The physics of low-dimensional semiconductors: an introduction

THE PHYSICS OF LOW-DIMENSIONAL SEMICONDUCTORS: AN INTRODUCTION......Page 1
Half-title......Page 2
Title Page......Page 3
Copyright Page......Page 4
Dedication......Page 6
Contents......Page 8
Acknowledgements......Page 14
Introduction......Page 16
Outline......Page 17
References and further reading......Page 18
1.1 Wave Mechanics and the Schrödinger Equation......Page 20
1.2 Free Particles......Page 22
1.3 Bound Particles: Quantum Well......Page 23
1.3.1 Optical Absorption in a Quantum Well......Page 25
1.4 Charge and Current Densities......Page 28
1.5.1 Operators......Page 32
1.5.2 Expectation Values......Page 33
1.5.3 Motion of a Wave Packet......Page 35
1.5.4 Further Properties of Operators......Page 38
1.6 Mathematical Properties of Eigenstates......Page 39
1.7.1 One Dimension......Page 41
1.7.2 Three Dimensions......Page 44
1.7.3 A General Definition of the Density of States......Page 46
1.7.4 Local Density of States......Page 48
1.8.1 Fermi–Dirac Occupation Function......Page 49
1.8.2 Occupation of States......Page 51
1.8.3 Classical Lightly Doped Semiconductors......Page 54
1.8.4 The Electron Gas......Page 56
1.8.5 Other Distribution Functions......Page 57
Further Reading......Page 59
Exercises......Page 60
2.1 Band Structure in One Dimension......Page 64
2.1.1 Formation of Band Gaps......Page 66
2.2 Motion of Electrons in Bands......Page 69
2.3 Density of States......Page 73
2.4 Band Structure in Two and Three Dimensions......Page 74
2.5 Crystal Structure of the Common Semiconductors......Page 76
2.6.1 Brillouin Zone......Page 80
2.6.2 General Features......Page 82
2.6.3 Valence Band......Page 83
2.6.4 Conduction Band......Page 85
2.6.4.2 X minima — Si and AlAs......Page 86
2.6.4.4 Higher minima......Page 87
2.7 Optical Measurement of Band Gaps......Page 88
2.8 Phonons......Page 89
2.8.1 Phonons in One Dimension......Page 90
2.8.3 Diatomic Chain......Page 92
2.8.4 Phonons in Three Dimensions......Page 94
Exercises......Page 95
3.1 General Properties of Heterostructures......Page 99
3.2.1 Molecular-Beam Epitaxy......Page 101
3.2.2 Metal–Organic Chemical Vapour Deposition......Page 103
3.3 Band Engineering......Page 104
3.4 Layered Structures: Quantum Wells and Barriers......Page 107
3.4.2 Quantum Well......Page 108
3.4.4 Superlattice......Page 109
3.4.5 Nature of the Conduction Band......Page 110
3.5.1 Modulation Doping......Page 111
3.5.2 Construction of Band Diagrams......Page 113
3.5.3 Modulation-Doped Field-Effect Transistor......Page 114
3.6 Strained Layers......Page 115
3.6.1 Structural Aspects of Strained Layers......Page 116
3.6.2 Effect of Strain on Band Structure......Page 117
3.7 Silicon–Germanium Heterostructures......Page 119
3.8 Wires and Dots......Page 121
3.9 Optical Confinement......Page 124
3.10 Effective-Mass Approximation......Page 126
3.11 Effective-Mass Theory in Heterostructures......Page 130
Exercises......Page 133
4.1 Infinitely Deep Square Well......Page 137
4.2 Square Well of Finite Depth......Page 138
4.3 Parabolic Well......Page 144
4.4 Triangular Well......Page 147
4.5 Low-Dimensional Systems......Page 149
4.6 Occupation of Subbands......Page 152
4.7 Two- and Three-Dimensional Potential Wells......Page 154
4.7.1 Cylindrical Well......Page 155
4.7.2 Two-Dimensional Parabolic Well......Page 156
4.7.4 Spherical Well......Page 158
4.8 Further Confinement Beyond Two Dimensions......Page 159
4.9 Quantum Wells in Heterostructures......Page 161
Exercises......Page 165
5.1 Potential Step......Page 169
5.2 T-Matrices......Page 172
5.2.1 Square Barrier......Page 174
5.3 More on T-Matrices......Page 177
5.4 Current and Conductance......Page 181
5.4.1 Current in One Dimension......Page 182
5.4.2 Current in Two and Three Dimensions......Page 184
5.5 Resonant Tunnelling......Page 186
5.5.1 Transmission Coefficient in One Dimension......Page 187
5.5.2 Partial Waves......Page 191
5.5.3 Current through a Resonant-Tunnelling Diode......Page 192
5.6 Superlattices and Minibands......Page 196
5.7 Coherent Transport with Many Channels......Page 202
5.7.1 Two Leads with Many Channels: The Quantized Conductance......Page 203
5.7.2 Systems with Many Leads......Page 207
5.8 Tunnelling in Heterostructures......Page 214
5.8.1 Intervalley Transfer......Page 216
5.9.2 Incoming Distribution of Electrons......Page 218
Further Reading......Page 219
Exercises......Page 220
6.1 The Schrödinger Equation with Electric and Magnetic Fields......Page 225
6.2 Uniform Electric Field......Page 227
6.2.1 Density of States......Page 229
6.2.2 Electric Field from a Vector Potential......Page 232
6.2.3 Narrow Band in an Electric Field......Page 233
6.3 Conductivity and Resistivity Tensors......Page 235
6.4.1 Solution in Landau Gauge......Page 238
6.4.1 Solution in Symmetric gauge......Page 241
6.4.4 Landau Levels......Page 242
6.4.5 Shubnikov–De Haas Effect......Page 246
6.4.6 Integer Quantum Hall Effect......Page 247
6.4.7 Crossed Electric and Magnetic Fields......Page 248
6.4.8 Gauge Invariance......Page 250
6.4.9 The Aharonov–Bohm Effect......Page 251
6.5 Magnetic Field in a Narrow Channel......Page 252
6.5.1 Quantum Dots and Superlattices......Page 256
6.6 The Quantum Hall Effect......Page 258
6.6.1 Edge States and Barriers......Page 261
6.6.2 The Fractional Quantum Hall Effect......Page 263
Further Reading......Page 264
Exercises......Page 265
7.1 The Matrix Formulation of Quantum Mechanics......Page 268
7.2 Time-Independent Perturbation Theory......Page 271
7.2.1 General Theory......Page 272
7.2.2 Quantum Well in an Electric Field......Page 276
7.2.3 A Cautionary Tale......Page 279
7.3 k · p Theory......Page 280
7.4 WKB Theory......Page 282
7.4.1 General Theory......Page 283
7.4.2 Bound States in a Triangular Well......Page 286
7.4.3 Tunnelling through a Schottky Barrier......Page 287
7.5.1 General Theory......Page 289
7.5.2 Bound State in a Triangular Well......Page 291
7.6 Degenerate Perturbation Theory......Page 292
7.7 Band Structure: Tight Binding......Page 294
7.7.1 Two Wells: Diatomic Molecule......Page 295
7.7.2 Row of Wells: Tight-Binding Solid......Page 297
7.8 Band Structure: Nearly Free Electrons......Page 299
Exercises......Page 303
8.1 Golden Rule for Static Potentials......Page 309
8.2 Impurity Scattering......Page 314
8.2.1 Scattering by a Short-Range Impurity......Page 318
8.3 Golden Rule for Oscillating Potentials......Page 320
8.4 Phonon Scattering......Page 321
8.4.1 Longitudinal Acoustic Phonons and the Deformation Potential......Page 322
8.4.2 Longitudinal Optic Phonons and Polar Coupling......Page 325
8.5 Optical Absorption......Page 327
8.5.1 Macroscopic Equations......Page 328
8.5.2 Electron–Photon Interaction......Page 329
8.6 Interband Absorption......Page 332
8.7 Absorption in a Quantum Well......Page 335
8.8 Diagrams and the Self-Energy......Page 340
Exercises......Page 343
9.1 Band Diagram of Modulation-Doped Layers......Page 348
9.1.1 Electrostatic Potential......Page 351
9.1.2 Conduction Band and Gate Bias......Page 352
9.1.3 Threshold Voltage......Page 353
9.1.4 Gate-Channel Capacitance......Page 354
9.2 Beyond the Simplest Model......Page 355
9.2.1 Donors in AlGaAs: DX Centres......Page 356
9.2.2 Persistent Photoconductivity......Page 358
9.2.4 Negative Hubbard U......Page 359
9.2.5 Doped or Biassed Substrate......Page 360
9.3.1 Triangular-Well Model......Page 361
9.3.2 The Quantum Mechanics of Many Electrons......Page 363
9.3.3 Variational Hartree Calculation of a 2DEG......Page 366
9.4 Screening by an Electron Gas......Page 368
9.4.1 Screening in Three Dimensions......Page 369
9.4.2 Screening in a 2DEG......Page 372
9.5 Scattering by Remote Impurities......Page 375
9.5.1 Matrix Element......Page 376
9.5.2 Scattering Rate......Page 377
9.6.1 Background Ionized Impurities......Page 381
9.6.3 Phonons......Page 382
Further Reading......Page 384
Exercises......Page 385
10.1 General Theory......Page 390
10.1.1 Kramers–Kronig Relations......Page 391
10.1.2 Optical-Response Functions......Page 392
10.1.3 Sum Rules......Page 394
10.2 Valence-Band Structure: The Kane Model......Page 396
10.2.1 The Kane Model without Spin......Page 397
10.2.2 Spin–Orbit Coupling......Page 400
10.3 Bands in a Quantum Well......Page 403
10.4 Interband Transitions in a Quantum Well......Page 406
10.4.1 Interband Matrix Element......Page 408
10.4.2 Matrix Element of Envelope Functions......Page 410
10.4.3 Absorption Spectrum......Page 411
10.5 Intersubband Transitions in a Quantum Well......Page 412
10.6 Optical Gain and Lasers......Page 414
10.7 Excitons......Page 416
10.7.1 Excitons in Three Dimensions......Page 417
10.7.2 Excitons in Two Dimensions......Page 419
10.7.3 Excitons in a Quantum Well......Page 420
10.7.4 Quantum-Confined Stark Effect......Page 423
10.7.5 Excitons, Fluctuations, and Phonons......Page 424
Exercises......Page 425
Appendix 1. Table of Physical Constants......Page 428
Appendix 2. Properties of Important Semiconductors......Page 429
Appendix 3. Properties of GaAs–AlAs Alloys at Room Temperature......Page 431
Appendix 4. Hermite\'s Equation: Harmonic Oscillator......Page 432
Appendix 5. Airy Functions: Triangular Well......Page 434
A6.1 Derivation of the Kramers–Kronig Relations......Page 436
A6.2.1 Free Electrons: The Drude Model......Page 438
A6.2.2 Bound Electrons: The Lorentz Model......Page 440
Data on Semiconductors......Page 442
References......Page 443
Index......Page 446
Back Cover......Page 458