Non Linear Piezoelectricity in Wurtzite Semiconductor Core ...

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Non Linear Piezoelectricity in Wurtzite Semiconductor

Core-Shell Nanowires: An Atomistic Modelling

Approach

A thesis submitted to the University of Manchester for the degree

of

Doctor of Philosophy (PhD)

In the Faculty of Engineering and Physical Science

Hanan Yahya Saeed Al-Zahrani

2016

School of Electrical and Electronic Engineering

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Table of Contents

1 Introduction ................................................................................................... 16

1.1 A Diverse and Multifunctional future to going beyond Moore’s Law ..... 16

1.2 Basic Physics Underlying Piezotronics ....................................................... 17

1.3 Emergence of Piezotronics ........................................................................... 20

1.4 One-Dimensional Wurtzite Nanostructures for Piezotronics ................... 22

1.5 Piezoelectric Semiconductors ....................................................................... 25

1.6 Piezoelectric Effect ........................................................................................ 27

1.6.1 Piezoelectric Polarization 28

1.6.2 Spontaneous Polarization 32

1.7 A Microscopic Approach .............................................................................. 34

2 Density Functional Theory ........................................................................... 37

2.1 Hohenberg-Kohn Theorem .......................................................................... 37

2.1.1 Exchange-Correlation 39

2.1.2 Local Density Approximation 40

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2.1.3 Pseudopotentials 42

2.1.4 Plane Wave Basis Sets and Bloch's Theorem 45

2.2 Stress and Strain ........................................................................................... 47

2.2.1 Elastic strain 48

2.2.2 Macroscopic strain and stress 51

2.3 CASTEP ......................................................................................................... 53

3 Previous Works ............................................................................................. 54

3.1 ZB III-V Semiconductors ............................................................................. 54

3.1.1 Introduction 54

3.1.2 Piezoelectric Quantum Well 55

3.1.3 Piezo coefficients with Harrison’s Model 56

3.1.4 Bond Polarity, Atomic Effective Charge 60

3.1.5 Compositional Disorder Effect 61

3.1.6 Evaluation of piezo coefficient 62

3.1.7 Review and Important Discussion 63

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3.2 Pseudomorphic growth of zinc blende semiconductors previous Work

Bond Polarity and Kleinman ........................................................................................ 65

3.3 Piezoelectric coefficients ............................................................................... 69

3.4 Non-linear piezoelectric coefficients in ZB materials ................................ 70

4 WZ Semiconductors ...................................................................................... 72

4.1 Wurtzite III-Nitride Semiconductors .......................................................... 72

4.1.1 DFT Calculations 72

4.1.2 Linear Piezoelectric coefficients 73

4.1.3 Spontaneous Polarization 75

4.1.4 Strain dependence of the polarization 76

4.1.5 Second Order Piezoelectric coefficients 79

5 Current Work on Wurtzite III-Vs Semiconductors .................................. 80

5.1 Evaluation of Linear and Non Linear Piezoelectric Coefficients ............. 81

5.2 Internal Displacement (u) and Effective Charge (Z*) ............................... 83

5.3 Spontaneous Polarization ............................................................................. 91

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5.4 Strain dependence of the polarization ........................................................ 92

5.5 ZnO Semiconductors .................................................................................... 96

5.6 Methodology .................................................................................................. 97

5.6.1 Piezoelectric Coefficients 98

6 Nanowires and III-V Core-Shell Nanowires ............................................ 102

6.1 Quantum Nanowire properties .................................................................. 102

6.1.1 Synthetic techniques of ZnO NWs 104

6.1.2 ZnO NWS applications 105

6.1.3 The Example of NWs 106

6.2 Core Shell Nanowires .................................................................................. 110

6.2.1 Core Shell Advantages 112

6.2.2 Modelling AFM Tip Lateral Deflection 112

6.2.3 Homogeneous and Core Shell Nanowires 116

7 Conclusion .................................................................................................... 125

8 References .................................................................................................... 129

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Total Words count, including foot notes and end notes: 29,723

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Table of Figures

Figure 1.1-1 A view of development drives in electronics after Moore’s law. The graph shows size

reduction, higher density of devices, memory and speed of CPU against range of functions and

variety in mobile electronic devices, showing the future trend to be towards integrating rapid CPU

with function. Piezotronics to create mechanical activity within electronics is predicted to be

essential in creating CMOS and human interface devices. Reproduced from Reference [Z.L.

Wang, Springer Berlin Heidelberg, (2012)1-17]. ....................................................................... 17

Figure 1.2-1 Piezopotential in wurtzite crystal. (a) Atomic model for wurtzite-structured ZnO.(b)

ZnO nanowire arrays aligned in a solution-based technique. Numerical calculation of

piezopotential distribution along ZnO nanowire subject to axial strain, with the nanowire growing

along the c axis. The nanowire is 600 nm long while a = 25 nm, with an external force of fy = 80

nN. Reproduced from Reference [Z.L. Wang, Springer Berlin Heidelberg, (2012)1-17]. ......... 20

Figure 1.3-1 Piezopotential generated within a nanostructure, shown through colour coding, as the

basis in physics of nanogenerators and piezotronics. (a) Nanogenerator relying on electron flow

through external loading stimulated by piezopotential. (b) Piezotronics concerns fabrication of

devices through transportation of charge carriers as controlled via piezopotential and occurring at

the p-n junction or interface of semiconductor and metal. Piezo-phototronics concerns fabrication

of devices with piezopotential controlling production of charge carrier and the process of

separating, transporting and recombining at the junction or interface. Reproduced from Reference

[Z.L. Wang, Springer Berlin Heidelberg, (2012)1-17]. .............................................................. 22

Figure 1.4-1 ZnO nanowire arrays produced through: (a) pulse laser deposit approach; and

(b)solution-based technique at lower temperature. Reproduced from Reference [Z.L. Wang,

Springer Berlin Heidelberg, (2012)1-17]. ................................................................................... 24

Figure 1.7-1 Crystal Structures of Wurtzite in its original and strained case ............................. 34

Figure 3.1.3-1 Dependence of the Kleinman internal displacement parameter of zinc blende InAs

circles and GaAs (squares) on shear strain (hollow symbols) and hydrostatic strain filled symbols.

Notice the opposite strain dependence of these two. Reproduced from Reference [M. A.

Migliorato, D. Powell, A. G. Cullis, T. Hammerschmidt and G. P. Srivastava, Phys. Rev. B 74

(2006) 245332]. ........................................................................................................................... 59

Figure 3.2-1Bond Polarity plots of GaAs and InAs. Dependence of the bond polarity on the

applied strain for GaAs and InAs. For each value of the perpendicular strain, each point

corresponds to a different value of the parallel strain that ranges from -0.01 to +0.1, (top to

bottom) as indicated by the arrows. Reproduced from Reference [R. Garg, A. Hüe, V. Haxha, M.

A. Migliorato, T. Hammerschmidt, and G. P. Srivastava, Appl. Phys. Lett. 95 (2009) 041912].67

Figure 3.2-2 Kleinman parameter plots of GaAs and InAs.Reproduced from Reference [R. Garg,

A. Hüe, V. Haxha, M. A. Migliorato, T. Hammerschmidt, and G. P. Srivastava, Appl. Phys. Lett.

95 (2009) 041912]. ...................................................................................................................... 69

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Figure 3.3-1Piezoelectric coefficients plot for GaAs and InAs. Reproduced from reference [R.

Garg, A. Hüe, V. Haxha, M. A. Migliorato, T. Hammerschmidt, and G. P. Srivastava, Appl. Phys.

Lett. 95 (2009) 041912]. ............................................................................................................. 70

Figure 4.1.4-1 Comparison of the total polarization as a function of perpendicular and parallel

strain calculated in III-N work (circles) and that calculated using the linear model with parameters

from Ref [30] (dashed lines). The perpendicular strain varies from -0.1 to 0.1 in steps of 0.02.

Reproduced from the work of Reference [J. Pal, G. Tse, V. Haxha, M. A. Migliorato and S.

Tomić, Journal of Physics: Conference Series 367,012006 (2012)]. .......................................... 77

Figure 5.2-1 Strain dependence of internal displacement parameter (u) as a function of in-plane

and perpendicular strain (from -0.08 to 0.08) for GaAs. ............................................................. 84


and perpendicular strain (from -0.08 to 0.08) for InAs. .............................................................. 85


and perpendicular strain (from -0.08 to 0.08) for GaP. ............................................................... 86


and perpendicular strain (from -0.08 to 0.08) for InP. ................................................................ 87

Figure 5.2-5 Z* as a function of plane strain (from -0.08 to 0.08) along with perpendicular strain,

for GaAs. ..................................................................................................................................... 88


for InAs. ...................................................................................................................................... 89


for InP. ......................................................................................................................................... 90


for GaP. ....................................................................................................................................... 91

Figure 5.4-1 Dependence of the total polarization (C/m2) of wurtzite GaP on combination of strain

in the range -0.1 to + 0.1 according to the classic linear model (LM) and our non-linear (quadratic)

model (NLM). ............................................................................................................................. 93

Figure 5.4-2 Dependence of the total polarization (C/m2) of wurtzite InP on combination of strain


model (NLM). ............................................................................................................................. 94

Figure 5.4-3 Dependence of the total polarization (C/m2) of wurtzite GaAs on combination of

strain in the range -0.1 to + 0.1 according to the classic linear model (LM) and our non-linear

(quadratic) model (NLM). ........................................................................................................... 95

Figure 5.4-4Dependence of the total polarization (C/m2) of wurtzite InAs on combination of strain


model (NLM). ............................................................................................................................. 96

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Figure 5.4-4Dependence of the total polarization (C/m2) of wurtzite InAs on combination of strain


model (NLM). ............................................................................................................................. 96

Figure 5.6-1 Dependence of the total polarization (C/m2) on strain in the range -0.08 to + 0.08

according to the classic linear model (LM) and our non-linear (quadratic) model (NLM). The red

square and blue dot resemble the NLM and LM prediction at -2% In-plane strain. Reproduced

from Reference [H.Y.S. Al-Zahrani, J. Pal and M.A. Migliorato, Nano Energy 2 (2013) 1214.].101

Figure 6.1.3-1 Variation of the polarization (C/m2) in a cross section of a ZnO nanowire. The

perpendicular (parallel) strain varies from -2.8% (+2.8%) to +2.8% (-2.8%). The calculated

polarization of the non-linear (quadratic) model (NLM) is on the left half and the classic linear

model (LM) on the right. Reproduced from the work of Reference [M.A. Migliorato, J. Pal, R.

Garg, G. Tse, H. Y. S. Al-Zahrani, U. Monteverde, S. Tomić, C-K. Li, Y-R. Wu, B. G. Crutchley,

I. P. Marko and S. J. Sweeney, AIP Conf. Proc. 1590 (2014) 32]. ........................................... 107

Figure 6.1.3-2 Variation of the polarization (C/m2) in a cross section of a GaN nanowire. The

perpendicular (parallel) strain varies from -4% (+4%) to +4% (-4%). The calculated polarization

using NLE parameters of the non-linear (quadratic) model (NLM) is on the left half and the classic

linear model (LM) on the right. Reproduced from the work of Reference [M.A. Migliorato, J. Pal,

R. Garg, G. Tse, H. Y. S. Al-Zahrani, U. Monteverde, S. Tomić, C-K. Li, Y-R. Wu, B. G.

Crutchley, I. P. Marko and S. J. Sweeney, AIP Conf. Proc. 1590 (2014) 32]. ......................... 108

Figure 6.1.3-3 Variation of the polarization (C/m2) in a cross section of a InN nanowire. The

perpendicular (parallel) strain varies from -4% (+4%) to +4% (-4%). The calculated polarization

using NLE parameters of the non-linear (quadratic) model (NLM) is on the left half and the the

classic linear model (LM) on the right. Reproduced from the work of Reference [M.A. Migliorato,

J. Pal, R. Garg, G. Tse, H. Y. S. Al-Zahrani, U. Monteverde, S. Tomić, C-K. Li, Y-R. Wu, B. G.

Crutchley, I. P. Marko and S. J. Sweeney, AIP Conf. Proc. 1590 (2014) 32]. ......................... 109

Figure 6.2-1 Schematic of nanowire and nanowire heterostructure growth.(a) Nanowire synthesis

through catalyst-mediated axial growth. (b,c) Switching of the source material results in nanowire

axial heterostructures and superlattices. (d,e) Conformal deposition of different materials leads to

the formation of core/shell and core/ multishell radial nanowire heterostructures reproduced from

Reference[O. Hayden, R. Agarwal and W. Lu, Nanotechnology 3 (2008) 12]. ....................... 111

Figure 6.2.2-1 AFM Tip Lateral Deflection of nanowire where (R), is the radius of curvature, (H)

is the length of the NW, (D) is the diameter of the NW and the deflection caused by the AFM tip is

(d). Ɵ is the angle that subtends the arch formed by the deformed NW, (H +) and (H-) lengths of

the NW on the tensile and compressed. Reproduced from Reference [H.Y.S. Al-Zahrani, J. Pal, M.

Migliorato, G. Tse, and D. Yu, Nano Energy, 14 (2015) 382-391]. ......................................... 114

Figure 6.2.3-1Comparison of the Total Polarization in Homogeneous and Core-Shell Nanowires

(CSNWs) when deflected by AFM tip. The first row resembles the homogeneous III-As and III-P

nanowires having 1µm length and 0.5µm diameter in dimensions with an AFM tip deflection

range of 0-360nm. While the second, third and fourth row are the different combinations CSNWs.

Typical CSNW dimensions are of 1µm length and core/shell diameter of 0.25µm/0.5µm with a

360nm deflection. Reproduced from Reference [H.Y.S. Al-Zahrani, J. Pal, M. Migliorato, G. Tse,

and D. Yu, Nano Energy, 14 (2015) 382-391]. ......................................................................... 117

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List of Journal Publications

H.Y.S. Al-Zahrani, J. Pal, M. Migliorato, G. Tse, and D. Yu, Nano Energy, “Piezoelectric

Field Enhancement in III-V Core-Shell Nanowires“ 14 (2015) 382-391

H.Y.S. Al-Zahrani, J. Pal and M. A. Migliorato, Nano Energy, “Non Linear Piezoelectricity

in Wurtzite ZnO Semiconductors “ 2 (2013) 1214.

List of Conference Publications

M.A. Migliorato, J. Pal, R. Garg, G. Tse, H.Y.S. Al-Zahrani, C-K. Li and Y R. Wu, B. G.

Crutchley, I. P. Marko and S. J. Sweeney, S. Tomić, “A Review of Non Linear Piezoelectricity in

Semiconductors” AIP Conf Proc. 2014.

List of Presentations

H. Y.S. Al-Zahrani “Non linear Piezoelectricity in III-V Wurtzite Nanostructures and

Devices”, at Materials Research Society (MRS) Spring Meeting and Exhibit 2014, San Francisco,

USA.

H. Y.S. Al-Zahrani, “Non linear piezoelectric effects in polar semiconductors and Wurtzite

III-N semiconductors", Theory, Modelling and Computational Methods for Semiconductors

TMCS IV, Manchester, UK, 2014.

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List of Abbreviations

PL Photoluminescence

DFT Density Functional Theory

LDA Local Density Approximation

DFPT Density Functional Perturbation Theory

LED Light Emitting Diode

HFET Heterostructure Field-Effect Transistor

DEG Dimensional Electron Gas

GGA Generalized Gradient Approximation

PZC Piezoelectric Coefficients

LM Linear Model

NLM Non-linear Model

NLE Non-linear Elasticity

WZ Wurtzite

ZB Zinc Blende

QW Quantum Well

QD Quantum Dot

NW Nanowires

CS Core-Shell

NW Nanowire

AFM Atomic Force Microscope

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Abstract

Piezotronics is a new field, as first explored by Professor Zhong Lin Wang (Georgia Institute of

Technology, Atlanta, USA), which describes the exploitation of the piezoelectric polarization and

internal electric field inside semiconductor nanostructures by applying strain, to develop electronic

devices with new functionality. Such concepts find applications in both III-V and II-VI

semiconductor compounds, in optics, optoelectronics, catalysis, and piezoelectricity, sensors,

piezoelectric transducers, transparent conductor and nanogenerators. In this work I explore the

strain dependence of the piezoelectric effect in wurtzite ZnO crystals. The Linear and quadratic

piezoelectric coefficients of III-V (GaP, InP, GaAs and InAs) wurtzite semiconductors are also

calculated using ab-initio density functional theory.

The polarization in terms of the internal anion–cation displacement, the ionic and dipole charges is

written and the ab initio Density Functional Theory is used to evaluate the dependence of all

quantities on the strain tensor. The piezoelectric effect of III–V semiconductors are nonlinear in

the strain tensor. The quadratic piezoelectric coefficients and a revised value of the spontaneous

polarization are reported. Furthermore, the ZnO nanowires is found to be non-linear piezoelectric

effect and leads to predictions in some cases opposite to those obtained using the widely used

linear model.

The predicted magnitude of such coefficients are much larger than previously reported and of the

same order of magnitude as those of III-N semiconductors. We also model the bending distortion

created on a III-V wurtzite nanowire by an atomic force microscope tip induced deflection to

calculate the piezoelectric properties of both homogenous and core shell structures. A number of

combinations of III-V materials for the core and the shell of the nanowires, are shown a favour

much increased voltage generation.

The largest core voltages in core/shell combinations of InAs/GaP, InP/GaP, GaP/ InAs and

GaP/InP are observed which can be theoretically 3 orders of magnitude larger than the typical

values of ±3V in homogenous nanowires. Also considering properties such as bandgap

discontinuity and mobility, III-V wurtzite core shell nanowires are candidates for high

performance components in piezotronics and nanogeneration.

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Declaration

The author of this thesis declares that no portion of the work referred to in the thesis has been

submitted in support of an application for another degree or qualification of this or any other

university or other institute of learning.

Hanan Yahya Saeed Al-Zahrani

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Copyright Statements

The author of this thesis (including any appendices and/or schedules to this thesis) owns

certain copyright or related rights in it (the “Copyright”) and s/he has given The University of

Manchester certain rights to use such Copyright, including for administrative purposes.

Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may

be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and

regulations issued under it or, where appropriate, in accordance with licensing agreements which

the University has from time to time. This page must form part of any such copies made.

The ownership of certain Copyright, patents, designs, trademarks and other intellectual

property (the “Intellectual Property”) and any reproductions of copyright works in the thesis, for

example graphs and tables (“Reproductions”), which may be described in this thesis, may not be

owned by the author and may be owned by third parties. Such Intellectual Property and

Reproductions cannot and must not be made available for use without the prior written permission

of the owner(s) of the relevant Intellectual Property and/or Reproductions.

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Acknowledgements

First and foremost, I would like to extend my sincere gratitude to my supervisor Dr. Max

Migliorato for his excellent supervision and invaluable guidance and help throughout the lifetime

of the project. His wide knowledge and logical way of thinking have been of great value for me.

I am deeply indebted to Dr. Joydeep Pal for his advice and support throughout the research work.

I would also like to acknowledge the financial support of the Saudi Arabia Ministry of higher

education and King Abdul-Aziz University in Jeddah for making this work possible.

Finally, I owe sincere and earnest thankfulness to my Mother and Father, Yahya and Alwa who

have always encouraged and supported me all throughout my life. A special thanks to My Sisters

and Brothers, Sharifa, Ali, Muteb, Layla, Areej, Fahad and Saeed for their love and affection.

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1 Introduction

1.1 A Diverse and Multifunctional future to going beyond

Moore’s Law

Over the past 40 years, Moore’s law has held broadly true in considering technological

advances in IT. Thus, the number of items which can be held on one computer chip has

increased twofold every 1.5 years with faster CPUs and the capacity for systems to be based

on one chip being principal areas for development. As the width of lithographically defined

lines approaches 10 nm, the issue of how far miniaturisation of devices can be continued is a

key issue of current computing technology, in addition to the need to explore the

consequences of this miniaturisation for stable and reliable technologies. It is unclear whether

other factors in addition to speed will become important in these developments, in light of the

fact that it is accepted that Moore’s law has a definite limitation of applicability, i.e. the

diffraction limit of UV light, there is a need to explore which factors or drives may replace it.

The field of individual healthcare will likely be a key driver of the IT sector for the near

future, as will that of sensor networks. The current trend is for individual devices, movable

devices and flexible equipment based on polymers. Variety and the inclusion of multiple

functions within a device are key considerations. For instance, although superfast computing

power is not necessarily a major future trend in mobile phones, the availability of new

functions will drive developments, which might include sensor devices to monitor users’

temperature, blood pressure or blood sugar, as well as tracking environmental conditions and

warning of the presence of harmful substances and gas, or measuring UV sunlight. If this

prediction is correct, IT development will therefore take a new direction, with function

joining speed as a prime concern (see Figure 1.1-1). So, individualised mobile devices

containing organic or flexible polymer-based electronics which integrate sensor devices and

are capable of powering themselves. The aim here is to create an interface with both humans

and environmental elements. The mixing of rapid CPU function, high memory capacity, logic

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and multiple functions suggests that a self-powering, smart system will form the goal of

future developments in the field[1].

Figure 1.1-1 A view of development drives in electronics after Moore’s law. The graph shows

size reduction, higher density of devices, memory and speed of CPU against range of

functions and variety in mobile electronic devices, showing the future trend to be towards

integrating rapid CPU with function. Piezotronics to create mechanical activity within

electronics is predicted to be essential in creating CMOS and human interface devices.

Reproduced from Reference [Z.L. Wang, Springer Berlin Heidelberg, (2012)1-17].

1.2 Basic Physics Underlying Piezotronics

Piezoelectricity has been identified for over 100 years and concerns the potential to produce

an electric charge when a material is subjected to alterations in pressure. This phenomenon is

best known in perovskite structured Pb(Zr, Ti)O3 (PZT), and has led to a wide range of uses

for electromechanical sensory devices, in energy generation and in actuators. Despite this

PZT does not lend itself to use in electronics due to its electrical insulation properties. The

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area of Piezoelectronics is studied as an individual academic area and in particular has

attracted strong interest from the ceramic field. Piezoelectric effects are observed for

structures of Wurtzite, including ZnO, GaN, InN, and ZnS, but have not found such extensive

use for actuators and sensor equipment as has PZT, because of their comparatively lower

piezoelectric coefficients. Thus wurtzite structures draw interest mainly from the fields of

photonics and electronics. Control of CMOS technologies based on silicon occurs through

electrical transportation and in order to allow mechanical control, that control must therefore

produce electrical charge, for which piezoelectric charge is a suitable candidate. The need for

Piezoelectricity for this purpose suggest Wurtzite structures rather than PZT, as ZnO, GaN,

InN, and ZnS for example possess both piezoelectric characteristics and act as

semiconductors. Within this, ZnO’s crystalline structure is symmetrical but not central, and

this structure leads to piezoelectricity being generated when under strain. Crystal Wurtzite is

structured hexagonally and displays significant anisotropic features along and at right angles

to the c axis. Therefore tetrahedral co-ordination between O2− anions and Zn2+ cations is

present and the central part of positive and negative ions are overlapping. As a consequence,

mechanical stress placed on the tetrahedron’s top causes the centres of both anions and

cations to displace relative to each other, and create a dipole moment (see Figure 1.2-1(a)).

The dipole moments from each unit within a crystal are constructively added to create a drop

in potential at macroscopic level in the crystal’s direction of strain, known as piezopotential

(see Figure 1.2-1 (b)) [2]. This can drive electron flow in external load after mechanical

deformation has taken place, and creates a basis for nanogeneration [3,4,5,6]. Calculation of

piezopotential distribution on a nanowire (NW) of c-axis ZnO can be made through the

Lippman theory which described the bent piezoelectric NW in the case of extremely low

donor concentration that can be neglected and therefore no reference to ZnO doping [7,8,9].

A 1200 nm long NW with 100nm long hexagonal sides, an 85 nN tensile force generates

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around 0.4 V of drop potential between each end, and the +c side is positive (seeFigure 1.2-1

(b)). Reversal of piezopotential occurs as compressive strain comes to be applied, and while

the potential difference is still 0.4 V, −c-axis has greater potential, creating the central

element of piezotronics. Newly-emerging research areas have sprung from exploration of

crystal piezopotential and the development of a nanogeneration device to transform energy

from mechanical to electric form has taken place [10,11,12,13]. On connecting piezoelectric

crystal with strain to an electricity source externally, via the crystal’s end poles,

piezopotential means that Fermi levels fall in the regions of contact, driving free electron

flow from end to end seeking a state of equilibrium. This electron flow creates a current

within the load. Frequently changing applied mechanical stress can create an alternating

electron flow as the piezopotential continually alters, thus providing continuing energy output

in the nanogenerator (see Figure 1.3-1(a)). Development of this nanogeneration device has

led to ∼3 V outputs, with sufficient energy to power mobile phone sized LCD displays, as

well as light-emitting and laser diodes [14,15,16,17]. Thus, nanogeneration has the potential

to be a key element for harvesting of energy to provide nano or micro systems with power

sustainability and self-sufficiency.

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Figure 1.2-1 Piezopotential in wurtzite crystal. (a) Atomic model for wurtzite-structured

ZnO.(b) ZnO nanowire arrays aligned in a solution-based technique. Numerical calculation

of piezopotential distribution along ZnO nanowire subject to axial strain, with the nanowire

growing along the c axis. The nanowire is 600 nm long while a = 25 nm, with an external

force of fy = 80 nN. Reproduced from Reference [Z.L. Wang, Springer Berlin Heidelberg,

(2012)1-17].

1.3 Emergence of Piezotronics

Wang and colleagues were the first to contribute to the field of piezotronics, beginning with

two separate experiments conducted in 2006 [18]. Firstly, measurement was made of the

transport of electricity in a high-length ∼100 μm ZnO wire with each end entirely enclosed

with electrodes, while bending the wire within a scanning electron microscope [18]. As the

wire was bent to a greater extent, conductance of electricity dropped sharply. This was

considered to mean that bending the wire produced a piezoelectric potential capable of

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creating a sufficiently large (gate) voltage with which to control charge carrier transportation

along the ZnO wire, termed the piezoelectric field effect transistor or PE-FET.

Other experimental work involved manipulating a ZnO nanowire held by two probes and

measuring transportation characteristics of the wire [19]. The first probe secured the NW by

one of its ends, while lying upon a substrate insulator, while the second was used at the

opposite end to push the nanowire via contact with its tensile surface. Ohmic contact was

made with the nanowire through tips made of tungsten. As the nanowire was bent further, an

alteration in I–V curve was seen, which was first linear and then altered to rectifying

behaviour. The conclusion drawn from this finding was that this had occurred due to the

creation of positive piezopotential at the metal semiconductor interface, creating a potential

obstacle to electron flow in a single direction, termed a piezoelectric-diode, or PE-diode.

Thus, PE-diode and PE-FET are both based on a nanowire having piezoelectric potential

induced through strain. The electron flow which is induced for external circuits through

piezoelectric potential represents an opportunity for generating energy. Piezoelectric potential

may have dramatic effects on transportation properties for an FET based on a nanowire.

Systematic representation was needed for piezoelectric semiconductors and their

characteristics, and thus Wang created the term nano-piezotronics for the new field, revealing

this in November 2006 [20]. Piezotronics was also first used by Wang, in a 2007 published

paper [21,22]. Piezotronics is based on using piezopotential in directing and managing

transport of carriers within the NW (see Figure 1.3-1(b)). From that time onwards, significant

advances have been achieved in the field, as discussed in later chapters of this thesis.

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Figure 1.3-1 Piezopotential generated within a nanostructure, shown through colour coding,

as the basis in physics of nanogenerators and piezotronics. (a) Nanogenerator relying on

electron flow through external loading stimulated by piezopotential. (b) Piezotronics

concerns fabrication of devices through transportation of charge carriers as controlled via

piezopotential and occurring at the p-n junction or interface of semiconductor and metal.

Piezo-phototronics concerns fabrication of devices with piezopotential controlling

production of charge carrier and the process of separating, transporting and recombining at

the junction or interface. Reproduced from Reference [Z.L. Wang, Springer Berlin

Heidelberg, (2012)1-17].

1.4 One-Dimensional Wurtzite Nanostructures for Piezotronics

Nanowires and thin film applications are governed by the piezotronics and piezo-phototronics

principles. However, nanowires have large advantages over thin film, due to a range of

factors. Firstly, nanowires based on ZnO may be chemically grown on substrates of any

material and shape at temperatures of lower than 100 ◦C, allowing production to be cheaply

23

scaled up, while there are practical challenges in fabricating good quality, single crystal thick

films at lower temperatures. Further, nanowires are strongly elastic because of their small

scale, and this means that they can be mechanically deformed without developing cracks or

fracturing, by as much as 6 % under tensile strain, based on theoretical extrapolation for

wires of extremely miniaturized scale [23]. At the same or less strain, thin film frequently

cracks. Thirdly, the fact that nanowires are so small allows a tougher and more robust

structure which is practically impervious to fatigue. In addition, NWs can be agitated using a

comparatively low force, which is highly useful for applications which require extremely

high levels of sensitivity. Fifth, it is possible that nanowires show greater piezoelectric

coefficients in comparison to thin films [24].

Nanowires as well as nanobelts, being structures in one dimension, are highly suitable for use

in piezo-photronic and piezotronic fields due to their high tolerance of mechanically induced

strains. Possible materials for piezotronics include InN, ZnO and GaN, as well potentially as

doped PZT. At present, ZnO nanowires are most commonly used. This is firstly because such

NWs are capable of being grown on a large-scale basis at low temperatures via chemical

techniques or by means of vapour-solid techniques. Second, ZnO nanowires are not harmful

to the environment and can be used alongside organic components. Their third benefit is that

it is possible to grow such NWs on substrates of any shape and material. Nanowires grown by

the vapour-solid technique are generally created through vapourisation of powdered ZnO

within a tube furnace alongside carbon, and this is carried out at ∼900 ◦C. Use of Au

catalysts enable growth to be patterned. NWs have also been grown via pulse laser deposition

(PLD). Ablative force was provided by a 248 nm KrF excimer laser and aimed at a ceramic

object with stacked powdered ZnO. Pressure management was able to provide good quality

nanowire arrays (see Figure 1.4-1(a)).

24

Figure 1.4-1 ZnO nanowire arrays produced through: (a) pulse laser deposit approach; and

(b)solution-based technique at lower temperature. Reproduced from Reference [Z.L. Wang,

Springer Berlin Heidelberg, (2012)1-17].

Hexamethylenetetramine and zinc nitrate hexahydrate are frequently utilised in

hydrothermally synthesising nanowires made of ZnO [25,26]. Zinc nitrate hexahydrate salt

offers the Zn2+ ions which are needed in order to construct such NWs, while molecules of

water within the solution offer O2− ions. While it is not yet apparent precisely how

hexamethylenetetramine contributes to growth of ZnO nanowires, it may provide a weak base

to gradually hydrolyze in a water solution, slowly generating OH−, and this slow pace has a

vital role, as otherwise, rapid precipitation of Zn2+ ions in the solution will occur under the

higher pH conditions. Based on patterns created by interference from a laser, it has been

possible to achieve good alignment for nanowire arrays at temperatures of approximately

85◦C (see Figure 1.4-1(b)). However, vapour phase approaches using higher temperatures

result in nanowires which have minimal flaws and are highly suitable for investigation of the

25

piezophotronic and piezotronic effects [27,28]. Oxygen plasma treatment is successful in

minimizing vacancy concentration. Meanwhile, although nanowires produced through lower

temperature chemical methods show comparatively higher defect rates, they are nevertheless

highly suitable for nanogenerators using piezoelectric effects.

This area of study has expanded dramatically in recent years, and this is especially the case

with regard to developing a range of techniques to utilise piezotronics to harvest energy and

in nanogeneration, leading also to commercially driven interest in development of sensors

which are self-powering and wireless. Advances in nanogenerators and piezotronics formed

the principle topic for the first international Xiangshan Science Conference in Beijing in

December 2012, followed in April 2013 by a symposium on piezotronics within the MRS

2013 Spring Meeting and Exhibit in San Francisco. A full conference called NGPT

(Nanogenerators and Piezotronics) was held in Atlanta (USA) in 2014, and will be followed

by a second one in Rome in 2016. These events are examples of the level of global interest in

both piezophototronics and piezotronics, signalling the drive to significantly improve existing

knowledge concerning the materials used and the characteristics which the field exploits, and

a discussion of this is given in the sections which follow [1].

1.5 Piezoelectric Semiconductors

Piezoelectric phenomena have been observed in quartz, cane sugar, sodium potassium tartrate

tetrahydrate, topaz and tourmaline. In addition, these effects have been shown to occur in

hexagonal (ZnO and GaN) and cubic (including InGaAs) classes of crystal. The piezoelectric

effect has been applied in technologies ranging from medical imagers, sonar ultrasound

transducers and actuators to energy harvesting devices, fuel injection systems and micro-

positioners.

26

The piezoelectric effect resembles electric dipole moment creation as seen for insulating

materials. Physical stress alters the crystal’s polarization with the breaking of charge

symmetry and creates an asymmetrical density in charge, thus generating the piezoelectric

field. While direct measurement of this field is challenging, it can be evaluated based on its

effects upon the crystal’s optical and electric characteristics.

In recent years, piezoelectric effects for semiconductors produced through epitaxial growth

have drawn considerable research attention. The potential to exploit piezoelectricity for

diverse functions and devices is clear from work in piezotronics, as discussed earlier in the

chapter. Further, Chapter 3 contains an in depth investigation of piezo effects for III-V

materials and particularly for III-Nitrides, as well as for II-VI materials.

Alterations in piezoelectric and spontaneous polarization enhance the electrostatic charge

density induced through polarisation. For bulk semiconductors which have structural purity,

polarisation field difference can be seen at surface level. However, mobile carriers or surface

states may neutralise the charges which are created at the surface. On the other hand,

polarisation in heterostructures and the charge which it induces markedly impacts upon the

electric field internally, as well as on charge distribution. Behaviour analogous to donors or

acceptors are induced variously by inducing charge polarisation which is positive or negative.

Charge induced through polarisation cannot be differentiated in electrostatic terms from

charge density, because of ionized dopants. The unit cells each have a low-level dipole factor

which means that volume charge density induced through polarisation is evenly distributed

throughout a layer of alloy.

While polarisation relies upon the properties of the semiconductor for WZ and ZB phases,

examples are given for a range of crystalline structures for II-VI, III-V and III-N

semiconductors. For pseudomorphically grown ZB InAs on GaAs (001), Beya-Wakata et

27

al.,[29] predict +0.069C/m2 PZ polarisation. In a WZ semiconductor, a fine layer of GaN

grown in the [0001] orientation on A1N creates 3% compressive strain and a prediction of

+0.095C/m2 polarisation is made. Meanwhile, for ZnO, predictions of +0.01C/m2 polarisation

are made under 3% compressive strain in the plane of growth according to the traditional

linear piezoelectricity model[30].

The section which follows gives a detailed account of the piezoelectric effect, with attention

to materials properties and founding concepts.

1.6 Piezoelectric Effect

Piezotronics is wholly based on piezoelectric effects, with the preceding sections

demonstrating the potential for using these effects to significantly impact the properties of

various devices. This section will discuss in greater depth the state of current knowledge

regarding piezoelectricity.

In terms of direct effect the piezoelectric phenomenon occurs when electric dipole moment is

generated within specific crystal structures on the application of stress and in proportion to

that stress[31]. The piezoelectric effect found its first demonstration in 1880 in the work of

Jacques and Pierre Curie. The term “piezo” comes from the Greek “piezen (πιέζειν)”,,

meaning “to press or squeeze”, and the piezoelectric effect is considered as the production of

electrical charge within crystal structures as a result of ionic displacement with application of

stress[32].

Converse piezoelectric effects are seen in the deformation of a crystal where an externally

produced electric field is introduced along the direction of polarisation, and it was Gabriel

Lippman who first set this effect out in mathematical terms based on the founding principles

of thermodynamics in 1881[33]. It was not until 1910 however that a number of categories of

28

naturally occurring crystals with piezoelectric properties were described by Woldemar Voigt,

in a work which also used tensor analysis to examine constants for the piezoelectric effect.

1.6.1 Piezoelectric Polarization

Both the cubic form of ZB and the hexagonal form of WZ are non-centrally symmetric, and

this leads to moduli for piezoelectricity which are values other than zero. While the two

materials each allow piezoelectric polarization, greater symmetry is given by ZB, as shown in

Figure 1.6.1-

Figure 1.6.1-1Crystal Structures of Wurtzite and Zinc blende

Tetrahedral structures form the building blocks for both crystal structures, but those in ZB are

not overlapping, whereas the less symmetric structure of WZ means that there is precise

overlap between atoms from separate tetrahedral structures, giving a force of repulsion in the

middle of each pair of tetrahedrons. The result of this is that there is a term of polarisation,

29

referred to as spontaneous polarisation, within WZ crystal which is not under any strain. This

will be examined in depth in the section which follows.

The following equation describes the relationships between piezoelectric polarization field

Ppz, piezoelectric moduli dijk and eijk , stress tensor σjk, and finally εjk as the strain tensor:

Ppz,i = dijkσjk = eijkεjk, σij = cijklεkl (1)

in which the elastic tensor is given by cijkl

Tensor symmetry between dijk,σjk, eijk and εjkrelates to the j and k indices; this leads to a

further indices range given as xx = 1, yy = 2, zz = 3, yz = 4, zx = 5, xy = 6 . Short

notation of matrices is set out equation(1) for tensors. Thus, εj and eij and matrix notation

is:

eij = (

e11 e12e21 e22e31 e32

e13 e14e23 e24e33 e34

e15 e16e25 e26e35 e36

) (2)

εjk = (

εxx εyx εzxεxy εyy εzyεxz εyz εzz

) =

(

ε1ε2ε3ε4ε5ε6)

(3)

Looking firstly at the structure of ZB there are just 3 independent coefficients which are not

equal to zero:

eij = ( 0 00 00 0

0 e140 00 0

0 0e25 00 e36

) (4)

30

In the absence of strain Nye (1957) demonstrated [31] that there is just a single coefficient

which is not equal to zero in ZB crystalline structures as seen in equation (4) where and

e14 = e25 = e36. Greater complexity is seen for the WZ phase as there are five separate non-

zero coefficients when under strain. This is seen in the following:

eij = ( 0 00 0e31 e32

0 00 e24e33 0

e15 00 00 0

)

(5)

Where there is zero strain upon WZ, there are no longer five but three independent

coefficients which are not zero, as seen in Eq. (5) where e31 = e32, e15 = e24, and an

analogous term is used to symbolise. dij

Thus, for ZB, there is a piezoelectric polarization field which has values not equal to zero in

the direction of growth (111) on a diagonal from the usual axis of growth (001). This

polarization occurs in the direction of the c axis for [0001] WZ II-VI and III-N WZ from

epitaxial growth and subject to coherent strain.

For ZB,

ε⊥ = −2c12c11ε∥ (6)

However, there is only piezoelectric polarization where there exist strain tensor components

for the off-diagonal terms

Ppz,xzb = e14ε4 , Ppz,y

zb = e25ε5 and Ppz,zzb = e36ε6 (7)

In WZ on the other hand, there should be a free-surface boundary condition for surface

charge (σzz ≡ σ33 = 0) , with [0001] or z- direction polarisation, so

31

ε⊥ = −2c13c33ε∥

(8)

Ppz,3 = 2(e31 −c13c33e33)ε1 = 2d31(c11 + c12 − 2

c132

c33e33)ε1 (9)

Where an alloy possessing a as a lattice constant is grown upon coherent-strain, the strain

created is given as:

ε1 = ε1 = a𝑠𝑢𝑏−a

a

in which substrate lattice constant is asub

TABLE I: Experimental values for piezoelectric coefficients of III-V, III-N and II-VI

semiconductors

Semiconductor Piezoelectric coefficients (C/m2)

e14=e25=e36 e31 e33 e15

ZB GaAs -0.16[34]

InAs -0.045,-0.12[34]

WZ

GaN -0.55 1.12

InN -0.55 0.95

AlN -0.60 1.50 -0.48

ZnO -0.62 0.96

GaP -0.15[34]

InP

GaSb -0.12[34]

InSb -0.071,0.097[34]

32

Experimental values for coefficients of various semiconductors of III-V, III-N and II-VI are

collected in the table below [35,36,37,38,39]. Data from experiment is only available

however for symmetry in the naturally occurring crystal.

Piezoelectric coefficients or PZCs will be discussed in greater depth further on in the chapter,

which will also consider effects of piezoelectricity which is not linear in nature.

1.6.2 Spontaneous Polarization

WZ semiconductors created through epitaxial growth in the [0001] direction as normal can

generate a significant piezoelectric field in the presence of strain. Significant spontaneous

polarization[30] additionally occurs for II-VI and III-N semiconductors. Similarly to

polarisation under strain, it is not possible to measure spontaneous polarization in a direct

way, and experimental data regarding electro-optical qualities and the way in which these

change under strain must be used to infer measurements. Furthermore, electrostatic charge

density generated under spontaneous polarization is analogous to densities which are created

by piezoelectric polarisation produced from strain [40].

Spontaneous polarization is generated as a result of the structure of bulk material at the level

of the atom. For WZ, adjacent dual tetrahedral structures are overlapping in the z-direction,

which does not occur in ZB, as discussed previously, meaning that those which are 2nd closest

to another are nearer in WZ than in ZB. Forces exerted between atoms in WZ then reduce

slightly the distance between the closest neighbours. Therefore, the crystal’s non-zero dipole

moment where there is not strain or electric field present [40] can be termed spontaneous

polarization. For bulk material, the way in which the charge is reorganised at surface level is

considered as cancelling uniform fields of polarization fields as a result of spontaneous

polarisation and piezoelectric effect. However, differences in the crystal structure within

33

heterostructures or non-homogeneous layering of alloys lead to a field which is non-

vanishing and changes in spatial terms because of both spontaneous and piezoelectric

polarisation. Compositional variation produces charge densities with significant impact for

material properties and the action of devices [40].

Thus, any investigation of piezo-devices and multi layered structures [41] should extend to

spontaneous polarization also.

III-N WZ structures show a peculiarity in that they display comparable piezoelectric

constants to materials from group II-V and which are clearly dissimilar to those of materials

in group III-V. Further, III-N materials are different to general III-V compounds, having

greater ionic charges as well as an ionic contribution from internal-strain which becomes

greater than the term for clamped ions. A collection of values of spontaneous polarisation for

a range of II-V and III-N semiconductors are given in the table below [30,42]:

TABLE II: Spontaneous polarization values of III-V and II-VI semiconductors

Semiconductor Spontaneous Polarization

(C/m2)

WZ GaN -0.029[30]

InN -0.032[30]

AlN -0.081[30]

ZnO -0.057[30]

GaP 0.003[42]

InP -0.001[42]

GaAs 0.002[42]

InAs 0.001[42]

34

1.7 A Microscopic Approach

Previous sections have considered piezoelectricity from macroscopic perspective. However,

Harrison[43] investigates polarisation through a different approach which is underlain by the

Bond-Orbital Approximation, as the basis of a microscopic view of piezoelectric phenomena.

This is the approach which will be taken in the current study. Overall polarization as caused

spontaneously and by inducing strain can be obtained through the sum of bond and direct

dipole contributions [43].

(10)

Figure 1.7-1 Crystal Structures of Wurtzite in its orginal and strained case

strained

original

35

In which Cartesian direction is given by �̂�𝑖 and δr represents the cation displacement vector

in respect of the anions away from an ideal structure in which tetrahedric bonds are all

identical. Also vectors of distance from the ideal and displacement from the ideal are given

by rq and δRq respectively for the closest atom q to that in the central position within the

tetrahedron, while bond polarity is given by αp and atomic volume by Ω. As utilised in tight

binding, atomic charge is given by ZH* and is distinct from transverse effective charge, as

this is given an equivalent by the Born or dynamic effective charge (Z*), which is obtained

through density functional perturbation theory (DFPT).

For ZB crystals, a lower shear strain applies to the tetrahedrons, which means that cohesive

energy is minimised as the sublattices of anions and cations are each displaced relative to the

other. Thus, Kleinman [44] states that if shear strain runs in the ij plane, relaxation occurs

along cross-wise direction k, at 90 degrees to this. Under internal displacement, tetrahedrons’

bond lengths are no longer equivalents and thus the two terms of eq. (11) are no longer zero.

Where the difference between these terms is non-zero, polarisation on a macroscopic level

should result. The ideas described here will be discussed in greater detail and formalised in

the third chapter of the thesis. Displacement relies upon shear strain and the assumption of its

linearity means that the Kleinman parameter ζ can be used to characterise it. This parameter

states how far apart the sublattices are, and for materials is generally a constant.

Displacement is shown by 𝑎 𝑖𝑗

4 , in which a, represents the lattice constant while 휀𝑖𝑗, gives

shear strain. For ZB, PZCs are therefore determined by

𝑃𝑠𝑡𝑟𝑎𝑖𝑛𝑍𝐵 = 𝑒14휀�̂��̂� = 𝑃�̂� =

𝑒

2Ω

𝑎휀�̂��̂�

4(𝑍𝐻

∗ −4

3𝛼𝑝(1 − 𝛼𝑝

2)(1 − 휁)

휁) (11)

in which strain component is shown by 휀�̂��̂� and the Kleinman parameter by 휁 . In structures

of WZ spontaneous polarization should also be accounted for:

36

𝑃𝑠𝑡𝑟𝑎𝑖𝑛𝑊𝑍 = 𝑒31휀∥ = 𝑃𝑇𝑜𝑡 − 𝑃𝑠𝑝 =

𝑍𝐻∗ (𝛿𝑟 − 𝛿𝑢) + 2𝛼𝑝(1 − 𝛼𝑝

2)∑ (𝑟𝑞 .⃗⃗⃗⃗ 𝑥�̂�)𝛿𝑅𝑞4𝑞=1

2Ω (12)

Subsequent chapters will present a more detailed discussion of the coefficients and

parameters of the calculations made in this study.

37

2 Density Functional Theory

Solid-state physics has for some time now relied on density functional theory (DFT) in

electronic structure calculations, due to the reasonably accurate results against computation

resource requirement presented by approximate functional equations. Thus, while still

accurate to within acceptable limits, the technique increased the scale of applicable systems

in comparison with previous approaches, and currently older perturbative or variation

wavefunction approaches are applied to gain accuracy in small-scale systems in order to

benchmark values from which density functionals can be created and used on larger scales

[45]. DFT is applied as a technique which has scientific rigour in treating interacting

problems and does this through creating a precise map to tie the problem to a non-interacting

one. DFT approaches are utilised across a broad range of areas, although most frequently

with problems of ground-state electronic structure.

2.1 Hohenberg-Kohn Theorem

Density functional theory can be summarised based on Hohenberg and Kohn[46], Kohn and

Sham as follows:

First Theorem: “For any system of interacting particles in an external potentil Vext(r), the

interparticle potential is uniquely determined by the ground state charge density”.

Therefore, system properties may be calculated based on ground-state charge density.

Second Theorem: “A universal functional of the energy F[n] in terms of density n(r) can be

defined, valid for any external potentia Vext(r). Conversely for any particular potential, the

exact ground state energy of the system is the global minimum value of this functional and the

density n(r) which minimizes this functional is the exact ground state density”.

38

Thus, this theorem supports the use of a functional F[n] in describing ground-state electron

density, and from this, the properties of the system. However, it does not establish a recipe

for obtaining the functional F[n].

As Kohn and Sham’s equations have consistency with themselves, a valid interpretation is

that electron density in the final ground-state creates unique potential. The Born-

Oppenheimer approximation can be utilised in solving the problem of ground-state electrons,

as in this approximation heavy nuclei take on the quality of being a fixed point [45]. In non-

degenerate spin systems, the Kohn-Sham equations are given thus:

[−ℏ2

2𝑚∇2 + 𝑉𝐻[𝑛(𝑟)] + 𝑉𝑖𝑜𝑛[𝑛(𝑟)] + 𝑉𝑥𝑐[𝑛(𝑟)]]𝜓𝑖(𝑟)=휀𝑖𝜓𝑖(𝑟) (13)

In which [−ℏ2

2𝑚∇2] represents the kinetic energy from non-interacting fictitious elements,

𝑉𝐻 [𝑛(𝑟)] gives the Hartree potential, 𝑉𝑖𝑜𝑛[𝑛(𝑟)] gives the ionic potential,

𝑉𝑥𝑐 [𝑛(𝑟)] gives exchange-correlation potential, 휀𝑖= eigenvalue,

and 𝜓𝑖(𝑟)= eigenfunction from non-interacting fictitious elements.

The potential of exchange-correlation potential 𝑉𝑥𝑐 is estimated through:

𝑉𝑥𝑐[𝑛(𝑟)] =𝛿𝐸𝑥𝑐[𝑛(𝑟)]

𝛿𝑛(𝑟) (14)

This potential may be determined using a density functional derivative where the functional

is 𝐸𝑥𝑐[𝑛(𝑟)]known, but without knowing the functional’s outline, approximation is sufficient

for reasonable accuracy.

39

The total density of the electronic charge density =[𝜓𝑖(𝑟)]2 to establish a first estimation,

basis set (𝜙𝑖) must be given equal priority.

While DFT has many current and potential uses, the following negatives exist for the theory:

1. The precise functional of exchange correlation, or ‘divine functional’ [47] is not

present.

2. There is no correct scheme for acquiring ground-state charge density.

The results given by DFT rely upon exchange-correlation functional approximations.

𝑉𝐻 and 𝑉𝑥𝑐in the Kohn-Sham equations give different facets in DFT, appearing in varied

form, and with a range of approaches, the best method in terms of results is unclear. It is

beneficial therefore to assess each of the approaches in terms of strengths and weaknesses to

gain a fuller picture.

2.1.1 Exchange-Correlation

The impact of both the Coulomb potential (which does not simply arise from electrons

interacting electrostatically) and Pauli Exclusion Principle are shown by the exchange-

correlation potential. An anti-symmetric wavefunction is given in the many-electron system,

leading to electrons with spin being separated in space: a distinctive characteristic which

ultimately leads repulsive Coulomb potential to decrease. Due to anti-symmetric waveform

therefore, exchange energy in the electronic system is termed energy minimization, with

spatially separated electrons spinning in opposing directions minimising Coulomb energy in

the electronic system. However Hartree-Fock approximations ignore electron correlation

effects [48].

40

Complex systems require a highly complex process for calculation of correlation energy.

Exchange-correlation energy is given as function which relies upon electron density. Kohn-

Sham (1965) put forward Local Density Approximation (LDA), and this is the simple basis

of work to calculate pseudopotential for total energy [47].

2.1.2 Local Density Approximation

LDA is the most basic approximation scheme. The assumption is made that the exchange-

correlation functional represents density of electron energy, and that this is similar to

homogenous electron gas within space, and changing at a slow rate is assumed to be the

electron energy density of the system resembling that of a slowly varying homogenous

electron gas in space. LDA:

“It uses only the electron density, n(r), at a spatial point r to determine the exchange-

correlation energy density at that point. The exchange-correlation energy density is taken to

be that of a uniform electron gas of the same density. The exchange part of the functional is

defined as the exact expression derived for a uniform electron gas. The available versions of

LDA differ only in their representation of correlation. All modern LDA correlation

functionals are based on Ceperly and Alder’s (CA’s) 1980 Monte Carlo calculation [49] of

the total energy of the uniform electron gas” [50].

If for LDA, exchange-correlation energy is the same as for homogenous electron gas, the

LDA functional can be given thus:

𝐸𝑋𝐶[𝑛(𝑟)]=∫ 휀𝑥𝑐ℎ𝑜𝑚 [𝑛(𝑟)]𝑛(𝑟)𝑑3𝑟 (15)

41

The derivative of this equation gives the related exchange-correlation potential. The relation

below maps exchange correlation as a function of density and exchange correlation as a

function of space between electrons:

n(𝑟)−1 =4𝜋

3𝑟𝑠3 (16)

Wigner gives a correlation energy estimate in the case of uniform electron gas, through

interpolation of (r<1) high density limit and (r>1) low-density limits.

ℰ𝑐 = 0.44 (𝑟𝑠 + 7.8)⁄ (hartree) (17)

Ceperly-Alder correlation data was given high and low density limit parameters by Perdew-

Wang [51] and Perdew-Zunger [52, 53].

Kohn-Sham equations are used for derivation of a pure exchange expression which relies on

the space from the first to the second electron:

휀𝑐 = 0.4582 𝑟𝑠⁄ (Hartree) (18)

LDA does not take account of exchange-correlation energy arising from lack of uniformity

for density of electrons. However, LDA calculations have been shown to provide acceptable

results despite this. Jones and Gunnarsson[54] demonstrate the ability of LDA to provide the

proper sum rule for the exchange-correlation hole. Therefore, in a non-spin-polarised system,

LDA can provide one global minimum energy value [47].

Variations in LDA differ solely in different correlation representations. For the various

functionals, LDA is chosen in place of GGA as giving more accurate results where there is a

requirement for calculating oxide properties and energy of a surface[55].

42

2.1.3 Pseudopotentials

A system’s electronic features and ground state are established on the basis of valence

electron charge. Bloch’s theorem describes use of unique planewave sets in determining

electronic wavefunctions. However, an enormous plane wave set is required for expansion of

electronic wavefunctions when considering core orbitals’ strong binding and quickly

deviating wavefunctions of the valence electron in that area. This leads to spiralling

computational costs with larger numbers of electrons. Valence electrons generally present

energy 3 orders of magnitude less than that of electrons of the core. Further, deduction of

variance in energy due to ionic configuration or bonding is made from valence electron

energy. While overall energy calculation is imprecise, energy variance gives rise to important

outcomes and the pseudo-atom becomes highly significant.

Pseudopotential comes from an assumption of pseudo- particles and wavefunctions as similar

to electrons and wavefunctions across all electrons. This leads to inclusion of pseudo-

particles and -wavefunctions for calculations, with pseudopotentials generally representing

Coulombic interactions in the ionic core as well as pseudo-electrons, adding another term in

the Kohn-Sham Hamiltonian.

Therefore one particle’s wavefunction can be shown thus:

𝜙 = 𝜓 +∑ 𝛼𝑐𝜓𝑐𝑐𝑜𝑟𝑒

(19)

where 𝜓 = wavefunction term related to the relevant valence electron,

𝜓𝑐= core electron wavefunction

43

𝛼𝑐allows orthogonality to be retained, avoiding the possibility of 𝜙 overlapping with 𝜓𝑐

following:

⟨𝜙|𝜓𝑐⟩=0 (20)

The Hamiltonian may be shown as the sum of pseudopotential and kinetic energy terms.

H=K𝐸0 + 𝑣𝑝𝑠𝑝 + 𝑣𝐻𝑎𝑟𝑡𝑟𝑒𝑒 + 𝑣𝑥𝑐 (21)

Pseudo- and real potential should be the same in terms of organisation in space and

magnitude of absolute charge density. Development of pseudopotential occurs so that both

phase shifts and scattering properties of pseudo-wavefunctions are maintained in line with

core and ion electron valence wavefunctions. The ion core’s phase-shift properties depend

upon angular momentum.

Therefore, pseudopotential which does not take reliance on terms of angular momentum into

account is responsible for defining local pseudopotential, which therefore shows a function

which relies upon nuclear distance, or on nuclear spacing or wave vector variance within

plane-wave-basis states.

Where pseudo wavefunctions and actual wavefunctions coincide away from a core area,

calculations of scattering of ions outwards from the ion core are demonstrated as accurate

[56]. Norm-conserving pseudopotentials belong to a class including both non-local and local

pseudopotentials and in which pseudo-wavefunctions are the same as actual ones away from

the area of the core.

Creation of the pseudopotential is achieved with use of an exchange-correlation functional

with a ground-state and excited-state isolated atom, for an all-electron calculation.

Eigenvalues and functions as well as electronic densities can be estimated via self-consistent

44

calculation. Calculations are then conducted using valence electrons and without change to

the exchange-correlation functional and altered parameters, with a convergence norm usually

used as the core cut-off radius. Results are adjusted for replication of results of all-electron

calculations for eigenvalues and functions. Values which have undergone the best

optimisation are utilised to give the atom’s pseudopotential, with better value optimisation

leading to a more precise estimation of wavefunction features.

Pseudopotential quality may be assessed by considering the ability to replicate all-electron

calculation results. This ability has led to development of various types of pseudopotential

including empirical, first-principle, model and semi-empirical.

Empirical pseudopotentials are fitted using data from experiment regarding bandstructure,

with the disadvantage of this being see in the fitting parameters and interchangeability used in

elucidating properties. Best results are given by fitted parameters where properties directly

relate to algorithms. Meanwhile, model and semi-empirical approaches are more flexible in

terms of fitting to algorithms.

Meanwhile, the original or first principle pseudopotentials are calculated based on DFT, as

previously explored. This approach creates pseudopotentials which first diverge with cut-off

radius near to the nucleus or r→0. Meanwhile, converged results are given for areas near the

nucleus by soft-core pseudopotentials [55].

GNCPPS, or generalized norm-conserving pseudopotentials (Hamman),[56] are used to

create pseudopotentials, as well as Troullier-Martin’s soft-core pseudopotentials [57] and

Vanderbilt’s non-norm-conserving ultrasoft pseudopotentials[58]. In analysing soft-core

pseudopotentials, there is an assumption that electron density is divided into valence and core

electrons, leading to any overlapping between core and valence being disregarded. However,

45

non-linear core corrections [59] add to the exchange-correlation functional a fractional core

correction charge along with semi-core charge density due to valence and core electrons

which are non-interacting. This may be applied in various pseudopotential approaches.

Therefore preference is given to ultrasoft pseudopotentials in providing better results,

notwithstanding greater computational cost.

2.1.4 Plane Wave Basis Sets and Bloch's Theorem

Bloch’s theorem is used to address the main challenge represented by infinite electron

numbers in movement within a non-moving field. Infinite crystal wavefunctions find

expression as domain wavefunctions for reciprocal space vectors (Bravais lattice). The

theorem uses crystals’ periodic property, to reduce wavefunction numbers from infinity to a

value matching electron numbers within a unit cell.

Wavefunction may therefore expand through treatment as a product from a term which is

similar to a wave in addition to a component which involves cell periodicity:

ψi,k(r) = fi(r)e(ik.r) (22)

The initial portion of this expression shows the wavefunction’s cell periodic term, which may

be expanded to a greater degree with use of plane waves. Finite planewaves reproduce the

crystal’s reciprocal lattice vectors (G) through their wave vectors. The expanded term is

given below:

fi(r) =∑ Ci,Ge

(iG.r)

G (23)

From this, it is seen that planewave summation can represent electronic wavefunction as

follows:

46

𝜓𝑖,𝑘(𝑟)=∑ 𝐶𝑖,𝑘+𝐺𝑒{𝑖(𝐾+𝐺).𝑟}

𝐺 (24)

Therefore, Bloch’s Theorem associates the two problems of infinite and reciprocal space

vectors within the first Brillouin zone. This is then solved through calculation of finite

wavefunctions in particular periodic cells, k, within the Brillouin zone.

A solution to the Kohn-Sham equation must be calculated for every k-point, converging into

a large k-point, and the computation required for this is costly. Imitations of wavefunctions

across the whole of the k-space may be created from wavefunctions located at certain k-

points so as to access identical wavefunctions located at reciprocal k-spaces next to this point.

Chadi-Cohen[60] and Monkhorst-Pack [61] outline differing approaches towards maximising

accuracy in space charge density, through sampling of reciprocal space, or later through

solving Kohn-Sham equations at single k points within the Brillouin Zone.

Margins of error are reduced under denser grids of k-points, while cost of computation is

increased with increased grid.

Wavefunctions of k-points are accumulated based on asset from unique plane waves with the

inclusion of infinite plane waves. Those plane waves possessing lower kinetic energy have

more impact than others, allowing finite reduction of the set. Theoretically, condensation of

the plane wave set occurs down to a particular energy point used as a cut-off parameter.

Kinetic energy is obtained via second order wavefunction derivatives expressed via Bloch’s

expanding of wavefunction as:

Kinetic Energy =ℏ2

2𝑚|𝐾 + 𝐺|2 ≤ Cut − off Energy

Constraining factors in terms of energy cut-off energy allows cost of computation to be kept

down due to reduction in plane waves for each k point. However, the lower G-vector

47

numbers used lead to errors regarding total system energy. This can be addressed by

increasing the energy cut-off point.

Kohn-Sham equations are simplified greatly from the planewave set and greater efficiency is

achieved. Reciprocal replication of space is achievable via the Fast Fourier transformation

approach for the change to reciprocal from actual space.

In addition, there are no Pulay terms [62] or stresses within the Hamiltonian, and this means

that Hellmann [63] Feynman [64] force is calculated as equal to the derivative from total

energy as related to ionic position. The simplicity of the convergence criterion relies only on

energy cut-off.

Strongly converged results require many plane waves, regardless of the advantages of the

plane wave approach. While planewaves do not precisely match their localised atomic

wavefunctions nonetheless, Wannier and Gaussian functions can be utilised in projection

analysis to yield information which is of benefit.

2.2 Stress and Strain

Stress is the force components per unit area that cause the distortion of the unit cell. While

strain is the ratio of extension per unit length. Both strain and stress have significance in

characterising condensed matter, and may each lead to defects or fracturing of crystals. In

addition, strain has impact on the way in which a device behaves, as well as band structure,

and this can be harnessed when designing equipment. Further, quantification of stress is made

possible via experimental microscopy. A discussion will be given later concerning strain,

stress and the relation between them as well as how they may be utilised in specific devices.

Variations in types of pressure, such as dilation and shear, applied to material are described

through stress tensors. Considering condensed matter, ultimate equilibrium conditions within

48

a system are determined via independent parameters including atomic force or macroscopic

stress [65]. The general conditions for simulation, including metric [66] and

Parrinello-Rahman [67] approaches, are classically:

“Condition (1): the total force vanishes on each atom

Condition (2): the macroscopic stress equals the externally applied stress.”

The conditions are currently accepted as an essential element [68] in calculating electronic

structure, [69]in which minimising of unit cell atomic positions, cell-size and form relax the

structure.

2.2.1 Elastic strain

The �̂�, �̂�, �̂� axes of the system without strain will be replaced by distortion from applied

external pressure with �́�, �́�and �́� axes. This may be as a result of various types of pressure,

through dilation or shear, and the result is minor strains within a system

Below is an exploration of the basic strain expression in crystals. Axes in the new set are

linked to unstrained axes thus:

�́�=(1+𝜖𝑥𝑥)�̂� + 𝜖𝑥𝑦�̂�+𝜖𝑥𝑧�̂� (25)

�́�=𝜖𝑦𝑥�̂� + (1 + 𝜖𝑦𝑦)�̂�+𝜖𝑦𝑧�̂� (26)

�́�=(𝜖𝑧𝑥)�̂� + 𝜖𝑧𝑦�̂�+(1 + 𝜖𝑥𝑧)�̂� (27)

In which 𝜖𝛼𝛽 gives system deformation. Notably, the resultant axes are no longer generally

orthogonal.

Space vector �́� is related to the new axes thus:

49

�́� = 𝑥�́� + 𝑦�́� + 𝑧�́� (28)

While in unstrained systems:

r=x�̂� + 𝑦�̂� + 𝑧�̂� (29)

Space vector difference is used to obtain deformation displacement,

R=�́� − 𝑟

=x(�́� + �̂�)+y(�́� − �̂�)+z(�́� − �̂�)

(30)

With reference to the 3 previous equations, the following shows the expression:

R=(x∈𝑥𝑥+ 𝑦 ∈𝑥𝑦+ 𝑧 ∈𝑥𝑧)𝑥+ (x∈𝑦𝑥+ 𝑦 ∈𝑦𝑦+ 𝑧 ∈𝑦𝑧)�̂�+ (x∈𝑧𝑥+ 𝑦 ∈𝑧𝑦+ 𝑧 ∈𝑧𝑧)�̂� (31)

From this u, v and was new quantities are given by

𝑢=(x∈𝑥𝑥+ 𝑦 ∈𝑥𝑦+ 𝑧 ∈𝑥𝑧) (32)

𝑣=(x∈𝑦𝑥+ 𝑦 ∈𝑦𝑦+ 𝑧 ∈𝑦𝑧) (33)

𝑤=(x∈𝑧𝑥+ 𝑦 ∈𝑧𝑦+ 𝑧 ∈𝑧𝑧) (34)

Within smaller strain constraints, strain as reliant on position may be determined in a general

distortion which is not uniform.

𝑋 ∈𝑥𝑥= 𝑋𝜕𝑢

𝜕𝑥; 𝑦 ∈𝑦𝑦= 𝑦

𝜕𝑣

𝜕𝑦; 𝑧 ∈𝑧𝑧= 𝑧

𝜕𝑤

𝜕𝑧 (35)

Diagonal strain elements are utilised in defining distortion:

𝑒𝑥𝑥 =∈𝑥𝑥=𝜕𝑢

𝜕𝑥; 𝑒𝑦𝑦 =∈𝑦𝑦=

𝜕𝑣

𝜕𝑦; 𝑒𝑧𝑧 =∈𝑧𝑧=

𝜕𝑤

𝜕𝑧 (36)

Meanwhile, off-diagonal terms which define either angular strain distortion or shear strain are

given as:

50

𝑒𝑥𝑦 = �́�. �́� ≈∈𝑦𝑥+∈𝑥𝑦=𝜕𝑢

𝜕𝑣+𝜕𝑣

𝜕𝑥 (37)

𝑒𝑦𝑧 = �́�. �́� ≈∈𝑧𝑦+∈𝑦𝑧=𝜕𝑣

𝜕𝑧+𝜕𝑤

𝜕𝑣 (38)

𝑒𝑥𝑧 = �́�. �́� ≈∈𝑧𝑥+∈𝑥𝑧=𝜕𝑢

𝜕𝑧+𝜕𝑤

𝜕𝑥 (39)

Dilation defines the distortion-caused net fractional volume alteration, expressed by

𝛿 =�́�−𝑣

𝑣≈ 𝑒𝑥𝑥 + 𝑒𝑦𝑦 + 𝑒𝑧𝑧 (40)

In which volume prior to and following distortion is represented by 𝑣 and �́�, while starting

cubic volume is unity.

Those stress components which lead to crystal unit cell distortion should be described,

before defining stress as force applied across area. Nine stress quantities are possible here,

but without torque, these may be decreased to six, represented thus

𝑋𝑋,𝑌𝑦,𝑍𝑧 , 𝑋𝑦, 𝑌𝑧 , 𝑍𝑥

In which the capital letter gives force direction whereas the small letter below the main text

shows direction at 90 degrees to the stress plane and meeting the condition below:

𝑋�́� = 𝑌𝑥; 𝑋�́� = 𝑍𝑥; 𝑌𝑧 = 𝑍𝑦 (41)

The sections below outline the means of obtaining an overall elastic constant expression with

the help of relations in stress and strain, which will be of benefit for investigating elastic

phenomena in crystals. Strain gives displacement terms, internal strain parameter terms and

estimations for piezoelectric constants.

51

2.2.2 Macroscopic strain and stress

Strain is identified where an atom is 𝑅 = 𝑟 − �́� displaced from position 𝑟 to �́� as a result of

being deformed through shear pressure or dilation. R as the displacement relies upon the r

spatial co-ordinate in determining deformation. For instance, where there is a connection

between two close points, vector alteration from 𝑑𝑟 to 𝑑�́� is caused by deformation, with

space between points being altered from 𝑑𝑙 = √𝑑𝑟12 + 𝑑𝑟2

2 + 𝑑𝑟32 to 𝑑𝑙.

The following shows the lowest order in 𝑢, 𝑑𝑙

(𝑑𝑙)́2 = 𝑑𝑙2 + 2𝑅𝛼,𝛽𝑑𝑟𝛼𝑑𝑟𝛽 (42)

In which assumption of summation over Cartesian coordinates α and β is made. Further,

𝑅𝛼,𝛽 =1

2(𝜕𝑅𝛼

𝜕𝑟𝛽+𝜕𝑅𝛽

𝜕𝑟𝛼) (𝑑𝑙 )2 (43)

Forms the strain tensor, which is analogous to the metric tensor giving length alterations for

the system under deformation based upon coordinates before deformation[70].

(𝑑𝑙)́2 = 𝛿𝛼,𝛽 + 2𝑅𝛼,𝛽𝑑𝑟𝛼𝑑𝑟𝛽 (44)

Further, strain tensor may be conveniently determined as 𝛿𝛼,𝛽 with unsymmetrized strain

tensor 𝜖𝛼,𝛽, a scaling in space:

𝑟𝛼 → (𝜖𝛼,𝛽 + 𝛿𝛼,𝛽)𝑟𝛽 (45)

Rotation and other terms do not impact upon internal coordinates: therefore, the internal

energy expression from Eq.(44) may be used, having validity for symmetric systems despite

the inclusion of terms which are antisymmetric.

52

The derivative of energy for strain tensor by volume unit defines the macroscopic average

stress tensor 𝜎𝛼,𝛽 and strain across macroscopic areas is assumed as uniform.

𝜎𝛼,𝛽 = −1

Ω

𝜕𝐸𝑡𝑜𝑡𝑎𝑙

𝜕𝜖𝛼𝛽 (46)

Enthalpy is reduced in positive strain: therefore negative applied forces determine

compressive strain.

Relations between stress and strain are used in defining elastic occurrences, with linear elastic

constants produced through:

𝐶𝛼𝛽,𝛾𝛿 =1

Ω

𝜕2𝐸𝑡𝑜𝑡𝑎𝑙

𝜕𝜖𝛼𝛽𝜕𝜖𝛾𝛿=𝜕𝜎𝛼𝛽

𝜕𝜖𝛾𝛿 (47)

In general symmetric crystals 𝐶𝛼𝛽,𝛾𝛿 may be given as 6x6 𝐶𝑖𝑗 array. [71,72] Meanwhile, i and

j are given through Voigt’s notation thus:

1 = 𝑥𝑥; 2 = 𝑦𝑦; 3 = 𝑧𝑧; 4 = 𝑥𝑦; 5 = 𝑦𝑧, 6 = 𝑥𝑧

Just 3 independent constants, namely C11, C12 and C44, give a cubic crystal. In contrast there

are 5, C11, C12, C13, C33 and C44, for a wurtzite crystal. Although linear elasticity constants are

considered in this discussion, higher order reliance on application of pressure is given by

non-linear models of elasticity [73].

Further, the relation of stress and strain to non-linear and linear terms is seen in computation

of stress, and derivatives describe both applied strain and stress. Non-zero forces, whether

applied internally or externally, fix the unit cell’s atomic positions in strain. Therefore, finite

strain theory may be considered based on simple theory. DFT was used for calculation of

stress/strain effects in the semiconductors investigated here, based on the theory outlined

above.

53

2.3 CASTEP

CASTEP (Cambridge Serial Total Energy Package) [74] is a cutting edge programme based

on quantum mechanics which is aimed at the requirements of solid-state materials science.

The programme uses DFT plane-wave pseudopotential approach, facilitating use of

calculations based on first principles of quantum mechanics to investigate crystal and surface

properties for a variety of materials, including ceramics, semiconductors, minerals, zeolites

and metals.

In general, CASTEP is used in studying features of structure, surface chemistry, band

structure, optical characteristics and state density. In addition, it is utilised in examining

wavefunction and charge density distribution in space within systems. Further uses in

crystalline materials include calculation of full tensor elastic constants of the second order as

well as associated mechanical qualities, including Poisson coefficient, Lame constants and

bulk modulus. CASTEP’s applications for seeking transition states facilitate investigation of

chemical reactions, whether on the material’s surface or during the gaseous phase, and

includes linear and quadratic synchronous transit technologies, suitable for both surface

diffusion and bulk [75].

54

3 Previous Works

3.1 ZB III-V Semiconductors

3.1.1 Introduction

Good light absorption and emission are characteristic features of direct bandgap

semiconductors, which include for example InAs and GaAs, making such semiconductors

highly utilised in the manufacture of LEDs, sensors and photovoltaic solar cells. Within the

direct bandgap semiconductor, a pair of semiconductors grows one on top of the other in an

epitaxial manner, and the shear strain between the mismatched lattices is behind the

production of the piezoelectric effect [76,77]. This can be clearly seen where growth of QW

takes place using (111) substrates, NWs and QDs with a strain tensor on the off-diagonal.

Anisotropies identified in recent observations are shown to have come from piezoelectric

effects in QDs and QWs. [78,79,80] In reviewing the literature, it appears that first order

piezoelectric effects only are the focus of Davies [81] Grundman et al., [82] and Stier et al.,

[83]use of experimental PZCs for bulk GaAs as well as InAs. Meanwhile, in work with

InGaAs, in general PZCs come from linear interpolation from bulk values.

However, as Bester et al., [84] demonstrate, where PZ fields are calculated without reference

to effects of the second order, this leads to significant error. Migliorato et al., [85] confirm

that DFT (Density Functional Theory) can be used to construct a semi-empirical model in

order to realise estimations for piezoelectric tensors of both the first order and the second.

Bester et al., [84] estimate PZC values of -0.230 C/m2 for InAs and -0.115 C/m2 for GaAs,

although according to Migliorato et al., [85] no reason can be given for the fact that these

values differ from the -0.045 C/m2 and -0.16 C/m2 values recorded through experiment.

Migliorato et al.,[85] also present an enhanced method to Bester et al.’s[84] linear response

55

technique approach, and this is reflected in a closer match between estimations and data

recorded through experiment with InxGa1-xAs/GaAs QWs.

3.1.2 Piezoelectric Quantum Well

The elements for polarization can be obtained through Pi = eijkεjk, where εjk is the strain tensor

and eijk represents piezoelectric coefficients. Further, εij which gives the off-diagonal shear

for the strain tensor relates directly to PZCs e14, e25 and e36. As ZB symmetry is present it is

possible to reduce the PZC values to simply e14. Thus,

𝜌(𝑟 = −∇. (2𝑒14(𝑟). )[휀𝑦𝑧(𝑟)𝑖 + 휀𝑥𝑧(𝑟)𝑗 + 휀𝑥𝑦(𝑟)𝑘]) (48)

gives the piezoelectric charge produced as a result of the shear strain.

As the strain tensor demonstrates non-zero off-diagonal values are given by [111] direction of

strain. It is notable here that for a QW, PZ polarization in this direction are possible to infer

using known quantities. Herein lies the necessity of research into and models for III-V

semiconductors in terms of their polarisation, with a PZ field being induced in the direction

of growth when QW nanostructures grow upon (111) substrate, giving a less complex

modelling process for data from experimental work. Data recorded experimentally to date

measuring photocurrents show with alloy compositions where x<0.3, it is not possible to

match measured values for e14 [86,87,88 ]with linear interpolation for bulk values for InAs

and GaAs. From modelling, Hogg [86] and Sanchez-Rojas [87] measure low-temperature

photocurrent values of as much of 70% of that from interpolation. Meanwhile, Cho [88]

obtains 80% of interpolated value when modelling at room temperature. If an ideal structure

and sharp interfaces are assumed for QW, with alloy composition and width used for a fitting

parameter electro-optical data gave the reduction in e14 [86,86,87,89,90].

56

Ballet et al.,[91]added Indium segregation at interface points to the previously adopted model

in which Molecular beam Epitaxy was used to grow QW structures on a substrate of form

(111), and this showed 86% is correct as a percentage of interpolated value, where x=0.15 as

related to 0.124C/m2 as a value. This far lower reduction as compared to initial hypotheses is

supported by Migliorato et al.,[85] who replicated this work.

3.1.3 Piezo coefficients with Harrison’s Model

The range of Indium concentration within which the findings in the literature are relevant is

limited to 0 ≤x≤0.2 with an InxGa1-xAs semiconductor. The quality of crystals is negatively

affected by the significant mismatch in lattices where semiconductors grow epitaxially, and

as a result of this experimental work has never produced a structure where x>0.4. Further,

many researchers [85,86,87,88,89,90,92,93] consider an estimation for e14 to be unachievable

where x>0.4. However, Migliorato et al.,[85] achieved estimation for e14 where x>0.2, and

this was done by using Harrison’s model of piezoelectricity [43] to observe reducing

interpolated values in a detailed manner. This entailed circumvention of the issue of the low

agreement level between PZC 𝑒𝛽∗ and values obtained through experiment, which had

previously meant that piezoelectric charge was not calculated accurately. This was done by

selecting a single parameter for adjustment in order to raise the degree of agreement between

the values. The advantages presented by Harrison’s model include its simplicity and ease of

adjustment, and the fact that issues can be analysed through non-complex concepts in

description of major mechanisms in structures with stability.

III-V semiconductors have a tetrahedron consisting of five atoms as their basic unit, and from

this element the crystal is constructed. A single tetrahedron comes under stress at the level of

the atom, and this links to hydrostatic strain and diagonal strain tensor elements, causing

57

displacement dk at the atomic level. Where low level shear strain in εij<0.02, off-diagonal

quantity, is applied, insignificant or low levels of atomic displacement are seen.

Kleinman [44] posited the theory that relaxation is present at right angles to shear strain

direction. Atomic displacement dk may be considered linear regarding strain. Therefore, the

Kleinman parameter ζ, which is usually < 1, becomes a constant, and was assumed as such

for any particular material until it was demonstrated by Rideau et al.,[94] and Migliorato et

al.,[85] that this was not the case. Figure 3.1.3-1 illustrates the reflection of ζ in strain

parameters εij and εk, as shown by Migliorato et al.,[85] who illustrate the point that dk

depends significantly on εk while depending only slightly on εij, where εij <0.02. This study is

also the first instance of use of an evaluative methodology other than the linear response

technique in assessing non-linear impacts, and uses Harrison’s model [43] in conjunction-

with DFT and DFPT (density functional perturbation theory) in the calculation of parameters

which depend on the material. Polarisation along the �̂�𝑖 direction in line with Harrison’s

model,[43] can be represented through following equation for an atom in the tetrahedron’s

centre:

𝑃�̂� = 𝑍𝐻∗ 𝛿𝑘 + 𝛽∑ (𝑟 𝑞 . �̂�)𝛿𝑅𝑞

4𝑞=1 (49)

In which 𝑍𝐻∗ represents atomic effective charge related to anions and cations within the

crystal structure and 𝛿𝑘 represents separation of the same based on the sublattices relaxing

internally, with β representing parameter of transfer. Further, 𝛿𝑘 may be expressed via ζ

showing the Kleinman parameter,휀�̂��̂�as shear strain and a being the lattice parameter:

𝛿𝑘 =𝑎 �̂��̂�

4 (50)

Meanwhile, there is an association between transfer parameter and polarity αp which

measures bond-associated electric dipole, which is shown through β=2αp(1-αp2). Terms

58

within the equation generally give the opposite sign and polarization and represent the fragile

balancing of direct dipole (𝑧𝐻∗ 𝛿𝑘) with bond contributions (𝛽∑ (𝑟𝑞

→. �̂�)4𝑞=1 𝛿𝑅𝑞), based on the

identical distance rq of the 4 atoms bordering the tetrahedral centre.𝛿𝑅𝑞 gives variations of rq

as a result of atoms altering their pattern on the crystalline structure becoming relaxed

internally from shear strain present in the orthogonal plane to vector �̂�.

Equations (49)and(50) the possibility of calculating the linear coefficient of piezoelectricity

e14 through division of Equation (49)by 2Ω, double the volume for each atom, before

multiplication by electron charge e, then expressing the bond dipole through 𝛿𝑘;

𝑃�̂� =𝑒

2Ω

𝑎 �̂��̂�

4(𝑍𝐻

∗ −2

3𝛽(1− )

) = 𝑒14휀�̂��̂� (51)

In order to gain accuracy in results in this model, it is essential that precise values are

established for 𝑧𝐻∗ ζ and β [95].

According to Migliorato et al.,[85] DFPT and DFT may be used in evaluation of both bulk

values and the reliance of these upon diagonal elements within the strain tensor. Further,

pseudopotentials were taken from Troullier-Martin [57] in calculating plane waves by

Migliorato et al.,[85] and displacement of the internal sublattice ζ may be identified through

calculation if the strain tensor is known. Where the Monkhorst-Pack grid [61] is 4×4×4, with

50 Rybergs for plane wave cutoff energy, the authors record 11.37a.u for the lattice constant

in InAs and 10.48a.u for that of GaAs[85].

To the lattice, which has undergone hydrostatic compression, a simple strain tensor and small

shear strain (γ<0.01) are added. This is followed by a relaxation of anion and cation, thus

reducing cohesive energy to a minimal level. The following equation gives both strain tensor

and simple relation between strain and ζ:

59

Strain tensor =(

1 − 휀 𝛾 2⁄ 𝛾 2⁄

𝛾 2⁄ 1 − 휀 𝛾 2⁄

𝛾 2⁄ 𝛾 2⁄ 1 − 휀) (52)

𝛿𝑟 = √3𝑎휁𝛾

4(1 + 휀)

(53)

Figure 3.1.3-1 Dependence of the Kleinman internal displacement parameter of zinc blende

InAs circles and GaAs (squares) on shear strain (hollow symbols) and hydrostatic strain filled

symbols. Notice the opposite strain dependence of these two. Reproduced from Reference [M.

A. Migliorato, D. Powell, A. G. Cullis, T. Hammerschmidt and G. P. Srivastava, Phys. Rev. B

74 (2006) 245332].

The manner in which ζ behaves with hydrostatic strain and shear is worthy of note, where one

quantity is made variable while fixing the second. Figure 3.1.3-1 shows that a greater

hydrostatic strain ε leads to a rise in ζ, while greater strain γ lowers ζ, suggesting a reduction

0.00 0.02 0.04 0.06 0.08 0.10

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.00 0.02 0.04 0.06 0.08 0.10

hydrostatic distortion

shear distortion

su

bla

ttic

e d

isp

lacem

en

t

circles: InAs

squares: GaAs

60

in piezoelectric charge where ζ is decreased [85]. Further, Wang and Ye [96] record a

reliance of strain upon ζ. Additional calculation for DFT [97] is made to explore impacts

from shear strain and hydrostatic strain combined. Migliorato et al., [85] were first to give a

strain dependence for ζ. in III-V semconductors.

3.1.4 Bond Polarity, Atomic Effective Charge

The latest bond polarity calculation 𝛼𝑝as Wang and Ye [96] report, calculated transverse

effective charge using DFPT, followed by determining 𝛼𝑝 using a tight-binding expression to

connect the quantities. A value of 0.423 is given in GaAs and 0.49 for InAs. Shen et al., [98]

support these findings with approximation of bond orbital approximation. Further, Wang and

Ye [96] state that hydrostatic strain has impact in terms of linear polarity reduction, and gave

an estimation of polarity decay in GaAs and of -1.08ε in InAs. The means of calculation of

atomic effective charge 𝑧𝐻∗ depends upon Harrison’s expression:

𝑍𝐻∗ = 𝑍 − 4 + 𝛼𝑃 (54)

in which Z represents the column number for chemical species, with extrapolation from data

obtained through experiment by Falter et al.,[99] however,𝑍𝐻∗ is overestimated in this method.

Further work [99,100,100] gives a good level of agreement between Z* values and data from

experimental work. However, despite use of an approximate estimate through 𝑍∗ = 𝑍𝐻∗ +

4𝛽 3⁄ , 𝑍𝐻∗ values have not found broad agreement over a range of data from experiment and

theoretical approaches.

61

Table III: Parameters for calculations in Migliorato et al., [85]

Bulk

휂

distortion

Second Order:

𝑋 + 𝑋2 + 휂2 + 𝑋3휂3

휁 GaAs 0.455 휀𝛼𝛼 5.88 -28.99 75.03

2휀𝛼𝛽 -0.23 -19.98 102.75

휀𝛼𝛼 ∗2휀𝛼𝛽 1.87 47.55 255.05

InAs 0.58 휀𝛼𝛼 5.42 -25.84 51.67

2휀𝛼𝛽 -0.45 -11.86 70.78

휀𝛼𝛼 ∗2휀𝛼𝛽 1.73 31.37 166.61

𝛼𝑝 GaAs 0.42398 휀𝛼𝛼 -0.95 0 0

InAs 0.4998 휀𝛼𝛼 -1.08 0 0

𝑍𝐻∗ GaAs 0.43 휀𝛼𝛼 −4[𝛼𝑝(𝑏𝑢𝑙𝑘) − 𝛼𝑝(휀)]

InAs 0.54 휀𝛼𝛼

The first approximating expression in (54) this produces an overestimation of. Further, an

acceptable approximation of the impact of hydrostatic strain is given by 𝑍𝐻∗ - 𝑍𝐻

∗ (휀), allowing

derivation of strain dependence for 𝑍𝐻∗ using the 𝛼𝑝(휀)expression.

3.1.5 Compositional Disorder Effect

Incorrect estimations of quantities such as polarity are given by linear interpolation, and this

is demonstrated by Bouarissa [101] in study of empirical pseudopotentials in approximations

in virtual crystals, in which 8% overestimates are given for values in comparison with

interpolated values for In0.5Ga0.5As. Figure 3.1.7-1 presents this outcome, which is considered

62

to be linked with random alloy compositional disorder. As given by Migliorato et al.,[85]

composition-dependent polarity and derived bowing parameter can be expressed by:

𝛼𝑝 = 0.423 + 0.161𝑥 + 0.000148𝑥2 (55)

Vergard’s law is defined as the existence of a linear relation between the crystal lattice

constant of an alloy and the concentrations of the constituent components at constant

temperature [102]. Vergard’s law valid for the elastic properties of InxGa1-xAs, and therefore

there is no need for a bowing parameter for the Kleinman parameter ζ. In contrast, the

bowing parameter is required by 𝑍𝐻∗ in order to calculate strain dependence, which is done

with substitution of 𝛼𝑝from Equation(55)to Equation(54) [85].

3.1.6 Evaluation of piezo coefficient

In order to evaluate PZC where shear strain and hydrostatic strain have created deformation

in the crystalline structure, equation (51) must be amended, in that polarization should be

divided by (1-ε) in order to give the quantity needed. V’ gives cell volume equivalents in first

order strain, with V (1+3ε) found to be equivalent. Direct comparison is made between values

from theory and those from experimental work by Migliorato et al.,[85] Calculation via linear

interpolation is made for parameter ζ as well as the lattice constant 𝑎 with InxGa1-xAs in

intermediate composition, based on the approach given above for 𝑍𝐻∗ and 𝛼𝑃. Linear values

for PZC result from this, and this is shown in Figure 3.1.7-1 However, these differ from

values from simple interpolation of e14 bulk value. In addition, the effect of strain is included

in making comparison with experimental values of an even epitaxial growth layer for InxGa1-

xAs on top of a substrate of GaAs (111), which is a function of composition (x). Alteration

for ζ, 𝑍𝐻∗ and 𝛼𝑃 interpolated parameters when strain is exerted may be determined based on

63

the previously discussed relationship. An adapted version of Equation (51) can be utilised in

evaluating PZC e14, with findings obtained displayed in Table III.

3.1.7 Review and Important Discussion

A comparative analysis of the models developed by Bester et al.,[84] and Migliorato et

al.,[85] will be presented here. Within this, a notable difference is found in the photocurrent

spectrum fitted findings of Migliorato et al., [85] and linear terms of Bester et al., [84]. It is

clear from the previously presented discussion that a major alteration is made in PZC e14

values when strain is present. Further, the work of Migliorato et al., [85] shows that elastic

properties are impacted in a significant manner by both hydrostatic and shear strain, and that

this is particularly so in the case of the Kleinman parameter. In addition, Figure 3.1.7-1 PZC

can be seen as a function of strain where a particular thickness of an epitaxially grown,

uniformly composed layer (x) on top of GaAs (111) substrate is created, rather than being a

basic function of composition, as may be assumed in error.

64

0 20 40 60 80 100-0.25

-0.20

-0.15

-0.10

-0.05

0.00

Ref [17]

GaAs (exp)

Ref [12]Ref [14] (300°K)Ref [16]

InAs (exp)

This work

Pie

zo

ele

ctr

ic C

oe

ffic

ien

t (C

/m2)

In content (%)

Ref [13]

Figure 3.1.7-1 Piezoelectric coefficient dependence on In compositions. Reproduced from

Reference [M. A. Migliorato, D. Powell, A. G. Cullis, T. Hammerschmidt and G. P.

Srivastava, Phys. Rev. B 74 (2006) 245332].

Thus, there is no direct match between unstrained bulk values of InAs as shown by Figure

3.1.7-1 and those of strained InAs values in GaAs (111). Further significant difference in

Migliorato et al.,[85] and Bester et al.’s[84] models is seen through linear terms, as while

Migliorato et al.,[85] can replicate values of e14, this is not attempted in Bester et al.,[84]

where estimations are higher as given by the linear term.

A range from 120 to 165 kV/cm is reported for piezoelectric field where composition of QW

falls from x=0.15 to x=0.21[85,86,87,88,89,90]. The underlying assumption behind these

values is that the strongest oscillation occurs under flat band conditions where internal

piezoelectric field is compensated by applied. The assumption is also made of constant

bandgap over the area of the well, and where the stoichiometric profile is not uniform, with

65

varied applied bias, it is possible to record maximal values for oscillator strength and flat-

band. However, estimation of piezoelectric field is not consistent with identified issues

regarding such estimates in terms of composition variation [84]. Where QW is of width

10nm, and x=0.15, 220kV/cm is given as an approximation of piezoelectric field when

theoretically expressed [103]. Where e14 is reduced by as much as 30% as compared with

interpolated value, a field of 165kV/cm is recorded, and where e14 is reduced by only 16%,

using segregated diffused interfaces produces an estimated field of 190kV/cm. From this,

significance can be attributed to segregation of In when considering the large difference in e14

values. QW of width 11nm where x=0.15 has an estimated piezoelectric field reaching 80

kV/cm according to Bester et al.,[84], while this prediction is low in comparison with

experimental results[86,87,88] being just 42% of the highest realistic estimate [91]. In

contrast work by Migliorato et al.,[85] matches far more accurately with Ballet [91] in terms

of field estimates, due to the methodological similarities when assessing e14 and field values.

3.2 Pseudomorphic growth of zinc blende semiconductors

previous Work Bond Polarity and Kleinman

Garg et al.,[104] built upon Migliorato et al.,[85], investigating tetragonal distortion, which is

the presence of strain while growth occurs pseudomorphically upon substrate (001) rather

than only with crystals subjected to uniaxial strain.

Calculation is made of the value for bond polarity 𝛼𝑃via DFPT in CASTEP, through

application to the unit cell of a strain tensor, as shown in

𝑇𝑒𝑛𝑠𝑜𝑟 = (

1 − 휀𝑥𝑥 0 00 1 − 휀𝑦𝑦 0

0 0 1 + 휀𝑧𝑧

) (56)

66

The values of diagonal strain elements may not be zero, and fall between - 0.01 and 0.1. y-

and x-direction strain elements do not differ from each other, but difference is seen in

direction of z.

With variously combined cut-off energy and k point grid, use is made of 4x4x4 k-point grid,

ultrasoft pseudopotential [105] and LDA, as well as an energy cut-off of 1000eV, which set-

up results in a reasonable convergence, with error of 1%. Calculation of bond polarity is

made in line with Harrison [136] in the expression below, where in (∆𝑍 = 1) III-V

compounds, based on born charge obtained through calculation for DFPT in InAs and GaAs,

and is a function of perpendicular or parallel strain elements.

𝑍∗ = −∆𝑍 + 4𝛼𝑝 + 4𝛼𝑝(1 − 𝛼𝑝2) (57)

67

The bond polarity value 𝛼𝑝 strain dependence for InAs and GaAs is shown in Figure 3.2-1

and is in line with the repetitive tendency in Born effective charge Z*. Thus, bond

polarity𝛼𝑝reduces with greater hydrostatic pressure.

Figure 3.2-1 Bond Polarity plots of GaAs and InAs. Dependence of the bond polarity on the

applied strain for GaAs and InAs. For each value of the perpendicular strain, each point

corresponds to a different value of the parallel strain that ranges from -0.01 to +0.1, (top to

bottom) as indicated by the arrows. Reproduced from Reference [R. Garg, A. Hüe, V. Haxha,

M. A. Migliorato, T. Hammerschmidt, and G. P. Srivastava, Appl. Phys. Lett. 95 (2009)

041912].

The DFT calculations for the Kleinman parameter were performed by the CASTEP [106] was

used to perform Kleinman parameter DFT calculations, in which the strain tensor used was as

follows:

68

𝑇𝑒𝑛𝑠𝑜𝑟 = (

1 − 휀𝑥𝑥 𝛾 2⁄ 𝛾 2⁄

𝛾 2⁄ 1 − 휀𝑦𝑦 𝛾 2⁄

𝛾 2⁄ 𝛾 2⁄ 1 + 휀𝑧𝑧

) (58)

Fractional coordinates represent the atomic displacement calculated via DFT, and following

this data conversion is conducted to arrive at the Cartesian coordinate. Thereafter, calculation

is made of internal sub-lattice displacement ζ, and this acts as analogous to ζ of the Kleinman

parameter, assessed via the equation below:

𝑑𝑟 =1

(1 + 휀) √3

4𝑎 𝛾휁 (59)

Here, dr represents cation to anion displacement, while 𝑎 = lattice constant and γ/2 = shear

strain or off-diagonal tensor, with hydrostatic strain being represented by ε. The Figure 3.2-2

presents Kleinman parameter ζ against parallel strain and perpendicular strain εzz in InAs and

GaAs. Here the tendency for strong hydrostatic strain dependence for the Kleinman parameter

is unique.

69

Figure 3.2-2 Kleinman parameter plots of GaAs and InAs.Reproduced from Reference [R.

Garg, A. Hüe, V. Haxha, M. A. Migliorato, T. Hammerschmidt, and G. P. Srivastava, Appl.

Phys. Lett. 95 (2009) 041912].

3.3 Piezoelectric coefficients

Polynomials were matched to data from DFT by Garg et al., [104] for calculation of PZCs,

and the findings show that upon integration of the second order effects, within one PZC,

reproduction of a high strength of dependence on combined off-growth direction and in-plane

strain. To achieve randomly combined diagonal strain components exx, eyy and ezz, it is

necessary to use 3 coefficients for e14, e25 and e36. For the current study, consideration is made

of second order effects for three effective coefficients, but it remains possible to calculate

complete second order PZ tensor with use of polynomial fitting. This was not done by Garg

et al., [104] because strains were combined including exx=eyy. Thus, one of the intentions of

the study was extension of previously completed research to achieve greater generalisation in

the strain tensor and then calculate PZCs of the second order.

70

Figure 3.3-1Piezoelectric coefficients plot for GaAs and InAs. Reproduced from reference [R.

Garg, A. Hüe, V. Haxha, M. A. Migliorato, T. Hammerschmidt, and G. P. Srivastava, Appl.

Phys. Lett. 95 (2009) 041912].

Figure 3.3-1 shows PZCs with high strain dependence for directions x, y and z. In InAs with

epitaxial growth over GaAs, a misfit in the match of the lattice is present in the order of

between 5 and 7%, whereupon PZC is transforms into positive from negative, with a sign

reversal of 7%.

3.4 Non-linear piezoelectric coefficients in ZB materials

The nonlinear piezoelectric coefficients for GaAs and InAs ZB materials related to strain

tensor diagonal terms are determined by Tse et al., [107] in Table IV. After conditions are

imposed on the coefficients based on the cubic symmetry of the crystal, reduced number of

PZCs for the fitting equation is found:

�́�𝑙𝑚=𝑒𝑙𝑚 +∑ 𝑒𝑙𝑛𝑚휀𝑛 +∑ 𝑒𝑙𝑛𝑛′𝑚ℰ𝑛ℰ�́� +∑ 𝑒𝑙𝑛𝑛′𝑛′′𝑚ℰ𝑛ℰ𝑛′ ℰ𝑛′′3

𝑛<�́�<𝑛′′=1

3

𝑛<�̀�=1

3

𝑛=1 (60)

0.00 0.02 0.04 0.06 0.08 0.10

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.00 0.02 0.04 0.06 0.08 0.10

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

e1

4

ezz

exx

GaAs

-0.16

e1

4

ezz

exx

InAs

-0.045

7% sign reversal

71

An evaluation was made for all terms in the above equation. Overall even in the limit of

small strain, the linear and quadratic terms are likely to be included, therefore, cubic terms

should only be present if the material undergoes significant (around 10%) strain [125].

Table IV: Linear and non-linear coefficients obtained from DFT data. For second and

third order terms the parameters are invariant upon cyclic permutation of the n

indexes. Reproduced from Reference [125]

Parameter GaAs InAs

𝑒𝑙𝑚 -0.160 -0.045

𝑒𝑙𝑛𝑚 I=n e114=e225=e336 -0.666 -0.653

𝑰 ≠ 𝒏 e124=e235=e316 -1.646 -1.617

𝒏 ⟺ 𝒏′

𝑒𝑙𝑛𝑛′𝑚 l=n=n’ e1114=e2225= e3336 -0.669 -3.217

l=n≠n’ e1124=e1134=e2215=

e2235=e3316=e3326

-2.694 -5.098

l≠n=n’ e1224=e1334=e2115=

e2335=e3116=e3226

-1.019 1.590

l≠n≠n’ e1234=e2135=e3126 -5.636 -1.962

𝒏 ⟺ 𝒏′⟺𝒏′′

𝑒𝑙𝑛𝑛′𝑛′′𝑚 l=n=n’=n’’ e11114=e22225= e33336 -0.840 21.063

l=n=n’≠n’’ e11124=e11134=e22215=

e22235=e33316=e33326

-0.241 12.112

l≠n=n’=n’’ e12224=e13334=e21115=

e23335=e31116=e32226

-9.168 -15.072

l≠n=n’≠n’’ e12234=e21135= e31126 -1.471 -7.450

n≠n’≠n’’ e11234=e21235=e31236 -4.725 -4.909

72

4 WZ Semiconductors

4.1 Wurtzite III-Nitride Semiconductors

III-N semiconductors, within the III-V group, reveal greater effects on their electrical and

optical properties from piezoelectricity [108] as a result of the tendency among such

semiconductors to have greater PZCs than others in the III-V group by one order of

magnitude [30]. The difference is also due to the generally parallel layout of the polarization

vector with direction of growth [109]. Frequently, the possibility of incorrect piezoelectricity

coefficients affects the ability to calculate semiconductors’ PZ properties. Further, for

wurtzite crystals, there is also an issue concerning identification of the spontaneous

polarization (Psp) component [30]. Second order piezoelectric coefficients have not been

given for III-N materials, with the result that it is challenging to evaluate the impact of

piezoelectricity of second order on nanostructures.

4.1.1 DFT Calculations

Evaluation of Z* and elastic deformation in strained and bulk subjects was conducted with

reference to planewave pseudopotential, wherein the pseudopotentials were calculated

following the Troullier-Martin scheme,[57] as well as density functional theory within local

density approximation [52] and density functional perturbation theory (DFPT), wherein the

Hamann scheme was used to derive pseudopotentials.[110] CASTEP [106] code was used.

Convergence beneath approximately 1% remaining error was achieved in various

configurations of kinetic energy and k-point grids. For this, single-particle orbitals shown

through a planewave basis set, and as much as 103 eV of kinetic energy, with Brillouin zone

summations reaching a maximum of 10x10x6 Monkhorst-Pack k-point grids [61] were

sufficient to converge the simulations below a remaining error of about 1% for multiple

combinations of k-point grids and kinetic energy.

73

Born charge matrix was used in calculating dynamic effective charge, and the Berry phase

approach [111] was followed through application of a finite electric field perturbation within

periodic boundary conditions. Matrix diagonalisation took place and the effective charge was

calculated by averaging eigenvalues obtained. Dependence for both strain and bulk was

calculated following the same procedure.

Characterisation of wurtzite crystals is possible through a, in-plane and c, off-plane lattice

parameters as well as u, which represents the deformation of the structure from ideal. The Psp

effect comes from u, where tetrahedrons remain asymmetric notwithstanding any external

strain, and thus just three from four bonds are of identical length. Further alterations in the

positions of cations compared with anions result in polarisation induced from strain where

pressure is applied externally. The previous study presents the benefit of allowing description

of polarisation of spontaneous and strain-induced type in a single model, creating a novel

connection between findings regarding PZCs and Psp. In addition, inputs are taken solely

from e31 PZC, with calculation of e33 and e15.

4.1.2 Linear Piezoelectric coefficients

Table V gives equilibrium values obtained through DFT calculations for a, c and u in A1N,

GaN, InN and GaN, in addition to Z* as well as bond polarity αp. The findings gave

analogous values compared with previously reported work. [30,112] Table V also presents

values for e31 obtained through experiment and utilised in fitting ZH* for the model, as well as

values found for ZH*. The latter are significantly lower than values for Z*, which point was

discussed above. ZH* represents atomic charge, for which an in-depth explanation was given

in Section 3.1.4.

74

The models predictions reveal significant agreement levels in comparison with values given

through experiment in previous work for e33 in III-N in bulk cases. In the case of e15 however,

the estimated values are consistently a little higher in comparison to those previously

reported. Further, the previous model predicts the sign of e15 to be negative across each of the

III-N cases considered here. The literature reports positive [36,113] as well as negative [114]

values for e15 however, this appears to come from a typographic error when Muensit et

al.,[36] cite values from experimental work by Tsubouchi and Mikoshiba [37], and turn a

negative into a positive sign [114].

Table V: Physical parameters of Group-III Nitrides (GaN, AlN and InN) calculated in

this work. In brackets comparison with other calculated or experimental values.

Reproduced from Reference [115]

Parameters GaN AlN InN

a (Ǻ) 3.155 3.063 3.523

c (Ǻ) 5.149 4.906 5.725

u (Ǻ) 0.376 0.382 0.377

Z* 2.583 2.553 2.850

αp 0.517 0.511 0.578

Z*H 0.70 0.85 0.65

Psp (C/m2) -0.007(-0.029th)[30] -0.051 (-0.081th)[30] -0.012 (-0.032th)[30]

e31(C/m2) -0.55 (-0.55exp)[36] -0.6 (-0.6exp)[36] -0.55 (-0.55exp)[38]

75

e33 (C/m2) 1.05 (1.12exp)[36] 1.47 (1.50exp)[36] 1.07 (0.95exp)[38]

e15 (C/m2) -0.57(-0.38th)[114] -0.6 (-0.48exp)[37] -0.65 (-0.44th)[114]

e311(C/m2) 6.185 5.850 5.151

e333(C/m2) -8.090 -10.750 -6.680

e133(C/m2) 1.543 4.533 1.280

Bernardini and Fiorentini [30] then took values from Muensit et al., [36] and not the first

authors, with the result that others using this report to list piezoelectric coefficients [114].

Vurgaftman and Meyer [116] for example repeat the mistake. Previous work by the current

authors with zincblende InAs and GaAs [85] shows that the ZH* value must be approximately

25% of that of Z* so as to reach PZ polarization values which agree with experimental

values.

4.1.3 Spontaneous Polarization

The predictions made in the III-N work differ most significantly from the previously reported

values, which are given in Table V in parentheses, in their Psp values. There is no data from

experiment presently published for such values. Psp was calculated in an identical process to

that for identifying PZCs after establishing the ZH* value. Psp values were much smaller than

those in previous literature, being from 25%-65% of those previous values. [30,114] The

previous frequently used method of calculation for Psp in comparison with the one used here

accounts for this difference. Earlier models generally have a simple dipole model where Z*

and charges are considered to equal each other. This relates to the first term of the model

which is taken from Harrison’s expression,[43] and the smaller ZH* is used rather than Z*,

76

bringing lower values in proportion to the difference between ZH* and Z*. This is in response

to the issue of significant overestimation of results when Z* is used to calculate Psp and

PZCs, as stated in Bernardini and Fiorentini [30]. In addition, this ties in with the

significantly lower experimental extrapolation of values for Psp [117,118,119] compared with

values from literature [30,114].

4.1.4 Strain dependence of the polarization

Work conducted also included a consideration of induced strain and Psp, taken together to

form total polarization. The aim of this was to explore the possibility that Wurtzite crystals

are affected by second order piezoelectric effects with strain. While previous reports exist for

non-linear polarization effects in III-N semiconductors, [120,121] It has not found any

previous work which gives a complete description of second order PZCs. Earlier work by the

same authors concerning ZB InGaAs [85,104] demonstrates that the origin of second order

effects within strain are found in interpenetrating cation and anion fcc sublattices being

displaced in a non-linear manner, and this was found where DFT-LDA was used also.

Additional second order strain-dependence, albeit less strong, is also seemingly shown by

effective charge, from which it follows that this is also the case for bond polarity. Wurtzite

crystals also show evidence of this phenomenon, and this causes total polarization to show

non-linear tendencies. Figure 4.1.4-1 presents total polarization in terms of a function of

perpendicular and parallel strain combined and which may range between -0.1 and 0.1,

obtained through calculations using the model in circles, which is then considered in

comparison with a linear model with use of parameters taken from Bernardini and Fiorentini

[30] and shown by dashed lines.

77

Figure 4.1.4-1 Comparison of the total polarization as a function of perpendicular and parallel

strain calculated in III-N work (circles) and that calculated using the linear model with

parameters from Ref [30] (dashed lines). The perpendicular strain varies from -0.1 to 0.1 in

steps of 0.02. Reproduced from the work of Reference [J. Pal, G. Tse, V. Haxha, M. A.

Migliorato and S. Tomić, Journal of Physics: Conference Series 367,012006 (2012)].

In-plane Strain (e//)

Po

lari

zati

on

(C

/m2)

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

Linear Pz

This Work

b) AlN

e

-0.10 -0.05 0.00 0.05 0.10-0.3

-0.2

-0.1

0.0

0.1

0.2

Linear Pz

This Work

c) InN

e

-0.2

-0.1

0.0

0.1

0.2

0.3-0.10 -0.05 0.00 0.05 0.10

a) GaN Linear Pz

This Work

e

78

It is clear from the above that the two models differ significantly in the lower Psp values

offsetting the different constant stress lines seen on the c-axis. In addition, the second order

terms used in the current model act to bend the lines, and this is not surprising in a quadratic-

dependent second order model.

When contrasted with the linear model, the III-N model is consistent in predicting much

lower positive total polarisation in tensile cases and much higher in compression where

perpendicular strain is high and parallel 0001-plane strain is not larger than ±0.08. This can

be illustrated through the case of a layer of thin film GaN layer produced through

pseudomorphic growth on top of AlN. In this case, -3% strain would be observed in GaN

along the plane of growth ε// while +6% would be seen along ε┴ or the c axis. This creates a

value of +0.095 C/m2 for linear polarisation with a significantly smaller value for the second

order, +0.06 C/m2, which represents an important difference[122].

In fact, if Psp were ignored, and strain induced polarization only were examined, the models

described above would produce highly similar results in cases of small strain. However, as

strain grows, so does the effect of second order terms, bringing stronger variance between the

models. With introduction of Psp, it may be seen that the III-N models give similar results

where there is significant compression strain along the plane, but this should not be

considered as more than a coincidence, with no foundation in physical properties. The major

finding of note here is rather that when considering cases of up to ±10% strain, the current

model estimates significantly narrower positive ranges and significantly broader negative

ranges for obtainable PZ fields [122].

79

4.1.5 Second Order Piezoelectric coefficients

The decision to fit data with a polynomial of the second order it was possible to attain second

order PZCs related to quadratic terms within perpendicular and parallel strain combinations,

although strains with shear coefficients could not be acquired. Table V presents these

findings. From the table, the appearance of 133 in the subscript indicates that perpendicular

and parallel strain are combined, while 311 denotes double strain in the plane and 333

indicates perpendicular double strain. Table V presents coefficients which facilitate

expression of strain dependence to the magnitude of total PZ polarization orthogonal to the

plane in the following way:

PTot = PSp + e33ε⊥ + 2e31ε∥ + e311ε∥2+e333ε⊥

2 + e133ε∥ε⊥ (61)

The III-N work does not assess the impact of shear strain along the growth plane on second

order dependence in polarization, as related to PZC e15. However, while this dependence may

potentially influence some nanostructures: for example quantum dots, it is not considered

applicable to thin film in two dimensions, and thus the validity of equation above is supported

in the III-N investigation[122].

This section has reviewed the previous work on WZ III- N semiconductors and the results of

the recent work will be presented in the next section.

80

5 Current Work on Wurtzite III-Vs Semiconductors

Generally, III-V semiconductors in the WZ phase (e.g. III-N) have a larger PZ response

compared to ZB crystals (e.g. III-As). Furthermore the polar axis of WZ crystals is typically

parallel to the growth direction, unlike the [111] polar axis of ZB materials [85,104].

All WZ III-V semiconductors, under strain, would lead to the generation of electric dipoles

and a resultant PZ field along the polar axis [0001] of the crystal. One would assume that all

III-V semiconductors, if synthesis as stable WZ crystals rather than ZB was possible, would

result in large PZ coefficients. This was reported not to be true by Bernardini et al.,[113] who

found e.g. WZ InP, GaP, InAs and GaAs to have PZ coefficients one order of magnitude or

more smaller than those of the III-N materials.

The question that intend to answer in this project is whether such WZ NWs, in addition to

favourable optical properties, also possess electrical properties and a PZ response that would

be sufficiently large to allow the use of these materials in piezotronic devices and NGs.

Piezoelectricity in semiconductors has long been treated as a linear effect in the strain.

Non-linearity has instead been recognized to have a significant magnitude in ZB III-V

[85,104,] WZ III-N[122,123] and ZnO [124] semiconductors. However, in the WZ crystal

phase, for some III-V semiconductors, second-order PZCs have not yet been reported,

making it difficult to assess the influence of non-linear piezoelectricity in NWs and CSNWs.

To resolve the issue of calculating the spontaneous and strain-induced PZ effect in WZ

structures, It has been reported the quadratic piezoelectric coefficients (PZCs) of four III-V

semiconductors, namely GaP, InP, GaAs and InAs, and shown the magnitude of such

coefficients is vital and non-negligible in any calculation of the polarization field. It has been

also performed calculations on the properties of III-V CSNWs grown in (0001) direction.

81

5.1 Evaluation of Linear and Non Linear Piezoelectric

Coefficients

In order to evaluate the linear and non-linear piezoelectric coefficients (LPZCs and

NLPZCs) we use the same method [125] that was proved accurate when calculating the

NLPZCs of ZnO [124], III-N [122 ,123] and III-As [84,85] semiconductors. The method,

based on Harrison’s formalism [43] involves a semi-empirical approach where PZ charges

are the sum of a bond and a direct dipole contribution. In the model, Harrison’s effective

charge is always determined such that when the bond polarity and the elastic deformation are

computed within small strain limits, the PZCs tend towards known experimental values

[125]. Model data was used in the evaluation of the PZCs and obtained via plane-wave

pseudopotential (the Troullier–Martin approach [57] was used for pseudopotential), with

density functional theory with local density approximation (DFT-LDA) [52] and density

functional perturbation theory (DFPT) via the Hamann approach [56] in the CASTEP code

[106]. The Berry phase approach [126], applying a finite electric field perturbation within

periodic boundary conditions, was also used. DFT calculated equilibrium values are given in

Table VI for WZ III-P and WZ III-As along with the values of the effective charge (Z*), the

resulting bond polarity (αp) and the much smaller Harrison’s effective charge (𝑍𝐻∗ ).

The linear and quadratic parameters e33, e31, e311, e333 and e313 are obtained when the DFT data

for the strained crystal is fitted to the to equation (61).

For all four materials listed, the values of 𝑍𝐻∗ were fitted to experimentally known linear

parameter of the ZB phase, as no experimental values can be found for the WZ phase. The

NLPZCs that we obtained are listed in Table VI.

82

Table VI. Calculated and measured physical parameters for III-P and III-As used in the

calculations. Comparisons between our calculated values and other

calculated/experimental ones are given in brackets.

Parameters GaP InP GaAs InAs

a(Å) 3.789(3.759[127])exp. 4.115(4.054[128])th. 3.928(3.912[129])th. 4.248(4.192[128])th.

c (Å) 6.253(6.174[127])exp. 6.753(6.625[128]) th 6.482(6.440[129])th 6.969(6.844[128]) th

U 0.371(0.374[127]) 0.374 (0.375[128]) 0.3712(0.374[129]) 0.374(0.376 [128])

Z* 1.86 2.33 1.75 2.29

𝜶𝑷 0.39 0.47 0.37 0.46

𝒁𝑯∗ 0.51 0.35 0.43 0.54

C33(GPa) 1.722[130] 1.438[130] 1.602[130] 1.209[130]

C13(GPa) 0.468[130] 0.386[130] 0.334[130] 0.321[130]

Psp 0.004(0.003[42]th. 0.001(-0.001[42]) th. 0.002(0.002[42]) th. 0.001(0.001[42]) th.

e33 (C/m2) 0.48(-0.07[30]) th. 0.59 (0.04[30]) th. 0.32(-0.12[30]) th. 0.51(-0.03[30]) th.

e31 (C/m2) -0.26(0.03[30]) th. -0.24(-0.02[30]) th. -0.17(0.06[30]) th. -0.26(0.01[30] )th.

e311 (C/m2) 1.64453 1.64704 0.87939 1.83548

e333 (C/m2) -2.89973 -2.64647 -1.61433 -2.75858

e313 (C/m2) 1.08261 0.83724 0.5155 0.9449

83

5.2 Internal Displacement (u) and Effective Charge (Z*)

In order to observe the impact of combinations of perpendicular strain and isotropic parallel

strain, the strain dependence of the internal displacement parameter (u) is examined.

The strain combination consists of the in-plane strain i.e. the strain in the xy-plane

(ɛxx=ɛyy=ɛ∥) and the strain perpendicular to the xy-plane, which is in the z-direction (ɛ⊥). The

results show the movement of the atoms gets larger as the crystal is compressed more in the

plane. The results of all such simulations are plotted in the Figure 5.2-1, Figure 5.2-2, Figure

5.2-3 and Figure 5.2-4 the plots are showing the strain dependency of the atomic movements

in the crystal structure when performed the geometry optimization. The atomic movement in

the z-direction is relatively much larger than the displacements along the x and y-axis. The

WZ crystal structure of III-As and III-P are compressed in the plane when ɛ∥ is negative

while the positive values of the ɛ∥ represent the tensile strain of the crystal lattice.

84

Figure 5.2-1 Strain dependence of internal displacement parameter (u) as a function of in-

plane and perpendicular strain (from -0.08 to 0.08) for GaAs.

-0.08 -0.04 0.00 0.04 0.08

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

In-plane Strain (||)

Dis

pla

ce

me

nt

in Z

-ax

is Å

Internal displacement (u)

GaAs

85


plane and perpendicular strain (from -0.08 to 0.08) for InAs.

-0.08 -0.04 0.00 0.04 0.08-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3Internal displacement (u)

InAs


Dis

pla

cem

en

t in

Z-a

xis

Å

86


plane and perpendicular strain (from -0.08 to 0.08) for GaP.

-0.08 -0.04 0.00 0.04 0.08-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3GaP



Dis

pla

cem

en

t in

Z-a

xis

Å

87


plane and perpendicular strain (from -0.08 to 0.08) for InP.

This work explains the effects of the compressive and the tensile in-plane strain in the III-As

and III-P WZ crystal structure. Also the case of the perpendicular strain in z-direction,

described in the current work, considers the nature of the strain to be tensile.

The effect of the born effective charge (Z*) is also examined for the strain in case of the

compression and the tension in the plane. The results include several unique combination of

the strains applied in both in-plane and perpendicular to the plane.

Figure 5.2-5, Figure 5.2-6, Figure 5.2-7 and Figure 5.2-8 show the effects of strain on Z* for

strain along perpendicular ɛ⊥ and In-plane strain ɛ∥ varies by 10% on application of a unique

combination of strain.

-0.08 -0.04 0.00 0.04 0.08-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3


InP


Dis

pla

cem

en

t in

Z-a

xis

Å

88

Figure 5.2-5 Z* as a function of plane strain (from -0.08 to 0.08) along with perpendicular

strain, for GaAs.

-0.08 -0.04 0.00 0.04 0.081.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4GaAs


Z*

89


strain, for InAs.

-0.08 -0.04 0.00 0.04 0.081.8

2.0

2.2

2.4

2.6

2.8

InAs


Z*

90


strain, for InP.

-0.08 -0.04 0.00 0.04 0.082.0

2.2

2.4

2.6

2.8InP


Z*

91


strain, for GaP.

5.3 Spontaneous Polarization

The first notable prediction contained in the current model data is that smaller values of the

spontaneous polarization (Psp) are predicted compared to those of other WZ semiconductors

[125]. However, for the same materials, such reduced values are instead well matched to

calculated values [42] using the Berry phase (electrostatic) approach of polymorphic

structures with mixture of WZ and ZB [141]. Furthermore (Psp) is positive for the case of III-

P and III-As, in both the calculations of Belabbes et al., [42] and in the current work (with

the only exception of InP which has an opposite sign though is very small in magnitude). The

similarity between the values of (Psp) obtained through independent methods by Belabbes et

al., [42] and this work, provides further confidence in Harrison’s method. Very recent

experimental work on WZ GaAs NWs by Bauer et al., [131] reported the (Psp) value to be in

-0.08 -0.04 0.00 0.04 0.081.2

1.6

2.0

2.4

2.8

GaP


Z*

92

the same range (0.0027±0.0006 C/m2) as my predicted value (0.002 C/m2). Since in this

method the framework to evaluate (Psp) and the PZCs is consistent, it can also draw

confidence in the validity of the predictions for e31 and e33. The second notable prediction is

that the coefficients e33 and e31 (Table VI), which were originally predicted to be negligible

[30] are instead sizeable and not very dissimilar to those of the III-N systems (e.g. in GaN

e31= 0.55 C/m2, e33=1.05 C/m2) [125], albeit slightly smaller (roughly half). It is also worth

noting here that these results appear to always differ in sign from the published values by

Bernardini and Fiorentini [30], which is most likely due to having used a different convention

for the [0001] crystal direction.

5.4 Strain dependence of the polarization

Figure 5.4-1, Figure 5.4-2, Figure 5.4-3 and Figure 5.4-4 show the total polarization as a

function of combinations of parallel and perpendicular strain (varying from -0.1 to 0.1)

calculated with our model (tringle) and compared with the predictions from the linear model

(lines) using parameters compiled by Bernardini and Fiorentini [30].

The most notable conclusion is that contrary to what was commonly believed WZ III-As and

III-P semiconductors appear to possess linear PZ properties that are comparable with WZ

III-N semiconductors and that hence can too be exploited as materials for piezotronics and

NGs.

93

Figure 5.4-1 Dependence of the total polarization (C/m2) of wurtzite GaP on combination of


(quadratic) model (NLM).

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10

GaP

LM

NLM


Po

lari

zati

on

(in

C/m

2)

94

Figure 5.4-2 Dependence of the total polarization (C/m2) of wurtzite InP on combination of



-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

InP NLM

LM


Po

lari

za

tio

n (

in C

/m2)

95

Figure 5.4-3 Dependence of the total polarization (C/m2) of wurtzite GaAs on combination of



-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10

-0.12

-0.08

-0.04

0.00

0.04

0.08

0.12 GaAs LM

NLM

Po

lari

zati

on

(in

C/m

2)


96

Figure 5.4-4Dependence of the total polarization (C/m2) of wurtzite InAs on combination of



Thus, we have estimated the linear and quadratic piezoelectric coefficients of III-V wurtzite

crystals and the magnitude of the quadratic terms is significant and necessitates inclusion even

in the limit of small strain.

5.5 ZnO Semiconductors

The field of piezotronics originates in the possibility of using ZnO nanowires to act as

mechanical sensors to detect energy, only lately established. Following this, extensive

opportunities are seen arising from the potential combination of electronic and piezoelectric

properties through semiconductors, including equipment which powers itself, flexible

electronics and nanogenerators [3]. For II-VI semiconductors and in an analogous manner to

III-V semiconductors, where the strain contains an element which is in line with the crystal’s

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

InAs

LM

NLM

Po

lari

za

tio

n (

in C

/m2)


97

polar axis, electrical dipoles are generated. In the case of WZ crystals there is a connection

between these dipoles and components of the diagonal strain tensor, causing formation of a

piezoelectric field on polar axis [0001]. While much literature concerning piezoelectric fields

in semiconductors considers effects in strain as being linear, non-linear elements are

discussed within Chapter Three and Chapter 4 for semiconductors based on wurtzite III-N,

III-V and zincblende III-V.

A discussion is presented below concerning quadratic PZCs of ZnO, as recently established,

[124] which contends that these coefficients are significant to the extent that they are

essential in calculation of the polarisation field. As with III-V wurtzite semiconductor

polarisation, non-linear polarisation was seen for ZnO, with a total polarization equation

being the same as (61) where values were those as shown by Table VII.

It is significant that the model used gives lower values for spontaneous polarization in

comparison with those calculated previously [113,132] as is anticipated based on reasons

discussed previously in WZ III-N semiconductors [122].

5.6 Methodology

Similarly, the selected method is semiempirical in approach and involves Harrison’s

suggested calculation of piezoelectric charge based upon the sum of a bond contribution and

a direct dipole contribution[136]. This approach has precedents, with testing with the

semiconductors III-N [122] and III-As. [84,85] both non-linear and linear coefficients are

found via complete polarization in equation (10) and this is found by calculating the sum of

strain induced polarization and spontaneous polarization ; where gives Cartesian direction,

δr represents the displacement vector for cations in relation to ideally placed anions, rq is

the distance vector of q, the closest neighbour of the central atom in the tetrahedron, δRq is

98

the displacement vector of the same, while 𝛼𝑝 represents bond polarity and Ω represents

atomic volume. As in tight binding, the atomic charge is shown by ZH*, while the Born

charge, or dynamic effective charge, required for assessment of bond polarity, is given by

(54) In the model, 𝑍𝐻∗ must always be defined in order that following calculation of 𝛼𝑝and

elastic deformation within small strain limits, as in with Bulk crystals, correct reproduction of

PZC experimental values is achieved by the model.

In strain and bulk conditions, evaluation of elastic deformation and Z* took place by using

the same method that mentioned earlier via planewave pseudopotential (The Troullier–Martin

approach was used to derive pseudopotentials [57]), density functional theory for local

density approximation (DFT-LDA) [52] and density functional perturbation theory (DFPT) in

CASTEP[106].

The Born charge matrix was used to calculate dynamic effective charge, and this was then

analysed with use of the Berry phase approach [126], through application of a finite electric

field perturbation within periodic boundary conditions. Following this diagonalisation of the

matrix took place, and the effective charge was determined by taking an average from the

Eigen values. The bulk dependence and strain dependence were calculated in a similar

fashion.

It also shows Z* values, as well as resultant bond polarity 𝛼𝑝 and ZH*values: these last are

very small in comparison to Z* values, as previously discussed.

5.6.1 Piezoelectric Coefficients

The ab initio DFT data is easily combined using equation (10) with the only difficulty being

the calculation of the geometrical factor that multiplies the bond polarity. This requires

99

combining the strained positions of all the atoms in the tetrahedron under consideration,

which have already been obtained in the calculations of the internal distortion.

The result is the values of the total polarization for a given combination of a parallel and

perpendicular strain. In order to obtain the linear and quadratic parameters e33, e31, e333 and

e133, the data is then fitted to the same equation (61).

Since it needed at least one of the linear parameters to obtain the correct value of 𝑍𝐻∗ , in this

calculation, of the linear coefficients only e33 was fitted. For the coefficient e31 , I had to

make a suitable choice based on experimental values [133,134]. In choosing the value of e31 I

ensured that the resulting calculated value of e33 would also be in the range of the available

experimental data [133,134]. The values obtained are listed in TABLE VII. Note that in our

model smaller values of the spontaneous polarization are predicted compared to previous

calculations[30,135]. The reason behind these is explained in detail in our previous work on

wurtzite III-N semiconductors[122].

The dependence of the total polarization on strain in the range -0.8 to + 0.8 according to the

classic linear model (LM) and the current nonlinear (quadratic) model (NLM) is shown in

Figure 5.6.1-1. The main feature is that the NLM appear to always predict either less negative

or more positive values of the polarization compared to the LM. This is part a result of the

smaller values of the spontaneous polarization but also due the nonlinear effects which

manifest through the coefficients e333 and e133 when the strain is sufficiently large. It has also

observed that in same cases the LM and NLM predict opposite signs for the polarization, e.g.

around 2% compressive parallel strain gives a value of the total polarization of -0.04 C/m2 or

+0.01 C/m2 using the LM or NLM respectively.

100

Table VII. Calculated and measured physical parameters for ZnO used in the

calculations. Comparisons between our calculated values and other calculated and/or

experimental ones are given in brackets. Reproduced from Reference [H.Y.S. Al-

Zahrani, J. Pal and M.A. Migliorato, Nano Energy 2 (2013) 1214.]

Parameters

a (Å) 3.18 (3.25[30])th

c (Å) 5.16 (5.207[30])th

u 0.375 (0.375[30])th

Z* 2.164 (2.11[30])th

αp 0.67 (0.69[136])th

ZH* 0.23

C33 (GPa) 176[137]

C13 (GPa) 84[137]

Psp(C/m2) -0.01 (-0.057[30], -0.047[132])th.

e33 (C/m2) 1.15(1.22 ±0.04,[138] 0.96[39])exp

e31 (C/m2) -0.61(-0.51±0.04,[138] -0.62[39])exp

e311 (C/m2) 3.98

e333 (C/m2) -5.59

e313 (C/m2) 1.21

101

Figure 5.6.1-1 Dependence of the total polarization (C/m2) on strain in the range -0.08 to +

0.08 according to the classic linear model (LM) and our non-linear (quadratic) model (NLM).

The red square and blue dot resemble the NLM and LM prediction at -2% In-plane strain.

Reproduced from Reference [H.Y.S. Al-Zahrani, J. Pal and M.A. Migliorato, Nano Energy 2

(2013) 1214.].

Thus, the linear and quadratic piezoelectric coefficients of ZnO wurtzite crystals have

estimated and the magnitude of the quadratic terms is significant and necessitates inclusion

even in the limit of small strain. The next chapter discusses the NWs and Core-Shell NWs

that made from semiconductors WZ materials such as II-VI and III-V.

102

6 Nanowires and III-V Core-Shell Nanowires

This chapter shows the effect of II-VI WZ ZnO NWS and its potential applications in

addition to III-Vs WZ Core-shell NWs in enhancing the piezoelectric field.

6.1 Quantum Nanowire properties

NWs made from III-V and II-VI semiconductors [139], with excellent control over

dimensions and compositions, can be now routinely grown by metal organic chemical vapour

deposition MOCVD [140] or molecular beam epitaxy (MBE) using a vapour−liquid−solid

method with gold as catalyst, allowing for NWs containing many variations and

combinations of different materials. NWs in this case are vertically grown, epitaxially on a

lattice mismatched semiconductor substrate. Depending on the substrate orientation both

zincblende (ZB) and wurtzite (WZ) crystal structures are possible.

Moreover, NWs can be grown on other substrates such as glass, metals and polymers [141].

Glass substrates are a very attractive alternative to semiconductor substrates, because of

availability, transparency and cost. On the other hand, the amorphous state of the substrate

and the presence of impurities cause limitations to growth on a glass surface. Nevertheless

defect free ZB GaAs NWs grown on glass substrates by using a horizontal MOVPE process

involving trimethylgallium and tertiarybutylarsine at growth temperatures of 410-580°C have

been found to exhibit bright photoluminescence emission which implies good crystal quality

[141].

The stability of crystals varies notably when overall dimensions approach the nanometre

scale. In fact, with smaller sizes, the energy present on the nanostructure’s surface takes

greater significance in assessing the overall free energy change in the crystal formation[142].

Previous research indicates that lower surface energy is seen in WZ crystals as compared to

103

the ZB lattice [143]. From this, it can be postulated that once the surface area of the WZ

nanostructures reaches a particular proportion in relation to volume, thermodynamic

conditions may become favourable. In support of this view, a number of experimental works,

[144,145,146,147,148,149] using epitaxy or catalysis driven vapour-liquid-solid growth, has

resulted in WZ single-crystal NWs. The nanostructures created show optoelectronic

parameters different to those usually seen in the equivalent ZB bulk lattices [150,151,152].

As an example of this, NWs consisting of GaP in the WZ phase shows direct bandgap as

opposed to the indirect bandgap typical of the ZB phase [158]. Based on improvements in

terms of creating suitable alloys, there is potential for WZ NWs being used even in the silicon

system as effective and ecological emitters [143]. This is with the provision that surface-

based non-radiative recombination is avoided as far as possible by means of appropriate

passivation of the surface or with a sufficiently large NW diameter. A limitation exists

however in the maximum diameter that ensures crystal stability in the WZ phase. Instability

leads to the development of the nanostructure into mixed hexagonal and cubic lattices

[156,153,154] (also known as polymorphism) which create discontinuities in the band profile

and hence places restrictions in terms of electro-optical performance.

Work conducted with III-As and III-P shows that only if the material is synthesized as NWs,

a defect free WZ phase can be obtained [155]. Crystalline structural alteration accompanies

electronic structural change. Previous hypotheses suggested that the ZB phase would align as

a type II band with the WZ phase when utilised in III-V semiconductors[156]. In more recent

studies, there is evidence of type-II band alignment for InP NWs [157,158] giving rise to

crystal-phase quantum-dot formation[159]. A broad variation of band edges, from 1.43eV to

1.54eV is observed when measuring luminescence of GaAs NWs with a predominant WZ

phase [160,161,162,157, 172,163] showing how the interplay between the ZB and WZ

104

sections affects the homogeneity of the band edges, occasionally forming quantum confined

regions due to the different line-ups of the two crystal phases.

6.1.1 Synthetic techniques of ZnO NWs

Usual synthetic approaches in ZnO NWs rely on vapour phase synthesis, which takes place

under conditions of elevated heat. This is a catalytic reaction based on vapour dissolving

from the material of the semiconductor, for example ZnO, and forming nanoparticles of

metal: generally, gold. After this, super-saturation occurs, followed by crystal

formation[164]. ZnO nanostructure growth, which includes creation of NWs and nano-belts,

has been achieved through a range of approaches based on the basic method of vapour-to-

solid under extreme temperature [165,166,167]. However, in order to produce such growth in

a commercially applicable manner, it is necessary to create particular atmospheric and

temperature conditions which are costly to reproduce, as well as the high level of expenditure

represented by the insulation or substrate required to successfully direct nano-rod growth.

The technique is rendered more complex by the necessity of depositing a single gold

nanoparticle layer on top of the substrate so that epitaxial growth can be catalysed [168].

Alternative approaches to synthesising of NWs from ZnO are varied, and both template

assisted growth [169] and electrophoresis [170] are used. Approaches which do not require

high temperatures for metal oxide NWs are frequently based upon hydrothermal growth

which leads crystals in solution to grow epitaxially and in an anisotropic manner [171,172].

This approach generally does not depend on substrate [173], and in addition allows an

acceptable level of control of NW nanowire morphology. Considering the different synthetic

approaches to the production of ZnO NWs, hydrothermal ZnO growth based on sol-gel likely

105

presents the greatest energy efficiency as it does not require either high temperature or a

vacuum for the process.

As a result of the high surface to volume ratio and the size, these wires show significantly

different behaviour in optical, thermal, mechanical, magnetic and electrical properties from

the bulk material. The NWs provide an exciting framework to apply the “bottom up”

approach (Feynman, 1959) for the design and modelling of nanoscience applications. The

exploitation of these unique and novel properties of the NWs have resulted in widespread

applications from nanophotonics, piezotronics, thermoelectrics, energy harvesting.

6.1.2 ZnO NWS applications

Nanostructures based on ZnO present vast potential for application, including in flat screens,

field emission sources, chemical and gas sensors [174] and biological sensors, as well as

ultraviolet light switches and emitters.[187,175,176,177] Within the rage of potential

applications, individual crystals and epitaxially created layers have importance in creating

light detectors and emitters in the blue and ultraviolet ranges [178], as well as

piezoelectric[179] and spintronic [180] equipment. Further, such nanostructures might be

combined with GaN to create lighting to serve the current age [181]. Further potential is seen

in epitaxial ZnO in the form of a transparent thin-film semiconductor [182] in terms of

application in gas sensors, solar cells, wavelength specific applications and displays. Also,

transformations are taking place in extant technology as a result of the use of ZnO

nanoparticles. These changes include the areas of sun-protective skin cream, as well as

coatings and paints. Further potential is seen for space-based projects based on ZnO’s level of

imperviousness to MeV proton irradiation [183].

106

It is clear from the above discussion that while ZnO currently enjoys broad usage in a variety

of contexts, this usage is small in comparison to its potential social and industrial benefits.

This potential stems from unique features of ZnO which, while partially recognised and used

in terms of their application, remain under investigation currently. This is therefore an

exciting time in the development of ZnO applications, as potential uses are translated into

practical benefits [184].

6.1.3 The Example of NWs

In the current work on ZnO semiconductors, a calculations of the total polarization in ZnO

nanowires are shown and reported that for particular strains originating from an external

force, the non-linear model (NLM) of piezoelectricity predicts both positive and negative

polarizations in the nanostructure whereas the linear model (LM) only predicts negative

values. The Linear Elasticity in calculations have been considered for III-N model [125]. A

schematic diagram of a strained nanowire is shown in Figure 6.1.3-1. The LM and NLM is

tested by calculating the polarization in a ZnO nanowire subjected to a bending force

deforming the cylindrical shape into an arch. For simplicity it has been assumed that such

deformation would result in a polarization that is isotropic for each circular cross section of

the nanowire. It is also assumed that the resulting perpendicular strain 휀⊥is antisymmetric

along the section of the bent cylinder. Since in most materials compressibility is always lower

than the ability to withstand tensile deformation, this is a correct assumption only for small

strains. The perpendicular strain is related to the parallel strain 휀∥ through the elastic

constants of the material, given in TABLE VII: The combination of parallel and

perpendicular strain in equation (8) is then used in equation (61) to evaluate the polarization.

107

In Figure 6.1.3-1 it is shown the variation of the polarization for the case where the

perpendicular (parallel) strain 휀⊥(휀∥) varies from -2.8% (+2.8%) to +2.8% (-2.8%).

There are marked differences between the predictions of the LM and NLM. In particular, the

NLM predicts a gradient of the polarization ranging from -0.08 C/m2 at the compressed end

of the section, to +0.06 C/m2at the tensile end. The LM polarization instead ranges from -

0.12 C/m2 to 0.0 C/m2 within the same range of strains.

This demonstrates how the LM and NLM can produce opposite predictions.

Figure 6.1.3-1 Variation of the polarization (C/m2) in a cross section of a ZnO nanowire. The

perpendicular (parallel) strain varies from -2.8% (+2.8%) to +2.8% (-2.8%). The calculated

polarization of the non-linear (quadratic) model (NLM) is on the left half and the classic

linear model (LM) on the right. Reproduced from the work of Reference [M.A. Migliorato, J.

Pal, R. Garg, G. Tse, H. Y. S. Al-Zahrani, U. Monteverde, S. Tomić, C-K. Li, Y-R. Wu, B. G.

Crutchley, I. P. Marko and S. J. Sweeney, AIP Conf. Proc. 1590 (2014) 32].

ZnO

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

NLM

PTotal

= 0.0 C/m2

LM

108

The classic LM [30] and our NLM [126] is also used to calculate the polarization in Nitride

NWs i.e., GaN and InN subjected to a bending force deforming the cylindrical shape into an

arch. The effects of non-linear elasticity (NLE) [73] are also included, because of the

importance of having a precise calculation. Figure 6.1.3-2 and Figure 6.1.3-3 illustrate the

polarization difference of GaN and InN respectively, where the perpendicular (parallel) strain

휀⊥(휀∥)differs from -4% (+4%) to +4% (-4%).

Figure 6.1.3-2 Variation of the polarization (C/m2) in a cross section of a GaN nanowire. The

perpendicular (parallel) strain varies from -4% (+4%) to +4% (-4%). The calculated

polarization using NLE parameters of the non-linear (quadratic) model (NLM) is on the left

half and the classic linear model (LM) on the right. Reproduced from the work of Reference

[M.A. Migliorato, J. Pal, R. Garg, G. Tse, H. Y. S. Al-Zahrani, U. Monteverde, S. Tomić, C-

K. Li, Y-R. Wu, B. G. Crutchley, I. P. Marko and S. J. Sweeney, AIP Conf. Proc. 1590 (2014)

32].

By using the NLE parameters, there are differences between the predictions of the LM and

NLM. The NLM in GaN specifically, predicts a gradient of the polarization ranging

-0.04

+0.04

GaN

-0.12

-0.09

-0.06

-0.03

0.00

0.03

0.06

0.09

0.12

NLM+NLE LM+NLE

P

Total= 0.0 C/m

2

109

from -0.12 C/m2 at the compressed end of the section, to +0.12 C/m2 at the tensile end. While

in the case of the LM polarization it changes from -0.15 C/m2 to +0.09 C/m2 in the same

range of strains [125].

Similar effect is found in the case of InN NWs, the polarization gradient of the NLM predicts

to be in the range of -0.11 C/m2 at the compressed end of the section, and +0.08 C/m2 at the

tensile end. While the LM polarization ranges from -0.14 C/m2 to +0.05 C/m2 win the same

set of strains [125].

-0.08

-0.05

-0.02

0.01

0.04

0.070.08

-0.04

+0.04 NLM+NLE LM+NLE

InN

PTotal

= 0.0 C/m2

Figure 6.1.3-3 Variation of the polarization (C/m2) in a cross section of a InN nanowire. The

perpendicular (parallel) strain varies from -4% (+4%) to +4% (-4%). The calculated

polarization using NLE parameters of the non-linear (quadratic) model (NLM) is on the left

half and the the classic linear model (LM) on the right. Reproduced from the work of

Reference [M.A. Migliorato, J. Pal, R. Garg, G. Tse, H. Y. S. Al-Zahrani, U. Monteverde, S.

Tomić, C-K. Li, Y-R. Wu, B. G. Crutchley, I. P. Marko and S. J. Sweeney, AIP Conf. Proc.

1590 (2014) 32].

110

Unlike ZnO, the polarization in NLM compared to the LM in III-Nitride NWs are shown

better predictions. (Agrawal and Espinosa, 2011) found that non-linear piezoelectricity could

be explained the giant PZ coefficients in III-N NWs and pronounced dependence on NW

diameter[185]. Therefore, ZnO and III-Nitride NWs polarization calculations need to be done

in order to note the several prospective applications on the optical and electronic properties

effects of NWs.

6.2 Core Shell Nanowires

MBE or MOCVD has enabled the preparation of NWs combining two or more compounds,

leading to the formation of heterostructures. Such NWs fall into two main categories: radial

and axial, based on the chosen growth method of the compound [186].While axial NWs

[187,188,189] are easily synthesized by changing the growth conditions and material supply

during their formation (e.g. through one-dimensional modulation of the NW composition

[188, 189,190]and doping [187]), radial NWs, commonly referred to as core-shell NWs

(CSNWs) [186], require a two-step process involving the synthesis of the core first followed

by coating with the shell material. This process can be repeated several times to obtain multi-

shell structures.

CSNWs have a range of potential applications and are not restricted to III-V semiconductors.

Those based on e.g. Group IV [191,192,193,194,195,196]are motivated by the predicted

enhanced performance of nanophotonic and nanoelectronic devices [197], while II-VI

compounds like CdSe and ZnTe 198 are ideal for achieving high optical performance

together with energy harvesting ability. Finally III-V semiconductor based CSNWs

[199,200,201,202,203] have been explored as enhanced light-emitting and laser diodes

[204,205], photovoltaics [206 ]and high-current battery electrodes[207].

111

Figure 6.2-1 Schematic of nanowire and nanowire heterostructure growth.(a) Nanowire

synthesis through catalyst-mediated axial growth. (b,c) Switching of the source material

results in nanowire axial heterostructures and superlattices. (d,e) Conformal deposition of

different materials leads to the formation of core/shell and core/ multishell radial nanowire

heterostructures reproduced from Reference[O. Hayden, R. Agarwal and W. Lu,

Nanotechnology 3 (2008) 12].

The WZ CSNW system with an epitaxial interface provides e.g. an attractive way to explore

the effect of piezo-phototronic in energy harvesting system, GaN/InGaN CSNWs LEDs have

rivalled planar InGaN single quantum-well LEDs [208]. The composition and thickness of

InGaN in planar devices [208] can be limited by the lattice mismatch strain. However, a NW

structure provides pathways to lateral strain relaxation that often preserve coherence and

prevents defect formation. A core/multishell NW was reported by Qian et al., [209,210]

where both electron and hole carriers were confined into the InGaN shell permitting an

external quantum efficiency of 5.8% at 440 nm and 3.9% at 540 nm and a tuneable emission

in the range of 365 - 600 nm.

112

6.2.1 Core Shell Advantages

In general, CSNW systems offer many advantages when compared to their homogeneous

NW counterparts. The more notable advantages include; firstly, the ability to regulate surface

impurities and surface states, which are normally found in the vast majority of nanoscale

structures. [192,195] Secondly, the semiconducting NW core can be isolated from the

substrate inhomogeneity [187]. Lastly and mostly notably, the realizations of quantum

confinement of carriers within the core by cladding with a larger bandgap shell [191,192].

The rationale for using the shell material to significantly improve the optical and electrical

properties of the nanostructure through passivation of the free core surface, has been verified

in ZB GaAs/AlGaAs CSNWs [211] and wide band gap InP shells on InAs cores

[212,213,214]. Furthermore enhanced carrier mobility is observed in InAs NWs when

covered with InP shells [215]. In CSNWs strain effects are different from conventional 2D

heterostructured semiconductors. Yann-Michel Niquet [216,217] studied the electronic

properties of InAs NWs embedded in a GaAs shell, showing that the strain relaxation of the

InAs layers are limited by the GaAs shell yet the hydrostatic strain distribution is

homogenised. Consequently, the formation of strain-induced surface well in the conduction

band is prevented and the electron wave functions are more confined to the core material.

6.2.2 Modelling AFM Tip Lateral Deflection

Scanning probe microscopy, or SPM, is a fundamental part of the equipment used in

researching and developing nanotechnology. Within this range of tools, the scanning or

atomic force microscope represents the most widely used type of microscope [218]. The

AFM, created by Binning in 1986, utilises the ability to detect extremely small forces which

exist between surface atoms and the pointed tip of the devices. A cantilever arm holds this tip

113

and is used to move the tip to within a nanometre of surface atoms, whereupon forces at the

interatomic level are registered by the device through laser reflection of the bending of the

cantilevered arm to an optical sensor. This can be repeated as the tip moves across the surface

of the sample to create an image from the alteration in the cantilever, showing at a simple

level the 3D form of the surface. Where movement of the tip is along the x-y plane, forces of

attraction and repulsion act between it and the atomic surface to deflect the tip toward

direction Z (Sarid, 1991) [219]. There are two methods for the use of atomic force

microscopy. First is through the contact mode, in which the instrument’s tip is held within a

small number of angstroms of the surface, with tip and surface atoms therefore interacting.

This mode gives rise to high complexity in interacting forces, which must be considered via

simulations in molecular dynamics which include charge distribution and coulomb

interactions of charges, forces from quantum mechanics for interaction of electron orbitals

between individual pairs of tip and surface atoms, and induced dipole moment causing a

polarising effect. This approach is frequently used in analysing atomic-resolution

morphology of surface samples (Sarid and Elings, 1991)[220].

The second mode of use of AFM is through the non-contact or tapping approach. This mode

involves maintaining of a far greater distance of tip from surface: from 2-30 nanometres.

Here, the description of forces is made via interactions of entities on the macroscopic scale.

This technique requires a sample with a surface which is level, as well as a round-particled

tip. Here, forces involved are various and can include electrostatic force, taking into account

possible variation when comparing the surface and the tip, and where the material studied is

magnetic, magnetostatic forces are involved. These force present as much lower than those of

the contact mode, generally being lower by between two and four orders of magnitude.

Further, the non-contact use of the AFM does not give sub-nanometer data because the

interacting entities are no longer on the level of the individual atom but include large sections

114

of both sample and tip. This approach is utilised frequently when electronic equipment or

magnetic domains are being imaged.

From the above, it can be seen that AFM is useful in determining the surface features of a

sample on an atomic scale, including topographical analysis as well as analysis of electrical

and magnetic character. Further, the magnification presented by the 3-d image which AFM

can create means that a direct picture can be obtained for surface features as small as

individual atoms or molecules. Thus, nano-objects can be seen and measurement given to

their surface characteristics (Binning, Quanta and Gerber, 1986) [221].

Figure 6.2.2-1 AFM Tip Lateral Deflection of nanowire where (R), is the radius of curvature,

(H) is the length of the NW, (D) is the diameter of the NW and the deflection caused by the

AFM tip is (d). Ɵ is the angle that subtends the arch formed by the deformed NW, (𝐻+) and

(𝐻−) lengths of the NW on the tensile and compressed. Reproduced from Reference [H.Y.S. Al-

Zahrani, J. Pal, M. Migliorato, G. Tse, and D. Yu, Nano Energy, 14 (2015) 382-391].

From the calculation [222] of the linear and the quadratic piezoelectric coefficients of wurtzite

III-V (GaP, InP, GaAs and InAs) semiconductors, the predicted magnitude of even the first

115

order coefficients is much larger than previously reported and of the same order of magnitude as

those of III-N semiconductors. As a result, an atomic force microscope (AFM) tip is often used

to both provide deflection to a NW and measure its electrical properties. Tensile and

compressive strains on either side of a single NW are a result of the tip induced mechanical

force applied. By modelling the AFM tip induced deflection and predicting the resulting

piezoelectric polarization we can quantify the effect of the LPZCs and NLPZCs on the PZ field

and voltage. We show the polarization on a cross section of the NW, making the approximation

that subjecting the NW to a bending force deforms the cylindrical shape into an arch with

constant curvature. Figure 6.2.2-1 shows the quantities that are needed as input are: the radius of

curvature (R), the length of the NW (H), the diameter of the NW (D) and the deflection caused

by the AFM tip (d). If we assume that Ɵ is the angle that subtends the arch formed by the

deformed NW, then its unreformed length (H) is given by:

𝐻 = 𝑅. 휃 (62)

while the increased 𝐻+ and decreased 𝐻− lengths of the NW on the tensile and compressed

ends would be given by:

H𝐻+=(R±∆𝑅). 휃 (63)

Where ∆𝑅 is exactly equal to the radius of the nanowire (D/2). With reference to Figure 6.2.2-

1, and since 휃 is typically small (limit of small deflection), we can expand the expression for

the deflection (d) to , 𝑂(𝑁3)resulting in:

𝑑 = 𝑅 − 𝑅𝑐𝑜𝑠휃 = 𝑅(1 − 𝑐𝑜𝑠휃) ≅ 𝑅 (1 − 1 +휃2

2) = 𝑅

휃2

2=𝐻

휃.휃2

2=𝐻. 휃

2 (64)

From which:

116

휃 ≅2𝑑

𝐻 (65)

Which when used in combination with equation(62), gives:

𝑅 ≅𝐻2

2𝑑 (66)

Equations(65) and(66), in conjunction with equation(63) , give the expressions for the tensile

and compressive strain deformed lengths of the NW:

𝐻± ≅ (𝐻2

2𝑑±𝐷

2) .2𝑑

𝐻= (𝐻 ±

𝑑.𝐷

𝐻) (67)

the equation above depends solely on the diameter (D) and length (H) of the Nanowire and

the applied AFM tip deflection (d). The strain along the NW, in the direction [0001] is easily

calculated using Equation(67):

𝑒⊥ =𝐻±−𝐻

𝐻≅ (±

𝑑.𝐷

𝐻2) (68)

While the in-plane component (assuming that the unit cell preserves the volume elastically), is

given by:

𝑒∥ = −𝑒⊥.𝐶332𝐶13

(69)

where C33 and C13 are the elastic constants of the material, given in Table VI.

6.2.3 Homogeneous and Core Shell Nanowires

From the method described above, the NWs with dimensions of 1µm length and 0.5µm

diameter are subjected to an AFM tip deflection range of 0-360nm. The effect of varying

deflection on the polarization of a cross section of a NW or CSNW is shown in Figure 6.2.3-1

It is assumed that the polarization varies when moving from the compressive (bottom of each

117

graph) to the tensile side (top of each graph), but not in the orthogonal direction, so that a 1D

line of data is fully representative of the polarization over the 2D cross section.

Figure 6.2.2-1 Comparison of the Total Polarization in Homogeneous and Core-Shell

Nanowires (CSNWs) when deflected by AFM tip. The first row resembles the homogeneous

III-As and III-P nanowires having 1µm length and 0.5µm diameter in dimensions with an

AFM tip deflection range of 0-360nm. While the second, third and fourth row are the

118

different combinations CSNWs. Typical CSNW dimensions are of 1µm length and core/shell

diameter of 0.25µm/0.5µm with a 360nm deflection. Reproduced from Reference [H.Y.S. Al-


As expected, for all four cases of homogeneous III-As and III-P NWs shown, the PZ

polarization is enhanced with increasing deflection, as illustrated in the first row of Fig 6.2.3-

1 we have performed AFM tip deflection calculations for III-As and III-P homogeneous NWs

using Figure 6.2.3-1. For the largest deflection of 360 nm, the predicted maximum

polarization values for all combinations of GaAs, InAs, InP and GaP CSNWs range from -

0.23 C/m2 at the compressed end of the section, to +0.33 C/m2 at the tensile end and are in the

same order but smaller than those of III-N NWs of equivalent dimensions and for similar

deflections. As an example, when a homogeneous GaN nanowire undergoes a deflection of

360nm, polarization values of -0.27 C/m2 (compressed region) and +0.81C/m2 (tensile

regions) are predicted. In Table VIII shows the comparison between all the strain induced

polarizations in different III-As and III-P NWs when deflected by 360nm [222].

It is worth to mention that the abrupt interfaces in the core region for CSNWS, can improve

carrier confinement,[223] and narrow down the optical spectrum emitted by optically active

nanowires, such as (InGaN/GaN) multi-structure nanowires[224,225]. Abrupt interfaces in the

axial heterojunction NWs, are important to their use in tunneling field effect transistors,[225]

as well as thermoelectric devices[226].

One further aspect to mention is that while most applications of NWs in the field of

piezophototronics have been in the blue, violet or UV portion of the spectrum. III-V

semiconductor NWs such as the ones proposed here would have application in the lower

visible (red and green) and IR spectrum.

119

Table VIII. Calculated values of the total polarization (C/m2) in different homogeneous

nanowires comprising GaAs, InAs, GaP and InP in comparison with GaN nanowire when

subjected to an AFM lateral tip deflection of 360nm. Reproduced from Reference [H.Y.S.

Al-Zahrani, J. Pal, M. Migliorato, G. Tse, and D. Yu, Nano Energy, 14 (2015) 382-391].

Material

System

Total Polarization

(C/m2)

Tensile Compressed

InP 0.32 -0.22

GaP 0.28 -0.23

InAs 0.33 -0.20

GaAs 0.28 -0.13

GaN 0.81 -0.27

All possible combinations of GaAs, InAs, InP and GaP CSNWs, using typical NWs

dimensions (1µm in length and a core/shell diameter ratio of 0.25µm/0.5µm), are subjected to

the same deflections of 0-360nm as the homogeneous NWs. The predicted strain induced

polarizations are shown in the second, third and fourth row Figure 6.2.3-1. Considering the

case of an InP/GaAs CSNW, a hypothetical situation since very few defect free monolayers

can be typically grown due to strain, in the absence of any deflection, then the tensile

perpendicular strain would be around 4% with a typical compressive parallel strain of around

10%. Then the inherent strain due to the heterostructured growth of the CSNWs produces a

much stronger polarization compared to the case where the whole NW was made

homogeneously of the shell material. As an example, for a deflection of 360nm, the total

polarization of an InP/GaAs CSNW is increased by 29% at the tensile end of the shell and

reduced by 13% (making it less negative) at the compressed end of the shell. In Table IX, we

have compared the strain induced polarization and their difference compared to the case

120

where the whole NW was made homogeneously of the shell material (in %), for all the

different combinations of GaAs, InAs, InP and GaP CSN[222].

Table IX. Total polarization and their difference (in %) using CSNWs for all

combinations of GaAs, InAs, GaP and InP compared with homogeneous NWs at 360nm

deflection at both tensile and compressive ends. Reproduced from reference [H.Y.S. Al-


It is obvious that there are a number of combinations where there is an advantage in using a

CSNW structure, namely core/shell combinations of InAs/GaAs, InP/GaAs, InP/GaP,

InAs/InP and InAs/GaP where the total polarization is increased by 20-68% at the tensile end

Material

system

(Core|Shell)

Tensile End Compressed End

CSNW

Polarizatio

n (C/m2)

Homogene

ous

Polarizatio

n (C/m2)

Differen

ce (in

%)

CSNW

Polarizati

on (C/m2)

Homogene

ous

Polarizatio

n (C/m2)

Difference (in

%)

GaP|InAs 0.13 0.33 -60 -0.26 -0.20 29

GaP|InP 0.17 0.32 -45 -0.28 -0.22 28

GaAs|InAs 0.19 0.33 -44 -0.25 -0.20 22

GaAs|InP 0.24 0.32 -25 -0.25 -0.22 16

GaP|GaAs 0.21 0.28 -23 -0.14 -0.13 7

GaAs|GaP 0.22 0.28 -23 -0.28 -0.23 20

InP|InAs 0.26 0.33 -20 -0.23 -0.20 11

InAs|InP 0.38 0.32 21 -0.19 -0.22 -15

InP|GaAs 0.36 0.28 29 -0.11 -0.13 -13

InP|GaP 0.42 0.28 49 -0.13 -0.23 -43

InAs|GaAs 0.44 0.28 61 -0.09 -0.13 -32

InAs|GaP 0.48 0.28 68 -0.09 -0.23 -60

121

and reduced by 13-60% (making it less negative) at the compressed end of the shell. The rest

of the core/shell combinations (GaP/InAs, GaP/InP, GaAs/InAs, GaAs/InP, GaP/GaAs,

GaAs/GaP, InP/InAs) can provide even more negative polarization by 20-60% at the tensile

end and 7-30% at the compressed end. However not all are possible under experimental

conditions due to very high lattice mismatch/strain making it extremely challenging to grow

such structures. Alloy combinations in the shell are potentially more favourable as they

would allow for much reduced strain during growth. Experimentally a typical alloy

composition of 4% to 45% [227,228,229] for InGaAs/GaAs CSNW is reported in literature,

whose crystal quality is demonstrated by high optical performance. It has also calculated the

piezoelectric voltage in Figure 6.2.3-2, which is the important quantity for piezotronics

applications, and compared all combinations of GaAs, InAs, InP and GaP CSNWs (with

dimensions, 1µm in length and a core/shell diameter ratio of 0.25µm/0.5µm), at 4nm

deflection. The largest core voltages are predicted for the core/shell combinations InAs/GaP

(-1725V), InP/GaP (-1246V), GaP/InAs (+1034V) and GaP/InP (+912V), which are much

larger than the values for typical homogeneous NWs (±3V). While we observe strong

negative voltages (-1725V to -420V) in InAs/GaP, InP/GaP, InAs/GaAs, GaAs/GaP,

InP/GaAs and InAs/InP, swapping the core and shell materials yields strong positive voltages

(+335V to +1034V) in InP/InAs, GaP/GaAs, GaAs/InP, GaAs/InAs, GaP/InP and GaP/InAs

CSNW combinations. However the absolute values of the voltage in the core or shell are not

necessarily the important quantity when considering NWs as the power source in a NG. In

fact what is important is the aptitude to change such voltage when deformed. In Table X we

show both the core and shell voltages (in V) at a typical deflection of d=4nm, at the tensile

and compressed ends. We also show the differences in voltage for 4nm and 0 deflection.

The largest differences, which can related to voltage generation in NGs, are in the order of

±3V for InAs/GaP, InP/GaP and GaAs/GaP. The last four columns provide comparis between

122

the shell voltage of the core-shell structure and a homogenous nanowire of the same

dimensions (1µm in length and a diameter of 0.5µm) made of the shell or core material only.

The best improvements are found for InAs/GaP (+40.2% compared to InAs) and GaAs/GaP

(+48.3% compared to GaAs).

However, while considering CSNWs, we should also take into account the electron mobility

and bandgap of GaAs (8500cm2V-1s-1, 1.42eV), InAs (40000cm2V-1s-1, 0.35eV), GaP

(250cm2V-1s-1, 2.26eV) and InP (5400cm2V-1s-1, 1.34eV). All the values of electron mobility

and bandgaps are given for the ZB crystal phase as the WZ phase properties are not available

[230]. Materials with higher electron mobility and lower bandgap should be preferred in the

core compared to the shell material as they provide unique benefits of higher conductivity

along with electron confinement within the core. All the six core/shell combinations of

InAs/GaP, InP/GaP, InAs/GaAs, GaAs/GaP, InP/GaAs and InAs/InP conform to the above

criteria for optimal CSNW structures while if switching the core and shell material in such

combinations can be beneficial for applications seeking higher resistivity. The combinations

InAs/GaP and GaAs/GaP have both increased voltage generation, high conductivity in core

and confinement between shell and core[222].

123

Table X. Core and shell voltages (in V) at d=4nm deflection, at the tensile and

compressed ends. Differences are also given between voltages at 4nm and 0 deflection.

The last four columns provide comparison between the shell voltage of the core-shell

structure and a homogenous nanowire of the same dimensions (1µm in length and a

diameter of 0.5µm) made of the shell or core material only. Reproduced from reference

[H.Y.S. Al-Zahrani, J. Pal, M. Migliorato, G. Tse, and D. Yu, Nano Energy, 14 (2015)

382-391]

Material

System

Core Shell Voltage Difference between Shell and

Homogeneous NW At Deflection d=4nm Voltage

Difference

from

(d=0nm)

At Deflection d=4nm Voltage

Difference

from

(d=0nm) Compressed

End

Tensile

End

Compressed

End

Tensile

End

InAs GaAs InP GaP

Core/Shell

InAs/GaP -1724.45 -1722.17 1.15 -3.21 3.20 3.21 0.92

(40.2 %)

0.31

(10.7%)

InP/GaP -1245.85 -1243.17 1.34 -3.12 3.11 3.12 0.44

(16.4%)

0.22

(7.6%)

InAs/GaAs -959.65 -957.37 1.15 -2.67 2.67 2.67 0.38

(16.6 %)

0.64

(31.5%)

InP/GaAs -452.51 -452.25 1.34 -2.36 2.35 2.36 0.33

(16.3%)

-0.32

(-11.9%)

InAs/InP -420.29 -418.01 1.15 -2.85 2.84 2.85 0.56

(24.5%)

0.17

(6.3%)

GaAs/GaP -619.16 -617.12 1.02 -3.01 3.01 3.01 0.98

(48.3%)

0.11

(3.8%)

GaP/GaAs 399.61 402.51 1.45 -1.72 1.72 1.72 -0.31

(-5.3%)

-1.18

(-40.7%)

GaAs/InAs 741.06 743.09 1.02 -1.87 1.86 1.87 -0.42

(-18.3%)

-0.16

(-7.9%)

GaP/InAs 1031.17 1034.07 1.45 -1.68 1.68 1.68 -0.61

(-26.6%)

-1.22

(-42.1%)

InP/InAs 331.91 334.59 1.34 -2.11 2.10 2.11 -0.18

(-7.9%)

-0.57

(-21.3%)

GaAs/InP 526.57 528.60 1.02 -2.47 2.46 2.47 0.44

(21.7%)

-0.21

(-7.8%)

GaP/InP 908.94 911.83 1.45 -2.31 2.30 2.31 -0.37

(-13.8%)

-0.59

(-20.3%)

124

Figure 6.2.3-2 Comparison of calculated output piezoelectric voltage from Core-Shell

Nanowires (CSNWs) when laterally deflected 4nm by AFM tip. Typical CSNW dimensions

are of 1µm length and core/shell diameter of 0.25µm/0.5µm.Reproduced from Ref [H.Y.S.

Al-Zahrani, J. Pal, M. Migliorato, G. Tse, and D. Yu, Nano Energy, 14 (2015) 382-391].

125

7 Conclusion

To summarise, the dissertation has considered piezoelectric non-linear effects in the

semiconductors, and had arrived at a number of unanticipated predictions for the way in

which a range of nanostructures behave in terms of electro-optical properties. These

structures included III-Vs semiconductors and ZnO nanowires. A model to describe non-

linear effects and the validity of this model was checked through experimental findings and

genuine devices, as well as through review of other researchers’ findings. The thesis also

encompassed study of Piezotronics, concerning the range of devices constructed to utilise

piezopotential to form a ‘gate’ for voltage, allowing the tuning or management of charge-

carrier transport at junctions or contact points. Such devices have recognised potential for

human-machine interfaces, sensor equipment and systems which incorporate biological

components, as well as energy science, MEMS, and incorporation into CMOS technologies

based upon silicon to enhance functionality for the post-Moore time.

Piezoelectricity has been subject to different proposed models; however, these have not come

to be universally accepted, in particular because the findings they produce for PZCs vary, and

especially in terms of values for spontaneous polarization. The source of this difference has

presented a considerable challenge to establish, although it was long suspected that the

quantity Z* in comparison to 𝑍𝐻∗ was subject to error.

While the non-linear modelling for piezoelectricity presented here is internally consistent and

able to offer both spontaneous polarisation and PZC data in one regime, 𝑍𝐻∗ is adopted for a

fitting parameter, and is generally 25% - 65% of broadly utilised effective charge (Z*). There

is the possibility that the model will be criticised for this approach: however, accurate

126

predictions for field estimation were generated through the model, which were closely in line

with the findings from experiment.

This thesis has shown the value of the novel non-linear to second order model with smaller

spontaneous polarisation in comparison with earlier linear models with greater spontaneous

polarisation, as it creates more accurate projections when considered against data from

experiments concerning PZ fields within quantum wells across a range of III-N materials and

alloys of these.

The current thesis presents near-total attention to non-linear piezoelectric effects for III-Vs

(GaP, InP, GaAs and InAs) and ZnO semiconductors of Wurtzite phase, with strain-reliant

non-linear piezoelectric coefficients (PZCs) being a novel finding of the study. Further,

parameters have been calculated for spontaneous polarisation, as well as 1st and 2nd orders of

PZCs for groups III-V and II-VI, within an ab initio DFPT and DFT framework and

alongside Harrison’s semi-empirical formulation.

For ZnO wurtzite crystals, I have estimated the linear and quadratic piezoelectric coefficients.

The magnitude of the quadratic terms is significant and necessitates inclusion even in the

limit of small strain. I showed calculations of the total polarization in ZnO nanowires and

report that for particular strains originating from an external force the non-linear model of

piezoelectricity predicts both positive and negative polarizations in the nanostructure whereas

the linear model only predicts negative values.

Future work on Non-linearity found within ZnO NWs may be utilised for design of

composite heterostructured combinations to enhance piezoelectric effects, to potentially

create engines to drive piezotronics equipment, including nanogeneration devices and

pressure sensors.

127

Piezoelectric wurtzite III-Vs semiconductors, including laterally heterostructured

semiconductors, as in core shell nanowires, are now routinely grown using catalyst enabled

MBE growth. In order to assess the piezoelectric properties of core shell nanowires made of

wurtzite III-Vs. The author has reported values of the linear and quadratic piezoelectric

coefficients of wurtzite GaP, InP, GaAs and InAs and show the magnitude of such

coefficients is much larger than previously reported and comparable with those of III-N

semiconductors. A model has been developed of the bending distortion created on a nanowire

by an atomic force microscope tip induced deflection to evaluate the piezoelectric properties

of wurtzite III–Vs core shell nanowires. The author has then analyzed a series of cross

sections of the NW, with increasing tip induced deflection, assuming that subjecting to a

bending force deforms the cylindrical shape into an arch with constant curvature. For the core

shell nanowires, it has been shown that a number of combinations of III–Vs semiconductors

are favourable for much increased voltage in the nanowire. The largest core voltages for a 4

nm deflection are predicted for the core/shell combinations InAs/GaP (-1725 V), InP/GaP (-

1246 V), GaP/ InAs (+1034 V) and GaP/InP (+912 V), which are much larger than the values

for a typical homogeneous nanowires (73V). Since materials with higher electron mobility

and lower bandgap would increase the nanowire conductivity in the core region, the six

core/shell combinations of InAs/GaP, InP/GaP, InAs/GaAs, GaAs/GaP, InP/GaAs and

InAs/InP satisfy these criteria. Of these, also considering which ones are predicted to have the

largest voltage generation ability, InAs/GaP (an increase of 40.2% compared to InAs) and

GaAs/GaP (an increase of 48.3% compared to GaAs) are predicted to be optimal candidates

for highly conductive piezotronics and nanogeneration elements.

128

Future Work

The thesis has discussed the importance of piezoelectricity in III–Vs semiconductors

nanowires for applications in piezotronics and nanogeneration devices. Having said this, non-

linear piezoelectric effects of the III-Vs wurtzite semiconductors potentially expand the

possibilities for semiconductor equipment design, and thus future work in this field is highly

recommended.

There remains a large amount of investigation required in the areas studied here, including

for example in II-Sb and hexagonal ZnS together with their alloys, and different wurtzite and

zincblende crystal phases.

While the methodology gives better agreement with the experimental data in comparison to

the linear model of piezoelectricity, the effect of nonlinear piezoelectricity has not been

studied and can enhance the output.

Moreover, only a small quantity of PZCs has been identified as yet in the Wurtzite phase,

restricted to some deformation contexts. While work to generalise results outwards should be

comparatively simple, it will require considerable time resources and more materials can

provide the required functionalities to demonstrate even more novel devices in future years.

129

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