Theory of Electron Transport in Semiconductors (eBook)

PhD in Solid-state Physics at Purdue University (Indiana, USA) in 1969. Since then at the University of Modena with successive positions from research-assistant to Dean of the School of Sciences. Published 3 books and about 170 papers in the field of semiclassical and quantum theory of electron transport in semiconductors. His main contributions are related to the Monte Carlo simulation of electron transport in materials and devices, and to the application of the Wigner-function formalism to quantum transport.

Theory of Electron Transport in Semiconductors 
3 
Preface 7
Contents 9
Symbols and Abbreviations 19
Part I Basic Concepts in Semiconductor Physics 27
1 Survey of Classical Physics 28
1.1 Newton Dynamics 28
Linear Momentum 28
Angular Momentum 29
1.2 Work and Energy 29
1.3 Hamiltonian Formulation of Dynamics 31
1.4 Canonical Transformations 32
1.5 Small Oscillations 33
1.6 Maxwell Equations 35
1.7 Electromagnetic Potentials and Gauge Transformations 36
1.8 Hamiltonian of a Charged Particle in an ElectromagneticField 38
2 Fundamentals of Quantum Mechanics 39
2.1 The First Postulates 40
2.2 Equations of Motion 42
2.2.1 Pictures and Representations 42
2.2.2 Evolution Operator and Its Equation of Motion 43
2.2.3 Equation of Motion in Schrödinger 
44 
2.2.4 Interaction Picture 44
2.3 Heisenberg Uncertainty Relations 45
2.4 How to Deal with a General Quantum-Mechanical Problem in a System with a Constant Hamiltonian 46
2.5 The {q} Representation: Wave Mechanics 47
Hamiltonian and Observables 47
Wavefunction and Schrödinger Equation 47
Eigenfunctions of the Momentum and the {p} Representation 48
2.6 Identical Particles and Pauli Exclusion Principle 49
3 Fundamentals of Statistical Physics 51
3.1 Introduction 51
3.2 Liouville Theorem 52
3.3 The Fundamental Hypotheses of Statistical Mechanics 54
3.4 Main Definitions and Results of Statistical Mechanics 55
Entropy 55
Temperature and Thermal Equilibrium 55
Chemical Potential and Particle Equilibrium 56
3.5 Thermal Bath 58
3.6 The Three Fundamental Statistical Ensembles 58
3.6.1 Microcanonical Ensemble 59
3.6.2 Canonical Ensemble 59
3.6.3 Grand Canonical Ensemble 60
3.7 Equilibrium Particle Distributions in Ideal Gases 61
3.7.1 Classical Gas: Maxwell–Boltzmann Distribution 62
3.7.2 Bose Distributions 62
3.7.3 Fermi Distribution 63
3.7.4 Classical Limit 63
4 Crystal Structures 64
4.1 Crystals 64
4.2 Lattices 65
Wigner–Seitz Primitive Cell 66
Points, Lines, and Planes in Crystals 66
Diamond and Zincblende Structures 68
4.3 Crystal Bonding 69
Electrostatic Interaction, Ionic Crystals 69
Homopolar Bond, Covalent Crystals 69
Other Types of Bonds 70
4.4 Reciprocal Lattice 70
5 Phonons 72
5.1 The Vibrating String 73
5.2 The Simplest Linear Chain 75
Traveling Waves 75
Periodicity of (q), Brillouin Zone 77
5.3 Monatomic Linear Chain with Multiple Coupling 78
5.4 Diatomic Linear Chain 79
Acoustic and Optical Modes 80
5.5 Three-Dimensional Lattice Vibrations 82
Density of States 83
5.6 Normal Coordinates and Quantization – Phonons 86
5.7 Phonon Momentum and Crystal Momentum 89
5.8 Experimental Determination of Phonon Dispersions 89
6 Bloch States and Band Theory 92
6.1 Bloch Theorem 92
6.2 Density of States 95
6.3 Tight-Binding Approach 96
6.4 Band-Structure Calculations 97
6.4.1 LCAO Method 98
6.4.2 kp Method 99
6.4.3 Pseudopotential Method 99
6.5 Band Structures of Most Important Semiconductors 103
6.6 Effective-Mass Approximation 103
6.7 Bloch Wavepackets 104
6.7.1 Group Velocity 106
7 Effective-Mass Theorems, Envelope Function,and Semiclassical Dynamics 107
7.1 Effective-Mass Theorem for Bloch States 107
7.2 Effective-Mass Theorem in Presence of a Scalar Potential 109
Constant Potential 109
Time-Dependent Potential 110
7.3 Accelerated Waves 111
7.3.1 Accelerated Classical Electrons in Free Space 111
7.3.2 Accelerated Quantum Electrons in Free Space 111
7.3.3 Accelerated Bloch States 112
7.4 Envelope Function for Steady States 113
7.5 Effective-Mass Theorem for a Wavepacket in Slow-Varying Electric and Magnetic Fields 117
7.6 Time-Dependent Envelope Function 120
7.7 Semiclassical Dynamics 121
8 Semiconductors 124
8.1 Free Dynamics of Bloch Electrons 124
8.2 A Fully Occupied Band Cannot Carry Current 125
8.3 Holes 126
8.4 Insulators, Conductors, Semiconductors 127
8.5 Intrinsic and Doped Semiconductors 129
8.5.1 Donors and Acceptors 129
8.5.2 n-Doped and p-Doped Semiconductors 131
8.6 Charge-Carrier Statistics 131
8.6.1 Metals 132
8.6.2 Semiconductors 135
Nondegenerate Semiconductors – Law of Mass Action 136
Intrinsic, Nondegenerate Semiconductors 138
8.7 General Models of Bands for Cubic Semiconductors 138
8.7.1 Different Types of Effective Masses 141
Acceleration Effective Mass 141
Conductivity Effective Mass 141
Density-of-States Effective Mass 142
8.7.2 Herring–Vogt Transformation 143
8.7.3 Nonparabolicity 144
Part II Semiclassical Transport in Bulk Semiconductors 145
9 Electronic Interactions 146
9.1 Classification 146
9.2 Fundamentals of Scattering – Crystal-MomentumConservation 148
Overlap Integral 150
9.3 Electron–Phonon Scattering Rates – Deformation Potential 151
9.3.1 Electron Intravalley Scattering by Acoustic Phonons 152
Quasi Elasticity 152
A - Elastic, Energy Equipartition, Spherical and Parabolic Bands 154
B - Elastic, Energy Equipartition, Ellipsoidal, Nonparabolic Bands 155
C - Inelastic Acoustic Scattering, Spherical, Parabolic Bands 157
D - Inelastic Acoustic Scattering, Ellipsoidal, Nonparabolic Bands 158
9.3.2 Electron Intravalley Scattering by Optical Phonons 159
A - Spherical, Parabolic Bands 159
B - Ellipsoidal Nonparabolic Bands 160
9.3.3 Electron Intervalley Scattering 161
9.3.4 Hole Intraband Scattering by Acoustic Phonons 162
9.3.5 Hole Intraband Scattering by Optical Phonons 163
9.3.6 Hole Interband Scattering 163
9.4 Electron–Phonon Scattering Rates – Electrostatic Interaction 163
9.4.1 Acoustic Phonons – Piezoelectric Interaction 164
9.4.2 Optical Phonons – Polar Interaction 167
9.5 Selection Rules 171
9.6 Impurity Scattering 172
9.6.1 Ionized Impurities 172
9.6.2 Neutral Impurities 177
9.7 Alloy Scattering 177
9.8 Carrier–Carrier Interaction 178
9.9 Relative Importance of the Different Scattering Mechanisms 179
10 Boltzmann Equation 181
10.1 The Distribution Function 181
10.1.1 Mean Quantities 182
10.2 Elementary Derivation of the Boltzmann Equation 183
10.3 The Collision Integral – Detailed Balance 185
10.4 Moment Method 186
10.4.1 Zero-Order Moment: Continuity Equation 187
10.4.2 First-Order Moment 189
10.4.3 Drift-Diffusion Equation 191
10.4.4 Higher-Order Moments: Hydrodynamic Equations 194
10.5 Chambers' Integral Equation 195
10.5.1 Path Variables 195
10.5.2 Chambers' Integral Equation 197
11 Linear Transport 199
11.1 Linearization of Boltzmann Equation 199
11.2 Relaxation-Time Approximation 201
11.3 Linear Transport Properties in a ``Simple Semiconductor'' 202
11.3.1 Ohmic Mobility 202
A Simple, Intuitive Derivation 204
11.3.2 Matthiessen Rule 205
11.3.3 Magnetotransport 206
11.3.4 Hall Effect 210
Elementary Theory 211
Kinetic Theory 211
Case of Constant 212
Case of Weak Magnetic Field 213
Hall Mobility and Hall Factor 214
11.4 High-Magnetic-Field Effects 215
11.5 Evaluation of the Momentum Relaxation Times 216
11.5.1 Relaxation Time for Velocity-Randomizing Collisions 216
11.5.2 Relaxation Time for Elastic Collisions 217
11.6 Mobilities 219
11.6.1 Acoustic–Phonon Scattering, Deformation Potential,Elastic 219
11.6.2 Optical–Phonon Scattering, Deformation Potential 220
11.6.3 Ionized–Impurity Scattering 221
11.6.4 Acoustic–Phonon Scattering, Piezoelectric, Elastic 222
11.6.5 Optical–Phonon Scattering, Polar Interaction 224
12 Diffusion, Fluctuations, and Noise 225
12.1 Fick Laws 225
12.2 Einstein Relation 227
12.3 Drift-Diffusion Equation and the Gaussian Solution 228
12.4 Moments and Correlations 230
12.5 Spectral Density and Wiener–Kintchine Theorem 232
12.6 Nyquist Theorem 234
13 Nonlinear Transport 237
13.1 Hot Electrons 238
13.1.1 The Warm Electron Region 242
13.2 Electron–Electron Collisions and the Heatedand Drifted Maxwell Distribution 243
13.3 Anisotropy of Transport Coefficients 243
13.4 Negative Differential Mobility and Gunn Effect 245
13.5 High-Field Diffusivity 247
13.5.1 Intervalley Diffusion 248
13.6 Transient Transport 251
13.7 Hot Phonons 253
13.8 Ultrafast Spectroscopy 253
14 Monte Carlo Simulation of Bulk Electron Transport 255
14.1 The Monte Carlo Method 255
14.2 Direct Monte Carlo Simulation 257
14.2.1 A Typical Monte Carlo Program for Homogeneous, Stationary Transport 258
Definition of the Physical System 259
Initial Conditions of Motion 259
Free-Flight Duration – Self-Scattering 259
Choice of the Scattering Mechanism 261
Choice of the State After Scattering 261
Time Averages 262
Synchronous Ensemble 262
14.2.2 Time- and Space-Dependent Phenomena –Ensemble MC 263
Transients 264
Periodic Fields 264
Space-Dependent Phenomena 265
14.2.3 Diffusion 265
14.2.4 Ohmic Mobility 266
14.2.5 Electron–Electron Interaction and DegenerateStatistics 267
14.2.6 Impact Ionization 269
14.2.7 Variance-Reducing Techniques 269
14.2.8 Full-Band Monte Carlo 270
14.3 Formal Monte Carlo Solution of the BE – WeightedMonte Carlo 272
14.3.1 Monte Carlo Evaluation of Sums and Integrals 272
MC Evaluation of a Sum 272
Generalization to a Number of Sums 273
Generalization to Integrals 274
Finally, a Function Defined as an Infinite Sum of Multiple Integrals 275
14.3.2 The Integral Boltzmann Equation with Approximate Total Scattering Rate 275
14.3.3 The Neumann Expansion 276
14.3.4 Sampling 278
15 Bulk Transport Properties of Main Semiconductors 282
15.1 Electrons in Silicon 282
15.2 Holes in Silicon 289
15.3 Electrons in Gallium Arsenide 291
15.4 Holes in Gallium Arsenide 296
15.5 Organic Semiconductors 298
Part III Quantum Transport in Bulk Semiconductors 300
16 Quantum Transport in Homogeneous Systems 301
16.1 Introduction to Quantum Transport 301
16.1.1 Semiclassical Transport and Quantum Physics 301
16.1.2 From Reversible Dynamics to Irreversible Boltzmann Equation 302
16.1.3 Coherence, Dephasing, and Entanglement 305
16.1.4 When is Quantum Transport Necessary? 306
Characteristic Times 306
Electrons are not Classical Particles 308
16.2 The Density Matrix 309
Time Evolution of the Density Matrix 310
Density Matrix at Equilibrium 311
16.3 Reduced Density Matrix 312
16.4 Kubo Formula 313
Linearization of the Von Neumann Equation 314
A General Identity 315
Kubo Formula 315
The Classical Limit 317
16.5 The Path-Integral Approach 317
17 The Wigner-Function Approach to Quantum Transport 320
17.1 Introduction 320
17.2 Definition and Main Properties 321
17.2.1 Weyl–Wigner Transformation 321
17.2.2 Transformation Between the Matrix Elements of an Operator and Its Weyl–Wigner Transform 322
17.2.3 Definition of the Wigner Function 323
17.2.4 Main Properties 324
Average Physical Quantities 325
17.3 Coherent Evolution of the Wigner Function 326
Free-Electron Evolution 326
Scattering States 327
17.4 Dynamical Equations of the Wigner Function 327
Free Electrons 328
Electrons Subject to a Potential V(r) 328
17.4.1 Moyal Expansion 330
17.5 Electron–Phonon Interaction 331
17.6 Wigner Paths and MC Simulation 335
17.6.1 Integral Equation 335
17.6.2 Neumann Expansion and Wigner Paths 337
17.6.3 Monte Carlo Simulation 339
Path Multiplicity 341
Phonon Average 341
17.7 Two-Time Wigner Function 342
17.8 Many-Particle Wigner Function 343
Part IV Transport in Semiconductor Structures 345
18 Inhomogeneous and Open Systems: Electronic Devices 346
18.1 Inhomogeneous, Open Systems 346
18.2 Self-Averaging Transport, Coherent Transport,and Intermediate Cases 347
18.3 pn Junctions 349
18.3.1 pn Junction at Equilibrium 349
18.3.2 pn Diode 353
18.3.3 Solar Cells 357
18.3.4 Light-Emitting Diodes 358
18.4 The Bipolar Junction Transistor 360
18.5 Metal–Semiconductor Junctions, Schottky Barrier Diode 362
18.6 Field-Effect Transistors 364
JFET 364
MESFET 365
MOSFET 366
18.7 Device Simulation 368
18.7.1 Drift-Diffusion Models 369
Solution of the Differential Equations 371
18.7.2 Hydrodynamic Models 372
18.7.3 Monte Carlo Simulations 373
19 Low-Dimensional Structures 375
19.1 Epitaxial Heterostructures 375
Alloys 377
19.2 Quantum Wells 378
19.2.1 Electron States 378
Subbands and Density of States 379
Orthogonal States and Self Consistency 380
19.2.2 Transport 382
Phonon Scattering 383
Ionized-Impurity Scattering 385
Modulation Doping 386
Surface-Roughness Scattering 387
Hot-Electron Effects 387
19.2.3 Multiple Quantum Wells 388
19.3 Quantum Wires 388
Electron States 389
Transport 390
19.4 Quantum Dots 391
19.4.1 Transport: Coulomb Blockade 392
19.5 Superlattices 394
19.5.1 Minibands 394
19.5.2 Transport: Bloch Oscillations 396
19.5.3 Wannier–Stark Ladder 397
19.5.4 Negative Differential Conductivity 398
19.6 Applications 398
High-Electron-Mobility Transistor 399
Single-Electron Transistor 399
Quantum-Well Laser 399
Quantum Cascade Laser 400
20 Carbon Nanotubes 401
20.1 Introduction 401
20.2 Structure 401
Graphite 401
Graphene 402
Nanotubes 403
20.3 Electron States: Bands 405
20.4 Electron Transport 408
Contacts 408
Coherent Transport 409
Dissipative Transport 409
Electron–Electron Interaction 410
Doping and Impurity Scattering 411
Magnetic Fields 411
Multiwalled Carbon Nanotubes 411
Devices 412
21 Coherent Transport in Mesoscopic Structures 413
21.1 Landauer–Büttiker Theory of Transport 413
Scattering Matrix and Transmission Coefficients 414
Conductance Coefficients 417
21.2 Point Contacts 420
21.3 Quantum Hall Effect 421
The Experimental Effect 422
Edge States 423
Fermi Energy and Landau-Level Filling 424
Integer QHE from Landauer–Büttiker Theory 425
Effect of Impurities in the Plateaux 427
An Important Note 428
Fractional Quantum Hall Effect 428
21.4 Aharonov–Bohm Oscillations 429
21.5 Localization 432
21.6 Weak Localization – Quantum Corrections 435
21.7 Universal Conduction Fluctuations 436
21.8 Resonant Tunneling Diode 439
22 Semiconductor Photo Gallery 443
Part V Quantum Transport with Nonequilibrium Green Functions 451
23 Second-Quantization Formalism 452
23.1 Many-Particle Wavefunctions 452
23.1.1 Expansion in Symmetric Wavefunctions 454
23.2 Vector Space of Many-Particle States 455
23.2.1 Creation and Annihilation Operators 456
Bosonic Operators 457
Fermionic Operators 457
23.2.2 Field Operators 457
23.3 From First to Second Quantization 459
23.4 Dynamics 460
23.5 Commutations at Different Timesfor Non-Interacting Particles 461
23.6 Field Operators in Momentum and Energy Space 462
24 Introduction to Green Functions 464
24.1 GFs from Differential Equations to Many-Body Theory 464
24.1.1 GF of Schrödinger Equation 464
24.1.2 The Evolution Operator as Greenian 466
24.1.3 Green Functions for a One-Particle System 467
24.1.4 Single-Particle Green Functions in Many-Particle Systems 469
24.2 Green Functions in Momentum and Energy Space 471
24.3 Equilibrium GFs for NonInteracting Particles 472
Noninteracting particles at equilibrium 472
Free particles at equilibrium 473
The equations of motion 475
24.4 Green Functions and Mean Quantities 476
24.5 Spectral Density 477
24.5.1 Relation Between G< and G>
25 Wick–Matsubara Theorems 480
25.1 Time-Ordered Products, Normal Products, and Contractions 480
25.2 Wick Theorem 482
25.2.1 Lemma 482
X is an Annihilation Operator 482
X is a Creation and All the Others are Annihilations 482
X is a Creation Operator and the Others are Creation and Annihilation 483
25.2.2 Wick Theorem 484
25.3 Wick–Matsubara Theorem 485
26 Perturbation Expansion of Green Functions: Feynman Diagrams and Dyson Equation 488
26.1 The Interaction Picture 488
26.2 Contour Integration 490
26.3 Perturbation Expansion and Feynman Diagrams, Potential Interaction 492
26.3.1 Cancellation of Disconnected Diagrams 494
26.3.2 Term Multiplicity 495
Impurity Scattering 495
26.4 Particle–Particle Interaction 496
26.5 Electron–Phonon Interaction 497
26.6 Self-Energy and Dyson Equation 499
26.6.1 Matrix Formulation of G and of Dyson Equation 500
26.6.2 Dyson Equations for Separate GFs 502
26.6.3 (k,) Representation 504
26.7 Electron–Phonon Self-Energy 507
27 Nonequilibrium Green Functions Applied to Transport: Quantum Boltzmann Equation 508
27.1 The Equations for G< (r,t,r',t') and Gr(r,t,r',t')
27.2 The Equations for G< (R,T,k,) and Gr(R,T,k,)
27.3 Gradient-Expansion Approximation 513
27.4 Equations for Linear Response in Homogeneous Systemsin Steady State 519
28 Nonequilibrium Green Functions Applied to Transport: Mesoscopic Systems 523
28.1 GFs for the Time-Independent Schrödinger Equation 524
28.2 GF for a Perfect, Infinite, Two-Dimensional Wire 526
28.3 From Green Function to S Matrix 527
28.4 Finite-Difference Scheme for the Conductor GF 529
28.5 The Effect of the Leads 531
Self-Energy 531
The Coupling Hamiltonian 533
The Lead GF 533
28.6 Conductance 535
Scattering Mechanisms 536
Part VI Appendices 537
A Vector Spaces and Fourier Analysis 538
Hilbert Spaces 538
Linear Operators 539
Orthonormal Basis 541
Unitary Transformations 542
Eigenvalues and Eigenvectors 542
Observables 542
Square-Integrable Functions 543
Equation of Harmonic Motion 544
Fourier Series 544
The Continuum Spectrum and the Dirac Delta Function 545
Integral Representation of the and the Fourier Integral 547
B One-Dimensional Potential Step, Barrier, and Well 550
One-Dimensional Free Particle 550
Wavepackets, Phase and Group Velocities 551
Potential Step and Infinite Barrier 552
Potential Barrier: Tunnel Effect 555
Infinite Potential Well 556
Finite Potential Well: Resonances 557
C Quantum Theory of Harmonic Oscillator 558
The Hamiltonian 558
Creation and Annihilation Operators 558
Eigenvalues and Eigenvectors 559
The {q} Representation, Eigenfunctions 561
D Landau Levels 562
E Perturbation Theory 566
The General Idea 566
E.1 Time-Independent Perturbations 567
Perturbation Expansion 567
Representation of the Unperturbed Hamiltonian 567
Zero Order: No Degeneracy 567
First-Order Eigenvalues: No Degeneracy 568
First-Order Eigenvectors: No Degeneracy 568
Removal of the Degeneracy 569
E.2 Time-Dependent Perturbations 569
Harmonic Perturbations: Fermi Golden Rule 570
Discussion on the Validity of the Fermi Golden Rule and Collisional Broadening 573
References 574
Index 589