Introduction To Modern Solid State Physics - Y. Galperin.pdf
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Introduction to Modern
Solid State Physics
Yuri M. Galperin
FYS 448
Department of Physics, P.O. Box 1048 Blindern, 0316 Oslo,
Room 427A
Phone: +47 22 85 64 95,
E-mail: iouri.galperinefys.uio.no
Contents
I Basic concepts
1
1 Geometry of Lattices ... 3
1.1 Periodicity: Crystal Structures . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The Reciprocal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 X-Ray Diraction in Periodic Structures . . . . . . . . . . . . . . . . . . . 10
1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Lattice Vibrations: Phonons 21
2.1 Interactions Between Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Lattice Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Quantum Mechanics of Atomic Vibrations . . . . . . . . . . . . . . . . . . 38
2.4 Phonon Dispersion Measurement . . . . . . . . . . . . . . . . . . . . . . . 43
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Electrons in a Lattice. 45
3.1 Electron in a Periodic Field . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.1 Electron in a Periodic Potential . . . . . . . . . . . . . . . . . . . . 46
3.2 Tight Binding Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 The Model of Near Free Electrons . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Main Properties of Bloch Electrons . . . . . . . . . . . . . . . . . . . . . . 52
3.4.1 Eective Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.2 Wannier Theorem ! Eective Mass Approach . . . . . . . . . . . . 53
3.5 Electron Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5.1 Electric current in a Bloch State. Concept of Holes. . . . . . . . . . 54
3.6 Classication of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.7 Dynamics of Bloch Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.7.1 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.7.2 Quantum Mechanics of Bloch Electron . . . . . . . . . . . . . . . . 63
3.8 Second Quantization of Bosons and Electrons . . . . . . . . . . . . . . . . 65
3.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
i
ii
CONTENTS
II Normal metals and semiconductors
69
4 Statistics and Thermodynamics ... 71
4.1 Specic Heat of Crystal Lattice . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Statistics of Electrons in Solids . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Specic Heat of the Electron System . . . . . . . . . . . . . . . . . . . . . 80
4.4 Magnetic Properties of Electron Gas. . . . . . . . . . . . . . . . . . . . . . 81
4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Summary of basic concepts
93
6 Classical dc Transport ... 97
6.1 The Boltzmann Equation for Electrons . . . . . . . . . . . . . . . . . . . . 97
6.2 Conductivity and Thermoelectric Phenomena. . . . . . . . . . . . . . . . . 101
6.3 Energy Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4 Neutral and Ionized Impurities . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.5 Electron-Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.6 Scattering by Lattice Vibrations . . . . . . . . . . . . . . . . . . . . . . . . 114
6.7 Electron-Phonon Interaction in Semiconductors . . . . . . . . . . . . . . . 125
6.8 Galvano- and Thermomagnetic .. . . . . . . . . . . . . . . . . . . . . . . . 130
6.9 Shubnikov-de Haas eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.10 Response to \slow" perturbations . . . . . . . . . . . . . . . . . . . . . . . 142
6.11 \Hot" electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.12 Impact ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.13 Few Words About Phonon Kinetics. . . . . . . . . . . . . . . . . . . . . . . 150
6.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7 Electrodynamics of Metals 155
7.1 Skin Eect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.2 Cyclotron Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.3 Time and Spatial Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.4 ... Waves in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8 Acoustical Properties... 171
8.1 Landau Attenuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.2 Geometric Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.3 Giant Quantum Oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.4 Acoustical properties of semicondictors . . . . . . . . . . . . . . . . . . . . 175
8.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
CONTENTS
iii
9 Optical Properties of Semiconductors 181
9.1 Preliminary discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
9.2 Photon-Material Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.3 Microscopic single-electron theory . . . . . . . . . . . . . . . . . . . . . . . 189
9.4 Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
9.5 Intraband Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
9.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9.7 Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9.7.1 Excitonic states in semiconductors . . . . . . . . . . . . . . . . . . 203
9.7.2 Excitonic eects in optical properties . . . . . . . . . . . . . . . . . 205
9.7.3 Excitonic states in quantum wells . . . . . . . . . . . . . . . . . . . 206
10 Doped semiconductors 211
10.1 Impurity states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
10.2 Localization of electronic states . . . . . . . . . . . . . . . . . . . . . . . . 215
10.3 Impurity band for lightly doped semiconductors. . . . . . . . . . . . . . . . 219
10.4 AC conductance due to localized states . . . . . . . . . . . . . . . . . . . . 225
10.5 Interband light absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
III Basics of quantum transport
237
11 Preliminary Concepts 239
11.1 Two-Dimensional Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . 239
11.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
11.3 Degenerate and non-degenerate electron gas . . . . . . . . . . . . . . . . . 250
11.4 Relevant length scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
12 Ballistic Transport 255
12.1 Landauer formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
12.2 Application of Landauer formula . . . . . . . . . . . . . . . . . . . . . . . 260
12.3 Additional aspects of ballistic transport . . . . . . . . . . . . . . . . . . . . 265
12.4 ee interaction in ballistic systems . . . . . . . . . . . . . . . . . . . . . . 266
13 Tunneling and Coulomb blockage 273
13.1 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
13.2 Coulomb blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
14 Quantum Hall Eect 285
14.1 Ordinary Hall eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
14.2 Integer Quantum Hall eect - General Picture . . . . . . . . . . . . . . . . 285
14.3 Edge Channels and Adiabatic Transport . . . . . . . . . . . . . . . . . . . 289
14.4 Fractional Quantum Hall Eect . . . . . . . . . . . . . . . . . . . . . . . . 294
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