Nitride Semiconductor Devices E-Book

Hadis Morkoc

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 720

Veröffentlichungsjahr: 2013

Beschreibung

This book gives a clear presentation of the necessary basics of semiconductor and device physics and engineering. It introduces readers to fundamental issues that will enable them to follow the latest technological research. It also covers important applications, including LED and lighting, semiconductor lasers, high power switching devices, and detectors. This balanced and up-to-date treatment makes the text an essential educational tool for both advanced students and professionals in the electronics industry.

Leseprobe

Contents

Cover

Related Titles

Title Page

Copyright

Dedication

Preface

Chapter 1: General Properties of Nitrides

1.1 Crystal Structure of Nitrides

1.2 Gallium Nitride

1.3 Aluminum Nitride

1.4 Indium Nitride

1.5 AlGaN Alloy

1.6 InGaN Alloy

1.7 AlInN Alloy

1.8 InAlGaN Quaternary Alloy

1.9 Electronic Band Structure and Polarization Effects

1.10 Polarization Effects

1.11 Nonpolar and Semipolar Orientations

Further Reading

Chapter 2: Doping: Determination of Impurity and Carrier Concentrations

2.1 Introduction

2.2 Doping

2.3 Formation Energy of Defects

2.4 Doping Candidates

2.5 Free Carriers

2.6 Binding Energy

2.7 Conductivity Type: Hot Probe and Hall Measurements

2.8 Measurement of Mobility

2.9 Semiconductor Statistics, Density of States, and Carrier Concentration

2.10 Charge Balance Equation and Carrier Concentration

2.11 Capacitance–Voltage Measurements

Appendix 2.A. Fermi Integral

Further Reading

Chapter 3: Metal Contacts

3.1 Metal–Semiconductor Band Alignment

3.2 Current Flow in Metal–Semiconductor Junctions

3.3 Ohmic Contact Resistance

3.4 Semiconductor Resistance

Further Reading

Chapter 4: Carrier Transport

4.1 Introduction

4.2 Carrier Scattering

4.3 Calculated Mobility of GaN

4.4 Scattering at High Fields

4.5 Delineation of Multiple Conduction Layer Mobilities

4.6 Carrier Transport in InN

4.7 Carrier Transport in AlN

4.8 Carrier Transport in Alloys

4.9 Two-Dimensional Transport in n-Type GaN

Further Reading

Chapter 5: The p–n Junction

5.1 Introduction

5.2 Band Alignment

5.3 Electrostatic Characteristics of p–n Heterojunctions

5.4 Current–Voltage Characteristics of p–n Junctions

Further Reading

Chapter 6: Optical Processes

6.1 Introduction

6.2 Einstein's A and B Coefficients

6.3 Absorption and Emission

6.4 Band-to-Band Transitions and Efficiency

6.5 Optical Transitions in GaN

6.6 Free-to-Bound Transitions

6.7 Donor–Acceptor Transitions

Further Reading

Chapter 7: Light-Emitting Diodes and Lighting

7.1 Introduction

7.2 Current Conduction Mechanism in LED-Like Structures

7.3 Optical Output Power and Efficiency

7.4 Effect of Surface Recombination

7.5 Effect of Threading Dislocation on LEDs

7.6 Current Crowding

7.7 Perception of Color

7.8 Chromaticity Coordinates and Color Temperature

7.9 LED Degradation

7.10 Packaging

7.11 Luminescence Conversion and White Light Generation

Further Reading

Chapter 8: Semiconductor Lasers: Light Amplification by Stimulated Emission of Radiation

8.1 Introduction

8.2 A Primer to the Principles of Lasers

8.3 Loss, Threshold, and Cavity Modes

8.4 Optical Gain

8.5 A Glossary for Semiconductor Lasers

8.6 Threshold Current

8.7 Analysis of Injection Lasers with Simplifying Assumptions

8.8 GaN-Based LD Design and Performance

8.9 Thermal Resistance

8.10 Nonpolar and Semipolar Orientations

8.11 Vertical Cavity Surface-Emitting Lasers (VCSELs)

8.12 Degradation

Appendix 8.A: Determination of the Photon Density and Photon Energy Density in a Cavity

Further Reading

Chapter 9: Field Effect Transistors

9.1 Introduction

9.2 Operation Principles of Heterojunction Field Effect Transistors

9.3 GaN and InGaN Channel HFETs

9.4 Equivalent Circuit Models: De-embedding and Cutoff Frequency

9.5 HFET Amplifier Classification and Efficiency

9.6 Drain Voltage and Drain Breakdown Mechanisms

9.7 Field Plate for Spreading Electric Field for Increasing Breakdown Voltage

9.8 Anomalies in GaN MESFETs and AlGaN/GaN HFETs

9.9 Electronic Noise

9.10 Self-Heating and Phonon Effects

9.11 HFET Degradation

9.12 HFETs for High-Power Switching

Appendix 9.A. Sheet Charge Calculation in AlGaN/GaN Structures with AlN Interface Layer (AlGaN/AlN/GaN)

Further Reading

Index

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To those who advance the frontiers of science and engineering

Preface

This book aims to describe the fundamentals of light emitters and field effect transistors based on GaN and related semiconductors with supporting material. The book is intended to provide the know-how for the reader to be well versed in the aforementioned devices with selective further reading material for additional material.

Chapter 1 deals with the structural properties of nitride-based semiconductors and their band structure and polarization with extensive tables. Chapter 2 discusses defects and doping, electron and hole concentrations along with applicable statistics as affected by temperature, and Hall and C–V measurements. Metal semiconductor junctions along with the current conduction mechanisms, contact resistivity and its determination are discussed in Chapter 3. Scattering and carrier transport at low and high electric fields are discussed in Chapter 4, which embodies ionized impurity, deformation potential, piezoelectric, optical phonon, and alloy scattering, among others, in bulk and to a lesser extent in two-dimensional systems. Hall effect/Hall factor and magneto transport along with delineation of mobility for each of the contributing layers in multichannel constructs are also included in the discussion. Chapter 5 is devoted to p–n junctions, beginning with the discussion of band lineups and 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 also covered in a concise form. Chapter 6 contains a succinct discussion of optical processes in semiconductors such as absorption and emission vis-à-vis Einstein's A and B coefficients to pave the way for discussion of light emitters in the follow-up chapters. Chapter 7 delves into the fundamentals and practice of light-emitting diodes, perception of vision and color by human eye and methodologies, both used and proposed, for generation of white light and presents an in-depth discussion of efficiency 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, threshold current, efficiency, polar- and nonpolar-specific processes, and microcavity-based lasers are also discussed. The final chapter, Chapter 9, treats field effect transistor fundamentals, which are applicable to any semiconductor material with points specific to GaN-based varieties. The discussion primarily focuses on 2DEG channels formed at heterointerfaces and their use for FETs including polarization effects. A succinct analytical model is provided for calculating the carrier densities at the interfaces for various scenarios and current–voltage characteristics of FETs with several examples. Hot phonon effects responsible for this shortfall and attainable carrier velocity are uniquely discussed with sufficient theory and experimental data and design approaches to mitigate the problem, along with their effect on heat dissipation and reliability.

This book would not have been possible without the support of many of our colleagues, namely, Profs. Ü. Özgür, V. Avrutin, R. Shimada, and A. Matulionis and Drs. N. Izyumskaya, J. Xie, J. Leach, and C. Kayis, who helped with material, figures, proofreading, and extensive discussions during its production.

Hadis Morkoç

Richmond, VA USA

September 2012

1

General Properties of Nitrides

1.1 Crystal Structure of Nitrides

GaN and its binary cousins InN and AlN as well as their ternary and quaternary are considered one of the most important groups of semiconductors after Si. This follows from their ample applications in lighting and displays, consumer electronics, lasers, detectors, and high-power RF/switching devices owing to their excellent optical and electrical properties. The pertinent properties and materials parameters 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 for bulk AlN, GaN, and InN. The space grouping for the zincblende structure is in the Hermann–Mauguin notation and in the Schoenflies notation and has a cubic unit cell containing four group III elements and four nitrogen elements. (The term zincblende originated in describing compounds such as ZnS that could be cubic or hexagonal. But the term has been used ubiquitously for compound semiconductors with cubic symmetry. The correct term for the cubic phase is sphalerite.). The position of the atoms within the unit cell is identical to that in the diamond crystal structure. The stacking sequence for the (111) close-packed planes in this structure is AaBbCc. Small and large letters stand for the two different kinds of constituents. The rocksalt structure (with space group in the Hermann–Mauguin notation and in the Schoenflies notation) can be induced under very high 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 wurtzite structure is P63mc in the Hermann–Mauguin notation and in the Schoenflies notation. The point group symmetry is 6mm in the Hermann–Mauguin notation and C6v in the Schoenflies notation. The Wz structure consists of two interpenetrating hexagonal close-packed (hcp) sublattices, each with one type of atom, offset along the c-axis by 5/8 of the cell height (5c/8). The Wz structure consists of alternating biatomic close-packed (0001) planes of Ga and N pairs, thus the stacking sequence of the (0001) plane is AaBbAa in the (0001) direction.

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 the bond 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 is illustrated in Figure 1.1b. The nomenclature for various commonly used planes of hexagonal semiconductors in two- and three-dimensional versions is presented in Figure 1.2. The Wz group III nitrides lack an inversion plane perpendicular to the c-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 a N-polarity with a designation of () or ()B plane. The former notations for each are used here. The distinction between these two directions is essential in nitrides due to implications in the polarity of the polarization charge. Three surfaces and directions are of special importance in nitrides, which are (0001) c-, () a-, and () m-planes and the directions associated with them: 0001, , and .

Figure 1.1 Ball-and-stick stacking model of crystals with (a) (both top and bottom) 2H wurtzitic and (b) (both top and bottom) 3C zincblende polytypes. The bonds in an a-plane () are indicated with thicker lines to accentuate the stacking sequence. The figures on the top depict the three-dimensional view. The figures at the bottom indicate the projections on (0001) and (111) planes for wurtzitic and cubic phases, respectively. Note the rotation in the zincblende case along the 111 direction.

Figure 1.2 Labeling of planes in hexagonal symmetry (for sapphire), a telescopic view of labeling of planes in hexagonal symmetry in the (tuvw) coordinate system with w representing the unit vector in the c-direction is shown on the right. The lines are simply to show the symmetry only. If the lines connecting m-points among each other and a-points among each other were to be interpreted as the projection of those planes on the c-plane, the roles would be switched in that the lines connecting the m-points would actually represent the a-planes, and lines connecting the a-points would actually represent the m-planes that are normal to the plane of the page.

Delving further into the Wz structure, it can be represented by lattice parameters a in 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 c parameter depicts the unit cell height. The wurtzite structure is a hexagonal close-packed lattice, comprising vertically oriented M–N units at the lattice sites. The basal plane lattice parameter (the edge length of the basal plane hexagon) is universally depicted by a and the axial lattice parameter perpendicular to the basal plane is universally described by c. In an ideal wurtzite structure represented by four touching hard spheres, the values of the axial ratio and the internal parameter are and u = 3/8 = 0.375, respectively. The crystallographic vectors of wurtzite are , , and . In Cartesian coordinates, the basis atoms are (0, 0, 0), (0, 0, uc), a(1/2, , c/2a), and a(1/2, , [u + 1/2]c/a).

Figure 1.3 Schematic representation of a wurtzitic metal nitride structure having lattice constants a in the basal plane and c in the basal direction. u parameter is the bond length or the nearest-neighbor distance (b) divided by c (0.375 in ideal crystal), α and β (109.47° in ideal crystal) are the bond angles, and , , and are the three types of second-nearest-neighbor distances. M denotes metal (e.g., Ga) and N denotes N.

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

(1.1)

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

(1.2)

Most commonly used planes of nitride semiconductors, namely, the polar c-plane and 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.

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

Figure 1.5 Graphical representation of semipolar () plane and () plane.

Table 1.1 gives the calculated as well as the experimentally observed structural parameters discussed above, including the lattice parameters, the nearest- and second-nearest-neighbor distances, and the bond angles for three end binaries: GaN, AlN, and InN. The distances are in ångströms.

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

1.2 Gallium Nitride

The parameters associated with electrical and optical properties of wurtzitic GaN and AlN are given in Table 1.2.

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

Wurtzite polytype GaNParameter value/commentsBandgap energy Eg (eV), direct3.42 at 300 K3.505 at 1.6 KBreakdown field (cm−1)3–5 × 106 at 300 KElectron affinity (eV)4.1Energy separation between Γ valley and M–L valleys (eV)1–1.9 at 300 KEnergy separation between M–L valleys degeneracy (eV)0.6 at 300 K0.6 at 300 KEnergy separation between Γ valley and A valleys (eV)1.3–2.1 at 300 KIndex of refraction2.3 at 300 K away from band edgeDielectric constants, static10.4 (E||c), 9.5 () or 8.9 in c-direction (E||c) at 300 KDielectric constants, high frequency5.35 or 5.47 () at 300 K, 5.8 (E||c) at 300 KOptical LO phonon energy (meV)91.2A1 – LO, νA1(LO) (cm−1)710–744A1 – TO, νA1(TO||) (cm−1)533–534E1 – LO, νE1() (cm−1)741–742E1 – TO, νE1() (cm−1)556–559E2 (low) (cm−1)143–146E2 (high) (cm−1)560–579Energy of spin–orbit splitting Eso (meV)11(+5, −2) at 300 K calculated from the values of energy gap Eg,dir (given in this table)Energy of crystal field splitting Ecr (meV)40 at 300 K, 22 calculated from the values of energy gap Eg,dir (given in this table)Effective electron mass, me or 0.20m0 at 300 K, 0.27m0 by Faraday rotationEffective electron mass, or 0.20m0 at 300 K; 0.15–0.23m0 fit of reflectance spectrumEffective hole mass0.8m0 at 300 KEffective hole masses (heavy) mhhmhh = 1.4m0 at 300 Kmhhz = = 1.1m0, = = 1.6m0 at 300 K = 1.1–2.007m0 = 1.61–2.255m0Effective hole masses (light)mlh = 0.3m0 at 300 K, mlhz = = 1.1m0 at 300 K = = 0.15m0 at 300 K = 1.1–2.007m0, = 0.14–0.261m0Effective hole masses (split-off band) msmsh = 0.6m0 at 300 Kmshz = = 0.15m0, = = 1.1m0 at 300 K= = 0.12–0.16m0, = 0.252–1.96m0Effective mass of density of state mv1.4m0Effective conduction band density of states (cm−3)2.3 × 1018 at 300 KEffective valence band density of states (cm−3)4.6 × 1019 at 300 KElectron mobility [cm2/(V s)]~1400 experimental at 300 KHole mobility [cm2/(V s)]<20n-Doping range (cm−3)1016 cm−3–high 1019p-Doping range (cm−3)1016 cm−3–mid-1018Diffusion coefficient for electrons (cm2/s)25Diffusion coefficient for holes (cm2/s)Not well defined, but 5, 26, 94 have been reported

The elastic stiffness coefficients and the bulk modulus are compiled in Table 1.3.

Table 1.3 Experimental and calculated elastic coefficients (), bulk modulus () and its pressure derivative (B′, dB/dP), and Young's modulus (E or Y0) (in GPa) for WzGaN.

1.3 Aluminum Nitride

AlN has a molar mass of 40.9882 g/mol. Reported wurtzite lattice parameters range from 3.110 to 3.113 Å for the a parameter (3.1106 Å for bulk, 3.1130 Å for powder, and 3.110 Å for AlN on SiC) and from 4.978 to 4.982 Å for the c parameter. The c/a ratio thus varies between 1.600 and 1.602. The deviation from that of the ideal wurtzite crystal (c/a = 1.633) is plausibly due to lattice instability and ionicity. The u parameter for AlN is 0.3821, which is larger than the calculated value of 0.380. This means that the interatomic distance and angles differ by 0.01 Å and 3°, respectively, from the ideal parameters. Refer to Table 1.4 for electronic properties of Wz AlN.

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

Wurtzite polytype AlNParameterBandgap energy (eV)~6 at 300 K and ~6.1 at 5 KBreakdown field (V/cm)1.2–1.8 × 106dEg/dP (eV l/bar)3.6 × 10−3Conduction band energy separation between Γ valley and M–L valleys (eV)~0.7–1Conduction band energy separation between Γ valley and K valleys (eV)~1.0Valence band energy of spin–orbit splitting Eso (eV)0.019: 0.036 at 300 KValence band energy of crystal field splitting Ecr (eV), Γ7 on top of Γ9−0.225Effective conduction band density of states (cm−3)6.3 × 1018Effective valence band density of states (cm−3)4.8 × 1020Index of refractionn (3 eV) = 2.15 ± 0.05Dielectric constant, static7.34: 9.14 at 300 K9.32 for E//c (modeling)7.76 for (experiment)Dielectric constant, high frequency4.6–4.84 at 300 K4.35 from E//c (modeling)4.16 from (experiment)Infrared refractive index1.8–2.2 at 300 K3 from E//c (modeling)2.8 from (experiment)Effective electron mass me0.25–0.4m0 = 0231–0.35m0 = 0242–0.25m0Effective hole masses (heavy) = 3.53m0 at 300 Kfor kz direction mhz or 2.02–3.13m0 at 300 K = 10.42m0 at 300 Kfor kx direction mhx or = 1.869–4.41m0 = 2.18–11.14m0Effective hole masses (light)3.53m00.24m0for kz direction mlz or = 1.869–4.41m0for kx direction mlx or = 0.24–0.350m0Effective hole masses (split-off band)0.25m0 at 300 Kor kz direction msoz or 3.81m0 at 300 Kfor kx direction msox or = 0.209–0.27m0 = 1.204–4.41m0Effective mass of density of states mv7.26m0 at 300 KOptical phonon energy, meV99.2νTO(E1) phonon wavenumber (cm−1)a895, 614, 608νLO(E1) phonon wavenumber (cm−1)671.6, 821, 888.9νTO(A1) phonon wavenumber (cm−1)888, 514, 667.2νLO(A1) phonon wavenumber (cm−1)659.3, 663, 909ν(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)655–660a. Room temperature Raman, tentative.

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.

Table 1.5 Experimental bulk modulus and elastic coefficients (in GPa) of AlN.

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 E2 group representations.

Table 1.6 Optical phonon energies and phonon deformation potentials for AlN.

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 low compared to that above GaN that makes AlN easier to synthesize. The calculated temperatures at which the equilibrium N2 pressure reaches 1, 10, and 100 atm are 2836, 3088, and 3390 K, respectively. The thermal conductivity κ of AlN at room temperature has been predicted as κ = 3.19 W/(cm K) in O-free simulated material. The values of the refractive index n are in the range 1.99–2.25 with several groups reporting n = 2.15 ± 0.05. The dielectric constant of AlN (ε0) lies in the range 8.3–11.5 and most of the values fall within ε0 = 8.5 ± 0.2. Other measurements in the high-frequency range produced dielectric constants of 4.68 and ε∞ = 4.84. AlN has also been examined for its potential for second-harmonic generation.

1.4 Indium Nitride

The parameters associated with electrical and optical properties of wurtzitic InN are given in Table 1.7.

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

Wurtzitic InNValueBandgap energy, Eg (300 K)0.6–0.7 eVDielectric constant (static)15.3Dielectric constant (static, ordinary direction)ε0,ort = 13.1Dielectric constant (static, extraordinary direction)ε0, || = 14.4Dielectric constant (high frequency)8.4–9.3Infrared refractive indexReported range: 2.80–3.05Energy separation between Γ valley and M–L valleys (eV)2.9 ÷ 3.9~4.8Energy separation between Γ valley and A valleys (eV)0.7–2.7~4.5Energy separation between Γ valley and Γ1 valleys (eV)1.1–2.6Energy separation between Γ1 valley degeneracy (eV)1Effective conduction band density of states9 × 1017 cm−3Effective valence band density of states5.3 × 1019 cm−3Valence band crystal field splitting Ecr0.017 eVValence band spin–orbit splitting Eso0.003 eVIndex of refraction2.5–2.9 at 300 KEffective electron mass me*0.11m0 = 0.1–0/138m0, = 0.1–0.141m0Effective hole masses (heavy) mh1.63m0 at 300 K = 1.350–2.493m0, = 1.410–2.661m0Effective hole masses (light) mlp0.27m0 at 300 K = 1.350–2.493m0, = 0.11–0.196m0Effective hole masses (split-off band) ms0.65m0 at 300 K = 0.092–0.1′4m0, = 0.202–3.422Effective mass of density of state mv1.65m0 at 300 KOptical LO phonon energy (meV)73 at 300 K

The zincblende (cubic) form has been reported to occur in films containing both polytypes. The measured Wz InN lattice parameters using powder technique are in the range a = 3.530–3.548 Å and c = 5.704–5.960 Å with a consistent c/a ratio of about 1.615 ± 0.008. The density of InN deduced from Archimedean displacement measurements is 6.89 g/cm3 at 250 °C. Table 1.8 summarizes the measured and the calculated elastic coefficients for Wz InN.

Table 1.8 Theoretical and experimental elastic coefficients and bulk modulus (in GPa) of the various forms of 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 acoustic branches near 0 at k = 0. The infrared active modes are of the E1 (LO), El (TO), A1 (LO), and A1 (TO) type. Raman spectroscopy has yielded four optical phonons characteristic 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 transverse optical (TO) mode has been observed at 478 cm−1 (59.3 meV) by reflectance and 460 cm−1 (57.1 meV) 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 was deduced.

Thermal conductivity derived from the Leibfried–Schloman scaling parameter, assuming that the thermal conductivity is limited by intrinsic phonon–phonon scattering, is about 0.80 ± 0.20 W/(cm K). The estimated effective mass of and an index of refraction of n = 3.05 ± 0.05, while the long–wavelength limit of the refractive index has been reported to be 2.88 ± 0.15.

1.5 AlGaN Alloy

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

(1.3)

However, the bond lengths exhibit a nonlinear behavior, deviating from the virtual crystal approximation. Essentially, the nearest-neighbor bond lengths are not as dependent on the composition as might be expected.

The compositional dependence of the principal (fundamental) bandgap of can be calculated from the following empirical expression provided that the bowing parameter b is known accurately:

(1.4)

where Eg (GaN) = 3.4 eV, Eg (AlN) = 6.1 eV, x is the AlN molar fraction, and b is the bowing parameter. There is dispersion in the value of the bowing parameter owing to difficulties in retaining quality, particularly around the 50 : 50 composition, and the experimental procedures employed. This dispersion ranges from −0.8 eV (upward bowing) to +2.6 eV (downward bowing). More refined layers and the associated techniques seem to yield a bowing parameter of b = 1.0 eV for the entire range of alloy composition. It is still possible that as the quality of the films improves, 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 earlier controversy regarding its bandgap aggravated determination of the compositional dependence of its bandgap. As in the case of AlGaN, the calculated lattice parameter of this alloy follows Vegard's law:

(1.5)

The compositional dependence of InGaN bandgap is a crucial parameter in heterostructure design. As such, the topic has attracted a number of 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 be expressed by the empirical expression:

(1.6)

where .

The compositional dependence of the bandgap in the entire composition range can be well fit by a bowing parameter of b = 1.43 eV, assuming 0.7 eV for the bandgap of InN.

1.6 InGaN Alloy

InGaN alloy, together with allied nitride semiconductors, forms the backbone of emitters in wide wavelength range. With the crucial role InN plays in nitride devices comes the complexities associated with this ternary such as the great disparity between Ga and In could result in anomalies such as phase separation and instabilities. As in the case of AlGaN, the calculated lattice parameter of this alloy follows the Vegard's law:

(1.7)

By employing various tools such as high-resolution X-ray diffraction (HRXRD), the experimental data for various AlGaN support the applicability of Vegard's law in that the experimental data and are within about 2% of that predicted by linear interpolation, Vegard's' law. As in the case of AlGaN but to a larger extent, the bond lengths exhibit a nonlinear behavior, deviating from the virtual crystal approximation. Essentially, the nearest-neighbor bond lengths are not as dependent on composition 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 number of 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 be expressed by the empirical expression:

(1.8)

where = 3.40 eV and ≈ 0.7 eV. For the nomenclature GaxIn1−xN, the terms x and 1 − x in Equation 6.1 must be interchanged. Another point of caution is that the sign in front of the bowing parameter is changed to positive in some reports. When a comparison is made, the sign of the b parameter must be changed.

1.7 AlInN Alloy

In1−xAlxN can provide a lattice–matched barrier to GaN, low mole fraction of AlGaN, and InGaN and, consequently, can yield lattice-matched AlInN/AlGaN or AlInN/InGaN heterostructures, including Bragg reflectors. The lattice-matched composition naturally depends on the strain state of the GaN on which the InAlN layer is grown. As in the case of AlGaN and InGaN, the calculated lattice parameter of this alloy follows Vegard's law:

(1.9)

The compositional dependence of the bandgap of AlInN can be expressed with the following empirical expression using a bowing parameter bAlInN:

(1.10)

Using a bandgap of 6.0 eV for AlN, the bowing parameter value is about 2 eV.

1.8 InAlGaN Quaternary Alloy

By alloying InN together with GaN and AlN, the bandgap of the resulting alloy(s) can be increased from near 0.7 to 6 eV, which is critical for making high-efficiency visible light sources and detectors. In addition, the bandgap of this quaternary can be changed while keeping the lattice constant matched to GaN.

The relationships between composition and bandgap (or lattice constant) can be predicted by

(1.11)

The parameters x, y, and z represent the composition of GaN, InN, and AlN in the quaternary alloy. If GaN, InN, and AlN are represented by 1, 2, and 3, then T1,2 would represent GaxInyN. Furthermore, the term T12 can be expressed as , where b12 is the bowing parameter for the GaxInyN alloy, is the effective molar fraction for GaInN, InAlN, and AlGaN, respectively, B2 is the bandgap of InN, and B1 is the bandgap of GaN. Similar expressions can be constructed for T23 and T31 by appropriate permutations. An empirical expression similar to that used for the ternaries can also be constructed for the quaternary:

(1.12)

where the first three parameters on the right-hand side of the equation are contributions by the binaries to the extent of their presence in the lattice, the fourth term represents the bowing contribution related to Al, and the last term depicts the bowing contribution due to In. The bowing parameters bAlGaN and bInGaN in Equation 12.1 are the same as those discussed in conjunction with InGaN and AlInN.

The traditional bandgap versus the composition for the GaN family of semiconductors, including the appropriate bowing is shown in Figure 1.6. The locus indicates the ternaries with end points representing the binaries. The area within the locus represents the quaternary compounds of (Ga, In, Al)N.

Figure 1.6 The lattice bandgap versus the lattice parameter for AlGaN, InGaN, and InAlN using bowing parameters in the same order, 1, 1.43, and 3.1 eV, and bandgap values of 6 eV for AlN, 3.4 eV for GaN, and 0.8 eV for InN. The lattice constants used for the binary AlN, GaN, and InN are 3.11, 3.199, and 3.585 Å, respectively.

In nitride semiconductors, thermal, mechanical, electrical, and optical properties are interdependent. Changing one property affects one or more of the others. The Heckmann diagrams are typically used to describe the pathways between external forces, such as mechanical stress (σ), electric field (E), optical field (optical E), and thermal field, and the associated material properties, such as strain (ε) and electrical polarization (P). The trigonal diagram describing the above-mentioned interrelationships are shown in Figure 1.7. Nitride semiconductors exhibit spontaneous strain, pyroelectricity, and polarization, which can be switched hysteretically by applied stress, electric field, or heat (shown by arrows).

Figure 1.7 The well-known triangle diagrams, Heckman diagrams applied to the GaN family. (a) The interdependence of electrical, thermal, and mechanical properties. (b) The interdependence of the electrical, mechanical, and optical properties. Here, σ and ε represent stress and strain, respectively; D and E represent electrical polarization and electric field, respectively.

1.9 Electronic Band Structure and Polarization Effects

1.9.1 Introduction

The band structure of a given semiconductor is pivotal in the realm of devices. One group of nitride semiconductors pertains to stoichiometric systems where N represents 50% of the constituents, while the other 50% is made of metal constituents that can be in wurtzitic, which is the matter of discussion here, and zincblende forms. The other class of nitrides is the dilute compound semiconductors wherein very small amounts of N are added to the host lattice, such as GaAs, with resultant remarkably large negative bowing of the bandgap making the dilute nitride systems to compete in relatively long-wavelength applications. The latter group is not discussed here and one may review Chapter 2 of Morkoç (2008) for more details.

The structure and the first Brillouin zone of a wurtzite and zincblende crystal along with the irreducible wedges, calculated using the local density approximation (LDA) within the FP-LMTO method at the experimental lattice constant and optimized u value, are displayed in Figure 1.8a and b, respectively. In a crystal with Wz symmetry, the conduction band wavefunctions are formed of the atomic s-orbitals and transform the Γ point congruent with the Γ7 representation of the space group . The upper valence band states are constructed out of appropriate linear combinations of products of p3-like (px, py, and pz-like) orbitals with spin functions.

Figure 1.8 (a) Structure and the first Brillouin zone of a wurtzite crystal. Schematics of the irreducible wedges of WZ structure, indicating the high-symmetry points and lines. The Umin point of the WZ phase is located on the M–L line at two-thirds a distance away from the M point. (b). Structure and the first Brillouin zone of a zincblende crystal. Schematics of the irreducible wedges of ZB structure indicating the high-symmetry points and lines.

Under the influence of the crystal field and spin–orbit interactions, the hallmark of the wurtzite structure, the sixfold degenerate Γ15 level associated with the cubic system, splits into , upper , and lower levels (Figure 1.9).

Figure 1.9 Schematic representation of the splitting of the valence band in Wz crystals due to crystal field and spin–orbit interaction. From left to right, the crystal field splitting is considered first. From right to left, the spin–orbit splitting is considered first. Regardless of which is considered first, the end result is the same in that there are three valence bands that are sufficiently close to one another for band mixing to be nonnegligible.

The influence of the crystal field splitting, which is present only in the wurtzite structure, transforms the semiconductor from ZB to Wz, which is represented in the section on the left-hand side in Figure 1.9. The crystal field splits the Γ15 band of the ZB structure into Γ5 and Γ1 states of the wurtzite structure. These two states are further split into , upper , and lower levels by spin–orbit interactions. Application of the spin–orbit splitting, from right to left, splits the Γ15 band of the ZB crystal into Γ8 and Γ7 states, while the crystal possesses the zincblende symmetry. Application of a crystal field further splits these states into , upper , and lower levels, and the crystal now possesses the wurtzite symmetry.

We should mention that a carryover tradition from the zincblende nomenclature is still used for wurtzite symmetry by referring to the crystal filed split-off band with the nomenclature “SO” as if it is the spin–orbit split-off band because it happens to be the farthest from the HH band. In the zincblende symmetry, the crystal field splitting is nonexistent, making the top of the valence band degenerate, and the spin–orbit splitting is large. Portions of this book, unfortunately, participate in the misuse of this nomenclature. The solace is that the reader has been warned.

Figure 1.10 shows the dispersion of the uppermost valence band and lowermost conduction band structures in Wz GaN (a and b) and ZB GaN (c) ((a) near the band for WZ, (b) including M, L, and A minima in WZ, and (c) including , L, and X minima in ZB GaN).

Figure 1.10 (a) Schematic representation of the Γ point valence and conduction bands in crystal with wurtzite symmetry such as GaN where the spin–orbit splitting leads to the bands labeled as HH and LH. The band caused by splitting due to crystal field is labeled as XH. Courtesy of M. Suzuki. (b). Schematic representation of the band diagram for wurtzite GaN showing the separation between the Γ, A, and M–L band symmetry points at 300 K. The values with respect to the top of the valence band are EΓ = 3.4 eV, EM–L = 4.5–5.3 eV, EA = 4.7–5.5 eV, Eso = 0.008 eV, Ecr = 0.04 eV, EΓ = 3.4 eV, EM–L = 4.5–5.3 eV, EA = 4.7–5.5 eV, Eso = 0.008 eV, Ecr = 0.04 eV. The values of EΓ = 6 eV, EM–L = 7 eV, and EA = 8 eV. (c). Schematic representation of the band diagram for zincblende GaN showing the separation between the Γ, X, and L band symmetry points at 300 K. The values with respect to the top of the valence band are EΓ = 3.2 eV, EL = 4.8–5.1 eV, EX = 4.6 eV, Eso = 0.008 and 0.02 eV. Note that in the ZB structure, the valence band is degenerate. The values of EΓ = 3.2 eV, EL = 5.1 eV, and EX = 4.3 eV.

Without the spin–orbit interaction, the valence band would consist of three doubly degenerate bands: HH, LH, and CH bands. The spin–orbit interaction removes this degeneracy and yields six bands.

1.9.2 General Strain Considerations

Strain–stress relationship or Hooke's law describes the deformation of a crystal εkl due to external or internal forces or stresses σij,

(1.13)

where is the fourth ranked elastic tensor and represents the elastic stiffness coefficients in different directions in the crystal, which due to the C6v symmetry can be reduced to a 6 × 6 matrix using the Voigt notation: . The elements of the elastic tensor can be rewritten as with i, j, k, l = x, y, z and m, n = 1,. . ., 6. With this notation, Hooke's law can be reduced to

(1.14)

or in the expanded form

(1.15)

with . If the crystal is strained in the (0001) plane and allowed to expand and constrict in the [0001] direction, the σzz = σxy = σyz = σzx = 0, σxx ≠ 0 and σyy ≠ 0, and the strain tensor has only three nonvanishing terms, namely,

(1.16)

where a and a0 and c and c0 represent the in-plane and out-of-plane lattice constants of the epitaxial layer and the relaxed buffer (substrate), respectively. The above assumes that the in-plane strains in x- and y-directions are identical, namely, . When the crystal is uniaxially strained in the (0001) c-plane and free to expand and constrict in all other directions, σzz is the only nonvanishing stress term and the strain tensor is reduced to

(1.17)

Lack of any force in the growth direction and the fact that the crystal can relax freely in this direction lead to a biaxial strain , which in turn causes , with .

The internal strain is defined by the variation of the internal parameter under strain (u − u0)/u0. In the limit of small deviations from the equilibrium, Hooke's law gives the corresponding diagonal stress tensor with the elements:

(1.18)

In Equation 18.1, four of the five independent stiffness constants Cij of the wurtzite crystal are involved. The modification of Equation 18.1 by the built-in electric field due to the spontaneous and piezoelectric polarizations is neglected, as the effect is small.

In the case of uniaxial stress, for example, along the c-direction, there is an elastic relaxation of the lattice in the c-plane. The ratio of the resulting in-plane strain to deformation along the stress direction is expressed by the Poisson ratio, which in general can be anisotropic. For the wurtzite lattice subjected to a uniaxial stress parallel to the c-axis, holds. Then, Equation 18.1 gives the relation:

(1.19)

with being the Poisson's ratio.

1.9.3 k·p Theory and the Quasicubic Model

The conduction and valence bands of nitride semiconductors are comprised of s- and p-like states, respectively. Unlike the conventional ZB III–N semiconductors and the lack of a high degree of symmetry, the crystal field present removes the degeneracy at the top of the conduction band. The spin–orbit splitting is very small and makes all three bands in the valence band closely situated in energy, making an 8 × 8 k·p Hamiltonian imperative. Because the bandgaps of nitrides are very large, the coupling between the conduction and valence bands can be treated as a second-order perturbation that allows the 8 × 8 Hamiltonian to be split into one 6 × 6 Hamiltonian dealing with the valence band and another 2 × 2 Hamiltonian dealing with the conduction band. The conduction band dispersion relation is

(1.20)

where and represent the in-plane and out-of-plane deformation potentials, respectively. For an isotropic parabolic conduction band, Equation 20.1 reduces to

(1.21)

Ec0 is the conduction band energy at the k = 0 point, ε is the strain, and ac is the deformation potential for the conduction band. The other terms have their usual meanings. It should be pointed out the system under discussion is a linear. The details of the Hamiltonian mentioned above and related manipulations can be found in Morkoç (2008) and references therein.

Results of the above-mentioned calculations are shown in Figure 1.11 with Luttinger-like parameter A7 = 93 meV. The effective masses for all the three valence bands, both parallel and perpendicular, and , also ensue from such calculations.

Figure 1.11 Valence band structures of wurtzite GaN with the k·p theory fitting including the spin–orbit interaction with A7 = 93.7 meV Å in the solid line. The empirical pseudopotential method (EPM) calculation data. Courtesy of Y.C. Yeo, T.C. Chong, M.F. Li, G.B. Ren, Y.M. Liu, and P. Blood.

The effective masses can be expressed in terms of their dependence on the Luttinger-like parameters:

(1.22)

As indicated in the schematic of Figure 1.9, both the spin–orbit and the crystal field splitting affect the structure of the valence band in wurtzitic crystals. Typically the relevant parameters are correlated to one another as Δso= 3Δ2 = 3Δ3 (in spite of the fact that a small Δ2/Δ3 anisotropy has sometimes been reported and Δcr = Δ1). Experimentally, the splitting parameters are obtained from the energy differences of the A, B, and C free excitons, which have nonlinear dependencies on the various splittings. It should be pointed out that the nomenclature for the three valence bands for hexagonal system is A, B, and C for HH, LH, and SO (CH) bands when including A7 terms because spin splitting and strain can significantly alter which band of eigenstates is “heavy” or “light” at various k-values, particularly in the c-plane. Experiments led to values of Δcr = 16 meV and Δso = 12 meV, and Δcr = 25 meV and Δso = 17 meV, the latter set from a fit to exciton energies, A and B determined by PL and C determined by reflection, but with a geometry not fully ideal in terms that some error is introduced in the value of C exciton energy. There is a sizable dispersion in the calculations reported so far particularly in the value of Δcr. This treatment is provided for its simplicity; for more accurate data, the reader should refer to full band calculations provided.

A practical approach is to modify the well-established treatment developed for the cubic system and make it applicable to the Wz system, which is referred to as quasicubic approximation. The genesis of the quasicubic approximation relies on the fact that both the Wz and ZB structures are tetrahedrally coordinated and hence are closely related. The nearest-neighbor coordination is the same for Wz and ZB structures, but the next-nearest-neighbor positions differ between the two systems. The basal plane (0001) of the Wz structure corresponds to one of the (111) planes of the ZB. When the in-plane hexagons are lined up in Wz and ZB structures, the Wz [0001], [], and [] planes are parallel to ZB [111], [], and [] planes. This, in turn, leads to correlations between the symmetry direction and the k-points for the two polytypes. There are, however, twice as many atoms in the Wz unit cell as there are in the ZB one. In addition to the band structure similarities between the doubled ZB and Wz structures, one can establish a correlation between the Luttinger parameters in the ZB system and parameters of interest in the Wz system by taking the z-axis along the [111] direction and the x- and y-axes along the [] and [] directions, respectively. For details regarding the symmetry relations between the ZB and Wz polytypes, refer to Morkoç (2008) and references therein.

The large effective mass and the small dielectric constant of GaN, relative to more conventional group-III–V semiconductors, lead to relatively large exciton binding energies and make excitons, together with large exciton recombination rates, clearly observable even at room temperature. The bottom of the conduction band of GaN is predominantly formed from the s-levels of Ga and the upper valence band states from the p-levels of N. Even though sophisticated methods have been introduced and discussed, the method of Hopfield and Thomas, which treats the wurtzite energy levels as a perturbation to the zincblende structure, is discussed briefly as it provides a physical picture of band splitting in the valence band. Using the quasicubic model of Hopfield, one obtains

(1.23)

(1.24)

(1.25)

where δ and Δ represent the contributions of uniaxial field and spin–orbit interactions, respectively, to the splittings E1,2 and E2,3.

1.9.4 Temperature Dependence of Wurtzite GaN Bandgap

The temperature dependence of the bandgap in semiconductors is often described by an imperial expression (assuming no localization),

(1.26)

In the case of localization, which can also be construed as bandtail effect, the temperature dependence deviates from the above equation. In the framework of the bandtail model and Gaussian-like distribution of the density of states for the conduction and valence bands, the temperature-dependent emission energy could be described by the following modified expression, which is based on a model developed for Stokes shift in GaAs/AlGaAs QWs:

(1.27)

where the last term represents the localization component with indicating the extent of localization or bandtailing, which is nearly imperative for In-containing alloys. The parameters α is in units of energy over temperature and β is in units of temperature.

1.9.5 Sphalerite (Zincblende) GaN

The valence band of zincblende GaN has been the topic of various theoretical efforts. Although the hole effective masses in zincblende GaN have apparently not been measured, a number of theoretical predictions of Luttinger parameters are available in the literature (Table 1.9) (refer to Morkoç (2008) and references therein). Once the Luttinger parameters are known, the full picture in terms of the hole effective masses can be determined. First, it should be pointed out that in polar semiconductors such as the III–V compounds in general and GaN in particular, it is the nonresonant polaron mass that is actually measured. The polaron mass exceeds the bare electron mass by about 1–2%, the exact value of which depends on the strength of the electron–phonon interaction. Because the band structure is governed by the bare electron mass, this is the quantity that is typically reported whenever available.

Table 1.9 Luttinger parameters γ1, γ2, and γ3 for zincblende GaN obtained from a fit along the [110] direction.

ParameterValueγ12.67–3.07γ20.75–0.9 0γ31.07–1.26

At the valence band edge, the heavy hole (hh) effective masses in the different crystallographic directions are related to the free mass through the Luttinger parameters in the following manner:

(1.28)

Here, the z-direction is perpendicular to the growth plane of (001). These expressions described by Equation 28.1 show the relationship of the Luttinger parameters to the hh effective masses that can typically be measured in a more direct manner. The light hole (lh) and so hole effective masses are given by

(1.29)

(1.30)

To restate, although the hole effective masses in zincblende GaN have apparently not been measured, a number of theoretical predictions of Luttinger parameters are available in the literature (Table 1.10). The values are based on averages of the heavy hole and light hole masses along [001], as well as the degree of anisotropy in γ3–γ2. The parameter set used is γ1 = 2.70, γ2 = 0.76, and γ3 = 1.11. Similarly, averaging all the reported split-off masses leads to = 0.29m0. In its simplest form, the Luttinger parameters can be used to determine quickly the effective masses in various valence bands both in equilibrium and also under biaxial strain. In fact, with biaxial strain, the valence band degeneracy can be removed and, most strikingly, the heavy hole in-plane mass can be made smaller with compressive strain, a notion that has been exploited in the InGaAs/GaAs system very successfully, particularly for low-threshold lasers.

Table 1.10 Effective masses for electrons (e), and heavy holes (hh), light holes (lh) and spin–orbit split-off holes (so) in units of the free electron mass m0 along the [100], [111], and [110] directions for zincblende GaN.

1.9.6 AlN

AlN forms the larger bandgap binary used in conjunction with GaN for increasing the bandgap for heterostructures. As in the case of GaN, AlN also has wurtzitic and zincblende polytypes, the latter being very unstable and hard to synthesize. Owing to increasing interest in solar blind devices, UV emitters and detectors, and the expectation that AlGaN with large mole fractions of AlN would have relatively large breakdown properties, this material has been steadily gaining attention. It should also be mentioned that the N overpressure on Al is the smallest among those over Ga and In, paving the way for equilibrium growth of AlN bulk crystals, albeit not without O contamination.

1.9.6.1 Wurtzite AlN

Wurtzite AlN is a direct bandgap semiconductor with a bandgap near 6.1 eV and still considered to be semiconductor because it is dopable. The zincblende polytype is not stable with a predicted indirect bandgap. The AlN derive its technological importance from its providing the large bandgap binary component of the AlGaN alloy, which is commonly employed in both optoelectronic and electronic devices based on the GaN semiconductor system.

Absorption measurements carried out early on indicated a large energy gap of 6.1 eV for wurtzite AlN at 5 K and about 6 eV at room temperature. Averaging all the available theoretical crystal field splittings, one obtains a value of Δcr = −169 meV. Using optical reflectance data performed on a- and c-plane bulk AlN and a quasicubic model developed for the wurtzite crystal structure, the crystal field splitting was determined to be Δ = −225 meV. Note that the negative sign for the crystal field splitting has important implications, namely, that the Γ7valence band is on top of Γ9valence band, which is opposite of that in GaN. As for the spin–orbit splitting, the literature values range from 11 to 36 meV, the latter having been determined by optical reflectance spectra in high-quality bulk AlN. The bare mass values of = 0.30m0 and = 0.32m0 obtained by averaging the available theoretical masses may represent a good set of default values at this stage.

The effective masses for both the conduction band and various valence bands are compiled in Table 1.11.

Table 1.11 Effective masses and band parameters for wurtzitic AlN.

1.9.6.2 Zincblende AlN

Due to lack of sufficient experimental data, the treatment here primarily relies on theoretical projections. The only quantitative experimental study of the bandgap indicated a Γ valley indirect gap of 5.34 eV at room temperature. Assuming that the Varshni parameters for the wurtzitic AlN hold for the zincblende polytype, the aforementioned room-temperature bandgap translates to a low-temperature gap of 5.4 eV. The values for the X and L valley gaps are 4.9 and 9.3 eV. The spin–orbit splitting is expected to be nearly the same as in wurtzite AlN at 19 meV. Averaging the available theoretical results one arrives at a Γ valley effective mass of 0.25m0. The longitudinal and transverse masses for the X valley have been predicted to be 0.53m0 and 0.31m0, respectively. If the method used previously for the GaN is applied to zincblende AlN, one arrives at recommended Luttinger parameters of γ1 = 1.92, γ2 = 0.47, and γ3 = 0.85, and mso = 0.47m0. These as well as the other literature values of the Luttinger parameters are listed in Table 1.12.

Table 1.12 Luttinger parameters γ1, γ2, and γ3 for zincblende AlN obtained from a fit along the [110] direction in addition to those available in the literature.

The calculated effective masses for conduction and valence bands, the latter involving the light and heavy holes, as well as the spin–orbit split-off mass, and taking the anisotropy into account, are listed in Table 1.13 for zincblende AlN.

Table 1.13 Effective masses for electrons (e), heavy holes (hh), light holes (lh), and spin–orbit split-off holes (so) in units of the free electron mass m0 along the [100], [111], and [110] directions for zincblende AlN.

1.9.7 InN

As in the case of AlN, the main impetus in InN is due to the InGaN alloy that is used in lasers and LEDs operative in the visible and violet regions of the optical spectrum. The properties, particularly the fundamental parameters of InGaN for a given composition, depend very much on the InN parameters, particularly on its bandgap. The initially accepted bandgap value of 1.98 eV has given way to about 0.7 eV for Wz InN, which motivated some to consider this material system for photovoltaic cells. Estimates of the crystal field splitting in wurtzite InN range from 17 to 301 meV, but a value of 40 meV can be adopted. Based on the calculations, spin–orbit splittings vary from 1 to 13 meV, but a value of Δso = 5 meV can be assumed.

Turning our attention to other electronic properties affected by the band structure, measurements of the electron effective mass in wurtzitic InN produced values of 0.11–0.14m0. Accounting for the substantial nonparabolicity that can cause an overestimate of the mass because of high doping leads to a recommended band edge effective mass of 0.07m0. The dispersion in the effective mass for both the conduction band and various valence bands as obtained by various computational methods and the parameters used in the description of the bandgap for wurtzitic InN, particularly in the context of empirical pseudopotential method, are compiled in Table 1.14.

Table 1.14 Effective masses and band parameters for wurtzitic InN.

1.10 Polarization Effects

Group III–V nitride semiconductors exhibit highly pronounced polarization effects, the genesis of which has to do with large electronegativity in the case of single layers and with differential electronegativity in the case of heterostructures. Semiconductor nitrides lack center of inversion symmetry and exhibit large piezoelectric effects when strained along 0001. The strain-induced piezoelectric charge and spontaneous polarization, the latter caused by compositional gradient, have profound effects on device structures. Spontaneous polarization was understood fully only recently.

Polarization is dependent on the polarity of the crystal, namely, whether the bonds along the c-direction are from cation sites to anion sites, or vice versa. The convention is that when the single bonds are from cation (Ga) to anion (N) atoms, the [0001] axis points from the face of the N plane to the Ga plane and marks the positive z-direction, or c+- or +z-direction. When the single bonds are along the c-direction or from anion (N) to cation (Ga) atoms, the polarity is said to be the N-polarity, and the direction is said to be the c−- or −z-direction. A schematic representation of the spontaneous polarization in a model GaN/AlN/GaN wurtzitic crystal is shown in Figure 1.12.

Figure 1.12 Schematic depicting the convention used for determining the polarity and crystalline direction in wurtzitic nitride films. The diagram shows the case for a Ga-polarity film with its characteristic bonds parallel to the c-axis (horizontal in the figure) going from the cation (Ga or Al) to the anion (N). The spontaneous polarization components P0Ga and P0AlN for a periodic GaN/AlN structure are also indicated with that for AlN having a larger magnitude. The spontaneous polarization is negative and thus points in the [] direction. Caution must be exercised here as there is no long-range polarization field, just that it is limited to the interface. The polarization in AlN is larger in magnitude than that in GaN. There exists a difference in polarization at the interface, ΔP0 pointing in the [] direction for both GaN–AlN interfaces. The axial Born factor is defined as . Courtesy of V. Fiorentini.

The magnitude of the polarization charge, converted to number of electrons, can be in the mid 1013 cm−2 level for AlN–GaN heterointerfaces, which is huge by any standard, some 10 times larger than the doping-induced electron density in the GaAs/AlGaAs system. The magnitude of the polarization charge is compiled in Table 1.15 along with elastic coefficients taken from the literature.

Table 1.15 Elastic constants and spontaneous polarization charge in nitride semiconductors.

The data in bold letters are recommended.

1.10.1 Piezoelectric Polarization

In a polarizable medium, the displacement vector can be expressed in terms of two components due to both the dielectric and polarizability nature of the medium:

(1.31)

where and represent the electric field and polarization vectors, respectively. Considering only the piezoelectric component, the piezoelectric polarization vector is given by

(1.32)

where and are the piezoelectric and stress tensors, respectively. In hexagonal symmetry, electric polarization is related to strain through electric piezoelectric tensor:

(1.33)

where , , and represent the electric polarization, electric piezoelectric coefficient, and strain, respectively. Note that for hexagonal symmetry, .

Without shear, . For biaxial strain only,

(1.34)

Here, and represent the relaxed (equilibrium) in-plane lattice constants of the buffer layer or the substrate, depending on layers and their thicknesses, and of the epitaxial layer of interest, the strained epitaxial layer, respectively. The expression for the out-of-plane strain is

(1.35)

Similarly, and represent the relaxed and the out-of-plane lattice parameters that would correspond to the buffer layer and epitaxial layer, respectively. In case the in-plane strain is anisotropic, .

The components of the piezoelectric polarization tensor given by Equation 33.1 can be expressed in terms of a summation, using instead of , as

(1.36)

where is the ith component of the piezoelectric polarization.

The wurtzite symmetry reduces the number of independent components of the elastic tensor to three, namely, , , and , and the negligence of the shear strain makes . In this case the electric polarization can be expressed as

(1.37)

For isotropic basal plane strain, the strain component , and thus Equation 37.1 can be written as

(1.38)

In hexagonal symmetry, strain in the z-direction can be expressed in terms of the basal plane strain by using Poisson's ratio, which is expressed in terms of the elastic coefficients as . In the case of externally applied pressure in addition to mismatch strain, the out-of-plane strain can be related to the in-plane strain through , where p is the magnitude of compressive pressure (in the same unit as the elastic coefficients). In terms of the nomenclature again, it should also be noted that and in the other notation used in the literature and also in this text:

(1.39)

Knowing the piezoelectric parameters of the end binary points is generally sufficient, to a first order, to discern parameters for more complex alloys. For example, in the case of AlxGa1-xN, the piezoelectric polarization vector expression, using linear interpolation within the framework of Vegard's law, can be described:

(1.40)

The same argument can be extended to piezoelectric polarization in quaternary alloys such as AlxInyGa1−x−yN in a similar fashion:

(1.41)

The linear interpolation is very convenient and accurate. However, as will be discussed in the following, while the Vegard's law applies to the alloys, the polarization charge itself is not a linear function of composition.

1.10.2 Spontaneous Polarization

Spontaneous polarizations calculated for the binary nitride semiconductors are compiled in Table 1.15