The k p Method (eBook)

Dr. Lew Yan Voon has received recognitions for this teaching (e.g., advising awards, NSF CAREER award) and has taught the k.p method to MS students, PhD students and postdocs; this pedagogical experience has been invaluable for testing the manuscript as a textbook.

Foreword 6
Preface 9
Contents 11
Acronyms 18
Chapter 1 Introduction 19
What Is kp Theory? 19
Electronic Properties of Semiconductors 19
Other Books 21
Part I Homogeneous Crystals 22
Chapter 2 One-Band Model 23
Overview 23
kp Equation 23
Perturbation Theory 25
Canonical Transformation 25
Effective Masses 28
Electron 28
Light Hole 29
Heavy Hole 30
Nonparabolicity 30
Summary 31
Chapter 3 Perturbation Theory -- Valence Band 32
Overview 32
Dresselhaus--Kip--Kittel Model 32
Hamiltonian 32
Eigenvalues 36
L,M,N Parameters 37
Properties 45
Six-Band Model for Diamond 47
Hamiltonian 47
DKK Solution 55
Kane Solution 58
Wurtzite 60
Overview 60
Basis States 61
Chuang--Chang Hamiltonian 61
Gutsche--Jahne Hamiltonian 67
Summary 69
Chapter 4 Perturbation Theory -- Kane Models 70
Overview 70
First-Order Models 70
Four-Band Model 71
Eight-Band Model 72
Second-Order Kane Model 76
Löwdin Perturbation 76
Four-Band Model 77
Full-Zone kp Model 79
15-Band Model 79
Other Models 84
Wurtzite 84
Four-Band: Andreev-O'Reilly 85
Eight-Band: Chuang--Chang 86
Eight-Band: Gutsche--Jahne 86
Summary 92
Chapter 5 Method of Invariants 93
Overview 93
DKK Hamiltonian -- Hybrid Method 93
Formalism 98
Introduction 98
Spatial Symmetries 98
Spinor Representation 102
Valence Band of Diamond 102
No Spin 103
Magnetic Field 104
Spin-Orbit Interaction 107
Six-Band Model for Diamond 128
Spin-Orbit Interaction 129
k-Dependent Part 129
Four-Band Model for Zincblende 130
Eight-Band Model for Zincblende 131
Weiler Hamiltonian 131
14-Band Model for Zincblende 134
Symmetrized Matrices 135
Invariant Hamiltonian 137
T Basis Matrices 139
Parameters 142
Wurtzite 146
Six-Band Model 146
Quasi-Cubic Approximation 150
Eight-Band Model 151
Method of Invariants Revisited 154
Zincblende 154
Wurtzite 160
Summary 165
Chapter 6 Spin Splitting 166
Overview 166
Dresselhaus Effect in Zincblende 167
Conduction State 167
Valence States 167
Extended Kane Model 169
Sign of Spin-Splitting Coefficients 173
Linear Spin Splittings in Wurtzite 174
Lower Conduction-Band e States 176
A,B,C Valence States 177
Linear Spin Splitting 178
Summary 179
Chapter 7 Strain 180
Overview 180
Perturbation Theory 180
Strain Hamiltonian 180
Löwdin Renormalization 183
Valence Band of Diamond 183
DKK Hamiltonian 184
Four-Band Bir--Pikus Hamiltonian 184
Six-Band Hamiltonian 185
Method of Invariants 187
Strained Energies 190
Four-Band Model 190
Six-Band Model 192
Deformation Potentials 192
Eight-Band Model for Zincblende 193
Perturbation Theory 194
Method of Invariants 195
Wurtzite 196
Perturbation Theory 196
Method of Invariants 197
Examples 199
Summary 199
Part II Nonperiodic Problem 200
Chapter 8 Shallow Impurity States 201
Overview 201
Kittel--Mitchell Theory 202
Exact Theory 203
Wannier Equation 205
Donor States 206
Acceptor States 209
Luttinger--Kohn Theory 210
Simple Bands 211
Degenerate Bands 222
Spin-Orbit Coupling 225
Baldereschi--Lipari Model 226
Hamiltonian 228
Solution 229
Summary 231
Chapter 9 Magnetic Effects 232
Overview 232
Canonical Transformation 233
One-Band Model 233
Degenerate Bands 241
Spin-Orbit Coupling 243
Valence-Band Landau Levels 246
Exact Solution 246
General Solution 250
Extended Kane Model 251
Landé g-Factor 251
Zincblende 252
Wurtzite 254
Summary 255
Chapter 10 Electric Field 256
Overview 256
One-Band Model of Stark Effect 256
Multiband Stark Problem 257
Basis Functions 257
Matrix Elements of the Coordinate Operator 259
Multiband Hamiltonian 260
Explicit Form of Hamiltonian Matrix Contributions 264
Summary 266
Chapter 11 Excitons 267
Overview 267
Excitonic Hamiltonian 268
One-Band Model of Excitons 269
Multiband Theory of Excitons 271
Formalism 271
Results and Discussions 276
Zincblende 277
Magnetoexciton 278
Summary 280
Chapter 12 Heterostructures: Basic Formalism 282
Overview 282
Bastard's Theory 283
Envelope-Function Approximation 283
Solution 285
Example Models 286
General Properties 288
One-Band Models 289
Derivation 289
Burt--Foreman Theory 291
Overview 292
Envelope-Function Expansion 292
Envelope-Function Equation 296
Potential-Energy Term 303
Conventional Results 308
Boundary Conditions 314
Burt--Foreman Hamiltonian 315
Beyond Burt--Foreman Theory? 325
Sercel--Vahala Theory 327
Overview 327
Spherical Representation 328
Cylindrical Representation 333
Four-Band Hamiltonian in Cylindrical Polar Coordinates 338
Wurtzite Structure 345
Arbitrary Nanostructure Orientation 359
Overview 359
Rotation Matrix 359
General Theory 361
[110] Quantum Wires 362
Spurious Solutions 369
Summary 370
Chapter 13 Heterostructures: Further Topics 372
Overview 372
Spin Splitting 372
Zincblende Superlattices 372
Strain in Heterostructures 376
External Stress 376
Strained Heterostructures 378
Impurity States 380
Donor States 380
Acceptor States 381
Excitons 382
One-Band Model 382
Type-II Excitons 385
Multiband Theory of Excitons 386
Magnetic Problem 387
One-Band Model 388
Multiband Model 391
Static Electric Field 393
Transverse Stark Effect 393
Longitudinal Stark Effect 395
Multiband Theory 397
Chapter 14 Conclusion 399
Appendix A Quantum Mechanics and Group Theory 401
Löwdin Perturbation Theory 19
Variational Principle 401
Perturbation Formula 402
Group Representation Theory 19
Great Orthogonality Theorem 405
Characters 406
Angular-Momentum Theory 21
Angular Momenta 407
Spherical Tensors 407
Wigner-Eckart Theorem 408
Wigner 3j Symbols 408
Appendix B Symmetry Properties 409
Introduction 19
Zincblende 19
Point Group 405
Irreducible Representations 406
Diamond 21
Symmetry Operators 407
Irreducible Representations 407
Wurtzite 415
Irreducible Representations 418
Appendix C Hamiltonians 420
Basis Matrices 19
s=12 401
l=1 402
J=32 420
|JMJ"526930B States 19
Hamiltonians 21
Notations 407
Diamond 407
Zincblende 408
Wurtzite 408
Heterostructures 423
Summary of kp Parameters 415
References 438
Index 450