = − +idXdT

X X2 2 1η

X SignS S S= − −η η η( ) 2 1

η θ θ θS

BS BS S

HS HS

b b F

b C F F= − − − −( )sin( ) . ( )2 0 5 1 0Γ

Γ

θBS θF S0 FHS FHS

h k lΓ




(6.137)

where is the classical electron radius, , is the x-ray wave-length, and V is the crystal volume for which we have calculated the struc-ture factor. (For a cubic crystal, .) When the polarization of the incidentbeam is in the plane of incidence (π polarization), , and whenthe x-rays are polarized perpendicular to the plane of incidence (σ polariza-tion), .

Once the scattering amplitude has been calculated for the substrate, recur-sion equations are used N times for the N layers in the virtual structure. Forthe nth layer in the stack, the scattering amplitude at the top is relatedto the scattering amplitude at the bottom by

(6.138)

where

(6.139)

and

(6.140)

Here, the deviation parameter is the value for the nth layer, given by

(6.141)

and the thickness parameter for the nth layer is

(6.142)

where is the thickness of the nth layer and , , and are thestructure factors for the nth layer.

The substrate scattering amplitude is used as the scattering amplitude atthe bottom of layer 1, for the calculation of the scattering amplitude at the

Γ = rV

eλπ

2

re 2 818 10 5. × − Å λ

V a= 03

C Bπ θ= cos( )2

Cσ = 1

Xn

Xn−1

XS SS Sn n n

n n

n n

= + − +−

η η2 1 2

1 2

1( )

S X iTn n n n n n1 12 21 1= − + − − −−( )exp( )η η η

S X iTn n n n n n2 12 21 1= − + − −−( )exp( )η η η

η θ θ θn

Bn Bn n

Hn H n

b b F

b C F F= − − − −( )sin( ) . ( )2 0 5 1 0Γ

Γ

Tn

T hF F

n n

H n H n

H

=π

λ γ γ

Γ

0

hn F n0 FH n FH n




top of layer 1. This is the scattering amplitude at the bottom of layer 2, whichis used to obtain the scattering amplitude at the top of that layer, and so on.Once the scattering amplitude has been calculated for the top layer ofthe stack, the reflectivity is calculated from

(6.143)

This process is repeated for each angle in the range of interest to obtain thesimulated rocking curve.

Figure 6.51 shows the measured and simulated x-ray rocking curves for a10-period InGaAs/InP superlattice grown by MOVPE.134 The various ordersof the superlattice peaks are indexed in the figure. The InGaAs and InP layersin the superlattice have thicknesses of 2 and 140 nm, respectively. It can beseen that the simulated rocking curve matches the measured profile verywell, except in the low-intensity troughs where the experimental result isdominated by noise. To obtain good agreement with the measured rockingcurve, it was necessary to assume the presence of a InAsxP1–x graded layerat each interface where InP was grown after InGaAs. The graded layer was

FIGURE 6.51Measured and simulated x-ray rocking curves for a 10-period InGaAs/InP superlattice. Thevarious orders of the superlattice peaks are indexed in the figure. The InGaAs and InP layersin the superlattice have thicknesses of 2 and 140 nm, respectively. (Reprinted from Kim, I. etal., J. Appl. Phys., 83, 3932, 1998. With permission. Copyright 1998, American Institute of Physics.)

XN

PFF

XHHN

H NN=

2



105

104

103

102

101

100

X-ra

y int

ensit

y (cp

s)

InP (004) reflectionExponential grading

MeasuredSimulated

–900 –600 –300 0 300

–6–5

–4–3

–2

–1

0+1

+2

+3

Rocking angle (Arc sec)


assumed to have an arsenic composition x that decreased exponentially from7.4% at the nominal interface, with a characteristic depth of 18 nm. In thiscase, therefore, the rocking curve analysis was used to deduce this departurefrom the ideal growth structure.

Whereas the rocking curve simulation approach is generally applicable toheteroepitaxial multilayers, superlattice structures represent a special casefor which the superlattice period can be estimated without the need forrocking curve simulations. If the number of periods in the superlattice exceedthe number of atomic layers within each period, the interference peaks in thetales of the rocking curve will have a period dominated by the superlattice.Then if the superlattice period is D, the angular spacing of the superlatticepeaks in the rocking curve will be

(6.144)

For example, using the 004 reflection for an AlAs/GaAs superlattice withand , a superlattice period of 50 nm will result in

superlattice peaks separated by ~760 arc sec.

6.13 Growth Mode

It is important to be able to characterize the growth mode for the develop-ment of heteroepitaxial devices. In many applications, two-dimensionalgrowth (either Frank–van der Merwe or step flow growth) is necessary forthe attainment of flat interfaces in devices. On the other hand, quantum dotdevices require a Volmer–Weber (VW) or Stranski–Krastanov (SK) growthmode. In all of these cases, however, it is valuable to be able to characterizethe growth mode so the behavior can be understood and controlled.

In the case of MBE, the growth mode can be studied in situ using RHEED.A streaky RHEED pattern corresponds to an atomically smooth surface,whereas a spotty pattern indicates surface roughening, which is an indicationof Volmer–Weber or Stranski–Krastanov growth. If the growth mode is SK,layer-by-layer growth of the wetting layer will be followed by islanding. Inthis case, RHEED oscillations will be seen during the growth of the wettinglayer, but these oscillations will damp rapidly upon the change from two-dimensional to three-dimensional growth. This allows the determination ofthe thickness for the transition to islanding.

Many studies of the growth mode have relied on ex situ microscopy. Here,the morphology of the surface is examined using Nomarski interferencecontrast microscopy, SEM, or AFM. Films of various thicknesses can beexamined to distinguish between the SK and VW modes.

Δθ λθ

=D Bcos( )

λ ≈ 1 540594. Å θB ≈ °33 0.




Bean et al.100 studied the growth mode of Si1–xGex on Si (001) using Nomar-ski interference contrast microscopy. The objective of this study was to findthe conditions under which SiGe could be grown on Si by MBE withoutislanding, to enable the growth of heterostructures and superlattices. Theygrew a series of Si1–xGex layers, 100 nm thick, with various compositions andat various temperatures. Each layer was inspected using Nomarski interfer-ence contrast microscopy to determine the growth morphology. The map ofFigure 6.52 was produced, in which it can be seen that planar layers may begrown over the entire compositional range at low temperatures.

Haffouz et al.135 studied the growth mode for a GaN nucleation layerdeposited on sapphire (0001) using SEM. Prior to growth, the sapphirewafers were subjected to a high-temperature Si/N treatment, involvingexposure to SiH4 and NH3 at 1150°C, in the growth chamber. Then a 25- to50-nm GaN nucleation layer was grown on c-plane sapphire by low-pressureMOVPE at 525°C. Following the growth of the nucleation layer, some sam-ples were heated up to 1130 to 1150°C to simulate the temperature ramp thatwould be used prior to the growth of a thick GaN layer. Figure 6.53 showsSEM micrographs of the GaN samples. Figure 6.53a shows an as-grownnucleation layer, which has smooth surface morphology. The sample ofFigure 6.53b was 25 nm and heated to 1130 to 1150°C for 60 s; Figure 6.53c,50 nm thick and heated to 1130 to 1150°C for 60 s; and Figure 6.53d, 50 nm

FIGURE 6.52Film morphology for GexSi1–x on Si (001) grown with different growth temperatures and com-positions, with a total thickness of 100 nm. The filled circles represent layers exhibiting two-dimensional growth, and the open circles are for samples that had rough morphology indicativeof islanding. (Reprinted from Bean, J.C. et al., J. Vac. Sci. Technol. A, 2, 436, 1984. With permission.Copyright 1984, American Institute of Physics.)

Germanium fraction x

Gro

wth

tem

pera

ture

(°C) Three-dimensional growth

(islanding)

Two-dimensional growth(planar)

800

700

600

500

400

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0




thick and heated to 1130 to 1150°C for 30 s. It can be seen that the nucleationlayer thickness and its temperature treatment control the size and densityof the resulting GaN islands.

AFM has proven to be invaluable in studies of quantum dots and quantumdot assembly processes because of the atomic-scale resolution it affords.Refer to Chapter 4, in which several examples of AFM micrographs clearlyshow the sizes, shapes, and distributions of nanometer-scale islands.

Oliver et al.136 used AFM to study the growth modes of InGaN grown onGaN/sapphire (0001) by MOVPE. They studied the dependence of the growthmodes on the temperature and source flows. Figure 6.54 shows some represen-tative AFM micrographs obtained in this study that show the capabilities ofthe method. The InGaN layers shown in Figure 6.54 were 10 ml thick and grownat a temperature of 700 to 710°C, with a total pressure of 300 torr. The flows oftrimethylgallium and trimethylindium were fixed at 2 and 100 standard cubiccentimeters per minute (sccm), respectively, while the NH3 flow was varied.

FIGURE 6.53SEM micrographs of GaN nucleation layers grown on sapphire (0001) by low-pressure MOVPEfollowing a Si/N treatment: (a) as-grown nucleation layer; (b) 25-nm-thick nucleation layer thatwas heated to 1130 to 1150°C for 60 s; (c) 50-nm-thick nucleation layer that was heated to 1130to 1150°C for 60 s; (d) 50-nm-thick nucleation layer that was heated to 1130 to 1150°C for 30 s.(Reprinted from Haffouz, S. et al., Appl. Phys. Lett., 73, 1278, 1998. With permission. Copyright1998, American Institute of Physics.)

(a) (c)

(b) (d)

1 μm 1 μm

1 μm1 μm

0398 15kV 815.000

0398 15kV 815.000

0398 15kV 815.000

0398 15kV 815.000




The growth morphology changes markedly as the NH3 flow is varied from 1to 10 standard liters per minute (slm). In Figure 6.54a, the surface shows largespiral mounds, separated by flat terraces that do not have monolayer islands.This is indicative of a modified step flow growth mode. The samples of Figure5.54b and c exhibit island growth with clear second-layer nucleation, indicatingeither a VW or SK growth mode. On the surface of the sample of Figure 6.54d,only terraces are observed, revealing a step flow growth mode.

Problems

1. Suppose Al0.2Ga0.8As is grown on GaAs (001). (a) Assuming the layeris pseudomorphic, determine the 004 x-ray Bragg angle difference,the 115 Bragg angle difference, and the change in the inclination ofthe (115) planes due to the tetragonal distortion. (b) Repeat thecalculations with the assumption of a fully relaxed layer.

FIGURE 6.54AFM micrographs (1 × 1 μm) showing the surfaces of 10-ml InGaN layers grown on GaN/sapphire (0001) by MOVPE at various NH3 flow rates. All layers were grown at a temperatureof 700 to 710°C, with a total pressure of 300 torr. The flows of trimethylgallium and trimeth-ylindium were fixed at 2 and 100 sccm, respectively. The NH3 flow was (a) 1 slm, (b) 3 slm, (c)5 slm, and (d) 10 slm. (Reprinted from Oliver, R.A. et al., J. Appl. Phys., 97, 13707, 2005. Withpermission. Copyright 2005, American Institute of Physics.)

(a) (b)

(c) (d)




2. Calculate the 0002, 0004, and 0006 Bragg angles for sapphire assum-ing Fe kα1 radiation.

3. The 004 rocking curve is measured for a 1-μm-thick InxGa1–xAs layeron an InP (001) substrate at different azimuths. The peak separationvaries from –400 to –600 arc sec. The FWHM for the epitaxial layerrocking curve varies from 180 to 220 arc sec. Estimate the composi-tion of the layer, making reasonable assumptions about the state ofstrain in the InxGa1–xAs.

4. ZnSe is grown heteroepitaxially on GaAs (001). For a 460-nm-thicklayer, the measured rocking curve width (FWHM) is 314 arc sec. (a)What is the expected rocking curve width for a perfect crystal ofZnSe with this thickness? Is the 460-nm layer relaxed, and if so, whatis the approximate threading dislocation density? (b) Repeat for a95-nm-thick layer for which the measured 004 rocking curve widthis 304 arc sec.

5. A layer of ZnSxSe1–x is grown on GaAs (001) and characterized byHRXRD. With the azimuth, , set to zero, the projection of the x-rayincident beam is along the [110] direction in the surface of the sam-ple. The (004) rocking curve peak separation was –90 arc sec at ψ =0° and also at ψ = 180°. The 044 peak separations were determinedto be –150 arc sec at ψ = 45° and –120 arc sec at ψ = 225°, using the

incidence for both measurements. Determine the 004 and 044Bragg angles for the epitaxial layer and its in-plane and out-of-planelattice constants a and c. Assuming the elastic constants for the filmare the same as those for pure ZnSe, determine the relaxed latticeconstant a0. Using this value and Vegard’s law, estimate the compo-sition of the epitaxial layer.

6. The 004 rocking curve is measured for a vicinal Si (001) substrate atvarious azimuths. Determine the expected variation of the rockingcurve width (FWHM) assuming a perfect crystal with a 4° tilt andneglecting the instrumental broadening. Repeat, assuming theinstrument introduces a Gaussian broadening function with a widthof 6 arc sec.

7. A 40-period 80 nm/100 nm Al0.2Ga0.8As/GaAs superlattice is grownon a GaAs (001) substrate. 004 rocking curves are measured using Cukα radiation. If the entire structure is pseudomorphic, what is the posi-tion of the zero-order superlattice rocking curve peak in relation to thesubstrate peak? What is the expected spacing for the superlattice peaks?

8. Assuming 100-keV electrons, what is the expected spacing betweenthe 00 and 01 RHEED streaks for an unreconstructed GaAs surfaceif the sample-to-screen spacing is 30 cm?

9. An FTIR reflectance spectrum is measured for a 1-μm-thick layer ofInxGa1–xP on GaAs (001). What is the expected fringe spacing (inwavenumbers)?

ψ

θ φB +




References

1. W.L. Bragg, The diffraction of short electromagnetic waves by a crystal, Proc.Cambridge Phil. Soc., 17, 43 (1913).

2. N.W. Ashcroft and N.D. Mermin, Solid State Physics, Holt, Rinehart and Win-ston, Philadelphia, 1976.

3. International Tables for X-Ray Crystallography, Vol. IV, J.A. Ibers and W.C. Hamil-ton, Eds., Kynoch Press, Birmingham, England, 1974.

4. B.D. Cullity, Elements of X-Ray Diffraction, 2nd ed., Addison-Wesley PublishingCo., Reading, MA, 1978.

5. R.W. James, The Crystalline State, Vol. II, The Optical Principles of the Diffractionof X-Rays, Sir Lawrence Bragg, Ed., G. Bell and Sons, Ltd., London, 1948.

6. B.E. Warren, X-Ray Diffraction, Addison-Wesley, Reading, MA, 1969.7. V.S. Speriosu, H.L. Glass, and T. Kobayashi, X-ray determination of strain and

damage distributions in ion-implanted layers, Appl. Phys. Lett., 34, 539 (1979).8. V.S. Speriosu, Kinematical x-ray diffraction in nonuniform crystalline films:

strain and damage distributions in ion-implanted garnets, J. Appl. Phys., 52,6094 (1981).

9. V.S. Speriosu, B.M. Paine, M.A. Nicolet, and H.L. Glass, X-ray rocking curvestudy of Si-implanted GaAs, Si, and Ge, Appl. Phys. Lett., 40, 604 (1982).

10. V.S. Speriosu and C.H. Wilts, X-ray rocking curve and ferromagnetic resonanceinvestigations of ion-implanted magnetic garnet, J. Appl. Phys., 54, 3325 (1983).

11. V.S. Speriosu and T. Vreeland, Jr., X-ray rocking curve analysis of superlattices,J. Appl. Phys., 56, 3743 (1984).

12. V.S. Speriosu, M.A. Nicolet, J.L. Tandon, and Y.C.M. Yeh, Interfacial strain inAlxGa1–xAs layers on GaAs, J. Appl. Phys., 57, 1377 (1985).

13. C.G. Darwin, The theory of x-ray reflexion, Phil Mag., 27, 315 (1914); C.G.Darwin, The theory of x-ray reflexion, part II, Phil. Mag., 27, 675 (1914).

14. P.P. Ewald, Optics of crystals, Ann. Physik, 49, 1 (1916); Ann. Physik, 49, 117(1916).

15. P.P. Ewald, Foundations of the optics of crystals. III. The crystal-optics of x-rays, Ann. Physik, 54, 519 (1916).

16. P.P. Ewald, Group velocity and phase velocity in x-ray crystal optics, Acta Cryst.,11, 888 (1958).

17. M. von Laue, Dynamical theory of x-ray interference, Ergeb. Exakt. Naturw., 10,133 (1931).

18. B.W. Batterman and H. Cole, Dynamical diffraction of x rays by perfect crystals,Rev. Mod. Phys., 36, 681 (1964).

19. W.H. Zachariasen, Theory of X-Ray Diffraction in Crystals, John Wiley & Sons,New York, 1950.

20. B. Klar and F. Rustichelli, Dynamical neutron diffraction by ideally curvedcrystals, Il Nuovo Cimento, 13B, 249 (1973).

21. S. Takagi, A dynamical theory of diffraction for a distorted crystal, J. Phys. Soc.Jpn., 26, 1239 (1969).

22. S. Takagi, Dynamical theory of diffraction applicable to crystals with any kindof small distortion, Acta Cryst., 15, 1311 (1962).

23. D. Taupin, Dynamical theory of x-ray reflection by deformed crystals, C. R.Acad. Sci. (France), 256, 4881 (1963).




24. M.A.G. Halliwell, M.H. Lyons, and M.J. Hill, The interpretation of x-ray rockingcurves from III-V semiconductor device structures, J. Cryst. Growth, 68, 523 (1984).

25. C.R. Wie, T.A. Tombrello, and T. Vreeland, Jr., Dynamical x-ray diffraction fromnonuniform crystalline films: application to x-ray rocking curve analysis, J.Appl. Phys., 59, 3743 (1986).

26. W.J. Bartels, Characterization of thin layers on perfect crystals with a highresolution x-ray diffractometer, J. Vac. Sci. Technol. B, 1, 338 (1983).

27. J.A. Prins, Die Reflexion von Rontgenstrahlen an absorbierenden idealen Kri-stallen, Z. Phys., 63, 477 (1930).

28. P.B. Hirsch and G.N. Ramachandran, Intensity of x-ray reflexion from perfectand mosaic absorbing crystals, Acta Cryst., 3, 187 (1950).

29. B.K. Tanner and M.J. Hill, Double axis x-ray diffractometry at glancing angles,J. Phys. D, 19, L229 (1986).

30. B.D. Cullity, Elements of X-Ray Diffraction, 2nd ed., Addison-Wesley, Reading,MA, 1978, p. 511.

31. E.R. Cohen and B.N. Taylor, The 1986 Adjustment of the Fundamental PhysicalConstants, report of the Committee on Data for Science and Technology of theInternational Council of Scientific Unions (CODATA) Task Group on Funda-mental Constants, CODATA Bulletin 63, Pergamon, Elmsford, NY, 1986.

32. J. Ladell, W. Parish, and J. Taylor, Interpretation of diffractometer line profiles,Acta. Cryst., 12, 561 (1959).

33. H. Berger, Study of the Kα emission spectrum of copper, X-Ray Spectrom., 15,241 (1986).

34. J. Ayers and J. Ladell, Spectral widths of the Cu Kα lines, Phys. Rev. A, 37, 2404(1988).

35. J.W.M. DuMond, Theory of the use of more than two successive x-ray crystalreflections to obtain increased resolving power, Phys. Rev., 52, 872 (1937).

36. P.F. Fewster, A high-resolution multiple-crystal multiple-reflection diffractome-ter, J. Appl. Cryst., 22, 64 (1989).

37. P.F. Fewster, Combining high-resolution x-ray diffractometry and topography,J. Appl. Cryst., 24, 178 (1991).

38. J. Wang, H. Wu, R. So, Y. Liu, and M.H. Lie, Structure determination of indium-induced Si(111)-In-4×1 surface by LEED Patterson inversion, Phys. Rev. B, 72,245324 (2005).

39. G.M. Nomarski, Microinterferometre differential à ondes polarisées, J. Phys.Radium Paris, 16, 9S (1955).

40. D.B. Williams and C.B. Carter, Transmission Electron Microscopy: A Textbook forMaterials Science, Springer, Berlin, 2004.

41. K.R. Breen, P.N. Uppal, and J.S. Ahearn, Interface dislocation structures inInxGa1–xAs/GaAs mismatched epitaxy, J. Vac. Sci. Technol. B, 7, 758 (1989).

42. G. Binnig, C.F. Quate, and Ch. Gerber, Atomic force microscope, Phys. Rev. Lett.,56, 930 (1986).

43. A. Jasik, K. Kosiel, W. Strupinski, and M. Wesolowski, Influence of coveringon critical thickness of strained InxGa1–xAs layer, Thin Solid Films, 412, 50 (2002).

44. M. Tachikawa and H. Mori, Dislocation generation of GaAs on Si in the coolingstage, Appl. Phys. Lett., 56, 2225 (1990).

45. T. Ohno, A. Ohki, and T. Matsuoka, Surface cleaning with hydrogen plasmafor low-defect-density ZnSe homoepitaxial growth, J. Vac. Sci. Technol. A, 16,2539 (1998).




46. J.C. Campbell, S.M. Abbott, and A.G. Dentai, A comparison of “normal” lasersand lasers exhibiting light jumps, J. Appl. Phys., 51, 4010 (1980).

47. H. Kasano, Regular compositional steps generated in GaAs1–xPx VPE layers, J.Appl. Phys., 51, 4178 (1980).

48. P.L. Gourley, R.M. Biefeld, and L.R. Dawson, Elimination of dark line defectsin lattice-mismatched epilayers through use of strained-layer superlattices,Appl. Phys. Lett., 47, 482 (1985).

49. K. Matsuda, T. Saiki, H. Saito, and K. Nishi, Room-temperature photolumines-cence of self-assembled In0.5Ga0.5As single quantum dots by using highly sen-sitive near-field scanning optical microscope, Appl. Phys. Lett., 76, 73 (2000).

50. T. Saiki, S. Mononobe, M. Ohtsu, N. Saito, and J. Kusano, Tailoring a high-transmission fiber probe for photon scanning tunneling microscope, Appl. Phys.Lett., 68, 2612 (1996).

51. P.L. Gourley, T.J. Drummond, and B.L. Doyle, Dislocation filtering in semicon-ductor superlattices with lattice-matched and lattice-mismatched layer mate-rials, Appl. Phys. Lett., 49, 1101 (1986).

52. Y. Ohizumi, T. Tsuruoka, and S. Ushioda, Formation of misfit dislocations inGaAs/InGaAs multiquantum wells observed by photoluminescence microsco-py, J. Appl. Phys., 92, 2385 (2002).

53. E. Estop, A. Izrael, and M. Sauvage, Double-crystal spectrometer measurementsof lattice parameters and x-ray topography on heterojunctions GaAs-AlxGa1–xAs, Acta Cryst. A, 32, 627 (1976).

54. J. Hornstra and W.J. Bartels, Determination of the lattice constant of epitaxiallayers of III-V compounds, J. Cryst. Growth, 44, 513 (1978).

55. W.J. Bartels and W. Nijman, X-ray double-crystal diffractometry of Ga1–xAlxAsepitaxial layers, J. Cryst. Growth, 44, 518 (1978).

56. A.T. Macrander, G.P. Schwartz, and G.J. Gualtieri, X-ray and Raman character-ization of AlSb/GaSb strained layer superlattices and quasiperiodic Fibonaccilattices, J. Appl. Phys., 64, 6733 (1988).

57. A. Leiberich and J. Levkoff, A double crystal x-ray diffraction characterizationof AlxGa1–xAs grown on an offcut GaAs (100) substrate, J. Vac. Sci. Technol. B,8, 422 (1990).

58. A. Krost, G. Bauer, and J. Woitok, in Optical Characterization of EpitaxialSemiconductor Layers, G. Bauer and W. Richter, Eds., Springer, Berlin, 1996,p. 287.

59. B.E. Warren, X-Ray Diffraction, Addison-Wesley, Reading, MA, 1969, p. 353.60. J. Petruzzello, B.L. Greenberg, D.A. Cammack, and R. Dalby, Structural prop-

erties of the ZnSe/GaAs system grown by molecular beam epitaxy, J. Appl.Phys., 63, 2299 (1988).

61. A. Mazuelas, L. Gonzalez, F.A. Ponce, L. Tapfer, and F. Briones, Critical thick-ness determination of InAs, InP and GaP on GaAs by x-ray interference effectand transmission electron microscopy, J. Cryst. Growth, 131, 465 (1993).

62. H. Yamaguchi, J.G. Belk, X.M. Zhang, J.L. Sudijono, M.R. Fahy, T.S. Jones, D.W.Pashley, and B.A. Joyce, Atomic-scale imaging of strain relaxation via misfitdislocations in highly mismatched semiconductor heteroepitaxy: InAs/GaAs(111)A, Phys. Rev. B, 55, 1337 (1997).

63. T. Yao, The effect of lattice misfit on lattice parameters and photoluminescenceproperties of atomic layer epitaxy grown ZnSe on (100) GaAs substrates, Jpn.J. Appl. Phys., 25, L544 (1986).




64. M.J. Reed, N.A. El-Masry, C.A. Parker, J.C. Roberts, and S.M. Bedair, Criticallayer thickness determination of GaN/InGaN/GaN double heterostructures,Appl. Phys. Lett., 77, 4121 (2000).

65. P.L. Gourley, I.J. Fritz, and L.R. Dawson, Controversy of critical layer thicknessfor InGaAs/GaAs strained-layer epitaxy, Appl. Phys. Lett., 52, 377 (1988).

66. I. J. Fritz, P.L. Gourley, and L.R. Dawson, Dependence of critical layer thicknesson strain for InxGa1–xAs/GaAs strained-layer superlattices, Appl. Phys. Lett., 46,967 (1985).

67. I.J. Fritz, P.L. Gourley, and L.R. Dawson, Critical layer thickness in In0.2Ga0.8As/GaAs single strained quantum well structures, Appl. Phys. Lett., 51, 1004 (1987).

68. T. Reisinger, M.J. Kastner, K. Wolf, E. Steinkirchner, W. Hackl, H. Stanzl, andW. Gebhardt, Critical thickness determination of II-IV semicond. Mater. Sci.Forum, 182–184, 147 (1995).

69. P.J. Orders and B.F. Usher, Determination of critical layer thickness inInxGa1–xAs/GaAs heterostructures by x-ray diffraction, Appl. Phys. Lett., 50, 980(1987).

70. Y.C. Chen and P.K. Bhattacharya, Determination of critical layer thickness andstrain tensor in InxGa1–xAs/GaAs quantum-well structures by x-ray diffraction,J. Appl. Phys., 73, 7389 (1993).

71. Y. Kohama, Y. Fukuda, and M. Seki, Determination of the critical layer thicknessof Si1–xGex/Si heterostructures by direct observation of misfit dislocations, Appl.Phys. Lett., 52, 380 (1988).

72. H.-J. Gossman, G.P. Schwartz, B.A. Davidson, and G.J. Gultieri, Strain andcritical thickness in GaSb(001)/AlSb, J. Vac. Sci. Technol. B, 7, 764 (1989).

73. R. People and J.C. Bean, Calculation of critical layer thickness versus latticemismatch for GexSi1–x/Si strained-layer heterostructures, Appl. Phys. Lett., 47,322 (1985); 49, 229 (1986).

74. A. Fischer, H. Kuhne, and H. Richter, New approach in equilibrium theory forstrained layer relaxation, Phys. Rev. Lett., 73, 2712 (1994).

75. I.J. Fritz, Role of experimental resolution in measurements of critical layerthickness for strained-layer epitaxy, Appl. Phys. Lett., 51, 1080 (1987).

76. J.E. Ayers, S.K. Ghandhi, and L.J. Schowalter, Crystallographic tilting of het-eroepitaxial layers, J. Cryst. Growth, 113, 430 (1991).

77. S.K. Ghandhi and J.E. Ayers, Strain and misorientation in GaAs grown onSi(001) by organometallic epitaxy, Appl. Phys. Lett., 53, 1204 (1988).

78. X.G. Zhang, D.W. Parent, P. Li, A. Rodriguez, G. Zhao, J.E. Ayers, and F.C. Jain,X-ray rocking curve analysis of tetragonally distorted ternary semiconductorson mismatched (001) substrates, J. Vac. Sci. Technol. B, 18, 1375 (2000).

79. N.G. Anderson, W.D. Laidig, and Y.F. Lin, Photoluminescence of InxGa1–xAs–GaAs strained-layer superlattices, J. Electron. Mater., 14, 187 (1985).

80. K. Kamigaki, H. Sakashita, H. Kato, M. Nakayama, N. Sano, and H. Terauchi,X-ray study of misfit strain relaxation in lattice-mismatched heterojunctions,Appl. Phys. Lett., 49, 1071 (1986).

81. X.G. Zhang, P. Li, D.W. Parent, G. Zhao, J.E. Ayers, and F.C. Jain, Comparisonof x-ray diffraction methods for determination of the critical layer thicknessfor dislocation multiplication, J. Electron. Mater., 28, 553 (1999).

82. F. Scherrer, Bestimmung der Grosse und der Inneren Struktur von Kol-loidteilchen mittels Rontgenstrahlen, Nachr. Gottinger Ges., 98 (1918).

83. B.D. Cullity, Elements of X-Ray Diffraction, Addison Wesley, New York, 1978,p. 102.




84. B. Post, J. Nicolosi, and J. Ladell, Experimental procedures for the determina-tion of invariant phases of centrosymmetric crystals, Acta Cryst. A, 40, 684(1984).

85. C. Kim, I.K. Robinson, J. Myoung, K. Shim, M.-C. Yoo, and K. Kim, Criticalthickness of GaN thin films on sapphire (0001), Appl. Phys. Lett., 69, 2358(1996).

86. W.J. Bartels, J. Hornstra, and D.J.W. Lobeek, X-ray diffraction of multilayersand superlattices, Acta. Cryst. A, 42, 539 (1986).

87. L. Tapfer and K. Ploog, X-ray interference in ultrathin epitaxial layers: a ver-satile method for the structural analysis of single quantum wells and hetero-interfaces, Phys. Rev. B, 40, 9802 (1989).

88. P. Gay, P.B. Hirsch, and A. Kelly, The estimation of dislocation densities fromx-ray data, Acta. Met., 1, 315 (1953).

89. M.J. Hordon and B.L. Averbach, X-ray measurement of dislocation density indeformed copper and aluminum single crystals, Acta Met., 9, 237 (1961).

90. J.E. Ayers, The measurement of threading dislocation densities in semiconduc-tor crystals by x-ray diffraction, J. Cryst. Growth, 135, 71 (1994).

91. V.M. Kaganer, R. Kohler, M. Schmidbauer, R. Opitz, and B. Jenichen, X-raydiffraction peaks due to misfit dislocations in heteroepitaxial layers, Phys. Rev.B, 55, 1793 (1997).

92. A. Kamata and H. Mitsuhashi, Characterization of ZnSSe on GaAs by etchingand x-ray diffraction, J. Cryst. Growth, 142, 31 (1994).

93. C.B. O’Donnell, G. Lacey, G. Horsburgh, A.G. Cullis, C.R. Whitehouse, P.J.Parbrook, W. Meredith, I. Galbraith, P. Möck, K.A. Prior, and B.C. Cavenett,Measurements by x-ray topography of the critical thickness of ZnSe grown onGaAs, J. Cryst. Growth, 184/185, 95 (1998).

94. S.P. Edirisinghe, A.E. Staton-Bevan, and R. Grey, Relaxation mechanisms insingle InxGa1–xAs epilayers grown on misoriented GaAs B substrates, J.Appl. Phys., 82, 4870 (1997).

95. C.A. Parker, J.C. Roberts, S.M. Bedair, M.J. Reed, S.X. Liu, and N.A. El-Masry,Determination of the critical layer thickness in the InGaN/GaN heterostruc-tures, Appl. Phys. Lett., 75, 2776 (1999).

96. B. Elman, E.S. Koteles, P. Melman, C. Jagannath, J. Lee, and D. Dugger, In situmeasurements of critical layer thickness and optical studies of InGaAs quantumwells grown on GaAs substrates, Appl. Phys. Lett., 55, 1659 (1989).

97. C.W. Snyder, B.G. Orr, D. Kessler, and L.M. Sander, Effect of strain on surfacemorphology in highly strained InGaAs films, Phys. Rev. Lett., 66, 3032 (1991).

98. S. Ohkouchi, N. Ikoma, and M. Tamura, Growth mode transition processes ina GaAs/InP system studied by scanning tunneling microscopy, J. Electron.Mater., 25, 439 (1996).

99. D.K. Biegelsen, R.D. Bringans, J.E. Northrup, and L.E. Swartz, Early stages ofgrowth of GaAs on Si observed by scanning tunneling microscopy, Appl. Phys.Lett., 57, 2419 (1990).



102. F.L. Vogel, W.G. Pfann II, E. Corey, and E.E. Thomas, Observation of disloca-tions in lineage boundaries in germanium, Phys. Rev., 90, 489 (1953).

( )1 1 1




103. S. Ha, W.M. Vetter, M. Dudley, and M. Skowronski, A simple mapping methodfor elementary screw dislocations in heteroepitaxial SiC layers, Mater. Sci. Fo-rum, 389–393, 443 (2002).

104. Z. Zhang, Y. Gao, and T.S. Sudarshan, Delineating structural defects in highlydoped n-type 4H-SiC substrates using a combination of thermal diffusion andmolten KOH etching, Electrochem. Solid. State Lett., 7, G264 (2004).

105. K. Ishida, Proc. Mater. Res. Soc. Symp., 19, 133 (1987).106. K. Ishida, M. Akiyama, and S. Nishi, Misfit and threading dislocations in GaAs

layers grown on Si substrates by MOCVD, Jpn. J. Appl. Phys., 26, L163 (1987).107. D.J. Stirland, Quantitative defect etching of GaAs on Si: is it possible? Appl.

Phys. Lett., 53, 2432 (1988).108. W.C. Dash, Copper precipitation on dislocations in silicon, J. Appl. Phys., 27,

1193 (1956).109. E. Sirtl and A. Adler, Chromsäure-Flussäure als spezifisches System zur

Ätzgruben entwicklung auf Silizium, Z. Met., 52, 529 (1961).110. F. Secco d’Aragona, Dislocation etch for (100) planes in silicon, J. Electrochem.

Soc., 119, 948 (1972).111. M.W. Jenkins, A new preferential etch for defects in silicon crystals, J. Electro-

chem. Soc., 124, 757 (1977).112. J.G. Grabmaier and C.B. Watson, Dislocation etch pits in single crystal GaAs,

Phys. Status Solidi, 32, K13 (1969).113. J. Angilello, R.M. Potemski, and G.R. Woolhouse, Etch pits and dislocations in

{100} GaAs wafer, J. Appl. Phys., 46, 2315 (1975).114. K. Akita, T. Kusunoki, S. Komiya, and T. Kotani, Observation of etch pits in

InP by new etchants, J. Cryst. Growth, 46, 783 (1979).115. F. Kuhn-Kuhnenfeld, Polishing dislocation etch for GaP and GaAs, Inst. Phys.

Conf. Ser., 339, 158 (1977).116. H.C. Gatos and M.C. Lavine, Dislocation etch pits on the {111} and sur-

faces of InSb, J. Appl. Phys., 31, 743 (1960).117. Y.G. Chai and R. Chow, Molecular beam epitaxial growth of lattice-mismatched

In0.77Ga0.23As on InP, J. Appl. Phys., 53, 1229 (1982).118. T.J. de Lyon, D. Rajavel, S.M. Johnson, and C.A. Cockrum, Molecular-beam

epitaxial growth of CdTe(112) on Si(112) substrates, Appl. Phys. Lett., 66, 2119(1995).

119. W.J. Everson, C.K. Ard, J.L. Sepich, B.E. Dean, and G.T. Neugebauer, Etch pitcharacterization of CdTe and CdZnTe substrate for use in mercury cadmiumtelluride epitaxy, J. Electron. Mater., 24, 505 (1995)

120. J.-S. Chen, Etchant for Revealing Dislocations in II-VI Compounds, U.S. Patent4,897,152 (1990).

121. T. Koyama, T. Yodo, H. Oka, K. Yamashita, and T. Yamasaki, Growth of ZnSesingle crystals by iodine transport, J. Cryst. Growth, 91, 639 (1988).

122. G.D. U’Ren, M.S. Goorsky, G. Meis-Haugen, K.K. Law, T.J. Miller, and K.W.Haberern, Defect characterization of etch pits in ZnSe based epitaxial layers,Appl. Phys. Lett., 69, 1089 (1996).

123. C.C. Chu, T.B. Ng, J. Han, G.C. Hua, R.L. Gunshor, E. Ho, E.L. Warlick, andL.A. Kolodziejski, Defect characterization of etch pits in ZnSe based epitaxiallayers, Appl. Phys. Lett., 69, 1089 (1996).

124. L. Zeng, B.X. Yang, B. Shewareged, M.C. Tamargo, J.Z. Wan, F.H. Pollak, E.Snoeks, and L. Zhao, Determination of defect density in ZnCdMgSe layersgrown on InP using a chemical etch, J. Appl. Phys., 82, 3306 (1997).

{ }111




125. Y. Ono, Y. Iyechika, T. Takada, K. Inui, and T. Matsye, Reduction of etch pitdensity on GaN by InGaN-strained SQW, J. Cryst. Growth, 189, 133 (1998).

126. T. Kozawa, T. Kachi, T. Ohwaki, Y. Taga, N. Koide, and M. Koike, Dislocationetch pits in GaN epitaxial layers grown on sapphire substrates, J. Electrochem.Soc., 143, L17 (1996).

127. S.J. Rosner, E.C. Carr, M.J. Ludowise, G. Girolami, and H.I. Erikson, Correlationof cathodoluminescence inhomogeneity with microstructural defects in epitax-ial GaN grown by metalorganic chemical-vapor deposition, Appl. Phys. Lett.,70, 420 (1997).

128. J.E. Ayers, L.J. Schowalter, and S.K. Ghandhi, Post-growth thermal annealingof GaAs on Si (001) grown by organometallic vapor phase epitaxy, J. Cryst.Growth, 125, 329 (1992).

129. V.M. Kaganer, O. Brandt, A. Trampert, and K.H. Ploog, X-ray diffraction peakprofiles from threading dislocations in GaN epitaxial films, Phys. Rev. B, 72,45423 (2005).

130. C.O. Dunn and E.F. Koch, Comparison of dislocation densities of primary andsecondary recrystallization grains of Si-Fe, Acta Met., 5, 548 (1957).

131. B.E. Warren, Theory presented in the course on x-ray and crystal physics,Massachusetts Institute of Technology, Cambridge (1957).

132. W.F. Flanagan, Effect of Substructure on the Cleavage Fracture of Iron Crystals,Sc.D. dissertation, Massachusetts Institute of Technology, Cambridge (1959).

133. P.F. Fewster and C.J. Curling, Composition and lattice-mismatch measurementof thin semiconductor layers by x-ray diffraction, J. Appl. Phys., 62, 4154 (1987).

134. I. Kim, S.-W. Ryu, B.-D. Choe, H.-D. Kim, and W.G. Jeong, Matrix method forthe x-ray rocking curve simulation, J. Appl. Phys., 83, 3932 (1998).

135. S. Haffouz, H. Larèche, P. Vennéguès, P. de Mierry, B. Beaumont, F. Omnès,and P. Gibart, The effect of the Si/N treatment of a nitrided sapphire surfaceon the growth mode of GaN in low-pressure metalorganic vapor phase epitaxy,Appl. Phys. Lett., 73, 1278 (1998).

136. R.A. Oliver, M.J. Kappers, C.J. Humphreys, and G.A.D. Biggs, Growth modesin heteroepitaxy of InGaN on GaN, J. Appl. Phys., 97, 13707 (2005).

137. B.K. Tanner and M.J. Hill, X-ray double crystal diffractometry of multiple andvery thin heteroepitaxial layers, Adv. X-ray Anal., 29, 337 (1986).



355

7

Defect Engineering in Heteroepitaxial Layers

7.1 Introduction

Defect engineering

in heteroepitaxial layers refers to efforts to control thedensities, types, or arrangements of defects, especially dislocations. Thecommon approaches to defect engineering involve the use of buffer layers,patterned substrates, patterning and annealing, epitaxial lateral overgrowth,or compliant substrates. Some of these techniques, such as buffer layers,patterning with annealing, and epitaxial lateral overgrowth, are intended toremove existing defects from relaxed heteroepitaxial layers. Others, such asreduced area growth, nanoheteroepitaxy, and compliant substrates, aredesigned to prevent the introduction of dislocations in the first place. Thischapter presents the theory and practice of the most common defect engi-neering approaches, which were listed above.

7.2 Buffer Layer Approaches

Buffer layer approaches involve the insertion of an epitaxial layer or layersin between the substrate and the device layer, solely for the purpose ofreducing the dislocation density in the device layer. The buffer may be asingle, uniform layer, a graded composition layer, or a superlattice or othermultilayered structure, and all three types have been used with varyingdegrees of success.

7.2.1 Uniform Buffer Layers and Virtual Substrates

A uniform buffer layer has a constant composition throughout its thickness,and therefore the lattice mismatch with respect to the substrate is fixed at aconstant value. It is usually reported that the threading dislocation densityof such a buffer layer decreases with the reciprocal of its thickness. If the



356


buffer layer is designed to be lattice-matched to the device layer, then thisdevice layer may be grown on top of it without the introduction of newdislocations. In principle, then, the use of a sufficiently thick buffer layerwill allow the growth of a device layer with a low dislocation density on aconvenient lattice-mismatched substrate.

A thick, uniform buffer layer on a mismatched substrate is sometimes calleda virtual substrate (VS). For example, a thick epitaxial layer of GaN on asapphire substrate can serve as a virtual GaN substrate, even though conven-tional GaN substrates are not available in high quality at this time. If the GaNbuffer layer is very thick, it will behave as a conventional GaN substrate insome, but not all, respects. It is expected that the virtual substrate will berelaxed at the growth temperature, and that it will have a low dislocationdensity. But, unless the thick buffer is exfoliated from its substrate, it will beconstrained to mimic the thermal expansion of the substrate. All the same,virtual substrates open up possibilities for new materials, such as ternary orquaternary semiconductors, which cannot be readily manufactured in bulk.

The threading dislocation density at the surface of a uniform buffer layerdecreases monotonically with its thickness. Usually, a reciprocal relationshipis reported, and this is shown in the data of Figure 7.1. Moreover, the

FIGURE 7.1

Threading dislocation densities in uniform buffer layers vs. the buffer layer thickness. Thedata are from Sheldon et al.,

70

Ayers et al.,

71

Akram et al.,

72

and Kalisetty et al.,

73

as indicatedin the legend.

0.1 1 10Epitaxial layer thickness (μm)

InAs/GaAs Sheldon et al.GaAs/Ge/Si Sheldon et al.GaAs/InP Sheldon et al.InAs/InP Sheldon et al.GaAs/Si Ayers et al.ZnSe/GaAs Akram et al.ZnSe/GaAs Kalisetty et al.

1010

109

108

Thre

adin

g di

sloca

tion

dens

ity (c

m−2

)




357

dislocation densities depend only weakly on the lattice mismatch. Therefore,layers of ZnSe/GaAs (001) have dislocation densities similar to those ofInAs/GaAs (001) even though these systems differ in lattice mismatch by afactor of 1:30.

In some cases, the dislocation density in a uniform buffer layer can bereduced somewhat by postgrowth annealing. However, after annealing for asufficiently long time, the dislocation density saturates at a certain level andcannot be further reduced by additional annealing. In the case of GaAs on Si(001), it has been shown that there is a reciprocal relationship between thesaturated dislocation density and the layer thickness, as shown in Figure 7.2.

Several models have been proposed to explain these experimental results,based on dislocation–dislocation reactions, and are discussed in Section 5.10.Tachikawa and Yamaguchi

74

developed a semiempirical annihilation andcoalescence model. They assumed that both first-order and second-orderreactions are active, so that the equation governing the reduction of thedislocation density

D

with the thickness

h

is

(7.1)

FIGURE 7.2

Dislocation density vs. reciprocal of thickness for as-grown and postgrowth annealed layers ofGaAs/Si (001). (Reprinted from Ayers, J.E. et al.,

J. Cryst. Growth

, 125, 329, 1992. With permission.Copyright 1992, Elsevier.)

Reciprocal of thickness (μm−1)

As-grownPost-growth anneal

0

2

4

6

8

10

12

Thre

adin

g di

sloca

tion

dens

ity (1

08 cm−2

)

0 1 2 3

dDdh

C D C D= − −1 22



358


where and are constants. The solution gives the dislocation densityvs. thickness as

(7.2)

where is a constant. This model predicts a departure from the reciprocalrelationship between the dislocation density and the thickness for extremelythick layers.

Romanov et al.

1

extended the annihilation and coalescence model ofTachikawa and Yamaguchi to selective area growth and provided a physicalanalysis of the constants. Here, the starting equation was the same as thatgiven by Tachikawa and Yamaguchi:

(7.3)

However, it was assumed that the first-order reaction was due to the loss ofthreading dislocations to sidewalls in the case of selective area epitaxy. Forplanar (unpatterned) layers, this first-order reaction can be neglected so that

(7.4)

This model predicts an (approximately) inverse relationship between thedislocation density and thickness. However, it does not consider the depen-dence of the dislocation density on the lattice mismatch.

The glide model

2

was developed to account for the lattice mismatch depen-dence. Here, it was assumed that reaction (annihilation or coalescence)between two dislocations is limited by their ability to overcome the linetensions of their misfit segments so they can glide toward one another. Asshown in Section 5.10, this model predicts the dislocation density in a uni-form buffer layer to be

(7.5)

where

f

is the lattice mismatch, is the angle between the threading segmentsand the interface,

b

is the length of the Burgers vector,

h

is the layer thickness,and is the Poisson ratio. The glide model can be used to produce dislocationengineering curves for uniform buffer layers, as shown in Figure 7.3.

C1 C2

DD C C C h C C

=+ −

11 0 2 1 1 2 1( / / )exp( ) /

D0

dDdh

C D C D= − −1 22

DD

D C h h=

+ −0

0 2 01 ( )

Df

bh f=

−

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎛

⎝⎜

⎞

⎠⎟

cos

( )ln

φν16 1

14

φ

ν




359

An important conclusion that may be drawn from the experimental andmodeling studies is that

the threading dislocation density in a lattice relaxedheteroepitaxial layer will be of the order of 10

9

cm

–2

for 1-

μ

m thickness

. As aconsequence, a uniform buffer layer would have to be about 100

μ

m thickin order to achieve

D

< 10

7

cm

–2

. Therefore, the use of a uniform buffer layeralone is rarely adequate for the production of device quality material.

7.2.2 Graded Buffer Layers

Graded buffer layers can also be used to accommodate the lattice mismatchbetween the substrate crystal and the device layer. In the graded layer, thecomposition and therefore lattice constant vary continuously with distancefrom the substrate interface. Usually the composition is graded in linearfashion, so that the lattice mismatch is given by , where

y

is thedistance from the substrate interface and is the grading coefficient. It issometimes assumed that the linear profile is optimum, but this has not beenproven, and any arbitrary profile could be used. However, the material inthis section is confined to linearly graded layers, which have been the subjectof most of the theoretical and experimental work.

FIGURE 7.3

Dislocation engineering curves for uniform, relaxed buffer layers. The three curves show themismatch–thickness combinations that should result in threading dislocation densities of 10

7

,10

8

, and 10

9

cm

–2

, as indicated.

Laye

r thi

ckne

ss h

(μm

)

0 2 4 6 8

108 cm–2

107 cm–2

D = 109 cm–2

Lattice mismatch |f| (%)

100

10

1

0

f C yf=Cf



360


GaAs

1–x

P

x

/GaAs (001)

3–7

was one of the first graded material systems tobe studied, due to its importance for the production of LEDs. In their classicpaper, Abrahams et al. presented a series of transmission electron microscopy(TEM) micrographs to document the structures of dislocations in this materialsystem, and they developed the first quantitative models for threading andmisfit dislocation densities in graded layers. In their model, they assumedcomplete lattice relaxation in the graded layer; therefore, they neglected boththe equilibrium strain and the kinetic limitations to relaxation. This modelhas been discussed in Section 5.8.3 but will be reviewed briefly here.

Assuming a completely relaxed, linearly graded layer with a gradingcoefficient , the areal density of misfit dislocation segments inter-secting the {110} planes of the epitaxial layer will be

(7.6)

where is the mismatch-relieving component of the Burgers vector forthe misfit dislocation segments (the projection of the edge component intothe plane of the interface). Now, if it is assumed that the threading dislocationdensity increases to a constant value at a thickness equal to , and thatall dislocations are bent-over substrate dislocations, the (constant) threadingdislocation density in the top part of the graded layer will be

(7.7)

where

l

is the average length of the misfit segments. This length is assumedto be proportional to the separation of the misfit dislocations, with a constantof proportionality

m

, because of mutual repulsion. Then and

(7.8)

Therefore, the threading dislocation density at the top of the graded layerwill be proportional to the grading coefficient. This prediction was roughlyverified by the experimental results of Abrahams et al., as shown in Figure7.4. They found that the dislocation density increased in approximately linearfashion with the grading coefficient, from

D

= 8

×

10

5

cm

–2

with

C

f

=0.074%

μ

m

–1

to

D

= 4

×

10

7

cm

–2

for

C

f

= 0.185% cm

–1

.The limitation of the Abrahams et al. model is that it does not consider

kinetic factors and cannot predict the dependence of the threading disloca-tion density on the growth rate or temperature. To address this, Fitzgeraldet al. developed a model for dislocation flow in a linearly graded heteroepi-

C f yf = Δ Δ/

nC

bAf=

cos λ

b cos λ

nA−1 2/

Dn

lA=−2 1 2/

l m nA= −1 2/

DC

mbf=

2cos λ




361

taxial layer, which can be applied to determine the dislocation density atthe top of a linearly graded buffer. This dislocation dynamics model isapplicable as long as there are negligible impediments to dislocation glidein the graded layer.

In developing a dislocation dynamics model for graded layers, Fitzgeraldet al.

8

started with the idea that linear compositional grading during thegrowth of a heteroepitaxial layer is analogous to a constant-strain-rate exper-iment. In other words, if there is a sufficient number of threading dislocationsin the graded layer and these are gliding with sufficient velocity, then thestrain and threading dislocation density will be constant during grading.

It was assumed that the lattice mismatch varies linearly with distance fromthe interface, so that , where is the grading constant in cm

–1

and

y

is the distance from the interface. If the graded layer has a threadingdislocation density

D

, and each dislocation glides to create a length

l

of misfitdislocation, then the amount of strain relaxed will be approximately

(7.9)

FIGURE 7.4

Threading dislocation density vs. compositional gradient for GaAs

1–x

P

x

/GaAs (001) grown byvapor phase epitaxy. The grading coefficient is related to the compositional gradientby so that corresponds to .(Reprinted from

J. Mat. Sci.,

4, 223 (1969), Dislocation morphology in graded heterojunctions:GaAs

1–x

P

x

, M.S. Abrahams, L.R. Weisberg, C.J. Buiocchi, and J. Blanc, Figure 9. With kindpermission of Springer Science and Business Media.)

Compositional gradient ΔC/Δx (% P/μm)

105

106

107

108

Graded GaAs1−xPx/GaAs (001)

Thre

adin

g di

sloca

tion

dens

ity (c

m−2

)

0 1 10

C f x C xf = =Δ Δ Δ Δ/ . /0 037 Δ ΔC x/ %/= 10 μm C f = 0 37. %/μm

f C yf= Cf

δ ≈ Dbl4



362


The dislocation glide velocity is assumed to be given by the empiricalrelationship

(7.10)

where

B

is a constant (cm/s), is the effective stress, is a constanthaving units of stress, and

U

is the activation energy for dislocation glide.If the dislocation density is assumed to be constant, the time rate of strainrelaxation is

(7.11)

If the dislocations are all half-loops, then any particular misfit segmentwill grow by the glide of its associated threading segments in oppositedirections at a velocity

v

. Therefore,

(7.12)

where

Y

is the biaxial modulus and is the effective strain,

assumed to beconstant throughout the thickness of the graded layer

. Substituting this result intoEquation 7.11, we obtain

(7.13)

If it is assumed that the graded layer is much thicker than its critical layerthickness, and that the effective strain is constant with thickness, so that thestrain relief is a linear function of the thickness, then the threading disloca-tion density is found to be

(7.14)

where

g

is the growth rate. Therefore, the threading dislocation density atthe top of the graded layer will be proportional to the growth rate as wellas the grading coefficient.

Neither nucleation nor multiplication of dislocations was considered inthe development of this model. However, it is likely that these processes can

v BUkT

eff

m

=⎛

⎝⎜⎞

⎠⎟−

⎛⎝⎜

⎞⎠⎟

σσ0

exp

σ eff σ0

� �δ = Dbl

4

�l v BYUkT

meffm= = −

⎛⎝⎜

⎞⎠⎟

2 2 ε exp

εeff

�δ ε= −⎛⎝⎜

⎞⎠⎟

DbBY

UkT

meffm

2exp

DgC

bBYUkT

fm

effm

=⎛⎝⎜

⎞⎠⎟

2

εexp




363

act to produce a steady-state threading dislocation density in the earlierstages of film growth. The model should then be applicable to the growthof the remaining thickness.

Another key assumption of the dislocation dynamics model is that thereare no impediments to the glide of the dislocations. Often, this is not thecase. As will be demonstrated below, impediments to dislocation glide candrastically increase the defect densities in graded layers.

Fitzgerald et al.

9

applied the dislocation dynamics model to In

x

Ga

1–x

P/GaP(001) graded layers by lumping the parameter

B, the biaxial modulus, andthe effective strain together in an adjustable constant , yielding

(7.15)

This model was fit to experimental data for InxGa1–xP/GaP (001) gradedlayers grown to a final In composition of 10% by metalorganic vapor phaseepitaxy (MOVPE) in the temperature range of 650 to 800°C. The growth ratewas 3 μm/h (8.3 × 10–4 μm/s) and the grading coefficient was 0.4%/μm (4× 10–3/μm). Figure 7.5 shows the experimental data (filled circles) and the

FIGURE 7.5Threading dislocation density in graded InxGa1–xP/GaP (001) as a function of growth temper-ature. All layers were graded to a final composition of 10% with a grading rate of 0.4%/μm(total thickness, 1.9 μm). The filled circles represent experimental data. The curve was calculatedusing Equation 7.15 with U = 2 eV, , , and .

C1

DgC

bCUkT

f=⎛⎝⎜

⎞⎠⎟

2

1

exp

Growth temperature (°C)

104

105

106

107

108

109

1010

Graded InxGa1−xP/GaP (001)

Thre

adin

g di

sloca

tion

dens

ity (c

m−2

)

500 600 700 800 900

g = × −8 3 10 4. μm/s C f = × −4 10 3 μm C1610= cm/s




best fit based on Equation 7.15 using U = 2 eV and C1 = 106 cm/s. Theexcellent fit between the model and the experimental data suggests theabsence of impediments to dislocation glide in these graded buffer layers.

However, InxGa1–xP/GaP (001) graded layers grown in the temperaturerange from 500 to 650°C exhibit a deterioration of the surface morphologyand an anomalous increase in the threading dislocation density. This hasbeen attributed to the occurrence of branch defects, which can impede dis-location motion. Figure 7.6 shows plan view TEM micrographs of InxGa1–xP/GaP (001) graded layers with a top composition of 10% In grown by MOVPEat two different temperatures. The sample in Figure 7.6a, grown at 650°C,exhibits so-called branch defects, which are characterized by meanderinglines of strain contrast. The threading dislocations appear to have segregatedto the branch defects, indicating that the latter may be responsible for imped-ing the glide of the former. On the other hand, the sample of Figure 7.6bexhibits no visible branch defects or threading dislocation pileups. Giventhat the 650°C sample shows signs of dislocation pileups, it is surprisingthat its dislocation density lies so close to the curve in Figure 7.5. However,layers grown at still lower temperatures exhibit dislocation pileups to agreater degree and correspondingly higher threading dislocation densities.

At a fixed growth temperature of 760°C, the threading dislocation densityin a graded InxGa1–xP/GaP (001) buffer layer is a function of the final Inconcentration, even with a constant grading coefficient. This is shown inFigure 7.7. Here, the filled squares represent measured threading dislocationdensities for InxGa1–xP/GaP (001) graded layers, all of which were grownwith the same grading coefficient ( ) and temperature(760°C), but with different ending compositions. The sample with a final

FIGURE 7.6Plan view TEM micrographs of InxGa1–xP/GaP (001) graded layers with a top composition of10% In grown by MOVPE at two different temperatures: (a) 650°C and (b) 760°C. The branchdefects in (a) appear to impede the glide of threading dislocations, but these are absent in (b).(Reprinted from Fitzgerald, E.A. et al., Mater. Sci. Eng. B, 67, 53, 1999. With permission. Copy-right 1999, Elsevier.)

(a) (b)1 μm1 μm

C f = × −4 10 3 /μm



Defect Engineering in Heteroepitaxial Layers 365

composition of x = 0.10 can be modeled using the fit of Figure 7.5, shownhere with the flat line. Layers with a final indium composition of 0.2 orgreater exhibit an anomalous increase in the dislocation density, which isthought to be related to lower average dislocation mobility (impediments toglide) associated with branch defects.

In the Si1–xGex/Si (001) system it is also found that the dislocation densityincreases with the extent of the grading, as shown in Figure 7.8. Here, all ofthe samples were grown at the same temperature (750°C) and with the samegrading coefficient ( ). The total threading dislocationdensity includes the dislocations in the pileups. The field dislocation densityis the threading dislocation density in the areas between the pileups. Theline was calculated using the dislocation dynamics model (Equation 7.14)with , , U = 2.25 eV, ,m = 2, and . The biaxial modulus was estimated using thevalues for Si and Ge with a linear interpolation; this results in a slight upwardslope of the line. The data point with a final composition of x = 0.15 can befit with the dislocation dynamics model using this reasonable set of param-eters. However, the layers graded to higher values of x (0.3 and 0.5) exhibitanomalous high threading dislocation densities. They also have a greaterdisparity between the total threading dislocation density and the field dis-location density. This indicates a greater tendency toward dislocation pileupsand an associated reduction in the effective strain, which can explain theelevated dislocation density. In the case of Si1–xGex, the interaction between

FIGURE 7.7Threading dislocation densities in InxGa1–xP/GaP (001) graded layers as a function of the finalIn composition for layers grown with the same grading coefficient and tem-perature (760°C).9

0.0 0.1 0.2 0.3Final indium concentration x

105

106

107

108

109

1010

Graded InxGa1−xP/GaP (001)

Thre

adin

g di

sloca

tion

dens

ity (c

m–2

)

( )C f = × −4 10 3 μm

C f = × −4 24 10 3. /μm

g = × −1 1 10 3. μm/s C f = × −4 24 10 3. μm B = ×9 8 103. cm/sεeff = × −1 33 10 3.




dislocations and the surface undulations has been cited as the source ofdislocation drag.

In conclusion, graded buffer layers can effectively reduce the threadingdislocation density for lattice-mismatched heteroepitaxy. A dislocationdynamics model has been developed that can predict the threading disloca-tion densities in linearly graded layers, as long as impediments to dislocationglide are absent. This model predicts that the dislocation density in a gradedlayer will be proportional to the growth rate and grading coefficient. In mostreal graded layers, there are significant sources of dislocation drag, such asthe branching defects in InxGa1–xP/GaP (001) and InxGa1–xAs/GaAs (001)graded layers and the surface roughening in Si1–xGex/Si (001) graded layers.These impediments to glide decrease the effective strain and dramaticallyincrease the resulting threading dislocation density. Nonetheless, it is possi-ble to achieve threading dislocation densities of 105 to 106 cm–2 using linearlygraded buffer layers with practical growth rates and grading coefficients.This represents a significant improvement over the case of a uniform bufferlayer. This is why graded layers have been commonly applied in the fabri-cation of commercial devices on highly mismatched substrates, includingGaAs1–xPx LEDs on GaAs substrates and InxGa1–xAs high-electron-mobilitytransistors (HEMTs) on GaAs substrates.

FIGURE 7.8Threading dislocation densities in Si1–xGex/Si (001) graded layers as a function of the final Gecomposition for layers grown with the same grading coefficient and tem-perature (700°C).9 The total threading dislocation density includes the dislocations in the pile-ups. The field dislocation density is the threading dislocation density in the areas between thepileups.

Final germanium concentration x

FieldTotal

105

106

107

Graded Si1−xGex/Si (001)Th

read

ing

dislo

catio

n de

nsity

(cm

−2)

0.0 0.1 0.2 0.3 0.4 0.5

( .C f = × −4 24 10 3 μm)




7.2.3 Superlattice Buffer Layers

In a number of heteroepitaxial material systems, it has been reported thatthe insertion of a strained layer superlattice (SLS) can reduce the threadingdislocation density of the heteroepitaxial material, compared to the case ofdirect growth without an SLS. Experimental evidence obtained by cross-sectional TEM and etch pit density (EPD) characterization shows that someof the threading dislocations can bend over at the interfaces of the SLS. Thesemay reach the edge of the sample and thereby be removed. However, dis-locations that do not reach the wafer edge may serpentine back and forth inthe alternating layers of the SLS, which have mismatch strains of oppositesign. Even though these dislocations may not reach the sample edge, theywill have increased opportunity to participate in annihilation or coalescencereactions with other dislocations, thus reducing the dislocation density inthe overlying material. The efficacy of the edge removal mechanism isexpected to reduce with increasing wafer diameter. This is due to the finiteglide velocities for dislocations, and also the possibility of pinning or block-ing or dislocation motion by other dislocations or types of crystal defects.On the other hand, the enhancement of the coalescence/annihilation reac-tions can operate on wafers of arbitrary diameter. Sometimes this mechanismis referred to as dislocation filtering.10

Soga et al.11,12 used SLS buffers to produce GaAs on Si substrates withreduced threading dislocations compared to direct GaAs/Si (001) heteroepi-taxy. In their work, they used MOVPE to grow a GaP buffer on Si (001),followed by GaP/GaAs0.5P0.5 and GaAs0.5P0.5/GaAs superlattices, and finallya thick layer of GaAs. In one set of experiments they found that GaAs grownon Si (001) with a GaAs0.5P0.5/GaAs SLS buffer exhibited an order of mag-nitude higher PL emission intensity than GaAs grown under the same con-ditions but on a Ge-coated Si substrate. In another set of experiments, moltenKOH etching revealed a remarkably low EPD of 4 × 103 cm–2, compared to108 cm–2, which is typical for direct GaAs/Si (001) heteroepitaxy. It is wellestablished that molten KOH EPDs sometimes underestimate the truethreading dislocation density in GaAs on Si; nonetheless, these results sug-gested a significant reduction in the actual threading dislocation density.

Other workers have reported significant reductions in the dislocation den-sity for GaAs on Si (001) by using SLS dislocation filters. For example,Okamoto et al.13 used a Ga0.9In0.1As/GaAs SLS buffer to reduce the threadingdislocation density in GaAs grown on a Si substrate by MOVPE. The EPDobtained by KOH etching for a 3.5-μm-thick layer was 1.4 × 106 cm–2, abouttwo orders of magnitude better than that for direct heteroepitaxy.

One novel application of SLS dislocation filters to the growth of GaAs onSi (001) involved the use of (GaAs)1–x(Si2)x/GaAs superlattices, as reportedby Rao et al.14 In their work, MBE was used to grow GaAs on a vicinal Si(001) substrate with three SLSs, each comprising five periods of(GaAs)0.8(Si2)0.2/GaAs. The (GaAs)1–x(Si2)x material was grown by migration-enhanced epitaxy (MEE).15,16 Cross-sectional TEM micrographs showed that




all visible dislocations had been filtered by the three SLSs, so that the upperbound for the threading dislocation density in the upper layer of GaAs was5 × 105 cm–2, representing a reduction of the dislocation density by severalorders of magnitude.

Rao et al.14 also found that multiple SLS buffers could be more effectivethan a single SLS in filtering dislocations. In their structures, with three SLSsof five periods each, there was visible deflection of dislocations at each SLS.The middle SLS was most effective, however, affecting a two-order reductionin the dislocation density. It is likely that SLS buffers are less effective atfiltering dislocations when their density is either very high (due to the limitedmismatch) or very low (limited by the available dislocations).

Qualitatively similar results have been obtained in other material systems,such as GaSb on GaAs (001). Qian et al.17 investigated the use of SLS buffersto reduce the threading dislocation density in this material system usingMBE. They found that the insertion of a five-period GaSb/AlSb SLS reducedthe threading dislocation in a 1.1-μm-thick GaSb layer by more than an orderof magnitude compared to the case of direct heteroepitaxy. Here, the thread-ing dislocation densities were characterized using plan view TEM. The dis-location filtering action of the SLS is shown in Figure 7.9. In this bright-fieldtransmission electron micrograph it can be seen that most of the threadingdislocations are bent over in the SLS, resulting in a low dislocation densityin the top GaSb layer.

SLS dislocation filters have also been applied to the growth of II-VI semi-conductors on mismatched substrates. For example, Reno et al.18 used SLSbuffers in the growth of Cd0.955Zn0.045Te on GaAs (001) by MBE. The latticemismatch strain in his system is f = –13.7%. The samples investigated were

FIGURE 7.9Bright-field cross-sectional TEM image showing the dislocation filtering action of a GaSb/AlSbSLS inserted between a GaAs (001) substrate and the GaSb top layer grown by MBE. (FromQian, W. et al., J. Electrochem. Soc., 144, 1430, 1997. Reproduced by permission of ECS–TheElectrochemical Society.)

GaSb

SLS

GaAs200 nm




grown on GaAs (001) substrates, starting with 30 Å of ZnTe (to establish the(001) orientation), a 3-μm-thick Cd0.955Zn0.045Te buffer, an SLS buffer, andfinally a top layer of Cd0.955Zn0.045Te. The SLS comprised 15 periods ofCd0.91Zn0.09Te/CdTe. It was designed to have an average lattice constantmatching the top Cd0.955Zn0.045Te epitaxial layer, and the period of the SLSwas varied to find the optimum value. They found that the SLS was mosteffective in filtering dislocations when its period was approximately 2600 Å.In these structures, bright-field cross-sectional TEM images revealed that theCd0.955Zn0.045Te buffer had a threading dislocation density of 1010 to 1011 cm–2,but that the layer above the SLS had a much reduced threading dislocationdensity of <105 cm–2. The average threading dislocation density may havebeen higher, based on their minimum 004 x-ray rocking curve width of 175arc sec. Nonetheless, the SLS buffer affected a reduction in the dislocationdensity by orders of magnitude.

Early efforts to apply SLS dislocation filters to GaN on sapphire (0001) wereunfruitful. Using MOVPE, Qian et al.19 inserted three periods of 6 nm of GaN/6 nm of AlN on a 0.5-μm GaN layer grown on sapphire (0001), and then grewan additional 3.0 μm of GaN. Cross-sectional and plan view TEM investiga-tion showed no evidence of dislocation bending at the SLS. However, Qianet al. noted that the threading dislocations in GaN are on -type slipplanes. Therefore, the stresses of the SLS in the (0001) plane would not providea driving force for glide of dislocations on their slip planes.

More recent efforts to apply SLS dislocation filters to GaN on silicon andsapphire substrates have met with limited success. For example, Feltin etal.20 studied the dislocation filtering properties of GaN/AlN SLSs in theMOVPE growth of GaN on Si (111). They found that the threading dislocationdensity could be reduced by the insertion of strained layer superlattices;however, the reduction was only by a factor of π (from 1.6 × 1010 to 4 × 109

cm–2) when four SLSs were inserted.Sun et al.21 studied the use of AlN/Al0.85Ga0.15N SLSs grown by pulsed

atomic layer epitaxy (PALE) to filter dislocations in ~1.0-μm-thickAl0.55Ga0.45N layers grown on sapphire (0001). Based on cross-sectional TEMimaging, they found that the SLS could effectively block screw dislocations,but had a negligible effect on edge-type dislocations. A comparison betweenthe SLS sample and a control sample with no SLS showed that the densityof screw-type threading dislocations was reduced by more than an order ofmagnitude, from 4 × 109 to 3 × 108 cm–2. Of course, the effect on the overallthreading dislocation density was less dramatic.

Gourley et al.10 suggested that semiconductor superlattices might filterthreading dislocations even without built-in strains, provided that the indi-vidual layers in the superlattice differ in elastic stiffness. To test this concept,they carried out a set of experiments involving the growth of InxGa1–xAs onGaAs (001) substrates by MBE. In all experiments, an InzGa1–zAs buffer wasfirst grown on the GaAs (001) substrate, followed by an InxGa1–xAs/InyAl1–yAs superlattice, and then a top layer of InzGa1–zAs. The individuallayers in the superlattice were 100 Å thick. Two cases were investigated. For

{ }1100

{ }1100




structures having (case I), the superlattice was lattice-matched withthe buffer layer and top layer, and therefore expected to be essentially strain-free. For structures with and (case II), the superlatticelayers had equal and opposite strains. The strains, compositions, and super-lattice period were determined by XRD. The dislocations in the structureswere imaged using photoluminescence microscopy (PLM). Two differentexcitation wavelengths (647.1 and 406.7 nm, deep excitation and shallowexcitation, respectively) were used to probe different depths of the samples.It was found that both the lattice-matched and strained superlattices wereeffective in filtering dislocations, with a significant (at least an order ofmagnitude) reduction of the dislocation density in the top layer.

El-Masry et al.22 studied the mechanisms for filtering of dislocations inGaAs on Si (001) by strained layer superlattices. In this work, GaAs1–yPy/InxGa1–xAs SLSs (y = 2x) lattice-matched to GaAs were used, and cross-sectional TEM characterization was used to study the resulting dislocationinteractions. They described five different experimentally observed interac-tions between dislocations and the SLS, shown schematically in Figure 7.10.As shown in (a), an edge dislocation may experience a zero Peach–Koehlerforce, and therefore not bend at the SLS. In other cases, such as an edgedislocation on a vicinal substrate, there may be an insufficient Peach–Koehlerforce on the dislocation; it may therefore jog in the SLS as in (b), but therewill be no reduction of the dislocation density. On the other hand, a mixeddislocation such as that shown in (c) can bend over completely and beremoved from the upper epitaxial layer, as long as it can glide all the wayto the sample edge. Dislocations of opposite Burgers vectors may react asshown in (d), with the resultant removal of two threading dislocations fromthe upper layer. Finally, two dislocations may react to form a third dislocation,as in (e). This process removes one threading dislocation from the upper layer.

Based on the available experimental evidence, it appears that superlatticebuffer layers are generally applicable for the reduction of threading disloca-tion densities in mismatched heteroepitaxial zinc blende semiconductors.Both strained layer superlattices and superlattices with modulated elasticstiffness are effective as dislocation filters. In the former case, the strains inthe layers can cause dislocations to weave back and forth, thus promotingannihilation and coalescence reactions between dislocations. In superlatticeswith modulated elastic stiffness, dislocations will tend to bend over into thesofter material with the same end result. A single dislocation filter will bemost effective at a medium threading dislocation density, but less effectiveat both lower and higher dislocation densities. However, in materials withhigh threading dislocation densities, the effectiveness of the dislocation fil-tering can be enhanced by using multiple superlattices.

There is also some evidence to guide the design of superlattice dislocationfilters. If the strain filter mechanism is to be used, then the SLS must bedesigned to promote the bending over of existing dislocations without theintroduction of new ones. Thus, the individual layers of the SLS must growin a Frank–van der Merwe (layer-by-layer) mode, and they must exceed the

x y z= =

x y≠ z x y= +( )/2




Matthews and Blakeslee critical layer thickness for the bending over ofgrown-in dislocations. However, they must not have sufficient strain orthickness to promote significant dislocation multiplication or nucleation. Theoverall SLS stack must not exceed its Matthews and Blakeslee critical layerthickness, for this would introduce new dislocations. However, this problemcan usually be avoided by designing the SLS to be strain balanced. (Thealternate layers will have equal but opposite strains built in.) In practicalSLS dislocation filters, the individual layers of the SLS should have moderatestrain and thicknesses that are greater than the Matthews and Blakesleecritical layer thickness, but not by more than about a factor of 10. Qian etal.17 found that a GaSb/AlSb superlattice was most effective in reducing thedislocation density for a GaSb layer grown on a GaAs (001) substrate whenthe GaSb and AlSb layers were both about 1000 Å thick. The Matthews andBlakeslee critical layer thickness for GaSb on a thick AlSb substrate (or AlSbon a thick GaSb substrate) is about 500 Å, so the thicknesses in the optimumSLS were about twice hc. In the work of Reno et al., the optimum period for

FIGURE 7.10Five types of interactions between dislocations and an SLS buffer, as experimentally observedby El-Masry et al.22 (a) An edge dislocation may experience a zero Peach–Koehler force andtherefore not bend at the SLS. (b) Some dislocations may jog at the SLS without being removedfrom the top layer. (c) A mixed dislocation may bend over and glide all the way to the edge,resulting in the elimination of a threading dislocation in the top layer. (d) Dislocations ofopposite Burgers vectors may participate in an annihilation reaction, whereby a half-loop iscreated but two threading dislocations are eliminated from the top layer. (e) Two dislocationsmay coalesce to form a third dislocation. This process removes one threading dislocation fromthe upper layer.

(a) (b)No interaction Partial bending

(c) (d)Escape to edge Annihilation

(e)Coalescence




a Cd0.91Zn0.09Te/CdTe SLS (15 periods) to filter dislocations in aCd0.955Zn0.045Te layer grown on a GaAs (001) substrate was 2600 Å. Therefore,the individual layers were 1300 Å thick, or about six times the Matthewsand Blakeslee critical layer thickness for Cd0.91Zn0.09Te/CdTe = 0.55%(and Å).

The limited success of SLS buffers for the dislocation filtering in (0001)nitride materials appears to be related to the geometry of the slip systemsand specifically the inability to filter edge-type threading dislocations. It ispossible, however, that SLS dislocation filters may prove to be more effectivewhen used with other crystal orientations or when combined with otherdefect engineering techniques.

Despite the large body of experimental work, there remains a need for ageneral quantitative model that can be used to design SLS dislocation filters.Also, more experimental work is needed to determine if SLS buffers will beeffective in other material systems, such as heteroepitaxial silicon carbide onmismatched substrates.

7.3 Reduced Area Growth Using Patterned Substrates

In heteroepitaxial layers with moderate mismatch (|f| < 2%), the initialgrowth is pseudomorphic and the initiation of lattice relaxation occurs bythe bending over of substrate dislocations (the Matthews and Blakesleemechanism). However, the number of available threading dislocations N toparticipate in this lattice relaxation process depends on the growth areaaccording to N = DA, where D is the density of substrate threading disloca-tions (which are replicated in the epitaxial layer) and A is the growth area.Therefore, a reduction in the growth area can also reduce the density of misfitdislocations at the interface.23 In fact, if the growth area is reduced suffi-ciently, there may be no substrate threading dislocations available to partic-ipate in lattice relaxation. Other mechanisms involving dislocationnucleation can become active, but only at thicknesses much greater than thecritical layer thickness for the Matthews and Blakeslee mechanism. There-fore, a reduction in the growth area may even enable the achievement ofmetastable heteroepitaxial layers that are completely free from both misfitand threading dislocations, even though they are greater than the criticallayer thickness. Even if sources of heterogeneous dislocation nucleationbecome active, they are also expected to have a finite density so that reducedgrowth area can suppress relaxation by these as well.

To test these ideas, Fitzgerald et al.24,25 performed a series of experimentsinvolving the MBE growth of In0.05Ga0.95As layers on GaAs (001) substratesthat had been patterned with round or rectangular 2-μm-high mesas. Exper-iments were conducted with epitaxial layer thicknesses of 350, 700, and 825nm, all many times the expected critical layer thickness. Various substrates

fhc = 200




were utilized, with threading dislocation densities varying from 102 to 1.5× 105 cm–2.

Figure 7.11 shows cathodoluminescence (CL) images of 350-nm-thickIn0.05Ga0.95As grown on a GaAs (001) substrate with a dislocation density of1.5 × 105 cm–2. The layer of Figure 7.11a was grown over a large area, withoutmesa patterning. The samples of Figure 7.11b to d were grown on circularmesas having diameters of 200, 90, and 67 μm, respectively. It can be clearlyseen that the misfit dislocation density decreases with the size of the mesa.It is also evident that there are different numbers of dislocations along thetwo perpendicular [110] directions for the reduced area growth.

A quantitative analysis revealed that for the 350-nm-thick In0.05Ga0.95Aslayers, the linear misfit dislocation density increased linearly with the mesadiameter. And as expected, higher misfit dislocation densities were measuredon substrate having higher threading dislocation densities. The results areshown in Figure 7.12.

The data of Figure 7.12a correspond to a substrate threading dislocationdensity of 1.5 × 105 cm–2. For both the α misfit dislocations running alongthe [110] direction and the β dislocations running along the direction,the average misfit dislocation density (in cm–1) increased linearly with themesa diameter. Also, both dislocation densities extrapolate to zero at zero

FIGURE 7.11Cathodoluminescence images of 350-nm-thick In0.05Ga0.95As grown on a GaAs (001) substratewith a dislocation density of 1.5 × 105 cm–2: (a) large-area growth; (b) growth on a 200-μmcircular mesa; (c) growth on a 90-μm circular mesa; (d) growth on a 67-μm circular mesa.(Reprinted from Fitzgerald, E.A. et al., J. Appl. Phys., 65, 2220, 1989. With permission. Copyright1989, American Institute of Physics.)

(a)

(b) (d)

(c)10 μm

10 μm10 μm

10 μm

[ ]110




FIGURE 7.12Average linear densities of interfacial misfit dislocations for 350-nm-thick In0.05Ga0.95As layerson patterned GaAs (001) substrates as functions of the round mesa diameter.24 The filled squaresare for α misfit dislocations running along the [110] directions, whereas the open squares arefor β dislocations running along the directions. The substrate threading dislocation den-sity was (a) 1.5 × 105 cm–2, (b) 104 cm–2, and (c) 102 cm–2.

Mesa diameter (μm)

1500

1000

500

00

Line

ar m

isfit d

isloc

atio

nde

nsity

(cm

−1)

(b)

100 200 300 400 500

0

500

1000

1500

Line

ar m

isfit d

isloc

atio

nde

nsity

(cm

−1)

Mesa diameter (μm)0

(c)

100 200 300 400 500

Mesa diameter (μm)

3000

2000

1000

0

Line

ar m

isfit d

isloc

atio

nde

nsity

(cm

−1)

(a)

0 100 200 300 400 500

[ ]110




mesa diameter; this indicates that only area-dependent dislocation sourcesare active and eliminates the possibility of dislocation multiplication in thesesamples. However, there is a marked difference between the densities forthe two types of dislocations, and the α dislocations appear to nucleateroughly twice as much as the β dislocations.

For the samples of Figure 7.12b, the substrate dislocation density was 104

cm–2. The trends are qualitatively the same as in Figure 7.12a. The averagelinear densities of misfit dislocations are lower than those for the case of themore dislocated substrate, but the improvement is not as much as might beexpected if only the substrate dislocations are active as sources of misfitdislocations. This implies the existence of other fixed sources that scale withthe area.

This behavior is even more evident in Figure 7.12c, for which the substratethreading dislocation was 102 cm–2. Here the misfit dislocation densities aresimilar to the case shown in Figure 7.12b, even though the substrate thread-ing dislocation has been reduced by two orders of magnitude. Therefore,there are significant fixed sources of misfit dislocations in addition to thesubstrate threading dislocations. However, the misfit dislocation densitiesstill extrapolate to zero at zero mesa area, so a reduction in the growth areacan dramatically decrease the misfit dislocation density at the interface.

Fitzgerald et al.24 also investigated 700-nm-thick layers of In0.05Ga0.95As onGaAs (001) substrates that had been mesa patterned. For these thicker layers,the average linear misfit dislocation densities increased super linearly withthe mesa diameter for mesas larger than 200 μm. This shows that dislocationmultiplication was active in the thicker layers with large mesa diameters.Also, for mesas smaller than 200 μm, the misfit dislocation densities did notextrapolate to zero for zero mesa diameter. This indicates that there aredislocation sources that do not scale with the mesa size, and these arebelieved to be associated with the mesa edges.

In conclusion, it has been shown that a reduction in the growth area canreduce the densities of misfit dislocations in mismatched heteroepitaxiallayers that are greater than the critical layer thickness. For In0.05Ga0.95As layersthat are five times the experimentally determined critical layer thickness, itis possible to grow material entirely free from misfit dislocations if thegrowth area is reduced sufficiently. Only fixed dislocation sources that scalewith the mesa size are active, and no dislocation multiplication occurs inthese layers. For In0.05Ga0.95As layers that were 10 times the experimentallydetermined critical layer thickness, new sources of dislocations came intoplay. For small mesas, sources associated with the mesa edges, which do notscale with mesa area, become important. For large mesas, dislocation mul-tiplication becomes important. Nonetheless, in all cases the misfit dislocationdensity can be reduced by a reduction in the growth area. Qualitativelysimilar results should be expected for other material systems. Therefore,reduced growth area should be useful for the elimination of interfacialdefects in heterojunction devices if layers thicker than the critical layer thick-ness are required.




7.4 Patterning and Annealing

It was shown in the previous section that reduced area epitaxy can reduce,or even eliminate, the introduction of misfit dislocations at the interface ofa mismatched heteroepitaxial layer. That approach can extend the usablelayer thickness for heterojunction devices up to perhaps 10 times the criticallayer thickness. But many heteroepitaxial systems of interest exhibit highmismatch and a Volmer–Weber growth mode, so it is not possible to obtainpseudomorphic layers by a reduction in the growth area.

Instead, we can remove threading dislocations from the material after ithas relaxed through the use of patterning and annealing. This approach,proposed by Zhang et al.,26 is known as patterned heteroepitaxial processing(PHeP). The reduced lateral dimensions of the epitaxial material allowthreading dislocations to glide to the sidewalls, where they are removed. Ifthe pattern dimensions are small enough, sidewall image forces will attractthe threading dislocations and effectively getter them.

There are two embodiments of the PHeP approach. In the first, a planarheteroepitaxial layer is grown, then patterned by etching, and then annealedat an elevated temperature. In the second version, the substrate is patternedprior to growth. Either mesa patterning can be used or an oxide mask layercan be used with selective epitaxy. If the substrate is patterned prior togrowth, the annealing can occur during the growth itself, or during a post-growth annealing.

Zhang et al. reported a simple model for PHeP,27 which can be summarizedas follows. For a relaxed heteroepitaxial layer much greater than the criticallayer thickness, the linear density of misfit dislocations is

(7.16)

where f is the lattice mismatch, is the angle between the Burgers vectorand line vector for the misfit dislocations, is the angle between the inter-face and the normal to the slip plane, and is the misfit-relievingcomponent of the Burgers vector. The misfit dislocation density and thethreading dislocation density D are related by the mean length of the misfitdislocation segments, . If there are two orthogonal misfit dislocationarrays with the same value of and each misfit dislocation has n thread-ing segments associated with it, then

(7.17)

Here, n can range from 0 to 2, and for dislocation half-loops, .

ρMD

ρα φMD

cfb

hh

= −⎛⎝⎜

⎞⎠⎟cos cos

1

αφb cos cosα φ

LMD

LMD

ρMDMDDLn

=2

n = 2




Zhang et al. assumed that for an unpatterned (planar) heteroepitaxial layermuch greater than the critical layer thickness, dislocation multiplicationprocesses would have created plentiful threading dislocations, and so anyadditional lattice relaxation would occur by their glide rather than the cre-ation of new threading dislocations. Then, if the threading dislocation den-sity is fixed and the equilibrium strain is maintained,

(7.18)

For the planar layer, it can be assumed that n and D will remain unchanged,as long as threading dislocations do not move long enough distances toencounter an edge, and if there is negligible threading dislocation annihilation.

On the other hand, threading dislocations may reach the edges if theheteroepitaxial material is patterned into mesas, leading to a reduction in nand D. Suppose the mesas are square, with sides of length L parallel to the<110> directions in the (001) interface of a zinc blende or diamond semicon-ductor. Threading dislocations located within a distance Δ from a sidewallcan be removed by glide under the influence of the image forces. This leadsto a decrease in the average value of n, and therefore D, during thermalprocessing of such patterned layers. For a threading segment located at adistance r (along a (111) glide plane) from a sidewall, as shown in Figure7.13, the attractive image force28 is approximately

(7.19)

where G is the shear modulus, is the angle between the threading seg-ments and the interface, is the angle between the Burgers vector and theline vector for the threading segment, and is the Poisson ratio. Neglectingthe Peierls forces on the threading dislocation, the image force is opposedby the line tension in the misfit segment, which is given approximately by

(7.20)

FIGURE 7.13Removal of a threading dislocation from a patterned heteroepitaxial layer under the influenceof the image force. FL is the line tension in the misfit segment and FI is the attractive imageforce associated with the mesa sidewall.

FIFL

Lf

bhh

nDMD

c= −⎛⎝⎜

⎞⎠⎟cos cosα φ

12

FGb hrI = +

−⎛⎝⎜

⎞⎠⎟

2

4 1π λα α

νcoscos

sin( )

λα

ν

FGb R

bL = +−

⎛

⎝⎜⎞

⎠⎟⎛⎝⎜

⎞⎠⎟

22

2

4 4 1 2πα α

νcos

sin( )

ln




where R is one half the spacing between dislocations (perpendicular to theintersection of the glide plane and the interface) or the layer thickness,whichever is smaller. In the case of nearly complete relaxation (h >> hc), thespacing of the misfit dislocations is approximately . The con-dition for the glide of a dislocation to a sidewall is . If we consider theworst case of a threading dislocation located at the center of the mesa,then . The condition for removal of the threading dislocation by glideto the sidewall can therefore be written

(7.21)

It is expected that threading dislocations can be removed from the periph-ery of a mesa having arbitrary shape as long as the dislocations are withina distance Δ from the sidewall. Here, Δ can be considered the active rangeof the image forces and is equal to one half of the critical value of L calculatedabove. For the (001) heteroepitaxy of zinc blende or diamond semiconductorsit has been estimated as27

(7.22)

Therefore, neglecting dislocation–dislocation interactions and the Peierlsforce, threading dislocations can be removed completely from square pat-terned regions of size 2Δ.27 Zhang et al. calculated engineering curves forthe application of PHeP that predict the maximum mesa size for which allthreading dislocations may be removed by glide to the sidewalls; the resultsare shown in Figure 7.14.

Zhang et al. investigated the application of the PHeP process to ZnSe/GaAs (001) and ZnSe1–xSx/GaAs (001) grown by photoassisted MOVPE. Inthis work, planar layers were grown and, following growth, some of thelayers were patterned and annealed. The threading dislocation densities inthe heteroepitaxial material were determined using crystallographic etching(6 s in a 0.4% bromine-in-methanol solution at 300K).

In order to study the basic mechanism of PHeP, one ZnSe/GaAs (001) waferwas grown and cut into pieces that underwent different processes. The epi-taxial layer thickness was 600 nm. From this wafer four types of samples wereproduced: (1) as grown, (2) postgrowth annealed, (3) postgrowth patterned,and (4) PHeP prepared. The postgrowth annealing was conducted for 30 minat 600°C in flowing hydrogen. Figure 7.15 shows the etch pit morphology of(a) the as-grown layer and (b) the PHeP prepared material cut from the same

b fcos cos /α ϕF FI L>

r L= / 2

L

h

<

⎛⎝⎜

⎞⎠⎟

+−

⎛⎝⎜

⎞⎠⎟

+

21

2

coscos

sin( )

cossin

λα α

ν

α22

4 1 4αν

α ϕ( )

lncos cos

−⎛

⎝⎜⎞

⎠⎟⎛

⎝⎜

⎞

⎠⎟

f

Δ = 81 4

hfln( / )




FIGURE 7.14Dislocation engineering curves for patterned heteroepitaxial processing (PHeP). Lmax is themaximum mesa size for which all of the threading dislocations can be removed by glide to thesidewalls. (Adapted from Zhang, X.G. et al., J. Electron. Mater., 27, 1248, 1998. With permission.)

FIGURE 7.15Etch pit morphology for two 600-nm-thick ZnSe/GaAs (001) samples that were processeddifferently: (a) as grown and (b) patterned and annealed 30 min at 600°C. (Reprinted fromZhang, X.G. et al., J. Appl. Phys., 91, 3912, 2002. With permission. Copyright 2002, AmericanInstitute of Physics.)

Layer thickness h (μm)

f = 1%f = 2%f = 4%f = 8%

1

10

100

1000

0.1 1 10

L max

(μm

)

70 μm70 μm

(b)(a)




wafer. The as-grown layer had an EPD of 107 cm–2, whereas the PHeP materialwas completely free of threading dislocations even in the largest 70 × 70 μmpatterned regions. This corresponds to an EPD of less than 2.0 × 104 cm–2, andat least a 500-fold reduction compared to the as-grown layers. In fact, thisvalue is even lower than that of the GaAs substrate (EPD = 105 cm–2). Thesamples that underwent patterning alone or annealing alone exhibited thesame threading dislocation (TD) density as the as-grown layer, within exper-imental error. The result for layers that were annealed without patterning isconsistent with the early studies of Chand and Chu.29 This indicates that forpartially strained relaxed layers with large lateral dimensions, TDs may notmove long enough distances to encounter an edge easily, and annealing alonecauses very little TD annihilation or TD recombination. Patterning alone hasno effect on the TD density. Therefore, the removal of threading dislocations bypatterned heteroepitaxial processing involves thermally activated dislocation motionin the presence of sidewalls.

To investigate the behavior of the PHeP process for different layer thick-nesses, a series of ZnSe/GaAs (001) wafers were processed. The layer thick-nesses varied from 200 to 1200 nm. Each wafer was cut so that the EPD couldbe measured for the as-grown layer and also after patterning and annealing.In the case of PHeP processed wafers, the anneal was conducted for 30 minat 600°C in flowing hydrogen.

The EPDs for as-grown wafers were all of the order of 107 cm–2. The nearlyconstant threading dislocation density may indicate that there is little dislo-cation annihilation or coalescence. This may be a result of the low growthtemperature used for photoassisted MOVPE growth.

For the patterned and annealed wafers, the EPD decreased monotonicallywith increasing layer thickness, as shown in Figure 7.16 for the case of 70-μm-wide mesas. No etch pits were observed in the layers of thickness 300nm. Qualitatively, these results are consistent with the Zhang et al. model.Also, in the case of incomplete etching, it was found that it is the mesasidewall height, rather than the total epitaxial layer thickness, that deter-mines the effectiveness of PHeP. This is because the lateral forces acting onTDs are proportional to the sidewall height.

To study the effect of annealing temperature on the TD reduction by PHeP,a set of otherwise identically prepared 300-nm ZnSe/GaAs (001) patternedsamples were annealed at different temperatures in the range of 400 to 600°Cfor 30 min. Figure 7.17 shows the etch pit morphology of layers annealed at400, 450, 475, and 500°C. The sample in Figure 7.17a annealed at 400°C exhibitsthe same etch pit density as the as-grown sample, approximately 107 cm–2.Annealing at 450°C (Figure 7.17b) results in a reduction of the EPD to a valueof 3.5 × 106 cm–2, and at 475°C (Figure 7.17c), to a value of 9 × 105 cm–2. Whenthe annealing temperature is raised to 500°C (Figure 7.17d) or above, PHePresults in complete removal of TDs from 70 × 70 μm patterned regions. Zhanget al. plotted the results in the form of dD/dt on an Arrhenius plot andobtained an activation energy of 0.7 eV, which corresponds roughly to theactivation energy for dislocation glide, reported to be 1 eV for bulk ZnSe.30




The model of Zhang et al. predicts counterintuitively that PHeP shouldbe more effective for heteroepitaxial layers with higher mismatch. Zhang etal. reported a preliminary study of PHeP applied to ZnS0.02Se0.98 layers onGaAs that exhibit approximately 2/3 the lattice mismatch compared to ZnSeon GaAs (+0.18% vs. +0.27%). Both material systems have the same sign oflattice mismatch. They found that whereas for ZnSe/GaAs (001) all threadingdislocations could be removed from mesas having aspect ratios of W/h <250, the dislocations could not be removed completely from ZnS0.02Se0.98/GaAs (001), even with a mesa aspect ratio of W/h = 200. Further work isnecessary, however, to clarify the mismatch dependence.

7.5 Epitaxial Lateral Overgrowth (ELO)

Epitaxial lateral overgrowth (ELO),* now an important approach for mis-matched heteroepitaxy, was originally developed for the fabrication of high-

FIGURE 7.16EPD vs. layer thickness for mesa-etched and annealed ZnSe/GaAs (001). The data shown arefor 70 μm2 mesas. The annealing was conducted for 30 min at 600°C in flowing hydrogen.(Reprinted from Zhang, X.G. et al., J. Appl. Phys., 91, 3912, 2002. With permission. Copyright2002, American Institute of Physics.)

* This approach to heteroepitaxy also goes by the names lateral epitaxial overgrowth (LEO) andselective area lateral epitaxial overgrowth (SALEO).

Mesa height (nm)

105

106

107

108

104

103

0 200 400 600

Etch

pit

dens

ity (c

m−2

)




performance homoepitaxial devices in Si,31 GaAs,32,33 and InP.34 In theseapplications of ELO, growth proceeds from seed windows cut through amask layer (usually an oxide such as SiO2). Its successful implementationrequires selective growth (growth conditions that prevent nucleation and adeposition directly on the oxide). Then the growth over the oxide occursentirely as an extension of the seed regions, resulting in a single-crystal layer.It should also be noted that the achievement of a planar layer requires alateral growth rate that is much greater than the vertical rate (preferentialgrowth). In principle, ELO can be applied to a number of heteroepitaxialmaterial systems as long as the requirements of selective and preferentiallateral growth can be met.

For the ELO of Si on Si (001) substrates with patterned SiO2, selectiveepitaxial growth (SEG) is achieved by injecting HCl gas during growthfrom dichlorosilane (SiCl2H2).35 Unfortunately, this process has a unitylateral-to-vertical growth rate ratio, resulting in a nonplanar surface.36 (Themaximum thickness grows over the seed windows, and the minimumthickness grows midway between seed windows.) This necessitates the useof chemical-mechanical polishing (CMP) to planarize the ELO materialprior to device fabrication.37

The ELO growth of GaAs was first demonstrated by McClelland et al.32

using GaAs (110) substrates with a carbonized photoresist seed mask. The

FIGURE 7.17Etch pit morphology for 300-nm ZnSe/GaAs (001) samples that were patterned and annealedfor 30 min at different temperatures: (a) 400°C, (b) 450°C, (c) 475°C, and (d) 500°C. (Reprintedfrom Zhang, X.G. et al., J. Appl. Phys., 91, 3912, 2002. With permission. Copyright 2002, AmericanInstitute of Physics.)

70 μm

(a)

(d)

70 μm

(b)

(c)

70 μm 70 μm




AsCl3-GaAs-H2 growth process (chloride vapor phase epitaxy) was utilized,and the lateral-to-vertical growth rate ratio was approximately 25. Thisallowed the growth of GaAs layers having uniform thicknesses of 5 to 10μm, which could be cleaved from the substrate, thus allowing its reuse. (Thistechnique was termed the cleavage of lateral epitaxial films for transfer, orthe CLEFT process.32) Figure 7.18 shows this process in schematic fashion.Here, the use of chloride VPE gives preferential growth due to the differencein growth rates for different low-index faces, because the growth is kineti-cally controlled. It is therefore somewhat inflexible with regard to the choiceof substrate orientation.

Gale et al.33 demonstrated the ELO of GaAs using a SiO2 mask and MOVPE.Selective growth was achieved without the use of HCl. The growth rate waspreferential as well, with a lateral-to-vertical growth rate ratio of up to 5 on

FIGURE 7.18Epitaxial lateral overgrowth (ELO) of GaAs on a GaAs (110) substrate with a carbonizedphotoresist mask. (a) The mask is patterned with 2.5-μm-wide slots spaced 50 μm apart. (b andc) GaAs grows selectively in the slots (seed windows) and then grows laterally over the mask,with a lateral-to-vertical growth rate ratio of 25. (d) Adjacent areas of lateral growth merge toform a continuous layer of GaAs. (Reprinted from McClelland, R.W. et al., Appl. Phys. Lett., 37,560, 1980. With permission. Copyright 1980, American Institute of Physics.)



50 μm2.5 μm

Carbonizedphotoresist

GaAssinglecrystal

(a)

(b)

EpitaxialGaAs

(c)

~1 μm

(d)


(110) substrates, achieved when the seed openings were misaligned froma direction by 2 to 26°. Here, the growth conditions are such that thegrowth rate is minimum on low-index faces (facetted growth). Enhancedlateral growth occurs when the sidewalls are misoriented from a low-indexcrystal face. This can be understood as the consequence of a near-unitysticking coefficient for the Ga precursor. Surface diffusion of the adsorbedGa species then leads to enhanced growth on faces with high densities ofsteps and kinks.

Vohl et al.34 studied the ELO of InP using the PCl3-InP-H2 (chloride VPE)process on InP substrates of different orientations. The growth was selectiveusing a phosphosilicate glass (PSG) mask. Facetted growth resulted in thepreferential growth at high-index faces.

Nam et al.38 reported the first application of ELO for the attainment ofcontinuous layers of GaN on mismatched heteroepitaxial substrates. In theselective growth of GaN hexagonal pyramids for field emitters on 6H-SiC(0001) substrates, they had discovered that unintended lateral growthoccurred over the SiO2 mask layers with certain growth conditions.39 More-over, they found that the overgrown material contained a greatly reduceddensity of threading dislocations.40 The reduction in the dislocation densityin laterally overgrown GaN is shown dramatically in Figure 7.19. Here, GaNwas grown laterally by MOVPE from a stripe-geometry seed region of GaNusing a SiO2 mask. The seed GaN was grown on a 6H-SiC (0001) substratewith a 1000-Å AlN buffer. Above the seed region, the threading dislocationdensity is 108 to 109 cm–2, but there are no visible dislocations in the laterallygrown material.

FIGURE 7.19TEM micrograph in orientation showing the reduction of the dislocation density inlaterally grown GaN over a SiO2 mask. The GaN was grown laterally by MOVPE from a stripe-geometry seed region of GaN using a SiO2 mask. The seed GaN was grown on a 6H-SiC (0001)substrate with a 1000-Å AlN buffer. Above the seed region, the threading dislocation densityis 108 to 109 cm–2, but there are no visible dislocations in the laterally grown material. (Reprintedfrom Zheleva, T.S. et al., Appl. Phys. Lett., 71, 2472, 1997. With permission. Copyright 1997,American Institute of Physics.)

AlNGaN

GaN

SiO2

1 μm 6H-SiC

[ ]1120

[ ]110




Nam et al.38 then carried out a detailed investigation of ELO growth ofGaN on vicinal 6H-SiC (0001) substrates, misoriented by 3 to 4° towardthe . In this study, the oxide mask openings were stripes aligned withthe and directions. First, a 0.1-μm AlN buffer was grown onthe 6H-SiC (0001) substrate by MOVPE (TEAl + NH3) at 1100°C, followedby a 1.5- to 2.0-μm-thick layer of GaN grown by MOVPE (TEGa + NH3) at1000°C. The 0.1-μm-thick SiO2 mask layer was deposited by low-pressurechemical vapor deposition (CVD) at 410°C and patterned using photolithog-raphy and wet chemical etching in buffered HF. The stripe openings in theSiO2 were oriented along the and directions and were either3 or 5 μm wide. The parallel stripes were spaced by distances of 3 to 40 μm.Following the patterning of the SiO2 mask and a dip in 50% buffered HClto remove oxide from the exposed GaN surface, the lateral overgrowth ofGaN was carried out by MOVPE (TEGa + NH3) at 1000 to 1100°C.

The morphology of the ELO GaN was very different for theand stripe orientations. Figure 7.20 shows SEM micrographs of GaNgrown on 3-μm-wide stripe openings oriented along these two directionswith various growth times. After only 3 min of growth, the morphologylooks similar for the two stripe orientations. With additional growth, how-ever, the stripes oriented along the developed a triangular cross sec-tion with inclined side facets. The stripes oriented along the ,on the other hand, maintained a rectangular cross section with a (0001) topand {1120} sides.

Park et al.41 further studied the effect of stripe orientation, using 3-μm-wide stripe openings, 860 μm long, and indexed at 2° increments in a wagonwheel pattern, for the case of ELO GaN on sapphire (0001) substrates grownby MOVPE. They found the same cross sections and facets as Nam et al. forthe and stripe orientations.

The lateral-to-vertical growth rate ratio for ELO GaN is also quite differentfor the and stripe orientations. Nam et al.38 obtained a ratioless than unity for the -oriented stripes and approximately unity forthe stripes. Park et al.41 obtained a lateral-to-vertical growth rate ratioof up to 2 for -oriented stripes, as shown in Figure 7.21. They alsofound that the lateral-to-vertical growth rate ratio depends on the ratio ofthe open to masked stripe width (the fill factor) and the growth conditions,as well as the stripe orientation. As seen in Figure 7.21, the lateral-to-verticalgrowth ratio increases monotonically with the fill factor for the range inves-tigated.

Despite the relatively low lateral-to-vertical growth rate ratio, Nam et al.obtained smooth complete layers of ELO GaN by using stripes oriented alongthe . Figure 7.22 shows SEM micrographs of the cross section and thetop view for one such complete layer of ELO GaN, ~5 μm thick, grown using3-μm stripes spaced by 3 μm. The surface of the coalesced ELO layer isrelatively smooth. The 0.25-nm root mean square (rms) roughness is compa-rable to that for the underlying GaN layer. However, the process of coales-cence leaves small voids above the oxide stripes, as can be seen in Figure 7.22.

112 01100 112 0

1100 112 0

112 01100

112 0{ }1101 1100

112 0 1100

112 0 1100112 0

1100112 0

1100




FIGURE 7.20SEM micrographs of GaN grown on 3-μm-wide oxide stripe openings oriented alongthe and directions with the growth times of (a to d) 3 min, (e and f) 9 min, and(g and h) 20 min. (Reprinted from Nam, O.-H. et al., Appl. Phys. Lett., 71, 2638, 1997. Withpermission. Copyright 1997, American Institute of Physics.)

<112−0> <11−00>

Stripe Orientation

(a) (b)

(c) (d)

(e) (f )

(g) (h)

1 μm 1 μm

1 μm 1 μm

1 μm 1 μm

1 μm 1 μm

1120 1100




The complete layers of GaN formed by ELO in this manner show a dra-matic reduction in their threading dislocation density in the laterally grownregions, as shown in the SEM micrograph of Figure 7.23 obtained by Namet al. Above the seed stripe, threading dislocations thread from the GaN/AlN interface to the surface of the GaN layer. In the laterally grown material,above the oxide mask, only dislocations parallel to the (0001) plane werevisible. These are thought to be due to the bending over of threading dislo-

FIGURE 7.21Lateral-to-vertical growth rate ratio for ELO GaN/AlN/α-Al2O3 (0001) grown by MOVPE as afunction of the stripe orientation with fill factor as a parameter. The fill factor is the ratio ofopen to masked surface area. The masking layer is 0.1-μm SiO2. (Reprinted from Park, J. et al.,Appl. Phys. Lett., 73, 333, 1998. With permission. Copyright 1998, American Institute of Physics.)

FIGURE 7.22SEM micrographs of the (a) cross section and (b) top view of a 5-μm complete layer of ELOGaN grown on vicinal 6H-SiC (0001) using an AlN buffer layer. The layer was grown using 3-μm oxide openings spaced by 3 μm and oriented along the direction. (Reprinted fromNam, O.-H. et al., Appl. Phys. Lett., 71, 2638, 1997. With permission. Copyright 1997, AmericanInstitute of Physics.)

3

2

1

00 5 10 15 20 25 30

Late

ral-t

o-ve

rtic

al g

row

th ra

tio (L

TVG

R)

<11−20> Orientation of stripe opening (degree) <11−00>

LTVGR = (A−d)/2H

d

B

A

0.08

0.06

0.04

H

(a)

1 μm

(b)

1 μm

1100




cations from the highly defected region. However, they do not thread to thetop surface. Moreover, in an investigation of Si-doped ELO GaN on sapphire,Park et al. found that the laterally grown material exhibited two to threetimes the CL intensity of the material over the seed stripe (370-nm emission).

Chang et al.42 investigated the epitaxial lateral overgrowth of GaAs on Si(111) substrates. In their approach, which they called microchannel epitaxy(MCE), they first grew a thin layer of GaAs on the Si (111) by MBE. Then aSiO2 film was spun on and baked. A pattern of parallel line openings (micro-channels) was produced in the SiO2 by a photolithographic step. The channelopenings were 5 μm wide and separated by 200 to 1000 μm. Finally, the ELOgrowth of GaAs was accomplished by liquid phase epitaxy (LPE), resultingin the structure shown in Figure 7.24. Following ELO growth, the resulting

FIGURE 7.23Cross-sectional TEM micrograph of ELO GaN grown on vicinal 6H-SiC (0001) using an AlNbuffer layer. The layer was grown using 3-μm oxide openings spaced by 3 μm and orientedalong the direction. (Reprinted from Nam, O.-H. et al., Appl. Phys. Lett., 71, 2638, 1997.With permission. Copyright 1997, American Institute of Physics.)

FIGURE 7.24ELO (microchannel epitaxy (MCE)) of GaAs on Si (111). First a thin layer of GaAs is grown onthe Si (111) by MBE. Then a SiO2 film is spun on and baked. A pattern of parallel line openings(microchannels) is produced in the SiO2 by a photolithographic step. Finally, the ELO growth ofGaAs is accomplished by liquid phase epitaxy (LPE). A figure of merit for the process is the ratioof lateral to vertical growth. In their work, Chang et al. used the ratio W/h as defined in the figure.

GaN

GaN

SiO2

AlN6H-SiC 1 μm

1100

Wh

Si (111) substrate

MBE GaAs

LPE GaAs

SiO2 SiO2




material was characterized by crystallographic etching using molten KOH.In this study, it was found that the ratio of lateral to vertical growth couldbe as high as 17. As in the case of GaN, the ELO growth affected a dramaticreduction in the threading dislocation density in the laterally grown regions.This can be seen in the optical micrograph of Figure 7.25, which shows theEPD morphology of a microchannel epitaxy structure. In the central region,where the GaAs grew vertically over the Si substrate, there is a high EPD.But the laterally grown regions, 47 μm on either side, are virtually free frometch pits.

7.6 Pendeo-Epitaxy

Zheleva et al.43 proposed pendeo-epitaxy as a new approach for the lateralgrowth of III-nitrides on mismatched heteroepitaxial substrates. Thisapproach is similar to ELO and makes use of the difference in growth ratesfor the (0001) and planes. However, whereas ELO involves the lateralgrowth of GaN over a SiO2 mask, from seed openings, the pendeo-epitaxymethod involves lateral growth from mesa-patterned GaN and eliminatesthe need for a SiO2 mask. Figure 7.26 shows both approaches schematically.

Zheleva et al. have discussed two modes of pendeo-epitaxial growth. Inmode A, the growth on the seed pillars proceeds faster in the lateral direc-tions than in the vertical direction from its initiation. In mode B, the growth

FIGURE 7.25Etch pit morphology of GaAs on Si grown by ELO (microchannel epitaxy). The sample hasbeen etched in molten KOH to reveal etch pits associated with threading dislocations. In thecentral region, where the GaAs grew vertically over the Si substrate, there is a high EPD. Butthe laterally grown regions, 47 μm on either side, are virtually free from etch pits. (Reprintedfrom Chang, Y.S. et al., J. Cryst. Growth, 192, 18, 1998. With permission. Copyright 1998, Elsevier.)

25 μm

{ }112 0




is initially faster on the tops of the pillars, followed by rapid lateral growthfrom the newly formed side facets. Figure 7.27 shows SEM and TEMmicrographs of pendeo-epitaxial GaN grown in these two modes.

The primary advantage of pendeo-epitaxy compared to ELO is the elimi-nation of the thermal strain associated with the SiO2 mask. Like ELO,pendeo-epitaxy enables a dramatic decrease in the threading dislocationdensity (four to five orders of magnitude) in the laterally grown material,compared to direct growth.

FIGURE 7.26The ELO and pendeo-epitaxy approaches for the growth of high-quality GaN on SiC or sapphiresubstrates. (a) The ELO process: an AlN buffer is grown; a GaN layer is grown; SiO2 is depositedand patterned to open seed stripes; GaN is grown from the seed stripes laterally over theremaining SiO2 to produce a complete layer. (b) The pendeo-epitaxy process: an AlN buffer isgrown; a GaN layer is grown; GaN/AlN is mesa dry-etched to create seed pillars; GaN is grownlaterally from the pillars, to create a complete layer.

(a) (b)

6H SiC (0001)AlN

GaN

6H SiC (0001)AlN

GaN

6H SiC (0001)AlN

GaNGaN GaN GaN

SiO2

SiO2

6H SiC (0001)AlN

GaN SiO2

GaN

ELO

6H SiC (0001)

6H SiC (0001)

6H SiC (0001)

AlNGaN

GaN GaN GaN

GaN GaN GaN

6H SiC (0001)

GaN

AlN

AlN

AlN

Pendeo-epitaxy

{ }112 0




7.7 Nanoheteroepitaxy

Nanoheteroepitaxy (NHE) is a substrate-patterning approach that involvesthe growth of nanometer-scale islands on a mismatched heteroepitaxial sub-strate. In the implementation of NHE, selective epitaxial growth is carriedout on a substrate that has been patterned to have nanometer-scale seedpads. This may be achieved either by etching windows through a dielectricmask material or by mesa etching the substrate crystal. Typically, lateralepitaxial growth proceeds from the seed pads until the coalescence of thegrowing islands yields a complete layer of the heteroepitaxial material. NHEdiffers from epitaxial lateral overgrowth (ELO) in that the pattern involvesislands rather than stripes, and that the seed pads have dimensions on theorder of nanometers, not micrometers. This latter feature is not simply adifference in degree, but introduces new mechanisms of strain relaxationthat are of fundamental importance. This is illustrated in Figure 7.28 for thecase of a mesa-patterned substrate. In the planar growth of a pseudomorphicmismatched layer, as shown on the left, the only stress-relief mechanismother than the creation of dislocations is the vertical deformation of theepitaxial material (a). However, for the mismatched material on the mesa-patterned substrate, the stress can also be relieved by lateral deformationsin the epitaxial layer (b), along with vertical and lateral deformations in thesubstrate mesas (c and d). The stress-relief mechanisms (b to d) that are

FIGURE 7.27(a) SEM and (b) TEM micrographs of pendeo-epitaxial GaN grown in mode A; (c) SEM and(d) TEM micrographs of pendeo-epitaxial GaN grown in mode B. (Reprinted from T.S. Zhelevaet al., Pendeo-epitaxy — a new approach for lateral growth of gallium nitride structures, MRSInternet J. Nitride Semicond. Res., 451, G3.38 (1999).)

AlN

AlN

AlN

AlN

6H-SiC 6H-SiC

6H-SiC

3.0 kV10 μmX3, 300

PE-GaN

GaNcolumn

(a) (b)

(c) (d)

5 kV

PE-GaN GaN seed

GaN seedGaN

(112−0)

X12,000 16 mm1 μm

1 μm

1 μm

6H-SiC

PE-GaNGaN

column

(0001)




unique to patterned growth can make it possible to grow the heteroepitaxialmaterial much thicker than the Matthews and Blakeslee critical layer thick-ness for planar growth, without the introduction of misfit dislocations. Evenif stress relief should occur partly by the introduction of misfit dislocations,the associated threading dislocations can easily glide to the edge of the mesa,as in the PHeP approach. Whereas the planar layer will contain threadingdislocations such as (e), which are unable to glide to the edge of the sample,the NHE layer is likely to contain only misfit dislocations and their associatedsidewall steps, as in (f).

Luryi and Suhir44 first presented a theoretical treatment of strain relaxationin a heteroepitaxial layer on a nanopatterned substrate. They consideredgrowth of a mismatched semiconductor that makes rigid contact with thesubstrate only on round seed pads, the diameter of which is on the scale ofnanometers. They showed that in a pseudomorphic structure of this sort, thestrain in the heteroepitaxial layer decays exponentially with distance from theinterface. The characteristic length for this decay is on the order of the seedpad diameter. Because of this behavior, the critical layer thickness increases asthe seed pads are scaled down in size. For a given lattice mismatch strain, thereis a seed pad size below which the critical layer thickness diverges to infinity,so that structures entirely free from misfit dislocations may be produced.

In the next section, the Luryi and Suhir model is outlined for the case ofa noncompliant substrate. The following section describes the extension ofthis theory to include substrate compliancy, as developed by Zubia andHersee. Finally, experimental results for nanoheteroepitaxy are summarized.

7.7.1 Nanoheteroepitaxy on a Noncompliant Substrate

Luryi and Suhir44 developed the first theoretical model for the strain innanoheteroepitaxial material. They assumed that the lattice-mismatched het-eroepitaxial material makes rigid contact with a noncompliant substrate only

FIGURE 7.28Stress-relief mechanisms in a mismatched heteroepitaxial crystal. For a planar layer as shownon the left, the only stress-relief mechanism is the vertical deformation of the epitaxial material(a). But in the case of nanoheteroepitaxy, shown on the right, the epitaxial deposit can alsodeform in the lateral direction (b). In addition, the substrate mesas can deform both vertically(c) and laterally (d). For planar heteroepitaxy, a substrate dislocation (e) will thread throughthe epitaxial layer, but in the case of nanoheteroepitaxy, the threading dislocation can glide toa sidewall to create a sidewall step (f), similar to the case of PHeP.

Substrate

Substrate

a a

b

c

d

e

f

he




at round seed pads having a diameter 2l, as shown in Figure 7.29. Here, they-axis lies in the plane of the interface, along a major cord of a seed pad.The z-axis is perpendicular to the substrate and passes through the centerof this seed pad. The figure shows a heteroepitaxial layer that has coalescedinto a single layer by lateral growth, and the total thickness of the heteroepi-taxial layer is h. It is assumed that the areas between the seed pads are wideenough, so there is no interference of the strain fields from adjacent pads.In this situation, if the substrate is unstrained, then the in-plane stress in theepitaxial deposit is given by

(7.23)

where f is the lattice mismatch strain, E is the Young’s modulus, is thePoisson ratio, and

(7.24)

where is the effective range for the stress in the z direction, to be deter-mined below, and the interfacial compliance parameter k is given by

(7.25)

FIGURE 7.29Nanoheteroepitaxial growth on a patterned substrate. The substrate has been patterned withround seed pads having a diameter 2l. The y-axis lies in the plane of the interface, along amajor cord of one of the seed pads. The z-axis is perpendicular to the substrate and passesthrough the center of this seed pad. The heteroepitaxial layer may coalesce into a single layerby lateral growth, as shown. The total thickness of the heteroepitaxial layer is h. (Reprintedfrom Luryi, S. and Suhir, E., Appl. Phys. Lett., 49, 140, 1986. With permission. Copyright 1986,American Institute of Physics.)

O

z

ω

Substrate

Epitaxial layer

2l

heh

y

σν

χ π|| ( , )exp( / )=−

−fE

y z z l1

2

ν

χ( , )cosh( )cosh( )

;

;y z

kykl

z h

z h

e

e

=− ≤

≥

⎧⎨⎪

⎩⎪

1

1

he

kh he e

= −+

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ ≡3

211

11 2

νν

ζ/




The strain energy density per unit volume is

(7.26)

and is maximum at . The strain energy per unit area may be found byintegrating over the thickness of the epitaxial deposit and takes on a maxi-mum value at , which is

(7.27)

In this calculation, there is little contribution from , so that it is a goodapproximation to use the form of for . The right-hand side ofEquation 7.27 defines the characteristic thickness , which is then givenimplicitly by

(7.28)

The right-hand side of this equation defines the reduction factor, ,which is plotted in Figure 7.30. For , asymptotically, but for

, . In other words, for , and for ,

(7.29)

The strain energy per unit area from Equation 7.27 may be used in con-junction with an energy calculation for the critical layer thickness to find thecritical layer thickness for an island of radius l. The result is

In their work, Luryi and Suhir used the People and Bean model for thedetermination of the critical layer thickness, . However, theMatthews energy calculation of the critical layer thickness may also beused, with

(7.30)

ω ν σ( , ) ||y zE

= −1 2

y = 0

y = 0

E zE

f hs

h

e= ≡−∫ ω

ν( , )0

10

2 2

z he>χ( , )y z z he≤

he

h h hl

hh l

le

e

= −⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ − −1 1

2

sec [ exp( / )]ζ π

πhhh

lh

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪=

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥φ

2

φ( / )l hl h>> φ → 1

l h<< φ ∝ ( / ) /l h 1 2 h he ≈ l h>> l h<<

hlh

he ≈ −[ sec ( )]1 2ζπ

hcl

h h l h fcl

c cl= [ ( / ) ]φ

h x a xc[ ] . ( / )= 0 1 0

h xb h x b

xcc[ ]

( cos )[ln( [ ]/ ) ]( )cos

= − ++

1 18 1

2ν απ ν λλ




7.7.2 Nanoheteroepitaxy with a Compliant Substrate

Zubia and Hersee45 extended this theory to include the effect of substratecompliance and named the approach nanoheteroepitaxy. Here, the strain ispartitioned between the substrate and the epitaxial layer. If the epitaxiallayer is grown coherently (without misfit dislocations) on a compliant sub-strate with lattice mismatch strain f, then the substrate and epitaxial layerwill be strained in an opposite sense, such that

(7.31)

where and are the in-plane strains in the epitaxial layer and sub-strate, respectively. If we neglect the bending stresses, force balance in thestructure dictates that46

(7.32)

where and are the thicknesses of the epitaxial layer and substrate,respectively, and and are the corresponding in-plane stresses. Dueto the biaxial nature of the stress, the stress–strain relationships are

FIGURE 7.30Reduction factor φ as a function of l/h, where l is the radius of the seed pads and h is the epitaxiallayer thickness. (Reprinted from Luryi, S. and Suhir, E., Appl. Phys. Lett., 49, 140, 1986. Withpermission. Copyright 1986, American Institute of Physics.)

(l/h)

φ (l

/h)

1.0

0.5

0.00 1 2 3 4

ε εepi sub f− =

εepi εsub

σ σepi epi sub subh h+ = 0

hepi hsub

σ epi σsub

σν

εepiepi

epiepi

E=

−1




(7.33)

where and are the Young’s moduli and and are the Poissonratios. The simultaneous solution of these three equations yields

(7.34)

where K is given by

(7.35)

Now, combining the compliant substrate theory with the model of Luryiand Suhir, we have

(7.36)

and

(7.37)

The in-plane strains in the nanoheteroepitaxial material and substrate padsare given by

(7.38)

and

σν

εsubsub

subsub

E=−1

Eepi Esub νepi νsub

εepiepi

sub

f

Kh

h

=+

⎛

⎝⎜⎞

⎠⎟1

εsub

sub

epi

f

Khh

= −

+⎛

⎝⎜⎞

⎠⎟1

1

KE

Eepi

epi

sub

sub

=−

−( )

( )1

1ν

ν

σ εν

χ πepi epi

epi

epi

Ey z

zl

=−

−⎛⎝⎜

⎞⎠⎟1 2

( , )exp

σ εν

χ πsub sub

sub

sub

Ey z

zl

=−

−⎛⎝⎜

⎞⎠⎟1 2

( , )exp

εππ

epiepi

sub

f

Kh l

h l

=+

− −− −

11 21 2

( exp( / ))( exp( / )))

⎛

⎝⎜⎞

⎠⎟




(7.39)

Figure 7.31 shows the partitioning of strain between an epitaxial layer andcompliant substrate for the case of planar (unpatterned) growth. It can beseen that more of the strain is transferred to the substrate if the thicknessratio hepi/hsub is increased or if K is increased.

When the strain partitioning is accounted for, along with the stress reliefby lateral contraction/expansion at the sidewalls, it is expected that layerscan be grown coherently much greater than the critical layer thickness forone-dimensional (planar) growth. In fact, Zubia and Hersee predicted itshould be possible to grow layers with a lattice mismatch of 4.2% completelydislocation-free.

Zubia et al.47 investigated the nanoheteroepitaxy of GaN on patternedsilicon-on-insulator (SOI) by MOVPE. The SOI (111) wafers were producedusing the separation by ion implantation of oxygen (SIMOX) process. TheSOI wafers were patterned using interferometric photolithography and reac-tive ion etching in the manner reported by Zaidi et al.,48,49 forming a squaretwo-dimensional array of silicon islands on top of SiO2. These silicon islandshad a height of 40 nm and diameters of 80 to 300 nm, and their separation

FIGURE 7.31Strain partitioning between an epitaxial layer and compliant substrate for the case of planar(unpatterned) growth. (Reprinted from Zubia, D. and Hersee, S.D., J. Appl. Phys., 85, 6492, 1999.With permission. Copyright 1999, American Institute of Physics.)

εππ

sub

sub

epi

f

Kh lh

= −

+ − −− −

11 1 2

1( exp( / ))( exp( /22l))

⎛

⎝⎜

⎞

⎠⎟

0.0

0.2

0.4

0.6

0.8

1.0

0.01 0.1 1 10 100Layer thickness ratio hepi/hsub

Nor

mal

ized

stra

in |ε

/f|

Epilayer strain

Substrate strain

Decreasing KIncreasing K

K = 1.3K = 1.0K = 0.7




in the square array was 360 or 900 nm. An interesting aspect of this workwas the reduced melting point of the silicon islands. A comparison of theirmorphology before and after heating to 1110°C in the epitaxial reactor (Figure7.33) shows that the nanoscale silicon islands melted at or below this tem-perature, even though the bulk melting temperature is 1412°C.

The melting point reduction of the nanoscale islands can give rise to theirsoftening, and an enhanced compliant substrate effect for nanoheteroepitaxyon top of them. Zubia et al. referred to this effect as active compliance. This

FIGURE 7.32Strain partitioning for nanoheteroepitaxy. The nanoisland diameter is 2l and the substratethickness was assumed to be 500 Å. (Reprinted from Zubia, D. and Hersee, S.D., J. Appl. Phys.,85, 6492, 1999. With permission. Copyright 1999, American Institute of Physics.)

FIGURE 7.33Morphology of silicon nanoscale islands (a) before and (b) after heating to 1110°C in the epitaxialreactor. The change in shape shows that the nanoscale islands had melted at or below thistemperature. (Reprinted from Zubia, D. et al., Appl. Phys. Lett., 76, 858, 2000. With permission.Copyright 2000, American Institute of Physics.)

1

1 10 100 1000

0.8

0.6

0.4

0.2

0

Epitaxial layer thickness (Å)

l = 100 Å l = 300 Å

εepi0 ε/ε T

εsub0

K = 1.3

K = 1.0

(a) (b)

1.00 μm




effect was not included in the original model for nanoheteroepitaxy, but itcan be accounted for by the use of an effective value of K.

Figure 7.34 shows cross-sectional TEM micrographs of nanoheteroepitaxialGaN on SOI for the case of 80 and 280 nm islands. In both cases, thenanoheteroepitaxial GaN contained dislocations near the interface. However,the dislocation density decreased with distance from the interface.

7.8 Planar Compliant Substrates

In planar-mismatched heteroepitaxy on a thick substrate, all of the mismatchstrain resides in the epitaxial layer, which must be less than the critical layerthickness to avoid the introduction of dislocations. However, pseudomorphiclayers thicker than the critical layer thickness would be beneficial in manydevice applications. In order to lift the critical layer thickness constraint forpseudomorphic growth, Lo50 proposed the use of compliant substrates.

A compliant substrate is one thin enough so that it becomes strained bythe deposition of a mismatched heteroepitaxial layer.* The partitioning ofstrain between the epitaxial layer and substrate reduces the total strainenergy. If the substrate is sufficiently thin, the overall strain energy will neverbe large enough to cause the production of misfit dislocations. Then theeffective critical layer thickness will diverge to infinity so that a pseudomor-phic layer of any thickness may be grown.

FIGURE 7.34Cross-sectional TEM micrographs of nanoheteroepitaxial GaN grown on SOI islands withdiameters of (a) 80 nm and (b) 280 nm. (Reprinted from Zubia, D. et al., Appl. Phys. Lett., 76,858, 2000. With permission. Copyright 2000, American Institute of Physics.)

* In practical implementations of compliant substrates, the thin template layer is not free stand-ing, but mechanically decoupled from a thick handle wafer.

(a) (b)

GaN

SiO2Si

GaN(0002)

(NM)

50.00 nm

GaN

SiO2

Si50.00 nmSi(111)

GaN(010)Si(-111)




In practice, a membrane thin enough to act as a compliant substrate isdifficult to handle during processing. Another problem is that compliantmembranes are susceptible to bowing and other distortions during strainedheteroepitaxy. For these reasons, the use of compliant substrate technologyrequires the realization of a thin compliant layer on a rigid handle wafer. Insuch a realization, the handle wafer must restrain the compliant layer in thegrowth direction, to prevent buckling. However, the compliant layer mustbe mechanically decoupled from the substrate in the plane of the interface.No perfect scheme for a large-area compliant substrate on a handle layerhas been demonstrated. On the other hand, compliant substrate technologiesof this general type have been investigated with various degrees of success.These approaches include glass-bonded, metal-bonded, and twist-bondedwafers and silicon-on-insulator (SOI).

In the previous section, we considered briefly the theory of compliantsubstrates in the context of nanoheteroepitaxy. Here, we will consider com-pliant substrate theory in more detail, for its application to one-dimensional(unpatterned) heteroepitaxy. We will also review various schemes for com-pliant substrate realization, along with descriptions of the relevant experi-mental observations.

7.8.1 Compliant Substrate Theory

If an epitaxial layer is grown coherently (without misfit dislocations) on acompliant substrate with lattice mismatch strain f, then the substrate andepitaxial layer will be strained in the opposite sense, such that

(7.40)

where and are the in-plane strains in the epitaxial layer and thesubstrate, respectively. If we neglect the bending stresses, force balance inthe structure dictates that46

(7.41)

where and are the thicknesses of the epitaxial layer and substrate,respectively, and and are the corresponding in-plane stresses. Dueto the biaxial nature of the stress, the stress–strain relationships are

(7.42)

and

ε εepi sub f− =

εepi εsub

σ σepi epi sub subh h+ = 0

hepi hsub

σ epi σsub

σν

εepiepi

epiepi

E=

−1




(7.43)

where and are the Young’s moduli and and are the Poissonratios. The simultaneous solution of these three equations yields

(7.44)

and

(7.45)

where K is given by

(7.46)

The strain energy per unit area in the bilayer system is

(7.47)

With the approximations and , the strain energysimplifies to

(7.48)

If it is assumed that misfit dislocations will be introduced when the arealstrain energy exceeds the misfit dislocation energy per unit area , as deter-mined by Matthews, then the effective critical layer thickness for theepitaxial layer on the compliant substrate of thickness can be found from

σν

εsubsub

subsub

E=−1

Eepi Esub νepi νsub

εepiepi

sub

f

Kh

h

=+

⎛

⎝⎜⎞

⎠⎟1

εsub

sub

epi

f

Khh

= −

+⎛

⎝⎜⎞

⎠⎟1

1

KE

Eepi

epi

sub

sub

=−

−( )

( )1

1ν

ν

EE

hE

heepi

epiepi epi

sub

subsub s=

−+

−( ) ( )1 12

νε

νε uub

2

E E Eepi sub= = ν ν νepi sub= =

EE

fh h

h heepi sub

epi sub

=− +

⎛

⎝⎜

⎞

⎠⎟( )1

2

ν

Ed

heff

hsub




(7.49)

where is the Matthews and Blakeslee critical layer thickness. Figure 7.35shows the normalized critical layer thickness vs. the normalized sub-strate thickness . It is important to note that when , there existsno solution to Equation 7.49, so that .

Lo50 also showed that in partially relaxed epitaxial layers on compliantsubstrates, the modified image forces could help reduce the density ofthreading dislocations in the epitaxial layer. For the case of a thick substrate,the image force always attracts dislocations toward the free surface of theepitaxial layer. In that situation, the image force is associated with the freesurface of the epitaxial layer, and it is equal to the attractive force that wouldexist between the real dislocation and an image dislocation, with the oppositeBurgers vector, and located at an equal distance from the surface, but on theopposite side of it. With a compliant substrate, the image force can be greatlydecreased in magnitude, or may even change sign and drive the dislocationinto the substrate, away from the epitaxial layer surface. Here, both freesurfaces contribute to the overall image force. Lo calculated this image forcefor a 60° misfit dislocation along a direction for (001) heteroepitaxy of

FIGURE 7.35Normalized critical layer thickness, heff/hc , vs. the normalized substrate thickness, , forthe growth of a mismatched heteroepitaxial layer on a compliant substrate. heff is the effectivecritical layer thickness, hsub is the thickness of the compliant substrate, and hc is the Matthewsand Blakeslee critical layer thickness. (Reprinted from Lo, Y.H., Appl. Phys. Lett., 59, 2311, 2005.With permission. Copyright 2005, American Institute of Physics.)

1 1 1h h heff c sub

= −

hc

h heff c/h hsub c/ t tsub c<

heff → ∞

110

Normalized substrate thickness hsub/hc

Nor

mal

ized

effec

tive c

ritic

al th

ickn

ess (

h eff/

h c)

5

4

3

2

1

01 2 3 4 50

T eff i

s infi

nite

h hsub c/




a zinc blende material on a compliant substrate. The image force per unitlength of dislocation was found to be

(7.50)

The misfit dislocations experience the maximum attractive image force inthe case of an infinite (noncompliant) substrate. A reduction in the substratethickness decreases the attractive image force for a given thickness of theepitaxial layer. The sign of the image force may even change, indicating thatthe dislocations will be repelled from the surface of the epitaxial layer (orattracted to the surface of the compliant substrate). This behavior is illus-trated in Figure 7.36, which shows the image force (arbitrary units) vs. thenormalized epitaxial layer thickness , with the normalized substratethickness as a parameter.

7.8.2 Compliant Substrate Implementation

Several approaches have been invented for the implementation of compliantsubstrates. The first implementation of a compliant substrate was the canti-

FIGURE 7.36Image force (arbitrary units) for a 60° misfit dislocation at the interface between an epitaxiallayer and a compliant substrate vs. the normalized epitaxial layer thickness, , and withthe normalized substrate thickness, , as a parameter, where hc is the Matthews andBlakeslee critical layer thickness. (Reprinted from Lo, Y.H., Appl. Phys. Lett., 59, 2311, 2005. Withpermission. Copyright 2005, American Institute of Physics.)

−2

−1

0

1

2

3

Normalized epitaxial layer thickness hepi/hc

2

∞4

1 2 3 4 5 6

hsub/hc = 1.5

Imag

e for

ce p

er u

nit l

engt

h F 1

/L(A

rbitr

ary u

nits

)

h hepi c/h hsub c/

FL

Gbh h

I

epi sub

= +−( )

⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎞

⎠⎟2

414

12 1

1ν

cootπh

h hepi

epi sub+⎛

⎝⎜⎞

⎠⎟

h hepi c/h hsub c/




levered membrane, proposed by Teng and Lo51 and demonstrated by Chuaet al.52 Other approaches have involved the realization of a thin compliantlayer, which is on top of a thick handle wafer but mechanically decoupledfrom it. Along these lines, wafer bonding is the most studied method. Here,an etch stop layer and compliant layer are grown epitaxially on one wafer,which is then bonded to a handle wafer. The former wafer is then removedby lapping and etching, leaving just the compliant layer bonded to the handlewafer. Some degree of compliancy is achieved by the use of an intermediatelayer (e.g., metal or glass) between the compliant layer and the handle wafer,or by a twist bond. Carter-Coman et al.53,54 developed compliant substratesusing wafer bonding with an intermediate layer of indium. This indiumlayer melts at epitaxial growth temperatures, rendering the thin layer com-pliant. Moran et al.55,56 have developed bonded compliant substrates usingintermediate layers of borosilicate glass. In the case of twist-bonded compli-ant substrates, there is no need for the insertion of an intermediate layer.Instead, the twist boundary introduces an array of screw dislocations thatintroduces some level of compliancy. Silicon-on-insulator (SOI) wafers havealso been investigated as potential compliant substrates. For example, Powellet al.57 studied the epitaxy of SiGe alloys on an ultrathin SOI layer. Yang etal.58 extended this work to the growth of GaN on both SOI substrates andSiC-on-silicon-on-insulator substrates. However, Rehder et al.59 showed thatin the case of SiGe heteroepitaxy, a thin SOI layer does not act as a compliantsubstrate (in the sense of strain partitioning), even though it does alter thedislocation structure and dynamics.

In the following sections, the various approaches for realizing compliantsubstrates will be described in detail, including cantilevered membranes,glass-bonded, metal-bonded, and twist-bonded wafers, and silicon-on-insulator.

7.8.2.1 Cantilevered Membranes

Teng and Lo51 proposed the use of a cantilevered membrane as a compliantsubstrate. Their design, shown in Figure 7.37, could be created by an under-cutting etch. The membrane is supported at the four corners, but shouldbehave as a compliant substrate in the central region away from the supports.They were able to fabricate such cantilevered membranes using selective wetetching of GaAs/AlGaAs and InP/InGaAs epitaxial structures.

Chua et al.52 demonstrated the use of a cantilevered membrane havinga bench structure as shown in Figure 7.38. To create the 800-Å membrane,they first grew a 800-Å GaAs/1000-Å Al0.8Ga0.2As/GaAs (001) structure byMBE. Then, using photolithography and a nonselective etch, they mesa-etched stripes, 5 μm wide and with a 10-μm center-to-center spacing, inthe 800-Å GaAs/1000-Å Al0.8Ga0.2As. With a second photolithographicstep, they opened stripes perpendicular to the first set, having a width of10 μm, and etched the Al0.8Ga0.2As selectively through these openings with1 HF:5 H2O.




On the cantilevered membrane described above, Chua et al. grewIn0.14Ga0.86As, for which the room temperature mismatch strain isand the critical layer thickness is . They grew In0.14Ga0.86As, 2000 Åthick, simultaneously on the compliant platform and on a reference, unproc-essed GaAs substrate. The In0.14Ga0.86As is about 20 times the expected criticallayer thickness predicted by the Matthews and Blakeslee model. They foundthat the 004 x-ray diffraction peak separation (between the x-ray diffractionpeaks for GaAs and In0.14Ga0.86As) was significantly greater on the compliantplatform than on the reference substrate. This could be interpreted as anindication of tetragonal distortion in the GaAs compliant platform, whichwould be expected if it is compliant. Chua et al. also studied the surfacemorphology of In0.14Ga0.86As by atomic force microscopy (AFM). The layer

FIGURE 7.37A cantilevered membrane for use as a compliant substrate. (Reprinted from Teng, D. and Lo, Y.H.,Appl. Phys. Lett., 62, 43, 1993. With permission. Copyright 1993, American Institute of Physics.)

FIGURE 7.38Cantilevered membrane with a bench structure, for use as a compliant substrate. (Reprintedfrom Chua, C.L. et al., Appl. Phys. Lett., 64, 3640, 1994. With permission. Copyright 1994,American Institute of Physics.)

hsub

Corner supportedmembrane

GaAs substrate

5 μm

10 μ

m

80 nm GaAs100 nm AlGaAs

f = −0 94. %hc ≈ 100 Å




on the reference substrate showed a rough surface morphology associatedwith dislocations introduced during lattice relaxation. In contrast, the layergrown on the compliant platform exhibited a smooth surface texture.

Despite these results, cantilevered compliant membranes are expected tobe mechanically fragile and will have poor heat-removal performance. Otherapproaches, which produce a thin compliant layer on a handle wafer, aretherefore preferred.

7.8.2.2 Silicon-on-Insulator (SOI) as a Compliant Substrate

Silicon-on-insulator (SOI) has been investigated for use as a compliant sub-strate for the growth of SiGe alloys, GaN, and GaAs. In the case of an SOIwafer, the silicon layer may act as a compliant substrate if it is sufficientlythin and mechanically decoupled from the wafer by slippage at the Si/SiO2

interface. Experimental results with this type of compliant substrate havebeen mixed, however. Recent experiments by Rehder et al.59 involving thegrowth of SiGe on SOI wafers indicate that the silicon layer does not behaveas a compliant substrate in the usual sense. On the other hand, Rehder et al.and others have measured the existence of partial strain partitioning betweenthe thin Si layer and the SiGe and observed the preferential introduction ofdislocations in Si rather than SiGe.

Powell et al.57 performed initial experiments with the use of an ultrathinSOI layer as a compliant substrate for the epitaxy of SiGe alloys. In this work,they etched back an SOI wafer to leave a 50-nm layer of silicon. Then theygrew 10 nm of Si followed by 60 to 170 nm of Si0.85Ge0.15 by MBE with agrowth temperature of 500°C. (For Si0.85Ge0.15/Si, the room temperature mis-match strain is , corresponding to hc = 17 nm.) The various thick-nesses were obtained using shadow masking to keep all other growthconditions the same. It was found that for a 170-nm-thick layer of Si0.85Ge0.15

on the ultrathin SOI substrate, lattice relaxation occurred by the introductionof dislocations in the thin Si layer rather than the Si0.85Ge0.15. After a 1-hanneal at 700, 800, or 900°C, the layer had relaxed significantly compared tothe as-grown layer. Also, TEM analysis revealed that the structure annealedat 700°C contained misfit dislocations at the Si/Si0.85Ge0.15 interface, but thatthe associated threading segments were present only in the Si layer, not inSi0.85Ge0.15. They interpreted these results as evidence of compliance in theSi layer, associated with slippage at the Si/SiO2 interface. A conclusive testof compliant behavior in this material system could be made by growingvarious thicknesses of Si0.85Ge0.15 on an SOI layer less than hc = 17 nm inthickness. Then, no misfit dislocations would be expected to form at theinterface. In the work reported by Powell et al., however, the SOI layer was60 nm thick, so the observation of interfacial dislocations does not prove alack of compliancy in the thin Si layer.

LeGoues et al.60 further studied the ex situ relaxation of SiGe on SOIcompliant substrates. The SOI wafer used in this study was produced usingseparation by ion implantation of oxygen (SIMOX) and had a 65-nm-thick

f = −0 0062.




top Si layer with a dislocation density of about 105 cm–2. They grew a 180-nm-thick Si0.85Ge0.15 layer (and hc = 17 nm) by MBE at 400°C andobserved the relaxation and dislocation structure in the as-grown sampleand after thermal annealing. In the as-grown sample, no dislocations wereobserved by cross-sectional TEM inspection, indicating a threading disloca-tion density of less than the resolution of the technique (106 cm–2). Also, x-ray diffraction measurements revealed that the in-plane lattice constant ofSi0.85Ge0.15 matched that of the underlying Si. The as-grown Si0.85Ge0.15 wastherefore believed to be pseudomorphic. Upon annealing at 700 or 900°C inan inert ambient, Si0.85Ge0.15 relaxed by the formation of 60° dislocations atthe interface. However, the associated threading dislocations were observedonly in the thin Si layer and not in Si0.85Ge0.15. They interpreted these resultsas evidence of compliancy in the thin silicon layer on the SIMOX wafer.However, they did not compare this behavior to the case of growth onstandard silicon control wafers. Also, as with the previous study by Powellet al., they did not grow on a Si layer of less than 17 nm thickness to testthe ability to grow a pseudomorphic layer of any thickness.

Yang et al.58 demonstrated the growth of GaN on SiC on SOI. In their work,they produced a thin layer of SiC on a bonded and etched SOI (BESOI) waferby exposing the top silicon layer to a flux of carbon or acetylene at 900°C.Then they grew GaN on the SiC-on-SOI wafer using a 100-Å AlN nucleationlayer, a 100-Å GaN layer, and 10 periods of AlN/GaN superlattice with 40Å of periodicity. They grew a top GaN layer with 2000 Å thickness, but gavefew details of its material properties.

Seaford et al.61 compared the MBE growth of GaAs on Si (511) and SOI(511) wafers. The SOI (511) wafer was fabricated by bonding, and the toplayer of silicon was thinned to 100 nm. The GaAs grown on the SOI (511)wafer had a 25% reduction in the x-ray diffraction 004 FWHM compared togrowth on the control substrate. Also, the threading dislocation density onthe SOI wafer was lower by an order of magnitude, as determined by cross-sectional TEM characterization. Here, the SOI layer was insufficiently thinto provide a conclusive test of its compliancy.

Pei et al.62 also studied the growth of GaAs on SOI (511) wafers, with topsilicon layers having thicknesses of 100 and 200 nm. They showed by cross-sectional transmission electron microscopy (XTEM) that the GaAs on thethinner (100-nm) SOI layer had a lower threading dislocation density thanthe GaAs grown on the thicker (200-nm) SOI layer. Growth free from misfitdislocations could not be demonstrated on either wafer, however, becauseboth Si layers were thicker than the critical layer thickness for the GaAs/Siheteroepitaxial system.

Despite the aforementioned published results, the question remained asto whether a silicon-on-insulator layer could serve as a true compliant sub-strate, mechanically released from its handle substrate. In an attempt toanswer this question, Rehder et al.59 made a detailed experimental andmodeling study of SiGe relaxation on silicon-on-insulator substrates. TheSi0.82Ge0.18 layers (and hc = 14 nm) were grown by VPE to various

f = −0 0062.

f = −0 0074.




thicknesses and temperatures of 550, 630, and 670°C. Si0.82Ge0.18 was alsogrown at 700°C, but only to a thickness of 6 nm because it roughenedimmediately. The substrates included SOI wafers with various Si thicknesses(40, 70, 330, and 10,000 nm) as well as bulk Si control wafers. The resultingsamples were examined by XRD, AFM, and TEM.

Rehder et al. found that pseudomorphic Si0.82Ge0.18, 150 nm thick, could begrown on 40- or 70-nm-thick SOI layers. These metastable films could berelaxed ex situ by annealing in the range of 875 to 1050°C. Following annealingat 950°C, the SOI developed a strain (0.047 and 0.035% for the 40- and 70-nmSOI layers, respectively). However, these values of strain were only about onequarter of the values expected for an ideally compliant layer. In addition, thestrain in the SOI layer only appeared in conjunction with the broadening ofthe SiGe XRD peak and the emergence of surface crosshatch, both of whichare indirect indications of misfit dislocation production at the SiGe/Si interface.

Rehder et al. also studied the in situ relaxation of Si0.82Ge0.18 grown at 630°C.SiGe layer thicknesses of 150, 340, 765, and 1200 nm were chosen, resultingin a wide range of in situ strain relaxation; the 150-nm layer is unrelaxed,whereas 80% of the mismatch strain is relaxed in the thickest layer.

In order to understand whether the SOI behaved as a compliant substrate,Rehder et al. compared their experimental results to four equilibrium modelsfor the strain in the thin Si layer of the SOI. In the compliant substrate modelof Lo,50 if it is assumed that and , then the in-plane strainsin SiGe and Si will be

(7.51)

and

(7.52)

respectively.Rehder et al. developed three additional models by equating the line

tension on the misfit segment of a dislocation (at the SiGe/Si interface) withthe strain force exerted on the threading segment of the dislocation in thethin silicon-on-insulator layer. In model 1, the line tension of the misfitsegment of a dislocation at the interface was assumed to be the same as inthe case of growth on a thick, noncompliant substrate. Neglecting the coreparameter, this is given by

(7.53)

ν νSiGe Si= E ESiGe Si=

εSiGeSiGe Si

fh h

=+1 /

εSiSi SiGe

fh h

= −+1 /

FGb h

bLSiGe= −

−⎛⎝⎜

⎞⎠⎟

( cos )( )

ln14 1

2ν απ ν




where G has been assumed to be equal for the epitaxial layer and the sub-strate, is the angle between the Burgers vector and the line vector for thedislocations, and is the epitaxial layer thickness. However, they recog-nized that the line tension of the misfit segment would be reduced by thepresence of the SiO2 layer because of its lower shear modulus. In model 2,the line tension was calculated using Equation 7.53 above, but the averageshear modulus for Si and SiO2 was used. In developing model 3, theyassumed that the oxide acts as a free surface, leading to a modified linetension given by

(7.54)

where G has been assumed to be equal for the epitaxial layer and the sub-strate, is the angle between the Burgers vector and the line vector for thedislocations, is the epitaxial layer thickness, and is the silicon-on-insulator thickness. For all three models, the strain force on the threadingsegment of the dislocation in the silicon-on-insulator was calculated using

(7.55)

where is the in-plane strain in the silicon layer. The equilibrium strainin the Si is predicted to be

(model 1) (7.56)

(model 2) (7.57)

(model 3) (7.58)

(Lo model) (7.59)

Figure 7.39 shows the out-of-plane strain calculated using these four models,along with the experimental results of Rehder et al. The calculated resultsare shown for model 1 (dashed curve), model 2 (dotted curve), and model

αhSiGe

FGb h h

b h hLSi SiGe

Si SiGe

= −− +

( cos )( )

ln(

14 1

2ν απ ν ))

⎛⎝⎜

⎞⎠⎟

αhSiGe hSi

FGb h

TDSi Si= +

−ε ν

ν( )

( )1

1

εSi

ε ν απ νSi

Si

SiGebh

hb

= −+

⎛⎝⎜

⎞⎠⎟

( cos )( )

ln14 1

2

ε ν απ νSi

Si SiO

Si Si

G GG

bh

=+⎛

⎝⎜⎞

⎠⎟−

+2

214 1

2( cos )( ))

lnh

bSiGe⎛

⎝⎜⎞⎠⎟

ε ν απ νSi

Si

Si SiGe

Si S

bh

h hb h h

= −+ +

( cos )( )

ln(

14 1

2

iiGe)⎛⎝⎜

⎞⎠⎟

εSiSi SiGe

f

h h=

+1 /




3 (solid curve). The solid curve at the top labeled “compliant substrate” wascalculated using the Lo model. The experimental results for Si0.82Ge0.18 layerswith thicknesses of 1200 and 765 nm are plotted as well and can be fit verywell using model 3. However, the strain partitioning in the silicon-on-insu-lator layers does not follow the compliant substrate theory.

In summary, Rehder et al. found that the dependence of Si0.82Ge0.18 relax-ation on temperature and thickness was the same on bulk Si and SOI wafers.In all cases, relaxation of Si0.82Ge0.18 was accompanied by the introductionof misfit dislocations at the SiGe/Si interface. Tensile strain in the Si, pre-dicted by compliant substrate theory, only occurred with the introductionof interfacial misfit dislocations. Moreover, the amount of strain in the Siwas too small to be attributed to a compliant substrate mechanism. Theonly important effect of the SOI substrate is that the buried oxide layerreduces the line energies of misfit dislocations. Whereas a compliant sub-strate is supposed to increase the critical layer thickness for an epitaxialoverlayer, the reduction in the misfit dislocation line energy actuallydecreases the critical layer thickness. These results show that, in the workof Rehder et al., the SOI did not behave as a compliant substrate for theovergrowth of SiGe.

FIGURE 7.39Out-of-plane strain in a silicon-on-insulator layer as a function of the Si thickness. The calculatedresults are shown for model 1 (dashed curve), model 2 (dotted curve), model 3 (lower solidcurve), and the Lo compliant substrate model (upper solid curve). Also shown are experimentalresults for the growth of Si0.82Ge0.18 layers on SOI substrates, with Si0.82Ge0.18 thicknesses of 1200and 765 nm. (Reprinted from Rehder, E.M. et al., J. Appl. Phys., 94, 7892, 2003. With permission.Copyright 2003, American Institute of Physics.)

0 100 200 300SOI thickness (nm)

1200 nm film765 nm film

Compliant substrate−0.5

−0.4

−0.3

−0.2

−0.1

0

Out

-of-p

lane

stra

in (%

)




7.8.2.3 Twist-Bonded Compliant Substrates

Ejeckam et al.63,64 invented an approach involving the twist bonding of twowafers, followed by the thinning of the top wafer to render it compliant. Theycalled the twist-bonded structure a compliant universal (CU) substrate.65

The fabrication process developed by Ejeckam et al.64 for a twist-bondedcompliant substrate is shown schematically in Figure 7.40. The processbegins with two standard GaAs (001) wafers (Figure 7.40a). An AlAs etchstop layer and a 100-Å-thick compliant layer of GaAs are grown epitaxiallyon one of the wafers. Next, the two wafers are bonded together with a twistangle. The top GaAs substrate is etched away to the etch stop layer, and thenthe AlAs layer itself is removed by another selective etch step. This leavesonly the thin (compliant) GaAs layer twist-bonded to the bottom wafer.

At the twist boundary there is a large angular misalignment (~10°) betweenthe directions of the compliant layer and the substrate; however,the directions are parallel. The result is a dense square array of screwdislocations, with spacing d given by Frank’s rule:

(7.60)

where b is the length of the Burgers vector and is the twist angle. Figure7.41 shows a plan view TEM micrograph of such a twist boundary createdby bonding a 100-Å GaAs compliant layer to a GaAs (001) substrate. The

FIGURE 7.40Fabrication process for a twist-bonded compliant substrate. (Reprinted from Ejeckam, F.E. et al.,Appl. Phys. Lett., 70, 1685, 1997. With permission. Copyright 1997, American Institute of Physics.)

clean baresubstrate

bulksubstrate

after etch-stopremoval

after top substrateremoval

after twist bonding

etch-stop layerbulk substrate

compliant layer

110001

db=

2 2sin( / )θ

θ




twist angle is 4.2°. The spacing of the screw dislocations is d = 53 nm, whichis very close to the value predicted by Frank’s rule (d = 5.5 nm).

The atomic structure of the twist boundary is shown schematically inFigure 7.42 for the case of simple cubic crystals. The open circles representatoms in the thin, compliant layer, while the closed circles represent atomsin the substrate wafer. Inside the square regions, the atoms in the twist-bonded layer line up with the atoms in the underlying substrate. But in theboundaries between the square regions, the atoms in the compliant layer aredisplaced significantly by the array of screw dislocations.

Jesser et al.66 made a detailed study of the implementation of twist-bondedcompliant substrates. Two of their key findings were that the twist angleshould be large (greater than about 8°) and that coincidence angles shouldbe avoided. A large twist angle results in overlapping strain fields for thescrew dislocations at the boundary, rendering the thin layer more ideallycompliant. On the other hand, a coincidence angle (one that causes a largenumber of lattice sites to align on either side of the boundary) should beavoided because this locks the thin twist-bonded layer into a deep energyminimum with respect to the handle wafer and renders it noncompliant.

The design requirements for a twist-bonded compliant substrate, as enu-merated by Jesser et al.,66 are summarized below:

1. The compliant layer should be as thin as possible, but not so thin thatthe screw dislocations at the twist boundary are attracted to its surface.

2. The twist angle should be greater than about 8°.3. Coincidence angles should be avoided.4. Ideally, the compliant layer should be selected to have a small lattice

mismatch with the heteroepitaxial material that will be grown on

FIGURE 7.41Plan view dark-field weak-beam TEM micrograph of a twist boundary created by bonding a100-Å GaAs compliant layer to a GaAs (001) substrate. The twist angle is 4.2° and the spacingof the screw dislocations is d = 5.3 nm. (Reprinted from Ejeckam, F.E. et al., Appl. Phys. Lett.,70, 1685, 1997. With permission. Copyright 1997, American Institute of Physics.)

50 nm




top of it, but the compliant layer need not be lattice-matched withthe handle substrate.

5. The compliant layer should be chosen to induce layer-by-layer growthof the heteroepitaxial layer on top of it; island growth will lead togeometrically necessary dislocations where the islands coalesce.

6. Ideally, the compliant layer should have a smaller Young’s modulusthan the heteroepitaxial material that will be grown on top of it.

7. The handle wafer should have a large mismatch with respect to thecompliant layer, achieved through either a large twist angle or alarge lattice constant mismatch, but need not be of the same crystalstructure as the compliant layer.

Practical twist-bonded compliant substrates often satisfy several, but notall, of these design criteria. Nonetheless, twist-bonded GaAs compliant sub-strates have been used with varying levels of success for the growth ofheteroepitaxial InGaP, In0.22Ga0.78As, GaSb, and InSb.

Ejeckam et al.64 used twist-bonded GaAs compliant substrates to growIn0.35Ga0.65P. Their compliant substrates included a 100-Å top layer bondedto a GaAs substrate with a twist angle of 9, 17, or 32°. They grew 3000 Å of

FIGURE 7.42Schematic of a twist boundary between simple cubic crystals. The open circles represent atomsin the thin, compliant layer, while the closed circles represent atoms in the substrate wafer.(Reprinted from Ejeckam, F.E. et al., Appl. Phys. Lett., 70, 1685, 1997. With permission. Copyright1997, American Institute of Physics.)

Screw dislocation or stretched bonds




In0.35Ga0.65P on the twist-bonded compliant substrates by MOVPE. For thiscomposition, and . Though these layers are ~30 times thecritical layer thickness, they were found to be free from dislocations for allthree values of the twist angle investigated.

In another study, Ejeckam et al.65 used twist-bonded GaAs compliant sub-strates to grow InSb (f = –12.7%). The twist-bonded wafer had a 30-Å com-pliant layer bonded with a twist angle of 40 ± 5°. On these compliantsubstrates, they were able to grow pseudomorphic layers of InSb up to 6500Å thick, many times the critical layer thickness.

Other bonding approaches have also been tried in order to create compliantsubstrates. For example, Doolittle et al.67 demonstrated a compliant substratetechnology in which a GaN thin film was grown epitaxially on lithium gallate(LGO), LiGaO2, removed by selective etching, and then bonded on GaAs.Glass-bonded wafers have also been investigated for use as compliant sub-strates.56,57 Moran et al. studied the growth of In0.44Ga0.56As on GaAs compli-ant substrates that were glass-bonded using either 10% B2O3 or 30% B2O3

borosilicate glass. The 10 and 30% borosilicate glasses have viscosities of 1017

P and 1012 P, respectively, at the growth temperature of 650°C. In0.44Ga0.56Aslayers, 3 μm thick, were grown on conventional GaAs substrates, 12° twist-bonded compliant substrates, and both low- and high-viscosity glass-bondedcompliant substrates. They found that the material grown on the glass-bonded compliant substrates (either low or high viscosity) had the bestcrystal quality (as judged by the 004 x-ray rocking curve width). Interestingly,the material grown on the twist-bonded wafer appeared to be inferior to thatgrown on the conventional GaAs substrate.

7.9 Free-Standing Semiconductor Films

Due to the difficulties associated with the bulk growth of GaN substrates,III-nitride devices are implemented exclusively using heteroepitaxy on mis-matched substrates at the present time. This approach, while successful,gives rise to large threading dislocation densities (108 to 1010 cm–2) and alsosignificant thermal strains due to the mismatched thermal expansion coeffi-cients of the III-nitrides and their sapphire or SiC substrates. An alternativeapproach is the realization of free-standing GaN by its heteroepitaxy andsubsequent removal from the substrate.

Kelly et al.68 first demonstrated free-standing GaN films by growth onsapphire and laser-induced lift-off. They first grew 250- to 300-μm-thick GaNon a 2-inch c-face sapphire substrate by hydride vapor phase epitaxy(HVPE). Then, with the heteroepitaxial structure heated to 600°C, a pulsedlaser beam was scanned over the sample surface. The resulting decomposi-tion of a thin layer of GaN near the interface caused the lift-off of the GaN

f = 0 01. hc ≈ 100 Å




film. Nearly complete lift-off was achieved, and the resulting GaN film hadan area almost equal to that of the starting sapphire substrate.

Tsai et al.69 produced free-standing GaN films with EPDs lessthan . In this work, a 2-μm-thick GaN template layer was firstgrown on a sapphire (0001) substrate by MOVPE. Next, a thick layer (50 to200 μm) of GaN was grown on the template by HVPE. The resulting thickfilm of GaN was next separated from its sapphire substrate using laser-induced lift-off. Finally, an additional thickness of GaN was grown by HVPE.For the characterization of template layers and free-standing films,H3PO4:H2SO4 was used as the crystallographic etch, and the resulting etchpits were observed by AFM. In 2-μm-thick MOVPE template layers, the EPDswere found to be as high as . In a 500-μm-thick free-standing GaNfilm, however, no etch pits were observed in the field of view,therefore placing an upper limit of on the EPD.

While this approach appears to be promising, several problems need tobe solved before it can be exploited commercially. Two critical problems arethe incomplete separation of the heteroepitaxial layer from the substrate andthe retention of curvature following separation.

7.10 Conclusion

A number of defect engineering approaches for heteroepitaxial layers haveemerged. These include buffer layer approaches, patterned growth, pattern-ing and annealing, epitaxial lateral overgrowth, nanoheteroepitaxy, and com-pliant substrates, to name a few. All were designed to reduce the dislocationdensities of heteroepitaxial layers to practical levels for device applications.Some are intended to remove existing defects from lattice-relaxed heteroepi-taxial layers, such as patterning and annealing, epitaxial lateral overgrowth,or superlattice buffer layers (dislocation filters). Others were conceived toprevent lattice relaxation in the first place; these include patterned growth,nanoheteroepitaxy, and compliant substrates.

The proliferation of defect engineering methods could be taken as anindication that none of them are uniquely suited to the purpose for allmaterial systems. On the other hand, some of these approaches have beenhighly successful, to the point of being used in commercial devices. Gradedbuffer layers are the most important example of this and have been used incommercial GaAs1–xPx LEDs on GaAs substrates and InxGa1–xAs high-elec-tron-mobility transistors (HEMTs) on GaAs substrates. Epitaxial lateral over-growth (ELO) is an important method used to reduce the threadingdislocation densities in the active regions of III-nitride lasers. Other defectengineering approaches, such as the use of compliant substrates, show greatpromise, and yet their commercial exploitation is not yet in sight.

4 104 2× −cm

6 108 2× −cm50 50μ μm m×

4 104 2× −cm




In order to tap the great potential of heteroepitaxy, defect engineeringapproaches will continue to be important, not only in the applications listedabove, but in new ones as well. The commercial need to integrate devicesfrom different material systems will surely drive the development of newdefect engineering approaches. Many of these will probably draw on theprinciples outlined in this chapter.

Problems

1. Suppose a uniform GaAs0.5P05 layer is to be grown on a GaAs (001)substrate without any compositional grading. Determine theapproximate requirement on the thickness in order to achieve athreading dislocation density of <106 cm–2 at the top of the layer.

2. Suppose that a GaAs0.5P05 device layer is to be grown on a GaAs(001) substrate using a linearly graded buffer layer, with the samerequirement on the threading dislocation density as in Problem 1.Choose the thickness and grading coefficient for the graded layer toachieve this with the minimum total thickness.

3. In0.35Ga0.65As has been grown on InP with a thickness of 2 μm. If thematerial is to be patterned into square mesas and annealed, what isthe maximum width of the mesas such that all of the threadingdislocations may be removed by glide to the sidewalls?

4. If In0.05Ga0.95As is grown on nanopatterned GaAs (001), what is thecritical layer thickness for planar heteroepitaxy? What diameter ofthe seed pads will cause the critical layer thickness to double relativeto planar heteroepitaxy? At what diameter for the GaAs seed padswill the critical layer thickness diverge to infinity?

5. Si0.9Ge0.1 is to be grown on a truly compliant layer of Si having athickness of 100 nm. Determine the expected strains in the epitaxiallayer and compliant subtrate layer if the Si0.9Ge0.,1 is grown to athickness of 200 nm. Repeat this calculation assuming that the Silayer is not compliant.

References

1. A.E. Romanov, W. Pompe, G.E. Beltz, and J.S. Speck, An approach to threadingdislocation “reaction kinetics,” Appl. Phys. Lett., 69, 3342 (1996).

2. J.E. Ayers, New model for the thickness and mismatch dependencies of thread-ing dislocation densities in mismatched heteroepitaxial layers, J. Appl. Phys.,78, 3724 (1995).





4. J.J. Tietjen, J.I. Pankove, I.J. Hegyi, and H. Nelson, Vapor-phase growth ofGaAs1–xPx, room-temperature injection lasers, Trans. AIME, 239, 385 (1967).

5. C.J. Nuese, J.J. Tietjen, J.J. Gannon, and H.F. Gossenberger, Electroluminescenceof vapor-grown GaAs and Gas1–xPx diodes, Trans. AIME, 242, 400 (1968).

6. D. Richman and J.J. Tietjen, Rapid vapor phase growth of high-resistivity GaPfor electro-optic modulators, Trans. AIME, 239, 418 (1967).

7. R.H. Saul, Effect of a GaAsxP1–x transition zone on the perfection of GaP crystalsgrown by deposition onto GaAs substrates, J. Appl. Phys., 40, 3273 (1969).

8. E.A. Fitzgerald, Y.-H. Xie, D. Monroe, P.J. Silverman, J.M. Kuo, A.R. Kortan,F.A. Thiel, and B.E. Weir, Relaxed GexSi1–x structures for III-V integration withSi and high mobility two-dimensional electron gases in Si, J. Vac. Sci. Technol.B, 10, 1807 (1992).

9. E.A. Fitzgerald, A.Y. Kim, M.T. Currie, T.A. Langdo, G. Tarischi, and M.T.Bulsara, Dislocation dynamics in relaxed graded composition semiconductors,Mater. Sci. Eng. B, 67, 53 (1999).

10. P.L. Gourley, T.J. Drummond, and B.L. Doyle, Dislocation filtering in semicon-ductor superlattices with lattice-matched and lattice-mismatched layer mate-rials, Appl. Phys. Lett., 49, 1101 (1986).

11. T. Soga, S. Hattori, S. Sakai, M. Takeyasu, and M. Umeno, MOCVD growth ofGaAs on Si substrates with AlGaP and strained superlattice layers, Electron.Lett., 20, 916 (1984).

12. T. Soga, S. Hattori, S. Sakai, and M. Umeno, Epitaxial growth and materialproperties of GaAs on Si grown by MOCVD, J. Cryst. Growth, 77, 498 (1986).

13. H. Okamoto, Y. Watanabe, Y. Kadota, and Y. Ohmachi, Dislocation reductionin GaAs on Si by thermal cycles and InGaAs/GaAs strained-layer superlattices,Jpn. J. Appl. Phys., 26, L1950 (1987).

14. T.S. Rao, K. Nozawa, and Y. Horikoshi, Migration enhanced epitaxy growth ofGaAs on Si with (GaAs)1–x(Si2)x/GaAs strained layer superlattice buffer layers,Appl. Phys. Lett., 62, 154 (1993).

15. T.S. Rao, K. Nozawa, and Y. Horikoshi, Structural properties of (GaAs)1–x(Si2)x

layers on GaAs(100) substrates grown by migration enhanced epitaxy, Jpn. J.Appl. Phys., 30, L547 (1991).

16. T.S. Rao and Y. Horikoshi, Growth of (GaAs)1–x(Si2)x metastable alloys usingmigration-enhanced epitaxy, J. Cryst. Growth, 115, 328 (1991).

17. W. Qian, M. Skowronski, and R. Kaspi, Dislocation density reduction in GaSbfilms grown on GaAs substrates by molecular beam epitaxy, J. Electrochem. Soc.,144, 1430 (1997).

18. J.L. Reno, S. Chadda, and K. Malloy, Dislocation density reduction in CdZnTe(100) on GaAs using strained layer superlattices, Appl. Phys. Lett., 63, 1827 (1993).

19. W. Qian, M. Skowronski, M. De Graef, K. Doverspike, L.B. Rowland, and D.K.Gaskill, Microstructural characterization of α-GaN films grown on sapphire byorganometallic vapor phase epitaxy, Appl. Phys. Lett., 66, 1252 (1995).

20. E. Feltin, B. Beaumont, M. Laügt, P. de Mierry, P. Vennéguès, H. Lahrèche, M.Leroux, and P. Gibart, Stress control in GaN grown on silicon (111) by metal-organic vapor phase epitaxy, Appl. Phys. Lett., 79, 3230 (2001).

21. W.H. Sun, J.P. Zhang, J.W. Yang, H.P. Maruska, M.A. Khan, R. Liu, and F.A.Ponce, Fine structure of AlN/AlGaN superlattice grown by pulsed atomic-layer epitaxy for dislocation filtering, Appl. Phys. Lett., 87, 211915 (2005).




22. N.A. El-Masry, J.C. Tarn, and N.H. Karam, Interactions of dislocations in GaAsgrown on Si substrates with InGaAs-GaAsP strained layer superlattices, J. Appl.Phys., 64, 3672 (1988).

23. J.W. Matthews, S. Mader, and T.B. Light, Accommodation of misfit across theinterface between crystals of semiconducting elements or compounds, J. Appl.Phys., 41, 3800 (1970).

24. E.A. Fitzgerald, P.D. Kirchner, R. Proano, G.D. Pettit, J.M. Woodall, and D.G.Ast, Elimination of interface defects in mismatched epilayers by a reduction ingrowth area, Appl. Phys. Lett., 52, 1496 (1988).

25. E.A. Fitzgerald, G.P. Watson, R.E. Proano, D.G. Ast, P.D. Kirchner, G.D. Pettit,and J.M. Woodall, Nucleation mechanisms and the elimination of misfit dislo-cations at mismatched interfaces by reduction in growth area, J. Appl. Phys.,65, 2220 (1989).

26. X.G. Zhang, A. Rodriguez, X. Wang, P. Li, F.C. Jain, and J.E. Ayers, Completeremoval of threading dislocations from mismatched layers by patterned het-eroepitaxial processing, Appl. Phys. Lett., 77, 2524 (2000).

27. X.G. Zhang, P. Li, G. Zhao, D.W. Parent, F.C. Jain, and J.E. Ayers, Removal ofthreading dislocations from patterned heteroepitaxial semiconductors by glideto sidewalls, J. Electron. Mater., 27, 1248 (1998).

28. D. Hull and D.J. Bacon, Introduction to Dislocations, 3rd ed., Pergamon, NewYork, 1984.

29. N. Chand and S.N.G. Chu, Elimination of dark line defects in GaAs-on-Si bypost-growth patterning and thermal annealing, Appl. Phys. Lett., 58, 74 (1991).

30. I. Yonenaga, Dynamic behavior of dislocations in InAs:In comparison with III-V compounds and other semiconductors, J. Appl. Phys., 84, 4209 (1998).

31. L. Jastrzebski, SOI by CVD: epitaxial lateral overgrowth (ELO) process —review, J. Cryst. Growth, 63, 493 (1983).

32. R.W. McClelland, C.O. Bozler, and J.C.C. Fan, A technique for producing epi-taxial films on reusable substrates, Appl. Phys. Lett., 37, 560 (1980).

33. R.P. Gale, R.W. McClelland, J.C.C. Fan, and C.O. Bozler, Lateral epitaxial over-growth of GaAs by organometallic chemical vapor deposition, Appl. Phys. Lett.,41, 545 (1982).

34. P. Vohl, C.O. Bozler, R.W. McClelland, A. Chu, and A.J. Strauss, Lateral growthof single-crystal InP over dielectric films by orientation-dependent VPE, J.Cryst. Growth, 56, 410 (1982).

35. M. Kastelic, I. Oh, C.G. Takoudis, J.A. Friedrich, and G.W. Neudeck, Selectiveepitaxial growth of silicon in pancake reactors, Chem. Eng. Sci., 43, 2031 (1988).

36. J.L. Glenn, Jr., G.W. Neudeck, C.K. Subramanian, and J.P. Denton, A fully planarmethod for creating adjacent “self-isolating” silicon-on-insulator and epitaxiallayers by epitaxial lateral overgrowth, Appl. Phys. Lett., 60, 483 (1992).

37. G. Shahidi, B. Davari, Y. Taur, J. Warnock, M.R. Wordeman, P. McFarland, S.Mader, M. Rodriguez, R. Assenza, G. Bronner, B. Ginsberg, T. Lii, M. Polcari, andT.H. Ning, Fabrication of CMOS on ultrathin SOI obtained by epitaxial lateralovergrowth and chem-mechanical polishing, IEDM Technol. Dig., 587 (1990).

38. O.-H. Nam, M.D. Bremser, T.S. Zheleva, and R.F. Davis, Lateral epitaxy of lowdefect density GaN layers via organometallic vapor phase epitaxy, Appl. Phys.Lett., 71, 2638 (1997).

39. O.-H. Nam, M.D. Bremser, B.L. Ward, R.J. Memanich, and R.F. Davis, Growthof GaN and Al0.2Ga0.8N on patterned substrates via organometallic vapor phaseepitaxy, Jpn. J. Appl. Phys., 36, L532 (1997).




40. T.S. Zheleva, O.-H. Nam, M.D. Bremser, and R.F. Davis, Dislocation densityreduction via lateral epitaxy in selectively grown GaN structures, Appl. Phys.Lett., 71, 2472 (1997).

41. J. Park, P.A. Grudowski, C.J. Eiting, and R.D. Dupuis, Selective-area and lateralepitaxial overgrowth of III-N materials by metal organic chemical vapor dep-osition, Appl. Phys. Lett., 73, 333 (1998).

42. Y.S. Chang, S. Naritsuka, and T. Nishinaga, Optimization of growth conditionfor wide dislocation-free GaAs on Si substrate by microchannel epitaxy, J. Cryst.Growth, 192, 18 (1998).

43. T. Zheleva, S.A. Smith, D.B. Thomson, T. Gehrke, K.J. Linthicum, P. Rajagopal,E. Carlson, W.M. Ashmawi, and R.F. Davis, Pendeo-epitaxy: a new approachfor lateral growth of gallium nitride structures, MRS Internet J. Nitride Semicond.,4S1, G3.38 (1999).

44. S. Luryi and E. Suhir, New approach to the high quality epitaxial growth oflattice-mismatched materials, Appl. Phys. Lett., 49, 140 (1986).

45. D. Zubia and S.D. Hersee, Nanoheteroepitaxy: the application of nanostructur-ing and substrate compliance to the heteroepitaxy of mismatched semiconduc-tor materials, J. Appl. Lett., 85, 6492 (1999).

46. J.P. Hirth and A.G. Evans, Damage of coherent multilayer structures by injec-tion of dislocations or cracks, J. Appl. Phys., 60, 2372 (1986).

47. D. Zubia, S.H. Zaidi, S.R.J. Brueck, and S.D. Hersee, Nanoheteroepitaxialgrowth of GaN on Si by organometallic vapor phase epitaxy, Appl. Phys. Lett.,76, 858 (2000).

48. S.H. Zaidi, A.S. Chu, and S.R.J. Brueck, Scalable fabrication and optical char-acterization of nm Si structures, Mater. Res. Soc. Symp. Proc., 358, 957 (1995).

49. S.H. Zaidi, A.S. Chu, and S.R.J. Brueck, Optical properties of nanoscale one-dimensional silicon grating structures, J. Appl. Phys., 80, 6997 (1996).

50. Y.H. Lo, New approach to grow pseudomorphic structures over the criticallayer thickness, Appl. Phys. Lett., 59, 2311 (2005).

51. D. Teng and Y.H. Lo, Dynamic model for pseudomorphic structures grown oncompliant substrates: an approach to extend the critical thickness, Appl. Phys.Lett., 62, 43 (1993).

52. C.L. Chua, W.Y. Hsu, C.H. Lin, G. Christensen, and Y.H. Lo, Overcoming thepseudomorphic critical layer thickness limit using compliant substrates, Appl.Phys. Lett., 64, 3640 (1994).

53. C. Carter-Coman, A.S. Brown, N.M. Jokerst, D.E. Dawson, R. Bicknell-Tassius,Z.C. Feng, K.C. Rajikumar, and G. Dagnall, Strain accommodation in mis-matched layers by molecular beam epitaxy: introduction of a new compliantsubstrate technology, J. Electron. Mater., 25, 1044 (1996).

54. C. Carter-Coman, R. Bicknell-Tassius, R.G. Benz, A.S. Brown, and N.M. Jokerst,Analysis of GaAs substrate removal etching with citric acid:H2O2 andNH4OH:H2O2 for application to compliant substrates, J. Electrochem. Soc., 144,L29 (1997).

55. P.D. Moran, D.M. Hansen, R.J. Matyi, J.M. Redwing, and T.F. Kuech, Realizationand characterization of ultrathin GaAs-on-insulator structures, J. Electrochem.Soc., 146, 3506 (1999).

56. P.D. Moran, D.M. Hansen, R.J. Matyi, J.G. Cederberg, L.J. Mawst, and T.F.Kuech, InGaAs heteroepitaxy on GaAs compliant substrates: x-ray diffractionevidence of enhanced relaxation and improved structural quality, Appl. Phys.Lett., 75, 1559 (1999).




57. A.R. Powell, S.S. Iyer, and F.K. LeGoues, New approach to the growth of lowdislocation relaxed SiGe material, Appl. Phys. Lett., 64, 1856 (1994).

58. Z. Yang, F. Guarin, I.W. Tao, W.I. Wang, and S.S. Iyer, Approach to obtain highquality GaN on Si and SiC-on-silicon-on-insulator compliant substrate by mo-lecular-beam epitaxy, J. Vac. Sci. Technol. B, 13, 789 (1995).

59. E.M. Rehder, C.K. Inoki, T.S. Kuan, and T.F. Kuech, SiGe relaxation on silicon-on-insulator substrates: an experimental and modeling study, J. Appl. Phys., 94,7892 (2003).

60. F.K. LeGoues, A. Powell, and S.S. Iyer, Relaxation of SiGe thin films grown onSi/SiO2 substrates, J. Appl. Phys., 75, 7240 (1994).

61. M.L. Seaford, D.H. Tomich, K.G. Eyink, L. Grazulis, K. Mahalingham, Z. Yang,and W.I. Wang, Comparison of GaAs grown on standard Si (511) and compliantSOI (511), J. Electron. Mater., 29, 906 (2000).

62. C.W. Pei, J.B. Héroux, J. Sweet, W.I. Wang, J. Chen, and M.F. Chang, Highquality GaAs grown on Si-on-insulator compliant substrates, J. Vac. Sci. Technol.B, 20, 1196 (2002).

63. F.E. Ejeckam, Y. Qian, Z.H. Zhu, Y.H. Lo, S. Subramian, and S.L. Sass, Mis-aligned (or twist) wafer-bonding: A new technology for making III-V compliantsubstrates, Lasers and Electro-Optics Society Annual Meeting (IEEE/LEOS),Vol. 2, p. 352 (1996).

64. F.E. Ejeckam, Y.H. Lo, S. Subramanian, H.Q. Hou, and B.E. Hammons, Latticeengineered compliant substrate for defect-free heteroepitaxial growth, Appl.Phys. Lett., 70, 1685 (1997).

65. F.E. Ejeckam, M.L. Seaford, Y.-H. Lo, H.Q. Hou, and B.E. Hammons, Disloca-tion-free InSb grown on GaAs compliant universal substrates, Appl. Phys. Lett.,71, 776 (1997).

66. W.A. Jesser, J.H. van der Merwe, and P.M. Stoop, Misfit accommodation bycompliant substrates, J. Appl. Phys., 85, 2129 (1999).

67. W.A. Doolittle, T. Kropewnicki, C. Carter-Coman, S. Stock, P. Kohl, N.M. Jok-erst, R.A. Metzger, S. Kang, K.K. Lee, G. May, and A.S. Brown, Growth of GaNon lithium gallate substrates for development of a GaN thin compliant sub-strate, J. Vac. Sci. Technol. B, 16, 1300 (1998).

68. M.K. Kelly, R.P. Vaudo, V.M. Phanse, L. Gorgens, O. Ambacher, and M. Stutz-mann, Large free-standing GaN substrates by hydride vapor phase epitaxy andlaser-induced liftoff, Jpn. J. Appl. Phys., Part 2, 38, L217 (1999).

69. C.-C. Tsai, C.-S. Chang, and T.-Y. Chen, Low-etch-pit-density GaN substratesby regrowth on free-standing GaN films, Appl. Phys. Lett., 80, 3718 (2002).

70. P. Sheldon, K.M. Jones, M.M. Al-Jassim, and B.G. Yacobi, Dislocation densityreduction through annihilation in lattice-mismatched semiconductors grownby molecular beam epitaxy, J. Appl. Phys., 63, 5609 (1988).

71. J.E. Ayers, L.J. Schowalter, and S.K. Ghandhi, Post-growth thermal annealingof GaAs on Si(001) grown by organometallic vapor phase epitaxy, J. Cryst.Growth, 125, 329 (1992).

72. S. Akram, H. Ehsani, and I.B. Bhat, The effect of GaAs surface stabilization onthe properties of ZnSe grown by organometallic vapor phase epitaxy, J. Cryst.Growth, 124, 628 (1992).

73. S. Kalisetty, M. Gokhale, K. Bao, J.E. Ayers, and F.C. Jain, The influence ofimpurities on the dislocation behavior in heteroepitaxial ZnSe on GaAs, Appl.Phys. Lett., 68, 1693 (1996).

74. M. Tachikawa, and M. Yamaguchi, Film thickness dependence of dislocationdensity reduction in GaAs-on-Si substrates, Appl. Phys. Lett., 56, 484 (1990).



421

Appendix A

Bandgap Engineering Diagrams

FIGURE A.1

Energy gap as a function of lattice constant for cubic semiconductors. Room temperature valuesare given. Dashed lines indicate an indirect gap.

0

1

2

3


Ener

gy g

ap (e

V)

AlP

GaP

Si

ZnS

GaAs

Ge

ZnSe

ZnTe

CdSe

CdTe

InSbHgTe

GaSb

InAsHgSe

AlSb

AlAs

InP



422


References

1. Y.V. Shvyd’ko, M. Lucht, E. Gerdau, M. Lerche, E.E. Alp, W. Sturhahn, J. Sutter,and T.S. Toellner, Measuring wavelengths and lattice constants with the Möss-bauer wavelength standard,

J. Synchrotron Rad.

, 9, 17 (2002).

FIGURE A.2

Energy gap as a function of lattice constant a for hexagonal semiconductors. Room temperaturevalues are given. Sapphire, a commonly used substrate material for III-nitrides, has roomtemperature lattice constants of a = 4.7592 Å and c = 12.9916 Å.

1

0

2

4

6


Ener

gy g

ap (e

V)AlN

GaN

In N

6H-SiC

4H-SiC



423

Appendix B

Lattice Constants and Coefficients of

Thermal Expansion

Table B.1 and Table B.2 give the lattice constants and thermal expansioncoefficients of selected cubic and hexagonal crystals. The linear thermalcoefficient of expansion (TCE) is defined as

(B.1)

The variation of the lattice constant with temperature can also be fit to apolynomial:

(B.2)

where is in percent, with respect to 300K, and

T

is the absolute tem-perature in Kelvin. Thus, at a temperature

T

, the relaxed lattice constant forthe crystal is given by

(B.3)

The constants A, B, C, and D for cubic crystals are provided in Table B.3.

α

α ≡ ∂∂

1a

aT

Δaa

A BT CT DT= + + +2 3

Δa a/

a T a KA BT CT DT( ) = ( ) + + + +⎡

⎣⎢

⎤

⎦⎥300 1

100

2 3



424


TABLE B.1

Lattice Constants and Thermal Expansion Coefficients for

Cubic Semiconductor Crystals

a

(300K) (Å)

α

(300K)(10

–6

K

–1

)

α

(600K)(10

–6

K

–1

)

α

(1000K)(10

–6

K

–1

)

C 3.56684

1

1.0

2

2.8 4.4Si 5.43108

3

2.6

2

3.7 4.4Ge 5.6576

4

5.7 6.7 7.6

α

-Sn 6.4894

5

4.7 — —SiC (3C) 4.3596

6,7

— — —BN 3.615 1.8 3.7 5.9BP 4.538* — — —BAs 4.777 — — —AlP 5.467 — — —AlAs 5.660 — — —AlSb 6.1357 4.4 — —GaP 5.4512 4.7 5.8 —GaAs 5.6534 5.7 6.7 —GaSb 6.0960 6.1 7.3 —InP 5.8690 4.75 — —InAs 6.0584 5.19 — —InSb 6.4794 5.0 6.1 —BeS 4.865 — — —BeSe 5.139 — — —BeTe 5.626 — — —ZnS 5.4105 7.1 8.6 10.5ZnSe 5.6687

8

7.1 10.1 —ZnTe 6.1041 8.8 10.0 —CdTe 6.481 5.0 5.4 —

β

-HgS 5.851 — — —HgSe 6.084 — — —HgTe 6.461 5.1 — —

Note:

, in percent, where T is the abso-lute temperature.

* Low temperature.

Δa a A BT CT DT/ = + + +2 3



Appendix B: Lattice Constants and Coefficients of Thermal Expansion

425

TABLE B.2

Lattice Constants and Thermal Expansion Coefficients for Hexagonal Crystals

a

(Å)

c

(Å)

α

a

(300K)(10

–6

K

–1

)

α

c

(300K)(10

–6

K

–1

)

α

a

(600K)(10

–6

K

–1

)

α

c

(600K)(10

–6

K

–1

)

α

-l

2

O

39

4.7592 12.9916 4.3 3.9 5.6 7.4SiC (2H) 3.076 5.048 — — — —SiC (4H)

10

3.0730 10.053 — — — —SiC (6H)

6

3.0806 15.1173 — — — —AlN 3.112

11

4.978 — — — —GaN

12,13

3.1886(5) 5.1860(4) 3.1 2.8 4.7 4.2InN

14

3.533 5.693 3.4 2.7 5.7 3.7ZnS 3.8140 6.2576 — — — —ZnTe 4.27 6.99 — — — —CdS 4.1348 6.7490 — — — —CdSe 4.299 7.010 — — — —CdTe 4.57 7.47 — — — —

TABLE B.3

Temperature Dependence of Thermal Expansion for Cubic Crystals

A B

(10

–4

K

–1

)

C

(10

–7

K

–2

)

D

(10

–10

K

–3

)

C –0.010 –0.591 3.32 –0.5544 (25–1650K)Si –0.071 1.887 1.934 –0.4544 (293–1600K)Ge –0.1533 4.636 2.169 –0.4562 (293–1200K)

α

-Sn –0.525 13.54 15.87 –2.896 (100–500K)BN –0.0013 –1.278 4.911 –0.8635 (293–1300K)AlSb –0.049 –2.997 22.43 –22.34 (40–350K)GaP –0.110 2.611 4.445 –2.023 (293–850K)GaAs –0.147 4.239 2.916 –0.936 (200–1000K)GaSb –0.138 3.051 66.02 –3.380 (100–800K)InSb –0.099 1.249 8.773 –5.260 (50–750K)ZnS –0.0863) –3.386 30.18 –29.21 (60–335K)ZnSe –0.170 4.419 5.309 –2.158 (293–800K)ZnTe –0.200 5.104 6.811 –3.104 (100–725K)CdTe –0.0980 1.624 7.176 –4.445 (100–700K)HgTe –0.504 9.772 42.66 –59.22 (50–300K)



426


References

1. W. Kaiser and W.L. Bond, Nitrogen, a major impurity in common type I dia-mond,

Phys. Rev.

, 115, 857 (1959).2. Y.S. Touloukian, R.K. Kirby, R.E. Taylor, and P.D. Desai, Eds.,

ThermophysicalProperties of Matter

, Vol. 12,

Thermal Expansion, Metallic Elements and Alloys

,Plenum, New York, 1975.

3. E.R. Cohen and B.N. Taylor,

The 1986 Adjustment of the Fundamental PhysicalConstants

, report of the Committee on Data for Science and Technology of theInternational Council of Scientific Unions (CODATA) Task Group on Funda-mental Constants, CODATA Bulletin 63, Pergamon, Elmsford, NY, 1986.

4. J. Donahue,

The Structure of the Elements

, J. Wiley & Sons, New York, 1974.5. J. Thewlis and A.R. Davey, Thermal expansion of grey tin,

Nature

, 174, 1011,1954.

6. A. Taylor and R.M. Jones, in

Silicon Carbide: A High Temperature Semiconductor

,J.R. O’Connor and J. Smiltens, Eds., Pergamon Press, Oxford, 1960, p. 147.

7. Y.M. Tairov and V.F. Tsvetkov, in

Crystal Growth and Characterization of PolytypeStructures

, P. Krishna, Ed., Pergamon Press, Oxford, 1983, pp. 111–162.8. B. Greenberg, private communication.9. Y.V. Shvyd’ko, M. Lucht, E. Gerdau, M. Lerche, E.E. Alp, W. Sturhahn, J. Sutter,

and T.S. Toellner, Measuring wavelengths and lattice constants with the Möss-bauer wavelength standard,

J. Synchrotron Rad.

, 9, 17 (2002).10. R.W.G. Wyckoff,

Crystal Structure,

Vol. 1, Interscience, New York, 1963.11. C. Kim, I.K. Robinson, J. Myoung, K. Shim, M.-C. Yoo, and K. Kim, Critical

thickness of GaN thin films on sapphire (0001),

Appl. Phys. Lett.

, 69, 2358 (1996).12. S. Poroski, Bulk and homoepitaxial GaN-growth and characterization,

J. Cryst.Growth

, 189/190, 153 (1998).13. M. Leszczynski, T. Suski, H. Teisseyre, P. Perlin, I. Grzegory, J. Jun, S. Porowski,

and T.D. Moustakas, Thermal expansion of gallium nitride,

J. Appl. Phys.

, 76,4909 (1994).

14. K. Wang and R.R. Reeber, Thermal expansion and elastic properties of InN,

Appl. Phys. Lett.

, 79, 1602 (2001).



427

Appendix C

Elastic Constants

Table C.1 through Table C.5 provide the elastic stiffness constants of cubic andhexagonal crystals. Table C.6 and Table C.7 give values of the elastic moduli.

TABLE C.1

Elastic Stiffness Constants of Cubic Semiconductor Crystals at Room Temperature, in Units of GPa

(1 GPa = 10

10

dyn/cm

2

)

C

11

C

12

C

44

C

1

107.6 12.52(23) 57.74(14)Si

2

160.1 57.8 80.0Ge 124.0 41.3 68.3

α

-Sn 69.0 29.3 36.2SiC (3C)

3

352 120 232.9AlN (ZB)

4

322 156 138AlP 132 63.0 61.5AlAs 125 53.4 54.2AlSb 87.69(20) 43.41(20) 40.76(8)GaN (ZB)

5

325 142 147GaP

5

140.50(28) 62.03(24) 70.33(7)GaAs 118.4(3) 53.7(16) 59.1(2)GaSb 88.50 40.40 43.30InP 102.2 57.6 46.0InAs 83.29 45.26 39.59InSb 65.92(5) 35.63(6) 29.96(3)ZnS 104.62(5) 65.33(6) 46.50(12)ZnSe 87.2(8) 52.4(8) 39.2(4)ZnTe 71.3 40.7 31.2CdTe 53.3 36.5 20.44

β

-HgS 81.3 62.2 26.4HgSe 69.0 51.9 23.3HgTe 53.61 36.60 21.23



428


TABLE C.2

Elastic Stiffness Constants of 4H- and 6H-SiC at Room

Temperature, in Units of GPa (1 GPa = 10

10

dyn/cm

2

)

Elastic Constants4H-SiC

(Kamitani et al.

6

)6H-SiC

(Kamitani et al.

7

)

C

11

507(4) 501(4)

C

12

111(5) 111(5)

C

13

— 52(9)

C

33

547(7) 553(4)

C

44

159(4) 163(4)

C

66

198 195

TABLE C.3

Elastic Stiffness Constants of Wurtzite GaN at Room Temperature, in Units of GPa

(1 GPa = 10

10

dyn/cm

2

)

ElasticConstants

RecommendedValues

Polianet al.

7

Degeret al.

8

Deguchiet al.

9

V. Yu Davydovet al.

10

Savastenkoand Shelag

11

C

11

353

390(15) 370 373 315 296

C

12

135

145(20) 145 141 118 120

C

13

104

106(20) 110 80.4 96 158

C

33

367

398(20) 390 387 324 267

C

44

91

105(10) 90 93.6 88 24

C

66

110

123(10) 112 118 99 88

TABLE C.4

Elastic Stiffness Constants of Wurtzite AlN at Room Temperature, in Units of GPa

(1 GPa = 10

10

dyn/cm

2

)

ElasticConstants

RecommendedValues

Degeret al.

8

V. Yu Davydovet al.

10

McNeilet al.

12

Tsubouchiet al.

13

S. Yu Davydovet al.

4

C

11

397

410 419 411 345 369

C

12

145

140 177 149 125 145

C

13

113

100 140 99 120 120

C

33

392

390 392 389 395 395

C

44

118

120 110 125 125 96

C

66

128

135 121 131 131 112



Appendix C: Elastic Constants

429

TABLE C.5

Elastic Stiffness Constants of Wurtzite InN at Room Temperature, in Units of GPa

(1 GPa = 10

10

dyn/cm

2

)

ElasticConstants

RecommendedValues

Sheleg andSavastenko

14

Kimet al.

15

Wright

16

Marmalyuket al.

17

Chisholmet al.

18

C

11

250

190 271 223 257 297.5

C

12

109

104 124 115 92 107.4

C

13

98

121 94 92 70 108.7

C

33

225

182 200 224 278 250.5

C

44

54

9.9 46 48 68 89.4

C

66

70

43 74 54 82 95

TABLE C.6

Elastic Moduli of Cubic Semiconductor Crystals at 300K

G E

(001)

ν

(001)

Y

(001)

R

B

(001)

C 47 105 0.104 117 0.23Si 51 129 0.265 176 0.72Ge 41 103 0.25 138 0.67

α

-Sn 19.8 52 0.30 73 0.85SiC (3C) 116 290 0.25 390 0.68AlN (ZB) 83 220 0.33 330 0.97AlP 34 91 0.32 135 0.95AlAs 36 93 0.30 133 0.85AlSb 22 59 0.33 88 0.99GaN (ZB) 92 240 0.30 340 0.87GaP 39 102 0.31 148 0.88GaAs 32 85 0.31 124 0.91GaSb 24 63 0.31 92 0.91InP 22 61 0.36 95 1.13InAs 19.0 51 0.35 79 1.09InSb 15.1 41 0.35 63 1.08ZnS 19.6 54 0.38 88 1.25ZnSe 17.4 48 0.38 77 1.20ZnTe 15.3 42 0.36 66 1.14CdTe 8.4 24 0.41 40 1.37

β

-HgS 9.6 27 0.43 48 1.53HgSe 8.6 24 0.43 43 1.50HgTe 8.5 24 0.41 40 1.37

TABLE C.7

Elastic Moduli of Hexagonal Semiconductor

Crystals at 300K

E

(001)

ν

(001)

Y

(001)

R

B

(001)

6H-SiC 540 0.085 602 0.19GaN

7

320 0.21 430 0.57AlN 340 0.21 480 0.58InN 171 0.27 270 0.87



430


References

1. M.H. Grimsditch and A.K. Ramdas, Brillovin scattering in diamond,

Phys. Rev.B

, 11, 3139 (1975).2. S.P. Nikanorov, Yu. A. Burenkov, and A.V. Stepanov, Elastic properties of Si,

Sov. Phys. Solid State

, 13, 2516 (1972) (English translation).3. K.B. Tolpygo, Optical, elastic, and piczoelectric properties of ionic and valence

crystals with ZnS type lattice,

Sov. Phys. Solid State

, 2, 2367 (1961).4. S. Yu Davydov and A.V. Solomonov, Elastic properties of gallium and alumi-

num nitrides,

Tech. Phys. Lett.

, 25, 601 (1999) (translated from

Pis’ma Zh. Tekh.Fiz.

, 25, 23 (1999)).5. Y.K. Yogurtcu, A.J. Miller, and G.A. Saunders, Pressure dependence of elastic

behaviour and force constants of GaP,

J. Phys. Chem. Solids

, 42, 49 (1981).6. K. Kamitani, M. Grimsditch, J.C. Nipko, C.-K. Loong, M. Okada, and I. Kimura,

The elastic constants of silicon carbide: a Brillouin-scattering study of 4H and6H SiC single crystals,

J. Appl. Phys.

, 82, 3152 (1997).7. A. Polian, M. Grimsditch, and I. Grzegory, Elastic constants of gallium nitride,

J. Appl. Phys.

, 79, 3343 (1996).8. C. Deger, E. Born, H. Angerer, O. Ambacher, M. Stutzmann, J. Hornsteiner, E.

Riha, and G. Fischerauer, Sound velocity of Al

x

Ga

1–x

N thin films obtained bysurface acoustic-wave measurements,

Appl. Phys. Lett.

, 72, 2400 (1998).9. T. Deguchi, D. Ichiryu, K. Toshikawa, K. Segiguchi, T. Sota, R. Matsuo, T.

Azuhata, M. Yamaguchi, T. Yagi, S. Chichibu, and S. Nakamura, Structural andvibrational properties of GaN,

J. Appl. Phys.

, 86, 1860 (1999).10. V. Yu Davydov, Yu. E. Kitaev, I.N. Goncharuk, A.N. Smirnov, J. Graul, O.

Semchinova, D. Uffmann, M.B. Smirnov, A.P. Mirgorodsky, and R.A. Evarestov,Phonon dispersion and Raman scattering in hexagonal GaN and AlN,

Phys.Rev. B

, 58, 12899 (1998).11. V.A. Savastenko and A.U. Shelag, Study of the elastic properties of gallium

nitride,

Phys. Status Solidi A

, 48, K135 (1978).12. L.E. McNeil, M. Grimsditch, and R.H. French, Vibrational spectroscopy of

aluminum nitride,

J. Am. Ceram. Soc.

, 76, 1132 (1993).13. K. Tsubouchi, K. Sugai, and N. Mikoshiba, AlN material constants evaluation

and SAW properties on AlN/Al

2

O

3

and AlN/S,

1981 Ultrasonics SymposiumProceedings

, B.R. McAvoy, Ed., IEEE, New York, 1981, p. 375.14. K. Wang and R.R. Reeber, Thermal expansion and elastic properties of InN,

Appl. Phys. Lett.,

79, 1602 (2001).15. K. Kim, W.R.L. Lambrecht, and B. Segall, Elastic constants and related proper-

ties of tetrahedrally bonded BN, AlN, GaN, and InN,

Phys. Rev. B

, 53, 16310(1996).

16. A.F. Wright, Elastic properties of zinc-blende and wurtizite AlN, GaN, and InN,

J. Appl. Phys

., 82, 2833 (1997).17. A.A. Marmalyuk, R.K. Akchurin, and V.A. Gorbylev, Evaluation of elastic con-

stants of AlN, GaN, and InN,

Inorg. Mater.

, 34, 691 (1998) (translation of

Neorg.Mater.

).18. J.A. Chisholm, D.W. Lewis, and P.D. Bristowe, Classical simulations of the

properties of group-III nitrides,

J. Phys. Cond. Mater.

, 11, L235 (1999).



431

Appendix D

Critical Layer Thickness

References

1. J.W. Matheios, and A.E. Blakeslee, Defects in epitaxial multilayer, I. Misfitdislocations,

J. Cryst. Growth,

27, 118 (1974).2. R. People, and J.C. Bean, Calculation of critical layer thickness versus lattice

mismatch for Ge

x

Si

1–x

/Si strained-layer heterostructures,

Appl. Phys. Lett.,

47,322 (1985);

Appl. Phys. Lett.,

49, 229 (1986).3. J.H. van de Merwe, Crystal interfaces. II. Finite overgrowths,

J. Appl. Phys.,

34,123 (1962).

FIGURE D.1

Critical layer thickness as a function of lattice mismatch. The three curves were calculated usingthe models of Matthews and Blakeslee,

1

People and Bean,

2

and van der Merwe.

3

1

10

100

1000

0.01 0.1 1 10 |f| (%)

h c (n

m)

Matthews and Blakeslee1

People and Bean2

van der Merwe3



433

Appendix E

Crystallographic Etches

Table E.1 gives selected crystallographic etches for some semiconductormaterials.

TABLE E.1

Crystallographic Etches for Semiconductors

Semiconductor Etchant Remarks

Si 1 ml HF3 ml HNO

3

10 ml CH

3

COOH

Dash etch,

1

8 h

1 ml HF1 ml CrO

3

(5

M

in H

2

O)Sirtl etch,

2

for Si (111), 5 min

2 ml HF1 ml K

2

Cr

2

O

7

(0.15

M

in H

2

O)Secco etch,

3

for Si (001) and (111), 5 min

60 ml HF30 ml HNO

3

60 ml CH

3

COOH (glacial)60 ml H

2

O30 ml solution of 1 g CrO

3

in 2 ml H

2

O2 g (CuNO

3

)

2

:3H

2

O

Wright etch,

4

for Si (001) and (111), 5 min, long shelf-life

2 ml HF1 ml HNO

3

2 ml AgNO

3

(0.65

M

in H

2

O)

Silver etch, reveals stacking faults

Ge 50 ml HF100 ml HNO

3

110 ml CH

3

COOH (glacial)330 mg I

2

Reveals dislocations and stacking faults

5

GaAs 1 ml Br

2

100 ml CH

3

OHDistinguishes between (111)Ga and (111)As planes

1 ml HF2 ml H

2

O8 mg AgNO

3

1 g CrO

3

For GaAs (001) and (011)

Continued.



434


References

1. W.C. Dash, Copper precipitation on dislocations in silicon,

J. Appl. Phys.

, 27,1193 (1956).

2. E. Sirtl and A. Adler, Chromsäure-Flussäure als spezifisches System zurÄtzgruben entwicklung auf Silizium,

Z. Met.

, 52, 529 (1961).3. F.S. d’Aragona, Dislocation etch for (100) planes in silicon,

J. Electrochem. Soc.

,119, 948 (1972).

4. M.W. Jenkins, A new preferential etch for defects in silicon crystals,

J. Electro-chem. Soc.

, 124, 757 (1977).5. Q. Li, Y.-B. Jiang, H. Xu, S. Hersee, and S.M. Han, Heteroepitaxy of high-quality

Ge on Si by nanoscale Ge seeds grown through a thin layer of SiO

2

,

Appl. Phys.Lett.

, 85, 1928 (2004).6. M.S. Abrahams and C.J. Buicchi, Etching of dislocations on the low index planes

of GaAs,

J. Appl. Phys.

, 36, 2855 (1965).7. G.H. Olsen and M. Ettenberg, Universal stain/etchant for interfaces in III-V

compounds,

J. Appl. Phys.

, 36, 2855 (1965).8. J.G. Grabmaier and C.B. Watson, Dislocation etch pits in single-crystal gallium

arsenide,

Phys. Status Solidi

, 32, K13 (1967).

A: 40 ml HF 40 ml H

2

O 0.3 g AgNO

3

B: 40 ml H

2

O 40 g CrO

3

A-B etch,

6,7

separate parts store indefinitely, mix in a 1:1 ratio before use

1 g K

3

Fe(CN)

6

in 50 ml H

2

O12 ml NH

4

OH in 36 ml H

2

OMix in a 1:1 ratio before use

KOH Molten KOH,

8,9

250

°

C

InP HBrHF

Pyramidal pits

10

on (001)

HBrCH

3

COOHRectangular pits

10

on (001)

InSb 2 ml HNO

3

1 ml HF1 ml CH

3

COOH (glacial)

Round pits

11

on (001) and (111)

ZnSe 1 ml Br

2

25 ml CH

3

OHFast etch,

12,13

6 sec

GaN 1 ml H

3

PO

4

4 ml H

2

SO

4

230

°

C, 10 min

14

KOH Molten KOH, 250

°

C

TABLE E.1

(Continued)

Crystallographic Etches for Semiconductors

Semiconductor Etchant Remarks



Appendix E: Crystallographic Etches

435

9. J. Angilello, R.M. Potemski, and G.R. Woolhouse, Etch pits and dislocations in{100} GaAs wafer,

J. Appl. Phys.

, 46, 2315 (1975).10. K. Akita, T. Kusunoki, S. Komiya, and T. Kotani, Observation of etch pits in

InP by new etchants,

J. Cryst. Growth

, 46, 783 (1979).11. H.C. Gatos and M.C. Lavine, Dislocation etch pits on the {111} and sur-

faces of InSb,

J. Appl. Phys.

, 31, 743 (1960).12. A. Kamata and H. Mitsuhashi, Characterization of ZnSSe on GaAs by etching

and x-ray diffraction,

J. Cryst. Growth

, 142, 31 (1994).13. X.G. Zhang, A. Rodriguez, X. Wang, P. Li, F.C. Jain, and J.E. Ayers, Complete

removal of threading dislocations from mismatched layers by patterned hete-roepitaxial processing,

Appl. Phys. Lett.

, 77, 2524 (2000).14. Y. Ono, Y. Iyechika, T. Takada, K. Inui, and T. Matsye,

J. Cryst. Growth

, 189, 133(1998).

{ }111



437

Appendix F

Tables for X-Ray Diffraction

InP crystals with (001) and (111) orientations. The values included are themost important for the design and interpretation of x-ray diffraction exper-iments for a number of hkl reflections. They include the Bragg angle ;the inclination of the (hkl) planes with respect to the surface ; the theo-retical rocking curve full-width half maximums (FWHMs) forand incidence, and , respectively; the extinction depth

t

ext

; theabsorption depth

t

abs

; the penetration depth

t

p

; and the magnitude of thestructure factor |

F

h

|.

TABLE F.1

X-Ray Diffraction Data for Si (001) Crystals (a = 5.43108 Å) Using Cu k

α

Radiation

(

λ

= 1.540594 Å)

hkl

θ

B

(deg)

φ

(deg)

W

+

(sec)

W

–

(sec)

t

ext

(

μμμμ

m)

t

abs

(

μμμμ

m)

t

p

(

μμμμ

m) |

F

h

|

113 28.060 25.239 0.8 12.3 15.7 3.2 2.7 46004 34.564 0 3.5 3.5 34.3 19.8 12.6 60224 44.014 35.264 1.2 7.5 26.2 9.2 6.8 54115 47.474 15.793 1.5 2.6 68.3 23.1 17.3 36044 53.352 45.000 1.1 7.2 28.6 8.8 6.8 48135 57.044 32.312 1.3 3.1 71.6 20.6 16.0 33026 63.769 18.435 2.6 3.6 69.9 28.9 20.5 44335 68.443 40.316 1.7 3.4 81.6 22.0 17.3 30353 68.443 62.774 0.9 6.6 33.3 6.1 5.2 30444 79.307 54.736 4.6 7.9 50.1 18.4 13.5 40

θB

φ( )θ φB +

( )θ φB − W+ W−



Table F.1 through Table F.6 contain x-ray diffraction data for Si, GaAs, and

438


TABLE F.2

X-Ray Diffraction Data for Si (111) Crystals (a = 5.43108 Å) Using Cu k

α

Radiation

(

λ

= 1.540594 Å)

hkl

θ

B

(deg)

φ

(deg)

W

+

(sec)

W

–

(sec)

t

ext

(

μμμμ

m)

t

abs

(

μμμμ

m)

t

p

(

μμμμ

m) |

F

h

|

111 14.221 0 6.8 6.8 15.2 8.6 5.5 59224 44.014 19.471 2.0 4.3 41.3 19.8 13.4 54115 47.475 38.942 0.8 5.2 38.4 9.0 7.3 36333 47.475 0 2.0 2.0 73.5 25.8 19.1 36044 53.352 35.264 1.5 5.0 42.0 16.6 11.9 48135 57.044 28.561 1.4 2.9 76.3 22.5 17.4 33026 63.769 43.089 1.8 5.0 48.4 18.0 13.1 44335 68.443 14.420 2.2 2.6 109.5 31.2 24.2 30444 79.307 0 6.0 6.0 90 34.3 24.9 40

TABLE F.3

X-Ray Diffraction Data for GaAs (001) Crystals (a = 5.6534 Å) Using Cu k

α

Radiation (

λ

= 1.540594 Å)

hkl

θ

B

(deg)

φ

(deg)

W

+

(sec)

W

–

(sec)

t

ext

(

μμμμ

m)

t

abs

(

μμμμ

m)

t

p

(

μμμμ

m) |

F

h

|

002 15.814 0 0.5 0.5 200.4 4.0 3.9 6113 26.866 25.239 1.5 40.4 4.9 0.8 0.7 127004 33.026 0 8.7 8.7 13.7 8.0 5.0 163224 41.875 35.264 2.4 20.5 9.5 3.0 2.3 144115 45.072 15.793 3.5 6.3 27.4 9.1 6.9 98044 50.423 45.000 2.0 20.8 9.7 2.5 2.0 130135 53.715 32.312 2.7 7.5 28.1 7.8 6.1 89006 54.838 0 0.3 0.3 530.8 11.9 11.7 6026 59.513 18.435 5.4 8.0 28.0 11.5 8.1 118335 63.313 40.316 3.1 7.7 31.4 8.1 6.5 80117 76.667 11.422 7.7 8.5 52.7 13.9 11.0 75

TABLE F.4

X-Ray Diffraction Data for GaAs (111) Crystals (a = 5.6534 Å) Using Cu k

α

Radiation (

λ

= 1.540594 Å)

hkl

θ

B

(deg)

φ

(deg)

W

+

(sec)

W

–

(sec)

t

ext

(

μμμμ

m)

t

abs

(

μμμμ

m)

t

p

(

μμμμ

m) |

F

h

|

111 13.650 0 16.4 16.4 6.2 3.4 2.2 155224 41.875 19.471 4.6 10.7 16.5 7.8 5.3 144115 45.072 38.942 1.6 14.5 13.7 2.8 2.3 98333 45.072 0 4.8 4.8 29.7 10.3 7.7 98044 50.423 35.264 3.3 12.5 16.7 6.0 4.4 130135 53.715 28.561 3.0 6.9 30.2 8.7 6.7 89335 63.313 14.42 4.3 5.5 43.7 12.4 9.7 80444 70.733 0 8.4 8.4 35.9 13.8 10.0 108



Appendix F: Tables for X-Ray Diffraction

439

TABLE F.5

X-Ray Diffraction Data for InP (001) Crystals (a = 5.8690 Å) Using Cu k

α

Radiation

(

λ

= 1.540594 Å)

hkl

θ

B

(deg)

φ

(deg)

W

+

(sec)

W

–

(sec)

t

ext

(

μμμμ

m)

t

abs

(

μμμμ

m)

t

p

(

μμμμ

m) |

F

h

|

002 15.218 0 10.1 10.1 10.4 1.4 1.2 118113 25.804 25.239 0.9 70.5 2.8 0.1 0.1 143004 31.668 0 8.2 8.2 14.5 2.8 2.3 168224 40.015 35.264 2.0 22.8 8.7 0.8 0.7 151115 42.999 15.793 3.9 7.2 24.9 3.2 2.8 121044 47.941 45.000 1.4 26.8 7.6 0.5 0.5 139135 50.939 32.312 2.6 8.1 24.4 2.6 2.3 104006 51.952 0 3.2 3.2 53.3 4.2 3.9 70026 56.108 18.435 4.9 7.7 27.5 4.0 3.5 130335 59.390 40.316 3.0 9.1 26.2 2.6 2.4 106444 65.411 54.736 3.3 15.2 15.2 1.6 1.5 123117 69.603 11.422 5.6 6.6 44.8 4.9 4.4 92

TABLE F.6

X-Ray Diffraction Data for InP (111) Crystals (a = 5.8690 Å) Using Cu k

α

Radiation

(

λ

= 1.540594 Å)

hkl

θ

B

(deg)

φ

(deg)

W

+

(sec)

W

–

(sec)

t

ext

(

μμμμ

m)

t

abs

(

μμμμ

m)

t

p

(

μμμμ

m) |

F

h

|

111 13.140 0 18.0 18.0 5.9 1.2 1.0 184222 27.043 0 5.3 5.3 21.6 2.4 2.2 98224 40.015 19.471 4.3 10.4 16.9 2.7 2.3 151333 42.999 0.0 4.9 4.9 27.2 3.6 3.2 121044 47.941 35.264 2.9 12.9 15.6 1.6 1.7 139135 50.939 28.561 2.9 7.4 26.5 2.9 2.6 104444 65.411 0 7.0 7.0 34.6 4.8 4.2 123



Heteroepitaxy of Semiconductor

Documents