Theoretical Study of Optical Transition Matrix Elements in InGaN/GaN SQW Subject to Indium Surface Segregation

SEPTEMBER/OCTOBER 2011 VOLUME 17 NUMBER 5 IJSQEN (ISSN 1077-260X)

(Top right) Photograph of the EADFB laser array and schematics. (Top left) Photograph of theEADFB laser array module. (Bottom) Fabrication process of a module using

bridge-type RF circuit board (Kanazawa et al., p. 1191).

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SEPTEMBER/OCTOBER 2011 VOLUME 17 NUMBER 5 IJSQEN (ISSN 1077-260X)

SEMICONDUCTOR LASERS—PART 1

GUEST EDITORIAL

Introduction to the Issue on Semiconductor Lasers—Part 1 . . . . . . . L. F. Lester, V. Kovanis, E. P. O’Reilly, and Y. Tohmori 1136

PAPERS

High-Speed Lasers

Wide Temperature Range Operation of 25-Gb/s 1.3-μm InGaAlAs Directly Modulated Lasers (Invited Paper) . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Fukamachi, K. Adachi, K. Shinoda, T. Kitatani, S. Tanaka, M. Aoki, and S. Tsuji 1138

Distinguished RZ-OOK Performances Between DFBLD Pulsed Carriers Self-Started by Gain Switching and NonlinearAbsorption Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y.-C. Chi and G.-R. Lin 1146

Effects of Carrier Relaxation and Homogeneous Broadening on Dynamic and Modulation Behavior of Self-AssembledQuantum-Dot Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. H. Yavari and V. Ahmadi 1153

1550-nm High-Speed Short-Cavity VCSELs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Muller,W. Hofmann, T. Grundl, M. Horn, P. Wolf, R. D. Nagel, E. Ronneberg, G. Bohm, D. Bimberg, and M.-C. Amann 1158

Intrinsic Dynamics of Quantum-Dash Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Chen, Y. Wang, H. S. Djie, B. S. Ooi, L. F. Lester, T. L. Koch , and J. C. M. Hwang 1167

Lateral-Current-Injection Distributed Feedback Laser With Surface Grating Structure . . . . . . . . . . . . . . . . . . . . . . . T. Shindo,T. Okumura, H. Ito, T. Koguchi, D. Takahashi, Y. Atsumi, J. Kang, R. Osabe, T. Amemiya, N. Nishiyama, and S. Arai 1175

40-Gb/s Operation of a 1.3-/1.55-μm InGaAlAs Electroabsorption Modulator Integrated With DFB Laser in CompactTO-CAN Package (Invited Paper) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . W. Kobayashi, K. Tsuzuki, T. Tadokoro, T. Fujisawa, N. Fujiwara, T. Yamanaka, and F. Kano 1183

A Compact EADFB Laser Array Module for a Future 100-Gb/s Ethernet Transceiver (Invited Paper) . . . . . .S. Kanazawa,T. Fujisawa, A. Ohki, H. Ishii, N. Nunoya, Y. Kawaguchi, N. Fujiwara, K. Takahata, R. Iga, F. Kano, and H. Oohashi 1191

Laser Dynamics

Photonic Microwave Applications of the Dynamics of Semiconductor Lasers (Invited Paper) . . . . . X.-Q. Qi and J.-M. Liu 1198Quantifying Chaotic Unpredictability of Vertical-Cavity Surface-Emitting Lasers With Polarized Optical Feedback via

Permutation Entropy . . . . . . . . . . . . . . . . . . . . . S. Y. Xiang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, N. Jiang, and K. H. Wen 1212

(Contents Continued on Page 1134)

(Contents Continued from Page 1133)

Chaos Synchronization and Communication in Multiple Time-Delayed Coupling Semiconductor Lasers Driven by a ThirdLaser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Jiang, W. Pan, L. Yan, B. Luo, S. Xiang, L. Yang, D. Zheng, and N. Li 1220

Polarization-Resolved Nonlinear Dynamics Induced by Orthogonal Optical Injection in Long-Wavelength VCSELs. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Perez, A. Quirce, L. Pesquera, and A. Valle 1228

All Photonic Crystal DFB Lasers Robust Toward Optical Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . A. Larrue, J. Campos, O. Gauthier-Lafaye, A. Monmayrant, S. Bonnefont, and F. Lozes-Dupuy 1236

Dynamics of Polarized Optical Injection in 1550-nm VCSELs: Theory and Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .R. Al-Seyab, K. Schires, N. A. Khan, A. Hurtado, I. D. Henning, and M. J. Adams 1242

Characterizing the Hyperchaotic Dynamics of a Semiconductor Laser Subject to Optical Feedback Via PermutationEntropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Zunino, O. A. Rosso, and M. C. Soriano 1250

Low Hysteresis Threshold Current (39 mA) Active Multimode-Interferometer (MMI) Bistable Laser Diodes UsingLateral-Modes Bistability . . . . . . . . . . . . . . . . . . . . . . H. Jiang, H. A. Bastawrous, T. Hagio, S. Matsuo, and K. Hamamoto 1258

Mode-Locked Lasers

External-Cavity Mode-Locked Quantum-Dot Laser Diodes for Low Repetition Rate, Sub-Picosecond Pulse Generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Xia, M. G. Thompson, R. V. Penty, and I. H. White 1264

Dynamics of Quantum-Dot Mode-Locked Lasers With Optical Injection (Invited Paper) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Habruseva, G. Huyet, and S. P. Hegarty 1272

Toward Frequency-Domain Modeling of Mode Locking in Semiconductor Lasers (Invited Paper) . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Kreuter, B. Witzigmann, and W. Fichtner 1280

InAs/InP Quantum-Dot Passively Mode-Locked Lasers for 1.55-μm Applications (Invited Paper) . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . R. Rosales, K. Merghem, A. Martinez, A. Akrout, J.-P. Tourrenc, A. Accard, F. Lelarge, and A. Ramdane 1292

High-Power Versatile Picosecond Pulse Generation from Mode-Locked Quantum-Dot Laser Diodes (Invited Paper) . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. A. Cataluna, Y. Ding, D. I. Nikitichev, K. A. Fedorova, and E. U. Rafailov 1302

Microwave Characterization and Stabilization of Timing Jitter in a Quantum-Dot Passively Mode-Locked Laser viaExternal Optical Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.-Y. Lin, F. Grillot, Y. Li, R. Raghunathan, and L. F. Lester 1311

Quantum-Dot Devices

Time-Domain Traveling Wave Model of Quantum Dot DFB Lasers . . . . . . . . . . . . . . . . . . . . . M. Gioannini and M. Rossetti 1318Analysis of QD VCSEL Dynamic Characteristics Considering Homogeneous and Inhomogeneous Broadening . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .H. Abbaspour, V. Ahmadi, and M. H. Yavari 1327Toward 1550-nm GaAs-Based Lasers Using InAs/GaAs Quantum Dot Bilayers . . . . . . . . . . . M. A. Majid, D. T. D. Childs,

H. Shahid, S. Chen, K. Kennedy, R. J. Airey, R. A. Hogg, E. Clarke, P. Howe, P. D. Spencer, and R. Murray 1334Temperature-Dependent Threshold Current in InP Quantum-Dot Lasers (Invited Paper) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. M. Smowton, S. N. Elliott, S. Shutts, M. S. Al-Ghamdi, and A. B. Krysa 1343Will Quantum Dots Replace Quantum Wells As the Active Medium of Choice in Future Semiconductor Lasers? . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. W. Chow, M. Lorke, and F. Jahnke 1349

Quantum Well Materials and Devices

Fabry–Perot Laser Characterization Based on the Amplified Spontaneous Emission Spectrum and the Fourier SeriesExpansion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W.-H. Guo, D. C. Byrne, Q. Lu, B. Corbett, and J. F. Donegan 1356

Strained-Layer Quantum-Well Lasers (Invited Paper) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. R. Adams 1364Theoretical Study of Optical Transition Matrix Elements in InGaN/GaN SQW Subject to Indium Surface Segregation . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. V. Klymenko, O. V. Shulika, and I. A. Sukhoivanov 1374GaInAsP/InP Membrane Lasers for Optical Interconnects (Invited Paper) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Arai, N. Nishiyama, T. Maruyama, and T. Okumura 1381Carrier Transport in InGaN MQWs of Aquamarine- and Green-Laser Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. S. Sizov, R. Bhat, A. Zakharian, K. Song, D. E. Allen, S. Coleman, and C.-En Zah 1390Light Emitting and Laser Diodes in the Ultraviolet (Invited Paper) . . . . . . . . . . . . . . . . . . . . . . . . . P. J. Parbrook and T. Wang 1402Catastrophic-Optical-Damage-Free InGaN Laser Diodes With Epitaxially Formed Window Structure . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .M. Kawaguchi, H. Kasugai, K. Samonji, H. Hagino, K. Orita, K. Yamanaka, M. Yuri, and S. Takigawa 1412

Mid-IR Lasers

InP-Based Midinfrared Quantum Cascade Lasers for Wavelengths Below 4 μm (Invited Paper) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. G. Revin, J. P. Commin, S. Y. Zhang, A. B. Krysa, K. Kennedy, and J. W. Cockburn 1417

(Contents Continued on Page 1135)

(Contents Continued from Page 1134)

Type-I Diode Lasers for Spectral Region Above 3 μm (Invited Paper) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .G. Belenky, L. Shterengas, G. Kipshidze, and T. Hosoda 1426

Mid-IR Type-II Interband Cascade Lasers (Invited Paper) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . I. Vurgaftman, W. W. Bewley, C. L. Canedy, C. S. Kim, M. Kim, J. R. Lindle, C. D. Merritt, J. Abell, and J. R. Meyer 1435

Room Temperature CW Operation of Short Wavelength Quantum Cascade Lasers Made of Strain Balanced GaxIn1−xAs/AlyIn1−yAs Material on InP Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Xie, C. Caneau,H. P. LeBlanc, N. J. Visovsky, S. C. Chaparala, O. D. Deichmann, L. C. Hughes, C.-En Zah, D. P. Caffey, and T. Day 1445

Nonlinear Temperature Dependence of Resonant Pump Wavelengths in Optical Pumping Injection Cavity Lasers . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. J. Olafsen, L. D. Ice, and B. Ball 1453

ANNOUNCEMENTS

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1374 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 17, NO. 5, SEPTEMBER/OCTOBER 2011

Theoretical Study of Optical Transition MatrixElements in InGaN/GaN SQW Subject

to Indium Surface SegregationMykhailo V. Klymenko, Oleksiy V. Shulika, Member, IEEE,

and Igor A. Sukhoivanov, Senior Member, IEEE

Abstract—We investigate the dependence of dipole matrix ele-ments for InGaN/GaN single quantum well structures on the in-dium surface segregation (ISS). Obtained results show that theinfluence of the surface segregation on the dipole matrix elementis not equal for all optical transition. This effect results from thejoint action of the piezoelectric polarization and ISS that changeselection rules. In addition, surface segregation at each interfaceof the quantum well has different impact on optical characteristicsdepending on the direction of the piezoelectric polarization. Theeffect of the surface segregation has been estimated applying theglobal sensitivity analysis in the frame of six-band approximationfor the valence band and parabolic approximation for the conduc-tion band.

Index Terms—Envelope function, global sensitivity analysis, in-dium surface segregation (ISS), piezoelectric polarization, transi-tion matrix element.

I. INTRODUCTION

THE indium surface segregation (ISS) appears during thecrystal growth due to large difference between free bind-

ing enthalpies of GaN and InN semiconductor materials [1]. Amanifestation of the effect is dependent on the method of crystalgrowth. However, it is reported [2] that both the molecular beamepitaxy and metalorganic vapor phase epitaxy are accompaniedby the ISS.

The ISS can be observed using the transmission electron mi-croscopy (TEM) [3], reflection high-energy electron diffraction(RHEED) [4], and X-rays diffraction (XRD) [5]. However, allthese experimental methods suffer from shortcomings in thecase of ultrathin quantum wells (QWs). TEM induce an addi-tional local strain in the crystal lattice after a long duration ofthe electron beam exposition [6]. Therefore, this experimentaltechnique contains systematic errors. In addition, this methodrequires special sample preparation. RHEED can be applied

Manuscript received November 30, 2010; revised March 14, 2011; acceptedApril 17, 2011. Date of publication June 27, 2011; date of current versionOctober 5, 2011. This work was supported in part by the Direccion de Apoyoa la Investigacon y al Posgrado de la Universidad de Guanajuato under Grant000041/10.

M. V. Klymenko and O. V. Shulika are with the Lab. “Photonics,” KharkovNational University of Radio Electronics, Kharkov 61166, Ukraine (e-mail:[email protected]; [email protected]).

I. A. Sukhoivanov is with the DICIS, University of Guanajuato, Salamanca,36885, Mexico (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSTQE.2011.2151176

only during the crystal growth, and it is not applicable whenthe growth is over. The XRD technique has low sensitivity fordistances up to 2 nm and its experimental data are difficultto interpret. Thus, all these methods have many restrictions inthe case of ultrathin InGaN/GaN QWs with thickness smallerthan 5 nm. The optical spectroscopy allows to avoid most ofthese disadvantages. However, the application of this experi-mental technique requires the mathematical model giving aninterpretation of measured spectroscopic data. Such a model ofthe optical response should provide clear relationship betweenoptical spectroscopic data and parameters of structure imper-fections. In this paper, we investigate the influence of the ISS onthe dipole matrix elements defining the selection rules for opti-cal intraband transitions. Magnitudes of dipole matrix elementsreflect peak intensities of absorption spectra of semiconductorheterostructures.

The structure under consideration is 2 nm InGaN/GaN singleQW with strain effects, piezoelectric polarization and ISS. Inthis paper, the influence of the ISS is studied for each interfaceof the QW separately. For this purpose, we apply the globalsensitivity analysis to estimate the sensitivity of dipole matrixelements to variations of ISS parameters. The analyze is appliedfor optical transitions between all possible pairs of subbands inthe QW.

In Section II, we will consider two approaches to parametriza-tion and modeling of the indium distribution. Next, inSection III, we will describe the approach to modeling ofposition-dependent material parameters and piezoelectric po-larization in InGaN/GaN QW structure. Section IV containsresults of band structure computations. In Section V, we willpresent the theory of the dipole matrix element, its dependenceon the in-plane wave vector and results of the global sensitivityanalysis. Section VI contains conclusions.

II. INDIUM DISTRIBUTION PROFILE

The experimentally observed indium distribution in In-GaN/GaN QW structures is determined by the set of phenomena,which occur at interfaces of the QW. Approaches to model-ing and parameterizing of this indium distribution depends onwhich phenomenon is prevailed in considered structure underdefined physical conditions. For example, the interdiffusion ofindium, resulted from high-temperature annealing, is describedby the Fick’s law [7]. In this case, the indium distribution profile

1077-260X/$26.00 © 2011 IEEE

KLYMENKO et al.: THEORETICAL STUDY OF OPTICAL TRANSITION MATRIX ELEMENTS 1375

Fig. 1. Calculated indium distribution profiles and their parametrization.

expressed as a linear combination of the complementary errorfunctions.

It is possible to use the Gaussian function for the approxi-mation of QWs potential profile with the ISS. In this case, thewidth of the Gaussian function is a fitting parameter, whichcan be found from experimental data or theory treatment. TheGaussian approximation gives a symmetric distribution of in-dium. However, TEM images of QW structures suggest thatthe ISS leads to the asymmetric profile of the indium distribu-tion. Therefore, we use more accurate description based on thecombination of error function [1], which result from the kineticequations [8]:

nIn(z) =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

0, z ≤ z1

n0erf(

w0

L1

) [

1 − erf(

z − z2

L2

)]

, z ≥ z2

n0erf(

z − z1

L1

)

, otherwise

(1)here: nIn(z) is the indium distribution, n0 is the nominal indiummolar fraction in QW layer, L1 and L2 are the lengths of thesurface segregation, w0 = z2 − z1 , w0 is the nominal width ofQW.

This formula leads to the indium distribution, which is ingood agreement with published experimental results [9], [10].As one can see form Fig. 1 that the expression (1) gives asym-metric indium distribution. This approximation contains twofitting parameters L1 and L2 instead of single one in case of theGaussian approximation. This gives more freedom to provideaccurate fitting and allows to achieve an asymmetrical distri-bution. Hereafter, we name parameters L1 and L2 segregationlengths. As it follows from experimental data, fitting parametersL1 and L2 are not equal that means inequality of the segrega-tion effect for switch-ON and switch-OFF regimes of the MBEindium evaporator.

Real structures can demonstrate also alloy and well widthfluctuations besides the ISS. The fluctuations of the alloy, giv-ing rise to in-rich regions (In clusterization), and fluctuationsof the QW width can potentially influence the band diagram athydrogen temperatures, providing carrier localization and man-ifest in narrow photoluminescence lines. However, at room tem-

peratures, which are of primary interest for applications, thesemodifications to the potential profile could not localize carriersdue to thermal escape and energy spreading. The indium sur-face segregation, on the contrary, will influence carrier spectrumboth at low and room temperatures and thus will exert an impacton the transition matrix element. The alloy and well width fluc-tuations will manifest in additional broadening of the opticalspectral characteristics, as they provide additional mechanismof carrier scattering instead of additional mechanism of carrierlocalization.

III. POSITION-DEPENDENT MATERIAL PARAMETERS

AND INTERNAL ELECTRIC FIELDS

In this paper, we consider the single QW structure with lay-ers made of InxGa1−xN and GaN semiconductors. All position-dependent material parameters have been computed using linearinterpolation formulas except for the bandgap energy. For thebandgaps of alloys, we have used the second-order interpola-tion formula with the bowing parameter. Therewith, materialparameters for relevant binary semiconductors have been takenfrom [11].

A well-known peculiarity of the wurtzite crystal heterostruc-tures is strong internal electric fields caused by spontaneous po-larization and piezoelectric effects. Spontaneous and piezoelec-tric polarizations in quantum wells InGaN/GaN are of the sameorder. However, only the gradient of the polarization, whichgives the space charge, contributes to the potential relief [12].In contrast to the piezoelectric polarization, the spontaneouspolarization exists both in the quantum well and barrier lay-ers. Changes of this kind of polarization at interfaces are small.Therefore, if the approximation of the infinitely extended barri-ers is applied, we can neglect the spontaneous polarization.

The piezoelectric polarization Ppiezo is calculated using Ve-gard’s interpolation formula [13]:

Ppiezo(z) = xP InNpiezo [ε(z)] + (1 − x)PGaN

piezo [ε(z)] (2)

where the strain coefficient ε(z) is defined as

ε(z) =asubs − a(z)

a(z)(3)

here asubs is the lattice constant of the substrate and a(z) isthe lattice constant of the unstrained semiconductor alloy at apoint z.

The piezoelectric polarization of binary strained semiconduc-tors can be expressed as

P InNpiezo [ε(z)] = −1.373ε(z) + 7.559ε2(z) (4)

PGaNpiezo [ε(z)] = −0.918ε(z) + 9.541ε2(z). (5)

The negative divergence of the polarization gives the vol-ume charge density. Solving the Poisson equation with obtainedcharge distribution, one can get the potential profile of bandedges.

As follows from (2)–(5), the ISS leads to the positional de-pendence of the piezoelectric polarization. For the structureswith zero ISS, the piezoelectric polarization is approximatelyconstant within the QW. Fig. 2 contains results of computations


Fig. 2. Positional dependence of (a) piezoelectric polarization, (b) piezo-electric charge distribution, and (c) conduction band edge for the 2-nmIn0 .37 Ga0 .63 N/GaN quantum well.

of the piezoelectric polarization, piezoelectric charge distribu-tion, and conduction band potential profile for several possibleapproximations. Total neglecting of the ISS leads to the con-stant piezoelectric polarization and zigzag potential profile ofthe band edge. Such a shape of the band diagram is caused by thejoint action of the piezoelectric polarization in the QW and theelectric field of the space charge in doped semiconductor layers.The ISS influences both on the bandgap profile and configura-tion of the piezoelectric field. If only the effect on the bandgapis considered and the piezoelectric field is approximated by theconstant value, one gets the blueshift of the QW’s potential pro-file relative to its position in the case when the total ISS effectis taken into account. This energy shift is equal 40 meV for the2-nm In0.37Ga0.63N/GaN QW at L1 = L2 = 1 nm.

The envelope functions corresponding to the center of theBrillouin zone are shown in Fig. 3. Despite of strong modifi-cation of the potential profile [see Fig. 3(b)], the ISS has notsignificant influence on envelope functions for first conductionand valence subbands. In this case, a small delocalization is ob-served, and peaks of both envelope functions are slightly shiftedtoward the second interface of the QW [see Fig. 3(b)]. The ef-fect on the second valence subband is more pronounced andvaluable. For this subband, the probability density has two max-ima, which are nonequal in the structure without the ISS. In thiscase, ratio between maximal values amounts 3:2. The ISS effectequalizes peaks of the probability density having almost equal

Fig. 3. Influence of the ISS on envelope function corresponding to the firststate in the conduction band (E1), and first (H1) and second (H2) states inthe valence band. Shaded areas show nominal thickness of the QW layer.(a) Without surface segregation. (b) With surface segregation.

magnitudes of maxima. We will show further that this featuremanifests itself in optical dipole matrix elements.

IV. BAND STRUCTURE

The electron wave function can be obtained solving the Ben–Daniel–Duke equation that is resulted from joint action of thesingle-band and the envelope function approximation [14], [15].The valence band structure is computed separately using thesix-band model taking into account interactions between heavyholes, light holes and spin-orbit split-off holes states with allpossible directions of the spin [16]. This approach leads to 6 × 6Hamiltonian:

H =(

HU 0

0 HL

)

(6)

with HU and HL of the form

HU =

⎛

⎜⎝

F K −iH

K G Δ − iH

iH Δ + iH λ

⎞

⎟⎠ (7)

HL =

⎛

⎜⎝

F K iH

K G Δ + iH

−iH Δ − iH λ

⎞

⎟⎠ (8)

here

F = Δ1 + Δ2 + λ + θ, G = Δ1 − Δ2 + λ + θ

λ =h2

2m0

(A1k

2z + A2k

2t

)+ D1εzz + D2 (εxx + εyy )

θ =h2

2m0

(A3k

2z + A4k

2t

)+ D3εzz + D4 (εxx + εyy )

εxx = εyy =asubs − a

a, εzz = −2C13

C33εxx

εxy = εyz = εzx = 0

K =h2

2m0A5k

2t , H =

h2

2m0A6ktkz , Δ =

√2Δ3


Fig. 4. Band structure of the InGaN/GaN single quantum well without indiumsurface segregation (solid lines) and with indium surface segregation (dashedlines) for L1 = L2 = 1 nm.

kt = |k‖|, A1 , A2 , A3 , A4 , and A5 are the valence band structureparameters, D1 , D2 , D3 , and D4 are the deformation potentials,Δ1 , Δ2 , and Δ3 are the energy parameters, C13 , and C33 arethe elastic stiffness constants.

This problem is further solved numerically, applying the finitedifference method [16].

The band structure is presented in Fig. 4, where solid linesgive the band structure with no segregation, and dashed linesshow dispersion curves of the structure subject to the indiumsurface segregation with segregation lengths L1 = L2 = 1 nm.One can see energy shift of all states caused by the segregationeffect. The blueshift of the transition energy is in agreementwith experimental results of other authors [10]. However, it ishard to use this feature to detect and quantitatively estimate thesurface segregation, because the shift of all energy states can becaused by many other physical effects. For example, changes inthe width of the QW as well as in the depth affect the energyspectra in the same manner. So, one should search for anothermanifestations of the ISS to make it detectable and measurablevia optical spectroscopy.

V. INTERBAND OPTICAL TRANSITION MATRIX ELEMENTS

Interband optical transition matrix elements (MEs) are keyingredients in estimation of gain and absorption. Therefore, onecan expect that manifestation of ISS in MEs could be detectedusing absorption spectroscopy. Depending on geometry of ab-sorption measurement only TE or both TE and TM matrix ele-ments will contribute in absorption. Therefore, we analyze hereboth kinds of polarization.

The QW under consideration contains four hole subbandsand one electron subband. Matrix elements for optical transi-tions between these subbands are shown in Fig. 5 calculated fordifferent segregation lengths. MEs for wurtzite QW is calculatedas given by Chuang [17].

The ISS influences MEs in unequal degree for different in-plane wave vectors. In case of the TE polarization, the effect ofthe ISS is more observable for small in-plane wave vectors. In

Fig. 5. In-plane wave vector dependence of the dipole matrix elements for(a) TE polarization, transitions E1-HH1 and E1-LH1; (b) TM polarization,transitions E1-HH1 and E1-LH1; (c) TE polarization, transitions E1-HH2 andE1-LH2; (d) TM polarization, transitions E1-HH2 and E1-LH2. All curvesare obtained for segregation parameters L1 = L2 = 0.1, 0.2, 0.3, 0.4, 0.5,1.0 nm.

the case of the TM polarization, most effects appear at large in-plane wave vectors. Thus, these results suggest to use probe lightperpendicular to QW plane to increase measurement precision.

A. Global Sensitivity Analysis

Overall conclusion of the previous section is that increasingof segregation lengths leads to decreasing of the matrix element.However, segregation at each interface of a QW is not equal.Therefore, to resolve its influence much more data should beanalyzed than those presented in Fig. 5. To do that we use hereglobal sensitivity analysis [18]. This approach allow to estimatethe sensitivity of the matrix elements with respect to varia-tions of ISS parameters separately and without large number ofcomputations.

Using the global sensitivity analysis we try to clarify howstrong the response of the system on the ISS effect is, what MEthe most sensitive to the segregation is, and what segregationparameter having the strongest effect on the dipole matrix el-ement is. Final results are expressed as sensitivity coefficientswhich are defined as

sμi j

Lm=

σLm

σμi j

∂μij

∂Lm(9)

here σLmand σμi j

are standard deviations for segregationlengths and transition matrix elements respectively.

As we found earlier, TE matrix elements are more useful frompractical viewpoint. Therefore, here, we analyze MEs only forzero in-plane wave vector, i.e., at the center of the Brillouinzone. At the center of the Brillouin zone, shapes of the light andheavy hole envelope functions are almost identical. Therefore,we consider only heavy hole states, because results for light


Fig. 6. Global sensitivity analysis of the dipole matrix elements (TE polariza-tion) according to segregation parameters L1 and L2 . (a) μ11 . (b) μ13 .

TABLE ISENSITIVITY COEFFICIENTS

holes will be the same. Thus, we compute the dipole matrix ele-ments μ11 and μ13 for optical transitions between first subbandE1 in the conduction band and two heavy hole subbands H1 andH2 instead of full consideration.

In the frame of global sensitivity analysis, segregation lengthsL1 and L2 are changed simultaneously taking random values.That is main advantage of this approach, which leads to com-putational efficiency. The scatterplot in Fig. 6(a) reflects theparticular feature that the sensitivity of μ11 is not the samefor all values of L1 . The behavior of the scatterpoints distri-bution is changed significantly crossing the value L1 ≈ 0.7.This means that the sensitivity coefficient is dependent on thesegregation length L1 . In this connection, we use approxima-tion based on two linear regressions for ranges 0 ≤ L1 < 0.7and 0.7 ≤ L1 ≤ 1 and, consequently, we obtain two sensitivitycoefficients. The physical origin for sharp change of the μ11around the point L1 = 0.7 in Fig. 6(a) is delocalization of theelectron ground state. In this context, the term ”means that theelectron energy exceeds that one of the band offset.

Table I contains values of sensitivity coefficients. The ISSparameter L1 has stronger influence as compared with L2 . Thedipole matrix element μ13 is most sensitive to the indium sur-face segregation. This behavior can be interpreted looking onenvelope functions overlap, as shown in Fig. 3. The envelope

Fig. 7. Dependencies of the sensitivity coefficients on the intensity of theinternal electrostatic fields for dipole matrix elements (a) μ11 and (b) μ13 ofthe 2-nm In0 .37 Ga0 .63 N/GaN quantum well.

function for the second heavy hole state has two extrema, whichare located near the interfaces where the ISS occurs. It makesthis subband especially sensitive to the changes of the potentialat the interfaces of the QW.

Considered above, sensitivity coefficients are computed forthe 2-nm In0.1Ga0.9N/GaN QW with the piezoelectric fieldEpiezo = 1.2 MV/cm and without any other electrostatic fields.In real structures, the QW is surrounded by doped semiconduc-tor layers forming a p-i-n structure. Therefore, the QW layeris affected by the electric field of the space charge appeared atinterfaces in the doped layers. Sensitivity coefficients for thiscase are presented in Figs. 7–8. The Fig. 7 shows quantitativeestimation of the matrix element sensitivity to the segregationlengths under various internal electrostatic fields. As it followsfrom the figure, sensitivity coefficients are strongly dependenton the magnitude of the internal electrostatic field. This field isthe superposition of the fields caused by the piezoelectric po-larization and space charge in the doped layers. The sensitivitycoefficients sμ1 1

L1and sμ1 3

L1on the figure take double values in the

range from 0.8 to 3.25 MV/cm. In this range, the internal voltagedrop across the QW is slightly less than the potential of the bandoffsets. In this situation, the electron ground state can be local-ized or delocalized depending on the segregation lengths, andthe scatterplot of μij against L1 in such cases takes on the ap-pearance like that one in Fig. 6(a), which is characterized by twosensitivity coefficients (see Table I). Therefore, two additionalbranches in Fig. 7 are due to the attempt to apply piecewiselinear regression to significantly nonlinear distribution throughits fission on two parts.

The sensitivity coefficients sμ1 1L1

and sμ1 1L2

are nearly constantat the high internal fields while sμ1 3

L1and sμ1 3

L2have nontrivial

dependence on the internal field even at high magnitudes. Atvery low and very high intensities of internal fields, the dipolematrix element μ11 have opposite sensitivity relative to the ISSat each interface of the QW. This means that the ISS at oneinterface leads to increasing of the dipole matrix element, whilethe ISS at another interface leads to decreasing of this value.If the segregation lengths L1 and L2 are approximately equal,total changes in the dipole matrix element μ11 caused by theISS are very small due to the mutual compensation. It is not the


Fig. 8. Dependencies of the sensitivity coefficients sμ 1 1L 1

and sμ 1 1L 2

on the(a) quantum well width and (b) indium amount for the 2-nm Inx Ga1−x N/GaNquantum well.

case for the matrix element μ13 , which is very sensitive to theISS in these ranges.

Fig. 8 reflects weak dependence of the sensitivity on the QWwidth and indium amount. Increasing of the QW width leads tothe monotonic decreasing of the sensitivity to both segregationlengths. Indium amount determines the depth of the QW andinfluences on the magnitude of the piezoelectric polarization.As a result, this characteristic is very close to the dependence ofthe sμ1 1

L1and sμ1 1

L2on the internal fields.

As far as it is possible to change the magnitude of internalfields managing the doping profile and turning applied electricalbias, considered above sensitivity features could be of interestfor the experimental observation of the ISS. All presented re-sults are dependent on the polarity of the QW. Here, we haveconsidered only the case of the Ga-face growth.

VI. CONCLUSION

In summary, we have investigated the influence of the ISS onthe dipole matrix element for InGaN/GaN single QW structures.The joint action of the ISS and internal electrostatic fields leadsto pronounced sensitivity of dipole matrix elements to variationsof the segregation lengths L1 and L2 .

The global sensitivity analysis has shown that the transitionmatrix elements μ13 and μ14 , involving transitions E1-HH2 andE1-LH2, are more sensitive to the ISS comparing with opticaltransitions E1-HH1 and E1-LH1 subbands.

Dipole matrix elements are most sensitive for the internalfields limited in the ranges from 0.8 to 3.25 MV/cm. For the2-nm In0.37Ga0.63N/GaN QW, computed intensity of the inter-nal fields is equal Epiezo = 6.2 MV/cm. In this case, an effectiveobservation of the ISS requires decreasing of this magnitude thatcan be realized applying of the high-intense reverse voltage bias.

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[4] S. Martini, A. A. Quivy, T. E. Lamas, M. J. da Silva, and E. C. F. da Silva,“Influence of indium segregation on the RHEED oscillations during thegrowth of InGaAs layers on a GaAs(0 0 1) surface,” J. Crystal Growth,vol. 251, pp. 101–105, 2003.

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[12] R. Resta, “Macroscopic polarization in crystalline dielectrics: The geo-metric phase approach,” Rev. Mod. Phys., vol. 66, no. 3, pp. 899–915,1994.

[13] V. Fiorentini, F. Bernardini, and O. Ambacher, “Evidence for nonlinearmacroscopic polarization in III-V nitride alloy heterostructures,” Appl.Phys. Lett., vol. 80, pp. 1204a–1206, 2002.

[14] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures.Chichester, U.K: Wiley, 1991.

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[16] S. L. Chuang and C. S. Chang, “A band-structure model of strainedquantum-well wurtzite semiconductors,” Semicond. Sci. Technol., vol. 12,pp. 252–263, 1997.

[17] S. L. Chuang, “Optical gain of strained wurtzite GaN quamtum-welllasers,” IEEE J. Quantum Electron., vol. 32, no. 7, pp. 1791–1800, Jul.1996.

[18] A. Saltelli, M. Ratto, T. Andres, F. Compolongo, J. Cariboni, D. Gatelli,M. Saisana, and S. Tarantola, Global Sensitivity Analysis: The Primer.Chichester, U.K.: Wiley, 2008.

Mykhailo V. Klymenko received the B.S. degree(highest honors) in laser physics and optoelectronic in2004, and theM.S. degree in 2005, both from KharkivNational University of Radio Electronics, Kharkiv,Ukraine, where he has been working toward the Ph.D.degree at the Department of Physical Foundations ofElectronic Engineering.

His current research interests include semicon-ductor optics, quantum field theory, and physics ofsemiconductor nanostructures.

Mr. Klymenko was awarded by one-year scholar-ship from German Academic Exchange Service (DAAD) for research in Theo-retical Semiconductor Physics Group of Prof. S. W. Koch at Philipps-University,Marburg, in 2007.


Oleksiy V. Shulika (S’01–M’08) received the B.S.and M.S. degrees (highest honors) in electronic engi-neering from Kharkiv National University of RadioElectronics, Kharkiv, Ukraine, in 2000 and 2001, re-spectively, and the Ph.D. degree in optics and laserphysics from V. N. Karazin National University,Ukraine, in 2008.

He is currently an Associate Professor at KharkivNational University of Radio Electronics, Ukraine.From 2005 to 2006, he is a Visiting Researcher atthe Institute for High-Frequency and Quantum Elec-

tronics, University of Karlsruhe (TH), Germany. His research interests includesemiconductor optics, physics of semiconductor nanostructures, and optoelec-tronic devices simulation.

Dr. Shulika is a member of the IEEE Photonics Society, the Optical Societyof America, and the International Society for Optical Engineers, the OrganizingCommittee of the Conference Series Conference on Advanced Optoelectronicsand Lasers (CAOL), “NATO Advanced Research Workshop and Mid InfraredRadiation: Basic Research and Applications, TERA-MIR,” and “InternationalConference on Laser and Fiber-Optical Networks Modeling (LFNM).”

Igor A. Sukhoivanov (M’94–SM’00) received theM.S. degree in electronic engineering, the Ph.D.degree in quantum electronics and fiber optics,both from Kharkiv Institute of Radio Electronics,Kharkiv, Ukraine, and the Dr. Sc. degree in opticsand laser physics from V. N. Karazin National Uni-versity Kharkiv, Ukraine, in 1976, 1985 and 2002,respectively.

In 1985, he was an Assistant Professor, AssociateProfessor, and than a Full Professor of the Faculty ofElectronic Engineering, Kharkiv National University

of Radio Electronics, Kharkiv, Ukraine. From 1987 to 1988, he was a ResearchScientist at Humboldt University, Berlin, Germany. In 1994, 1997, and 2002,he was a Guest Scientist at the Institute of High Frequency Technique of theKarlsruhe University, Germany, and in 2001, he was with the Laboratory CEM2,University Montpellier, France. He is currently a Professor at the DICIS, Uni-versity of Guanajuato, Mexico. He is the author or coauthor of one monographand more than 70 papers published in refereed scientific journals. His researchinterests include fiber optics and semiconductor multiquantum-well lasers, pho-tonics crystals elements.

Prof. Sukhoivanov is a Senior Member of the IEEE Photonics Society, andmember of the Optical Society of America, the American Physical Society, andthe International Society for Optical Engineers, an Organizer and Chairmanof the Conference Series Conference on Advanced Optoelectronics and Lasers(CAOL),” NATO Advanced Research Workshop and Mid Infrared Radiation:Basic Research and Applications (TERA-MIR),” etc.

Theoretical Study of Optical Transition Matrix Elements in InGaN/GaN SQW Subject to Indium Surface Segregation

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