Hadis Morkoç

NitrideSemiconductorDevicesFundamentals and Applications

Hadis MorkoSc

Nitride SemiconductorDevices

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Hadis MorkoSc

Nitride Semiconductor Devices

Fundamentals and Applications

The Author

Prof. Dr. Hadis MorkocVirginia Commonwealth Univ.Dept. of Electric. Engineering601 W. Main St. Room 338Richmond, VA 23284-3072USA

CoverBackground: GaN based laser(courtesy of Prof. U. Schwarz).Insets: SEM images of a GaN based HFETwith80 nm gate length (courtesy of Prof. T. Palacio).

All books published byWiley-VCH are carefullyproduced. Nevertheless, authors, editors, andpublisher do not warrant the information containedin these books, including this book, to be free oferrors. Readers are advised to keep in mind thatstatements, data, illustrations, procedural details orother items may inadvertently be inaccurate.

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#2013 Wiley-VCH Verlag GmbH & Co. KGaA,Boschstr. 12, 69469 Weinheim, Germany

All rights reserved (including those of translation intoother languages). No part of this book may bereproduced in any form – by photoprinting,microfilm, or any other means – nor transmitted ortranslated into a machine language without writtenpermission from the publishers. Registered names,trademarks, etc. used in this book, even when notspecifically marked as such, are not to be consideredunprotected by law.

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To those who advance the frontiers of science and engineering

Contents

Preface XIII

1 General Properties of Nitrides 11.1 Crystal Structure of Nitrides 11.2 Gallium Nitride 51.3 Aluminum Nitride 61.4 Indium Nitride 101.5 AlGaN Alloy 131.6 InGaN Alloy 141.7 AlInN Alloy 141.8 InAlGaN Quaternary Alloy 151.9 Electronic Band Structure and Polarization Effects 181.9.1 Introduction 181.9.2 General Strain Considerations 221.9.3 k�p Theory and the Quasicubic Model 231.9.4 Temperature Dependence of Wurtzite GaN Bandgap 261.9.5 Sphalerite (Zincblende) GaN 261.9.6 AlN 281.9.6.1 Wurtzite AlN 281.9.6.2 Zincblende AlN 281.9.7 InN 291.10 Polarization Effects 311.10.1 Piezoelectric Polarization 321.10.2 Spontaneous Polarization 351.10.3 Nonlinearity of Polarization 351.10.3.1 Nonlinearities in Piezoelectric Polarization 421.10.4 Polarization in Heterostructures 461.10.4.1 Ga-Polarity Single AlGaN–GaN Interface 511.10.4.2 Polarization in QuantumWells 561.11 Nonpolar and Semipolar Orientations 59

Further Reading 61

jVII

2 Doping: Determination of Impurity and Carrier Concentrations 632.1 Introduction 632.2 Doping 632.3 Formation Energy of Defects 652.3.1 Hydrogen and Impurity Trapping at Extended Defects 672.4 Doping Candidates 692.5 Free Carriers 702.6 Binding Energy 702.7 Conductivity Type: Hot Probe and Hall Measurements 712.8 Measurement of Mobility 712.9 Semiconductor Statistics, Density of States, and Carrier

Concentration 742.10 Charge Balance Equation and Carrier Concentration 782.10.1 n-Type Semiconductor 792.10.2 p-Type Semiconductor 842.11 Capacitance–Voltage Measurements 87

Appendix 2.A. Fermi Integral 94Further Reading 95

3 Metal Contacts 973.1 Metal–Semiconductor Band Alignment 973.2 Current Flow in Metal–Semiconductor Junctions 1013.3 Ohmic Contact Resistance 1073.3.1 Specific Contact Resistivity 1073.4 Semiconductor Resistance 1083.4.1 Determination of the Contact Resistivity 109

Further Reading 113

4 Carrier Transport 1154.1 Introduction 1154.2 Carrier Scattering 1174.2.1 Impurity Scattering 1184.2.2 Acoustic Phonon Scattering 1204.2.2.1 Deformation Potential Scattering 1214.2.2.2 Piezoelectric Scattering 1244.2.3 Optical Phonon Scattering 1264.2.3.1 Nonpolar Optical Phonon Scattering 1264.2.3.2 Polar Optical Phonon Scattering 1274.2.4 Alloy Scattering and Dislocation Scattering 1344.3 Calculated Mobility of GaN 1434.4 Scattering at High Fields 1474.4.1 Transport at High Fields: Energy and Momentum Relaxation Times 1524.4.2 Energy-Dependent Relaxation Time and Large B 1534.4.3 Hall Factor 1554.5 Delineation of Multiple Conduction Layer Mobilities 156

VIIIj Contents

4.6 Carrier Transport in InN 1584.7 Carrier Transport in AlN 1594.8 Carrier Transport in Alloys 1614.9 Two-Dimensional Transport in n-Type GaN 1644.9.1 Scattering in 2D Systems 1664.9.1.1 Electron Mobility in AlGaN/GaN 2D System 1684.9.1.2 Numerical Two-Dimensional Electron Gas Mobility Calculations 1704.9.1.3 Magnetotransport and Mobility Spectrum 173

Further Reading 174

5 The p–n Junction 1775.1 Introduction 1775.2 Band Alignment 1775.3 Electrostatic Characteristics of p–n Heterojunctions 1795.4 Current–Voltage Characteristics of p–n Junctions 1855.4.1 Diode Current under Reverse Bias 1865.4.1.1 Poole–Frenkel and Schottky Effects 1875.4.1.2 Avalanching 1885.4.2 Diffusion Current 1895.4.2.1 Diffusion Current under Reverse Bias 1905.4.2.2 Diffusion Current under Forward Bias 190

Further Reading 191

6 Optical Processes 1936.1 Introduction 1936.2 Einstein’s A and B Coefficients 1946.3 Absorption and Emission 1966.4 Band-to-Band Transitions and Efficiency 1986.5 Optical Transitions in GaN 2006.5.1 Excitonic Transitions in GaN 2006.5.1.1 Strain Effects 2036.5.1.2 Bound Excitons 2046.6 Free-to-Bound Transitions 2056.7 Donor–Acceptor Transitions 206

Further Reading 207

7 Light-Emitting Diodes and Lighting 2097.1 Introduction 2097.2 Current Conduction Mechanism in LED-Like Structures 2117.3 Optical Output Power and Efficiency 2147.3.1 Efficiency and Other LED Relevant Terms 2157.3.2 Optical Power and External Efficiency 2177.3.3 Internal Quantum Efficiency 2187.3.3.1 Auger Recombination 2197.3.3.2 SRH Recombination 220

Contents jIX

7.3.3.3 Radiative Recombination 2227.3.3.4 Continuity or Rate Equations as Pertained to Efficiency 2237.3.3.5 Carrier Overflow (Spillover, Flyover, Leakage) 2317.4 Effect of Surface Recombination 2447.5 Effect of Threading Dislocation on LEDs 2477.6 Current Crowding 2477.7 Perception of Color 2507.8 Chromaticity Coordinates and Color Temperature 2517.9 LED Degradation 2537.10 Packaging 2557.11 Luminescence Conversion and White Light Generation 2577.11.1 Color-Rendering Index 2587.11.2 White Light from Multichip LEDs 2597.11.3 Combining LEDs and Phosphor(s) 262

Further Reading 266

8 Semiconductor Lasers: Light Amplification by StimulatedEmission of Radiation 267

8.1 Introduction 2678.2 A Primer to the Principles of Lasers 2688.2.1 Waveguiding 2708.2.2 Analytical Solution to the Waveguide Problem 2738.2.2.1 TE Mode 2748.2.2.2 TM Mode 2768.2.3 Far-Field Pattern 2808.3 Loss, Threshold, and Cavity Modes 2818.4 Optical Gain 2838.5 A Glossary for Semiconductor Lasers 2868.5.1 Optical Gain in Bulk Layers: a Semiconductor Approach 2898.5.1.1 Relating Absorption Rate to Absorption Coefficient 2908.5.1.2 Relating Stimulated Emission Rate to Absorption Coefficient 2908.5.1.3 Relating Spontaneous Emission Rate to Absorption Coefficient 2908.5.2 Semiconductor Realm 2918.5.3 Gain in QuantumWells 2998.5.3.1 Optical Gain 3028.5.3.2 Measurement of Gain in Nitride Lasers 3048.5.4 Gain Measurement via Optical Pumping 3048.6 Threshold Current 3068.7 Analysis of Injection Lasers with Simplifying Assumptions 3078.7.1 Recombination Lifetime 3098.7.2 Quantum Efficiency 3118.8 GaN-Based LD Design and Performance 3128.8.1 Gain Spectra of InGaN Injection Lasers 3178.8.2 Mode Hopping 3218.9 Thermal Resistance 322

Xj Contents

8.10 Nonpolar and Semipolar Orientations 3238.11 Vertical Cavity Surface-Emitting Lasers (VCSELs) 3258.11.1 Microcavity Fundamentals 3288.11.2 Polariton Lasers 3338.12 Degradation 337

Appendix 8.A: Determination of the Photon Density and PhotonEnergy Density in a Cavity 343Further Reading 348

9 Field Effect Transistors 3499.1 Introduction 3499.2 Operation Principles of Heterojunction Field Effect Transistors 3509.2.1 Heterointerface Charge 3509.2.2 Analytical Description of HFETs 3589.3 GaN and InGaN Channel HFETs 3649.4 Equivalent Circuit Models: De-embedding and Cutoff Frequency 3669.4.1 Small-Signal Equivalent Circuit Modeling 3679.4.2 Cutoff Frequency 3709.5 HFETAmplifier Classification and Efficiency 3739.6 Drain Voltage and Drain Breakdown Mechanisms 3789.7 Field Plate for Spreading Electric Field for Increasing Breakdown

Voltage 3839.8 Anomalies in GaN MESFETs and AlGaN/GaN HFETs 3849.8.1 Effect of the Traps in the Buffer Layer 3869.8.2 Effect of Barrier States 3929.8.3 Correlation between Current Collapse and Surface Charging 3939.9 Electronic Noise 3969.9.1 FET Equivalent Circuit with Noise 3989.9.2 High-Frequency Noise in Conjunction with GaN FETs 4029.10 Self-Heating and Phonon Effects 4059.10.1 Heat Dissipation and Junction Temperature 4069.10.2 Hot Phonon Effects 4099.10.2.1 Phonon Decay Channels and Decay Time 4119.10.2.2 Implications for FETs 4169.10.2.3 Heat Removal in View of Hot Phonons 4189.10.2.4 Tuning of the Hot Phonon Lifetime 4219.11 HFET Degradation 4279.11.1 Gated Structures: Reliability 4349.11.2 Reliability Tests 4389.12 HFETs for High-Power Switching 440

Appendix 9.A. Sheet Charge Calculation in AlGaN/GaNStructures with AlN Interface Layer (AlGaN/AlN/GaN) 444Further Reading 446

Index 449

Contents jXI

Preface

This book aims to describe the fundamentals of light emitters and field effecttransistors based on GaN and related semiconductors with supporting material. Thebook is intended to provide the know-how for the reader to be well versed in theaforementioned devices with selective further reading material for additionalmaterial.Chapter 1 deals with the structural properties of nitride-based semiconductors

and their band structure and polarization with extensive tables. Chapter 2 discussesdefects and doping, electron and hole concentrations along with applicable statisticsas affected by temperature, and Hall and C–Vmeasurements. Metal semiconductorjunctions along with the current conduction mechanisms, contact resistivity and itsdetermination are discussed in Chapter 3. Scattering and carrier transport at low andhigh electric fields are discussed in Chapter 4, which embodies ionized impurity,deformation potential, piezoelectric, optical phonon, and alloy scattering, amongothers, in bulk and to a lesser extent in two-dimensional systems. Hall effect/Hallfactor and magneto transport along with delineation of mobility for each of thecontributing layers in multichannel constructs are also included in the discussion.Chapter 5 is devoted to p–n junctions, beginning with the discussion of band lineupsand leading to consideration of current conduction mechanisms, such as diffusion,generation–recombination, and Poole–Frenkel current. Avalanche multiplication,pertinent to the high-field region of FETs and avalanche photodiodes is alsocovered in a concise form. Chapter 6 contains a succinct discussion of opticalprocesses in semiconductors such as absorption and emission vis-�a-vis Einstein’s Aand B coefficients to pave the way for discussion of light emitters in the follow-upchapters. Chapter 7 delves into the fundamentals and practice of light-emittingdiodes, perception of vision and color by human eye and methodologies, both usedand proposed, for generation of white light and presents an in-depth discussion ofefficiency and mechanisms responsible for its degradation at high injection levels.Chapter 8 focuses on lasers including the relevant theory and practical operation.Integral concepts such as gain and loss along with their measurement, thresholdcurrent, efficiency, polar- and nonpolar-specific processes, and microcavity-basedlasers are also discussed. The final chapter, Chapter 9, treats field effect transistorfundamentals, which are applicable to any semiconductor material with pointsspecific to GaN-based varieties. The discussion primarily focuses on 2DEG channels

jXIII

formed at heterointerfaces and their use for FETs including polarization effects. Asuccinct analytical model is provided for calculating the carrier densities at theinterfaces for various scenarios and current–voltage characteristics of FETs withseveral examples. Hot phonon effects responsible for this shortfall and attainablecarrier velocity are uniquely discussed with sufficient theory and experimental dataand design approaches to mitigate the problem, along with their effect on heatdissipation and reliability.This book would not have been possible without the support of many of our

colleagues, namely, Profs. €U. €Ozg€ur, V. Avrutin, R. Shimada, and A. Matulionis andDrs. N. Izyumskaya, J. Xie, J. Leach, and C. Kayis, who helped with material, figures,proofreading, and extensive discussions during its production.

Richmond, VA USA Hadis MorkoScSeptember 2012

XIVj Preface

1General Properties of Nitrides

1.1Crystal Structure of Nitrides

GaN and its binary cousins InN and AlN as well as their ternary and quaternary areconsidered one of the most important groups of semiconductors after Si. Thisfollows from their ample applications in lighting and displays, consumer elec-tronics, lasers, detectors, and high-power RF/switching devices owing to theirexcellent optical and electrical properties. The pertinent properties and materialsparameters upon which to build the chapters on devices are succinctly discussed.Group III nitrides can be of wurtzite (Wz), zincblende (ZB), and rocksalt structure.

Under ambient conditions, the thermodynamically stable structure is wurtzite forbulk AlN, GaN, and InN. The space grouping for the zincblende structure is F�43min the Hermann–Mauguin notation and T2

d in the Schoenflies notation and has acubic unit cell containing four group III elements and four nitrogen elements. (Theterm zincblende originated in describing compounds such as ZnS that could becubic or hexagonal. But the term has been used ubiquitously for compoundsemiconductors with cubic symmetry. The correct term for the cubic phase issphalerite.). The position of the atoms within the unit cell is identical to that in thediamond crystal structure. The stacking sequence for the (111) close-packed planesin this structure is AaBbCc. Small and large letters stand for the two different kindsof constituents. The rocksalt structure (with space group Fm�3m in the Hermann–Mauguin notation and O5

h in the Schoenflies notation) can be induced under veryhigh pressures, but not through epitaxial growth.The wurtzite structure has a hexagonal unit cell and thus two lattice constants c

and a. It contains six atoms of each type. The space grouping for the wurtzitestructure is P63mc in the Hermann–Mauguin notation and C4

6v in the Schoenfliesnotation. The point group symmetry is 6mm in the Hermann–Mauguin notationand C6v in the Schoenflies notation. The Wz structure consists of two interpene-trating hexagonal close-packed (hcp) sublattices, each with one type of atom, offsetalong the c-axis by 5/8 of the cell height (5c/8). The Wz structure consists ofalternating biatomic close-packed (0001) planes of Ga and N pairs, thus the stackingsequence of the (0001) plane is AaBbAa in the (0001) direction.

Nitride Semiconductor Devices: Fundamentals and Applications, First Edition. Hadis MorkoSc.# 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.

j1

The Wz and zincblende structures differ only in the bond angle of the second-nearest neighbor (Figure 1.1). The stacking order of the Wz along the [0001]c-direction is AaBb, meaning a mirror image but no in-plane rotation with thebond angles. In the zincblende structure along the [111] direction, there is a 60�

rotation that causes a stacking order of AaBbCc. The point with regard to rotation isillustrated in Figure 1.1b. The nomenclature for various commonly used planes ofhexagonal semiconductors in two- and three-dimensional versions is presentedin Figure 1.2. TheWz group III nitrides lack an inversion plane perpendicular to thec-axis; thus, nitride surfaces have either a group III element (Al, Ga, or In) polarity(referred to as the Ga-polarity) with a designation of (0001) or (0001)A plane or aN-polarity with a designation of (000�1) or (0001)B plane. The former notations foreach are used here. The distinction between these two directions is essential innitrides due to implications in the polarity of the polarization charge. Three surfacesand directions are of special importance in nitrides, which are (0001) c-, (11�20)a-, and (1�100) m-planes and the directions associated with them: k0001i, k11�20i,and k1�100i.

B

Wurtzitic Zincblende

[111]

View along

[0001] and [111]

Ga

N

A

B

A

C

(a)

[0001]

(b)

Figure 1.1 Ball-and-stick stacking model ofcrystals with (a) (both top and bottom) 2Hwurtzitic and (b) (both top and bottom) 3Czincblende polytypes. The bonds in an a-plane(11�20) are indicated with thicker lines toaccentuate the stacking sequence. The figures

on the top depict the three-dimensional view.The figures at the bottom indicate theprojections on (0001) and (111) planes forwurtzitic and cubic phases, respectively. Notethe rotation in the zincblende case along thek111i direction.

2j 1 General Properties of Nitrides

Delving further into theWz structure, it can be represented by lattice parameters ain the basal plane and c in the perpendicular direction and the internal parameter u,as shown in Figure 1.3. The u parameter is defined as the anion–cation bond length(also the nearest-neighbor distance) divided by the c lattice parameter. The cparameter depicts the unit cell height. The wurtzite structure is a hexagonal

cn

rn n

r

nnr

n

(m)

c

nr n n r

aa a

nn r

n

(c)

(m)

Common crystalograhicPlanes

Planename

MillerIndex

a (1120)m (1010)c (0001)r (1102)n (1123)s (1011)

Angles BetweenCommon Planes

(0001) ^ (1102) c ^ r 57o 35'(0001) ^ (1123) c ^ n 61o 11'(0001) ^ (1011) c ^ s 72o 23'(0001) ^ (1121) c ^ 79o 37'(0001) ^ (1120) c ^ a 90o 00'(0001) ^ (1010) c ^ m 90o 00'(1120) ^ (1010) a ^ m 30o 00'

<1120><1100>

30o

30o

61o

32.4 o

57.6 o

01111123

sn

rn s

n

r

n

s

n

r

c

d

d d

1012

2113 1101

n1213

1102

0001

0114 1104

1213

0112

11231014

1011

a

m

0110

1120

1010

2110

1100

0110

1120

1010

2110

1100

1210

2113

1210

t

u

v

m-planesa-planes

Figure 1.2 Labeling of planes in hexagonalsymmetry (for sapphire), a telescopic view oflabeling of planes in hexagonal symmetry in the(tuvw) coordinate system with w representingthe unit vector in the c-direction is shown on theright. The lines are simply to show thesymmetry only. If the lines connectingm-pointsamong each other and a-points among each

other were to be interpreted as the projection ofthose planes on the c-plane, the roles would beswitched in that the lines connecting them-points would actually represent the a-planes,and lines connecting the a-points would actuallyrepresent them-planes that are normal to theplane of the page.

1.1 Crystal Structure of Nitrides j3

close-packed lattice, comprising vertically orientedM–Nunits at the lattice sites. Thebasal plane lattice parameter (the edge length of the basal plane hexagon) isuniversally depicted by a and the axial lattice parameter perpendicular to the basalplane is universally described by c. In an ideal wurtzite structure represented by fourtouching hard spheres, the values of the axial ratio and the internal parameter arec=a ¼ ffiffiffiffiffiffiffiffi

8=3p ¼ 1:633 and u¼ 3/8¼ 0.375, respectively. The crystallographic vectors

of wurtzite are~a ¼ að1=2; ffiffiffi3

p=2; 0Þ,~b ¼ að1=2;� ffiffiffi

3p

=2; 0Þ, and~c ¼ að0; 0; c=aÞ. InCartesian coordinates, the basis atoms are (0, 0, 0), (0, 0, uc), a(1/2,

ffiffiffi3

p=6, c/2a), and

a(1/2,ffiffiffi3

p=6, [uþ 1/2]c/a).

In all Wz III nitrides, experimentally observed c/a ratios are smaller than idealparameters and a strong correlation exists between the c/a ratio and the u parametersuch that when c/a decreases, the u parameter increases in a manner that the fourtetrahedral distances remain nearly constant through a distortion of tetrahedralangles. For the equal bond length to prevail, the following relation must hold:

u ¼ ð1=3Þ a2=c2� �þ 1=4: ð1:1Þ

The nearest-neighbor bond length along the c-direction (expressed as b in Figure 1.3)and off c-axis (expressed as b1 in Figure 1.3) can be calculated as

b ¼ cu and b1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13a2 þ 1

2�u

� �2

c2

s: ð1:2Þ

Most commonly used planes of nitride semiconductors, namely, the polar c-planeand nonpolar a- and m-planes, are graphically shown in Figure 1.4. Other planes,semipolar planes, that are gaining some attention are shown in Figure 1.5.Table 1.1 gives the calculated as well as the experimentally observed structuralparameters discussed above, including the lattice parameters, the nearest- and

Figure 1.3 Schematic representation of awurtzitic metal nitride structure having latticeconstants a in the basal plane and c in the basaldirection. u parameter is the bond length or thenearest-neighbor distance (b) divided by c

(0.375 in ideal crystal), a and b (109.47� in idealcrystal) are the bond angles, and b01, b

02, and b03

are the three types of second-nearest-neighbordistances. M denotes metal (e.g., Ga) and Ndenotes N.

4j 1 General Properties of Nitrides

second-nearest-neighbor distances, and the bond angles for three end binaries:GaN, AlN, and InN. The distances are in a

�ngstr€oms.

1.2Gallium Nitride

The parameters associated with electrical and optical properties of wurtzitic GaNand AlN are given in Table 1.2.The elastic stiffness coefficients and the bulk modulus are compiled in Table 1.3.

Figure 1.4 Graphical representation of commonly used planes of nitride semiconductors,namely, the polar c-plane and nonpolar a- andm-planes, and r-plane.

Figure 1.5 Graphical representation of semipolar (10�10) plane and (11�22) plane.

1.2 Gallium Nitride j5

1.3Aluminum Nitride

AlN has a molar mass of 40.9882 g/mol. Reported wurtzite lattice parameters rangefrom 3.110 to 3.113A

�for the a parameter (3.1106A

�for bulk, 3.1130A

�for powder,

and 3.110A�for AlN on SiC) and from 4.978 to 4.982A

�for the c parameter. The c/a

ratio thus varies between 1.600 and 1.602. The deviation from that of the idealwurtzite crystal (c/a¼ 1.633) is plausibly due to lattice instability and ionicity. The uparameter for AlN is 0.3821, which is larger than the calculated value of 0.380. Thismeans that the interatomic distance and angles differ by 0.01A

�and 3�, respectively,

from the ideal parameters. Refer to Table 1.4 for electronic properties of Wz AlN.The measured bulk modulus B and Young’s modulus E or Y0 are compiled in

Table 1.5 along with the entire set of elastic stiffness coefficients.The phonon energies measured by Raman scattering apply to Raman active

modes (Table 1.6). Raman-active optical phonon modes belong to the A1, E1, and E2group representations.The thermal expansion of AlN is isotropic with a room temperature value of

2.56� 10�6 K�1. The equilibrium N2 vapor pressure above AlN is relatively lowcompared to that above GaN that makes AlN easier to synthesize. The calculatedtemperatures at which the equilibrium N2 pressure reaches 1, 10, and 100 atm are2836, 3088, and 3390K, respectively. The thermal conductivity k of AlN at roomtemperature has been predicted as k¼ 3.19W/(cm K) in O-free simulated material.The values of the refractive index n are in the range 1.99–2.25 with several groupsreporting n¼ 2.15� 0.05. The dielectric constant of AlN (e0) lies in the range 8.3–11.5and most of the values fall within e0¼ 8.5� 0.2. Other measurements in the high-frequency range produced dielectric constants of 4.68 and e1¼ 4.84. AlN has alsobeen examined for its potential for second-harmonic generation.

Table 1.1 Calculated (for ideal crystal) and experimentally observed structural parameters forwurtzitic GaN, AlN, and InN.

GaN AlN InN

Ideal Exp Ideal Exp Ideal Exp

u 0.375 0.377 0.375 0.382 0.375 0.379a (A

�) 3.199 3.199 3.110 3.110 3.585 3.585

c/a 1.633 1.634 1.633 1.606 1.633 1.618b (A

�) 1.959 1.971 1.904 1.907 2.195 2.200

b1 (A�) 1.959 1.955 1.904 1.890 2.195 2.185

b01 (A�) 3.265 3.255 3.174 3.087 3.659 3.600

b02 (A�) 3.751 3.757 3.646 3.648 4.204 4.206

b03 (A�) 3.751 3.749 3.646 3.648 4.204 4.198

a 109.47 109.17 109.47 108.19 109.47 108.69b 109.47 109.78 109.47 110.73 109.47 110.24

6j 1 General Properties of Nitrides

Table 1.2 Parameters related with electrical and optical properties of wurtzitic GaN.

Wurtzite polytype GaN Parameter value/comments

Bandgap energy Eg (eV), direct 3.42 at 300K3.505 at 1.6 K

Breakdown field (cm�1) 3–5� 106 at 300KElectron affinity (eV) 4.1Energy separation between C valley andM–L valleys (eV)

1–1.9 at 300K

Energy separation between M–L valleysdegeneracy (eV)

0.6 at 300K0.6 at 300K

Energy separation between C valleyand A valleys (eV)

1.3–2.1 at 300K

Index of refraction 2.3 at 300 K away from band edgeDielectric constants, static 10.4 (Ejjc), 9.5 (E ? c) or 8.9 in c-direction

(Ejjc) at 300KDielectric constants, high frequency 5.35 or 5.47 (E ? c) at 300K, 5.8 (Ejjc) at 300KOptical LO phonon energy (meV) 91.2A1 – LO, nA1(LO) (cm

�1) 710–744A1 – TO, nA1(TOjj) (cm�1) 533–534E1 – LO, nE1(LO?) (cm�1) 741–742E1 – TO, nE1(TO?) (cm�1) 556–559E2 (low) (cm

�1) 143–146E2 (high) (cm

�1) 560–579Energy of spin–orbit splitting Eso (meV) 11(þ5, �2) at 300K calculated from the values

of energy gap Eg,dir (given in this table)Energy of crystal field splitting Ecr (meV) 40 at 300K, 22 calculated from the values of

energy gap Eg,dir (given in this table)Effective electron mass,me orm

==e 0.20m0 at 300K, 0.27m0 by Faraday rotation

Effective electron mass,me? orm?e 0.20m0 at 300K; 0.15–0.23m0 fit of reflectance

spectrumEffective hole mass 0.8m0 at 300KEffective hole masses (heavy)mhh mhh¼ 1.4m0 at 300K

mhhz¼m==hh ¼ 1.1m0,mhh? ¼m?

hh ¼ 1.6m0 at 300K

m==hh ¼ 1.1–2.007m0

m?hh ¼ 1.61–2.255m0

Effective hole masses (light) mlh¼ 0.3m0 at 300 K,mlhz¼m==lh ¼ 1.1m0 at 300K

m?lh ¼mlh? ¼ 0.15m0 at 300K

m==lh ¼ 1.1–2.007m0,mlh? ¼ 0.14–0.261m0

Effective hole masses(split-off band) ms

msh¼ 0.6m0 at 300Kmshz¼m==

ch ¼ 0.15m0,msh? ¼m?ch ¼ 1.1m0 at 300K

¼m==ch ¼ 0.12–0.16m0,m?

ch ¼ 0.252–1.96m0

Effective mass of density of state mv 1.4m0

Effective conduction band density of states(cm�3)

2.3� 1018 at 300K

Effective valence band density of states (cm�3) 4.6� 1019 at 300KElectron mobility [cm2/(V s)] �1400 experimental at 300KHole mobility [cm2/(V s)] <20n-Doping range (cm�3) 1016 cm�3–high 1019

p-Doping range (cm�3) 1016 cm�3–mid-1018

Diffusion coefficient for electrons (cm2/s) 25Diffusion coefficient for holes (cm2/s) Not well defined, but 5, 26, 94 have been reported

1.3 Aluminum Nitride j7

Table 1.4 Parameters related to optical and electrical properties of wurtzitic AlN.

Wurtzite polytype AlN Parameter

Bandgap energy (eV) �6 at 300K and �6.1 at 5 K

Breakdown field (V/cm) 1.2–1.8� 106

dEg/dP (eV l/bar) 3.6� 10�3

Conduction band energy separation between C

valley and M–L valleys (eV)�0.7–1

Conduction band energy separation between C

valley and K valleys (eV)�1.0

Valence band energy of spin–orbit splitting Eso (eV) 0.019: 0.036 at 300K

Valence band energy of crystal field splitting Ecr (eV), C7

on top of C9

�0.225

Effective conduction band density of states (cm�3) 6.3� 1018

Effective valence band density of states (cm�3) 4.8� 1020

Index of refraction n (3 eV)¼ 2.15� 0.05

Dielectric constant, static 7.34: 9.14 at 300 K

9.32 for E//c (modeling)

7.76 for E ? c (experiment)

Dielectric constant, high frequency 4.6–4.84 at 300K

4.35 from E//c (modeling)

4.16 from E ? c (experiment)

Infrared refractive index 1.8–2.2 at 300K

3 from E//c (modeling)

2.8 from E ? c (experiment)

Effective electron massme 0.25–0.4m0

m==e ¼ 0231–0.35m0

m?e ¼ 0242–0.25m0

Effective hole masses (heavy) m==hh ¼ 3.53m0 at 300K

for kz directionmhz orm==hh 2.02–3.13m0 at 300K

m?hh ¼ 10.42m0 at 300K

for kx directionmhx orm?hh m==

hh ¼ 1.869–4.41m0

m?hh ¼ 2.18–11.14m0

Effective hole masses (light) 3.53m0

0.24m0

for kz directionmlz orm==lh m==

lh ¼ 1.869–4.41m0

for kx directionmlx orm?lh m?

lh ¼ 0.24–0.350m0

Effective hole masses (split-off band) 0.25m0 at 300Kor kz directionmsoz orm

==ch 3.81m0 at 300K

for kx directionmsox orm?ch m==

ch ¼ 0.209–0.27m0

m?ch ¼ 1.204–4.41m0

(continued )

1.3 Aluminum Nitride j9

1.4Indium Nitride

The parameters associated with electrical and optical properties of wurtzitic InN aregiven in Table 1.7.The zincblende (cubic) form has been reported to occur in films containing both

polytypes. The measured Wz InN lattice parameters using powder technique are inthe range a¼ 3.530–3.548A

�and c¼ 5.704–5.960A

�with a consistent c/a ratio of

about 1.615� 0.008. The density of InN deduced from Archimedean displacementmeasurements is 6.89 g/cm3 at 250 �C. Table 1.8 summarizes the measured and thecalculated elastic coefficients for Wz InN.As in the cases of Wz GaN and AlN, Wz InN has 12 phonon modes at the zone

center (symmetry group: C6v), 3 acoustic and 9 optical ones with the acousticbranches near 0 at k¼ 0. The infrared active modes are of the E1 (LO), El (TO),A1 (LO), and A1 (TO) type. Raman spectroscopy has yielded four optical phononscharacteristic for InN with wavenumbers 190 (E2), 400 (A1), 490 (E1), and 590 (E2)cm�1 in InN layers grown by atomic layer epitaxy (ALE). Moreover, a transverseoptical (TO) mode has been observed at 478 cm�1 (59.3meV) by reflectance and460 cm�1 (57.1meV) by transmission measurements. From other reflectance data,the existence of a TO phonon mode at 478 cm�1 and an LO mode at 694 cm�1 wasdeduced.Thermal conductivity derived from the Leibfried–Schloman scaling parameter,

assuming that the thermal conductivity is limited by intrinsic phonon–phononscattering, is about 0.80� 0.20W/(cmK). The estimated effective mass of m�

e ¼0:11m0 and an index of refraction of n¼ 3.05� 0.05, while the long–wavelengthlimit of the refractive index has been reported to be 2.88� 0.15.

Table 1.4 (Continued)

Wurtzite polytype AlN Parameter

Effective mass of density of statesmv 7.26m0 at 300KOptical phonon energy, meV 99.2nTO(E1) phonon wavenumber (cm�1)a 895, 614, 608nLO(E1) phonon wavenumber (cm�1) 671.6, 821, 888.9nTO(A1) phonon wavenumber (cm�1) 888, 514, 667.2nLO(A1) phonon wavenumber (cm�1) 659.3, 663, 909n(E2) phonon wavenumber (cm�1) 303,a) 426nTO (E1) phonon wavenumber (cm�1) 657–673nTO (A1) phonon wavenumber (cm�1) 607–614 or 659–667nLO (E1) phonon wavenumber (cm�1) 895–924nLO (A1) phonon wavenumber (cm�1) 888–910n(1) (E2) phonon wavenumber (cm�1) 241–252n(2) (E2) phonon wavenumber (cm�1) 655–660

a) Room temperature Raman, tentative.

10j 1 General Properties of Nitrides

Table 1.7 Parameters related to electrical and optical properties of wurtzitic InN.

Wurtzitic InN Value

Bandgap energy, Eg (300K) 0.6–0.7 eVDielectric constant (static) 15.3Dielectric constant (static, ordinary direction) e0,ort¼ 13.1Dielectric constant (static,extraordinary direction)

e0, ||¼ 14.4

Dielectric constant (high frequency) 8.4–9.3Infrared refractive index Reported range: 2.80–3.05Energy separation between C valley andM–L valleys (eV)

2.9 � 3.9�4.8

Energy separation between C valley andA valleys (eV)

0.7–2.7�4.5

Energy separation between C valley and C1

valleys (eV)1.1–2.6

Energy separation between C1 valleydegeneracy (eV)

1

Effective conduction band density of states 9� 1017 cm�3

Effective valence band density of states 5.3� 1019 cm�3

Valence band crystal field splitting Ecr 0.017 eVValence band spin–orbit splitting Eso 0.003 eVIndex of refraction 2.5–2.9 at 300KEffective electron massme

� 0.11m0

m==e ¼ 0.1–0/138m0,m?

e ¼ 0.1–0.141m0

Effective hole masses (heavy)mh 1.63m0 at 300Km==

hh ¼ 1.350–2.493m0,m?hh ¼ 1.410–2.661m0

Effective hole masses (light)mlp 0.27m0 at 300Km==

lh ¼ 1.350–2.493m0,m?lh ¼ 0.11–0.196m0

Effective hole masses (split-off band) ms 0.65m0 at 300Km==

ch ¼ 0.092–0.104m0,m?ch ¼ 0.202–3.422

Effective mass of density of state mv 1.65m0 at 300KOptical LO phonon energy (meV) 73 at 300K

Table 1.8 Theoretical and experimental elastic coefficients and bulk modulus (in GPa) of thevarious forms of InN.

Method Structure C11 C12 C13 C33 C44 B dB/dP¼B0

X-ray Wz 190 104 121 182 10 139, 140, 165LMTO Wz 271 124 94 200 46 165PWPP Wz 223 115 92 224 48 138, 141, 165 3.8, 3.9LMTO Wz 165

PWPP: plane wave pseudopotential; LMTO: linear muffin-tin orbital.

12j 1 General Properties of Nitrides

1.5AlGaN Alloy

The ternary alloy of GaN with AlN forms a continuous system with a wide range ofbandgap and a relatively small change in the lattice constant. An accurate knowledgeof the compositional dependence of AlGaN, which is often used as barrier materialin devices and to a lesser extent as active layer in, for example, UV detectors andemitters, is a prerequisite for analyzing heterostructures in general and quantumwells (QWs) and superlattices in particular. The compositional dependence of thelattice parameters follows the Vegard’s law:

aAlðxÞGað1�xÞN¼½ð3:189� 0:002Þ � ð0:086� 0:004ÞxA� ;cAlðxÞGað1�xÞN ¼½ð5:188� 0:003Þ � ð0:208� 0:005ÞxA� : ð1:3Þ

However, the bond lengths exhibit a nonlinear behavior, deviating from the virtualcrystal approximation. Essentially, the nearest-neighbor bond lengths are not asdependent on the composition as might be expected.The compositional dependence of the principal (fundamental) bandgap of

AlxGa1�xN can be calculated from the following empirical expression providedthat the bowing parameter b is known accurately:

EgðxÞ ¼ xEgðAlNÞ þ ð1� xÞEgðGaNÞ � bxð1� xÞ; ð1:4Þwhere Eg (GaN)¼ 3.4 eV, Eg (AlN)¼ 6.1 eV, x is the AlN molar fraction, and b is thebowing parameter. There is dispersion in the value of the bowing parameter owingto difficulties in retaining quality, particularly around the 50 : 50 composition, andthe experimental procedures employed. This dispersion ranges from �0.8 eV(upward bowing) to þ2.6 eV (downward bowing). More refined layers and theassociated techniques seem to yield a bowing parameter of b¼ 1.0 eV for the entirerange of alloy composition. It is still possible that as the quality of the filmsimproves, the bowing parameters may have to be revised.As for the InGaN alloy, the most celebrated one among the nitride family,

difficulties/challenges associated with the growth of high-quality InN and the earliercontroversy regarding its bandgap aggravated determination of the compositionaldependence of its bandgap. As in the case of AlGaN, the calculated lattice parameterof this alloy follows Vegard’s law:

AInðxÞGað1�xÞN ¼ ð3:1986þ 03862xÞA� ;cInðxÞGað1�xÞN ¼ ð5:2262þ 0:574xÞA� : ð1:5Þ

The compositional dependence of InGaN bandgap is a crucial parameter inheterostructure design. As such, the topic has attracted a number of theoreticaland experimental (to be discussed below) investigations and reports. Similar to thecase of AlGaN, the energy bandgap of InxGa1�xN over 0 x1 can be expressed bythe empirical expression:

EgInxGa1�xN

¼ xEgInN þ ð1� xÞEg

GaN � bInGaNxð1� xÞ¼ 0:7x þ 3:4ð1� xÞ � bInGaNxð1� xÞ eV; ð1:6Þ

where EgGaN ¼ 3:40 eV andEg

InN � 0:7 eV.

1.5 AlGaN Alloy j13

The compositional dependence of the bandgap in the entire composition rangecan be well fit by a bowing parameter of b¼ 1.43 eV, assuming 0.7 eV for thebandgap of InN.

1.6InGaN Alloy

InGaN alloy, together with allied nitride semiconductors, forms the backbone ofemitters in wide wavelength range. With the crucial role InN plays in nitride devicescomes the complexities associated with this ternary such as the great disparitybetween Ga and In could result in anomalies such as phase separation andinstabilities. As in the case of AlGaN, the calculated lattice parameter of this alloyfollows the Vegard’s law:

AInðxÞGað1�xÞN ¼ ð3:189� 0:3862xÞA� and cInðxÞGað1�xÞNð5:2262þ 0:574xÞA� :ð1:7Þ

By employing various tools such as high-resolution X-ray diffraction (HRXRD), theexperimental data for various AlGaN support the applicability of Vegard’s law in thatthe experimental data aAlðxÞGað1�xÞN ¼ ½ð3:560� 0:019Þð0:449� 0:019ÞxA� andcAlðxÞGað1�xÞN ¼ ½ð5:195� 0:002Þ þ ð0:512� 0:006ÞxA� are within about 2% of thatpredicted by linear interpolation, Vegard’s’ law. As in the case of AlGaN but to a largerextent, the bond lengths exhibit a nonlinear behavior, deviating from the virtual crystalapproximation. Essentially, the nearest-neighbor bond lengths are not as dependent oncomposition as might be expected from the virtual crystal approximation.The compositional dependence of InGaN bandgap is a crucial parameter in

designs of any heterostructure utilizing it. As such, the topic has attracted a numberof theoretical and experimental (to be discussed below) investigations and reports.Similar to the case of AlGaN, the energy bandgap of InxGa1�xN over 0 x 1 can beexpressed by the empirical expression:

EgInxGa1�xN

¼ xEgInN þ ð1� xÞEg

GaN � bInGaNxð1� xÞ¼ 0:7x þ 3:4ð1� xÞ � bInGaNxð1� xÞ eV; ð1:8Þ

where EgGaN ¼ 3.40 eV and Eg

InN � 0.7 eV. For the nomenclature GaxIn1�xN, theterms x and 1� x in Equation 1.6 must be interchanged. Another point of cautionis that the sign in front of the bowing parameter is changed to positive in somereports. When a comparison is made, the sign of the b parameter must be changed.

1.7AlInN Alloy

In1�xAlxN can provide a lattice–matched barrier to GaN, low mole fraction ofAlGaN, and InGaN and, consequently, can yield lattice-matched AlInN/AlGaN orAlInN/InGaN heterostructures, including Bragg reflectors. The lattice-matched

14j 1 General Properties of Nitrides