QUINN - YI

Lectures on Solid State Physics SPIN Springer’s internal project number, if known

– Monograph – February 29, 2008

Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo

Your dedication goes here

Preface

Please write your preface here Knoxville, August 2008

John J. Quinn Kyung-Soo Yi

Contents

Part I Basic Concepts in Solid State Physics 1

Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Crystal Structure and Symmetry Groups . . . . . . . . . . . . . . . . . . . 1.2 Common Crystal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Reciprocal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Diffraction of X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Bragg Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Laue Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Ewald Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Atomic Scattering Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Geometric Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.6 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Classification of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Crystal Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Binding Energy of Ionic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 10 16 17 17 18 20 21 22 23 25 25 27 34

2

LATTICE VIBRATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Monatomic Linear Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 M¨ossbauer Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Optical Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Lattice Vibrations in Three-Dimensions . . . . . . . . . . . . . . . . . . . . 2.5.1 Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Heat Capacity of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Einstein Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Modern Theory of the Specific Heat of Solids . . . . . . . . . 2.6.3 Debye Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 41 44 48 50 52 53 54 55 57 59

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2.6.4 Evaluation of Summations over Normal Modes for the Debye Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5 Estimate of Recoil Free Fraction in M¨ossbauer Effect . . 2.6.6 Lindemann Melting Formula . . . . . . . . . . . . . . . . . . . . . . . . 2.6.7 Critical Points in the Phonon Spectrum . . . . . . . . . . . . . . 2.7 Qualitative Description of Thermal Expansion . . . . . . . . . . . . . . 2.8 Anharmonic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Thermal Conductivity of an Insulator . . . . . . . . . . . . . . . . . . . . . . 2.10 Phonon Collision Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Phonon Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

61 63 63 64 67 68 70 72 73 75

FREE ELECTRON THEORY OF METALS . . . . . . . . . . . . . . . 77 3.1 Drude Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.4 Wiedemann–Franz Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.5 Criticisms of Drude Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.6 Lorentz Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.6.1 Boltzmann Distribution Function . . . . . . . . . . . . . . . . . . . . 81 3.6.2 Relaxation Time Approximation . . . . . . . . . . . . . . . . . . . . 81 3.6.3 Solution of Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . 81 3.6.4 Maxwell–Boltzmann Distribution . . . . . . . . . . . . . . . . . . . . 82 3.7 Sommerfeld Theory of Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.8 Review of Elementary Statistical Mechanics . . . . . . . . . . . . . . . . . 83 3.8.1 Fermi–Dirac Distribution Function . . . . . . . . . . . . . . . . . . 85 3.8.2 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.8.3 Thermodynamic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.8.4 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.9 Fermi Function Integration Formula . . . . . . . . . . . . . . . . . . . . . . . 89 3.10 Heat Capacity of a Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.11 Equation of State of a Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.12 Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.13 Electrical and Thermal Conductivities . . . . . . . . . . . . . . . . . . . . . 94 3.13.1 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.13.2 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.14 Critique of Sommerfeld Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.15 Magnetoconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.16 Hall Effect and Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.17 Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

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4

ELEMENTS OF BAND THEORY . . . . . . . . . . . . . . . . . . . . . . . . 105 4.1 Energy Band Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.2 Translation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.3 Bloch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.4 Calculation of Energy Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.4.1 Tight Binding Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.4.2 Tight Binding in Second Quantization Representation . . 111 4.5 Periodic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.6 Free Electron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.7 Nearly Free Electron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.7.1 Degenerate Perturbation Theory . . . . . . . . . . . . . . . . . . . . 116 4.8 Metals–Semimetals–Semiconductors–Insulators . . . . . . . . . . . . . . 117 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5

USE OF ELEMENTARY GROUP THEORY IN CALCULATING BAND STRUCTURE . . . . . . . . . . . . . . . . . . . 123 5.1 Band Representation of Empty Lattice States . . . . . . . . . . . . . . . 123 5.2 Review of Elementary Group Theory . . . . . . . . . . . . . . . . . . . . . . . 123 5.2.1 Some Examples of Simple Groups . . . . . . . . . . . . . . . . . . . 124 5.2.2 Group Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.2.3 Examples of Representations of the Group 4mm . . . . . . 126 5.2.4 Faithful Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.2.5 Regular Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.2.6 Reducible and Irreducible Representations . . . . . . . . . . . . 128 5.2.7 Important Theorems of Representation Theory (without proof) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.2.8 Character of a Representation . . . . . . . . . . . . . . . . . . . . . . . 129 5.2.9 Orthogonality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.3 Empty Lattice Bands, Degeneracies and IR’s at High Symmetry Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.3.1 Group of the Wave Vector k . . . . . . . . . . . . . . . . . . . . . . . . 132 5.4 Use of Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.4.1 Determining the Linear Combinations of Plane Waves Belonging to Different IR’s . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.4.2 Compatibility Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.5 Using the Irreducible Representations in Evaluating Energy Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.6 Empty Lattice Bands for Cubic Structure . . . . . . . . . . . . . . . . . . 142 5.6.1 Point Group of a Cubic Structure . . . . . . . . . . . . . . . . . . . 142 5.6.2 Face Centered Cubic Lattice . . . . . . . . . . . . . . . . . . . . . . . . 144 5.6.3 Body Centered Cubic Lattice . . . . . . . . . . . . . . . . . . . . . . . 148 5.7 Energy Bands of Common Semiconductors . . . . . . . . . . . . . . . . . . 149 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

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MORE BAND THEORY AND THE SEMICLASSICAL APPROXIMATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.1 Orthogonalized Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.2 Pseudopotential Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.3 k · p Method and Effective Mass Theory . . . . . . . . . . . . . . . . . . . . 158 6.4 Semiclassical Approximation for Bloch Electrons . . . . . . . . . . . . 162 6.4.1 Effective Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.4.2 Concept of a Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.4.3 Effective Hamiltonian of Bloch Electron . . . . . . . . . . . . . . 166 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7

SEMICONDUCTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.1 General Properties of Semiconducting Material . . . . . . . . . . . . . . 171 7.2 Typical Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.3 Temperature Dependence of the Carrier Concentration . . . . . . . 174 7.3.1 Carrier Concentration: Intrinsic Case . . . . . . . . . . . . . . . . 176 7.4 Donor and Acceptor Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 7.4.1 Population of Donor Levels . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.4.2 Thermal Equilibrium in a Doped Semiconductor . . . . . . 179 7.4.3 High Impurity Concentration . . . . . . . . . . . . . . . . . . . . . . . 180 7.5 p–n Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.5.1 Semiclassical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.5.2 Rectification of a p–n Junction . . . . . . . . . . . . . . . . . . . . . . 185 7.5.3 Tunnel Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.6 Surface Space Charge Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.6.1 Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 7.6.2 Quantum Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 7.6.3 Modulation Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.6.4 Minibands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 7.7 Electrons in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.7.1 Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.8 Amorphous Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 7.8.1 Types of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.8.2 Anderson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 7.8.3 Impurity Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 7.8.4 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

8

Dielectric Properties of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.1 Review of Some Ideas of Electricity and Magnetism . . . . . . . . . . 205 8.2 Dipole Moment Per Unit Volume . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.3 Atomic Polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.4 Local Field in a Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.5 Macroscopic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 8.5.1 Depolarization Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

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8.6 Lorentz Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.7 Clausius–Mossotti Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 8.8 Polarizability and Dielectric Functions of Some Simple Systems212 8.8.1 Evaluation of the Dipole Polarizability . . . . . . . . . . . . . . . 212 8.8.2 Polarizability of Bound Electrons . . . . . . . . . . . . . . . . . . . . 214 8.8.3 Dielectric Function of a Metal . . . . . . . . . . . . . . . . . . . . . . . 214 8.8.4 Dielectric Function of a Polar Crystal . . . . . . . . . . . . . . . . 215 8.9 Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 8.9.1 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8.10 Bulk Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8.10.1 Longitudinal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 8.10.2 Transverse Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 8.11 Reflectivity of a Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 8.11.1 Optical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 8.11.2 Skin Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.12 Surface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.12.1 Plasmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 8.12.2 Surface Phonon–Polariton . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 9

MAGNETISM IN SOLIDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 9.1 Review of Some Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . 235 9.1.1 Magnetic Moment and Torque . . . . . . . . . . . . . . . . . . . . . . 235 9.1.2 Vector Potential of a Magnetic Dipole . . . . . . . . . . . . . . . . 236 9.2 Magnetic Moment of an Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 9.2.1 Orbital Magnetic Moment . . . . . . . . . . . . . . . . . . . . . . . . . . 238 9.2.2 Spin Magnetic Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 9.2.3 Total Angular Momentum and Total Magnetic Moment 239 9.2.4 Hund’s Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 9.3 Paramagnetism and Diamagnetism of an Atom . . . . . . . . . . . . . . 240 9.4 Paramagnetism of Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 9.5 Pauli Spin Paramagnetism of Metals . . . . . . . . . . . . . . . . . . . . . . . 245 9.6 Diamagnetism of Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 9.7 de Haas–van Alphen Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 9.8 Cooling by Adiabatic Demagnetization of a Paramagnetic Salt 253 9.9 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

Part II Advanced Topics in Solid State Physics

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10 MAGNETIC ORDERING AND SPIN WAVES . . . . . . . . . . . 261 10.1 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 10.1.1 Heisenberg Exchange Interaction . . . . . . . . . . . . . . . . . . . . 261 10.1.2 Spontaneous Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . 263 10.1.3 Domain Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 10.1.4 Domain Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 10.1.5 Anisotropy Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 10.2 Antiferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 10.3 Ferrimagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 10.4 Zero-temperature Heisenberg Ferromagnet . . . . . . . . . . . . . . . . . . 269 10.5 Zero-temperature Heisenberg Antiferromagnet . . . . . . . . . . . . . . 272 10.6 Spin Waves in Ferromagnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 10.6.1 Holstein–Primakoff Transformation . . . . . . . . . . . . . . . . . . 273 10.6.2 Dispersion Relation for Magnons . . . . . . . . . . . . . . . . . . . . 277 10.6.3 Magnon–Magnon Interactions . . . . . . . . . . . . . . . . . . . . . . . 278 10.6.4 Magnon Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 10.6.5 Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 10.6.6 Experiments Revealing Magnons . . . . . . . . . . . . . . . . . . . . 280 10.6.7 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 10.7 Spin Waves in Antiferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 10.7.1 Ground State Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 10.7.2 Zero Point Sublattice Magnetization . . . . . . . . . . . . . . . . . 286 10.7.3 Finite Temperature Sublattice Magnetization . . . . . . . . . 287 10.7.4 Heat Capacity due to Antiferromagnetic Magnons . . . . . 288 10.8 Exchange Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 10.9 Itinerant Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 10.9.1 Stoner Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 10.9.2 Stoner Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 10.10Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 11 MANY BODY INTERACTIONS–Introduction . . . . . . . . . . . . 297 11.1 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 11.2 Hartree–Fock Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 11.2.1 Ferromagnetism of a degenerate electron gas in Hartree–Fock Approximation . . . . . . . . . . . . . . . . . . . . . . . 302 11.3 Spin Density Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 11.3.1 Comparison with Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 11.4 Correlation Effects–Divergence of Perturbation Theory . . . . . . . 311 11.5 Linear Response Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 11.5.1 Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 11.5.2 Properties of Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . 314 11.5.3 Change of Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 315 11.5.4 Equation of Motion of Density Matrix . . . . . . . . . . . . . . . 316 11.5.5 Single Particle Density Matrix of a Fermi Gas . . . . . . . . . 317

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11.5.6 Linear Response Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 11.5.7 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 11.6 Lindhard Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 11.6.1 Longitudinal Dielectric Constant . . . . . . . . . . . . . . . . . . . . 323 11.6.2 Kramers–Kronig Relation . . . . . . . . . . . . . . . . . . . . . . . . . . 327 11.7 Effect of Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 11.8 Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 11.8.1 Friedel Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 11.8.2 Kohn Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 12 MANY BODY INTERACTIONS–Green’s Function Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 12.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 12.1.1 Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 12.1.2 Interaction Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 343 12.2 Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 12.3 Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 12.3.1 Averages of Time Ordered Products of Operators . . . . . . 347 12.3.2 Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 12.3.3 Linked Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 12.4 Dyson’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 12.5 Green’s Function Approach to the Electron–Phonon Interaction352 12.6 Electron Self Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 12.7 Quasiparticle Interactions and Fermi Liquid Theory . . . . . . . . . . 360 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 13 SEMICLASSICAL THEORY OF ELECTRONS . . . . . . . . . . . 365 13.1 Bloch Electrons in a dc Magnetic Field . . . . . . . . . . . . . . . . . . . . . 365 13.1.1 Energy Levels of Bloch Electrons in a Magnetic Field . . 366 13.1.2 Quantization of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 13.1.3 Cyclotron Effective Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 13.1.4 Velocity Parallel to B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 13.2 Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 13.3 Two-Band Model and Magnetoresistance . . . . . . . . . . . . . . . . . . . 370 13.4 Magnetoconductivity of Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 13.4.1 Free Electron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 13.4.2 Propagation Parallel to B0 . . . . . . . . . . . . . . . . . . . . . . . . . 381 13.4.3 Propagation Perpendicular to B0 . . . . . . . . . . . . . . . . . . . . 382 13.4.4 Local vs. Nonlocal Conduction . . . . . . . . . . . . . . . . . . . . . . 382 13.5 Quantum Theory of Magnetoconductivity of an Electron Gas . 383 13.5.1 Propagation Perpendicular to B0 . . . . . . . . . . . . . . . . . . . . 385 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

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14 ELECTRODYNAMICS OF METALS . . . . . . . . . . . . . . . . . . . . . 389 14.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 14.2 Skin Effect in the Absence of a dc Magnetic Field . . . . . . . . . . . 390 14.3 Azbel–Kaner Cyclotron Resonance . . . . . . . . . . . . . . . . . . . . . . . . . 392 14.4 Azbel–Kaner Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 14.5 Magnetoplasma Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 14.6 Discussion of the Nonlocal Theory . . . . . . . . . . . . . . . . . . . . . . . . . 399 14.7 Cyclotron Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 14.8 Surface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 14.9 Magnetoplasma Surface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 14.10Propagation of Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 14.10.1Propagation Parallel to B0 . . . . . . . . . . . . . . . . . . . . . . . . . 409 14.10.2Helicon–Phonon Interaction . . . . . . . . . . . . . . . . . . . . . . . . 410 14.10.3Propagation Perpendicular to B0 . . . . . . . . . . . . . . . . . . . . 411 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 15 SUPERCONDUCTIVITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 15.1 Some Phenomenological Observations of Superconductors . . . . . 417 15.2 London Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 15.3 Microscopic Theory–An Introduction . . . . . . . . . . . . . . . . . . . . . . . 423 15.3.1 Electron–Phonon Interaction . . . . . . . . . . . . . . . . . . . . . . . . 424 15.3.2 Cooper Pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 15.4 The BCS Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 15.4.1 Bogoliubov–Valatin Transformation . . . . . . . . . . . . . . . . . . 429 15.4.2 Condensation Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 15.5 Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 15.6 Type I and Type II Superconductors . . . . . . . . . . . . . . . . . . . . . . . 435 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 16 The Fractional Quantum Hall Effect: the Paradigm for Strongly Interacting Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 16.1 Electrons Confined to a Two Dimensional Surface in a Perpendicular Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 16.2 Integral Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 16.3 Fractional Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 16.4 Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 16.5 Statistics of Identical Particles in Two Dimension . . . . . . . . . . . . 446 16.6 Chern–Simons Gauge Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 16.7 Composite Fermion Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 16.8 Fermi Liquid Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 16.9 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 16.10Angular Momentum Eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 16.11Correlations in Quantum Hall States . . . . . . . . . . . . . . . . . . . . . . . 460 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

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A

Operator Method for the Harmonic Oscillator Problem . . . . 465 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

B

Neutron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

Part I

Basic Concepts in Solid State Physics

1 Crystal Structure

1.1 Crystal Structure and Symmetry Groups Although everyone has an intuitive idea of what a solid is, we will consider (in this book) only materials with a well defined crystal structure. What we mean by a well defined crystal structure is an arrangement of atoms in a lattice such that the atomic arrangement looks absolutely identical when viewed from two different points that are separated by a lattice translation vector. A few definitions are useful: Lattice A lattice is an array of points obtained from three primitive translation vectors a1 , a2 , a3 . Any point on the lattice is given by n = n1 a1 + n2 a2 + n3 a3

(1.1)

Translation Vector Any pair of lattice points can be connected by a vector of the form Tn 1 n 2 n 3 = n1 a1 + n2 a2 + n3 a3 .

(1.2)

The set of translation vectors form a group called the translation group of the lattice. Group A set of operations that satisfies the following requirements is called a group: • • • •

The product (under group multiplication) of two elements of the group belongs to the group. The associative law holds for group multiplication. The identity element belongs to the group. Every element in the group has an inverse which belongs to the group.

4

1 Crystal Structure

{XZ

{WZ

{XW Fig. 1.1. Translation operations in a two-dimensional lattice

Translation Group The set of translations through any translation vector Tn1 n2 n3 forms a group. Group multiplication consists in simply performing the translation operations consecutively. For example, as is shown in Figure 1.1, we have T13 = T03 + T10 . For the simple translation group the operations commute, i.e., Tij Tkl = Tkl Tij for every pair of translation vectors. This property makes the group an Abelian group. Point Group There are other symmetry operations which leave the lattice unchanged. These are rotations, reflections, and the inversion operations. These operations form the point group of the lattice. As an example, consider the two-dimensional square lattice. (Figure 1.2.) The following operations (performed about any lattice point) leave the lattice unchanged.

y

yv{h{pvu

x

Fig. 1.2. The two-dimensional square lattice

1.1 Crystal Structure and Symmetry Groups

• • • • •

5

E : identity R1 , R3 : rotations by ±90o R2 : rotation by 180o mx , my : reflections about x-axis and y-axis, respectively m+ , m− : reflections about the lines x = ±y

The multiplication table for this point group is given in Table 1.1. The operTable 1.1. Multiplication table for the group 4mm. The first operations, such as m+ in R1 m+ = my , are listed in the first column, and the second operations, such as R1 in R1 m+ = my , are listed in the first row OPERATION E−1 = E R−1 1 = R3 R−1 2 = R2 R−1 3 = R1 m−1 x = mx m−1 y = my m−1 + = m+ m−1 − = m−

E E R3 R2 R1 mx my m+ m−

R1 R1 E R3 R2 m+ m− my mx

R2 R2 R1 E R3 my mx m− m+

R3 R3 R2 R1 E m− m+ mx my

mx mx m+ my m− E R2 R3 R1

my my m− mx m+ R2 E R1 R3

m+ m+ my m− mx R1 R3 E R2

m− m− mx m+ my R3 R1 R2 E

ations in the first column are the first operations, such as m+ in R1 m+ = my , and the operations listed in the first row are the second operations, such as R1 in R1 m+ = my . The multiplication table can be obtained as follows: •

label the corners of the square. (Figure 1.3) 2

1

3

4

Fig. 1.3. Identity operation on a two-dimensional square

•

operating with a symmetry operation simply reorders the labeling. For example, see Figure 1.4 for symmetry operations of m+ , R1 , and mx .

Therefore, R1 m+ = my . One can do exactly the same for all other products, for example, such as my R1 = m+ . It is also very useful to note what happens to a point (x, y) under the operations of the point group. (See Table 1.2.) Note that under every group operation x → ±x or ±y and y → ±y or ±x. The point group of the two-dimensional square lattice is called 4mm. The notation denotes the fact that it contains a four fold axis of rotation and two mirror planes (mx and my ); the m+ and m− planes are required by the existence of the other operations. Another simple example is the symmerty

6

1 Crystal Structure

Y

X

[

X

Z

[

Z

Y

X

X

Y

Z

Y

[

Z

Y

X

X

Y

Z

[

[

Z

P+ [

51

π 2

Py Fig. 1.4. Point symmetry operations on a two-dimensional square Table 1.2. Point group operations on a point (x, y) OPERATION E R1 R2 R3 mx my m+ m− x x y −x −y x −x y −y y y −x −y x −y y x −x

group of a two-dimensional rectangular lattice. (Figure 1.5) The symmetry

Fig. 1.5. The two-dimensional rectangular lattice

operations are E, R2 , mx , my , and the multiplication table is easily obtained from that of 4mm. This point group is called 2mm, and it is a subgroup of 4mm. Allowed Rotations Because of the requirement of translational invariance under operations of the translation group, the allowed rotations of the point group are restricted to certain angles. Consider a rotation through an angle φ about an axis through some lattice point.(Figure 1.6) If A and B are lattice points separated by a primitive translation a1 , then A0 (and B0 ) must be a lattice point obtained by a rotation through angle φ about B (or −φ about A). Since A0 and B0 are lattice points, the vector B0 A0 must be a translation vector. Therefore |B0 A0 | = pa1 ,

(1.3)

1.1 Crystal Structure and Symmetry Groups

7

hN

iN φ

−φ h

aX

i

Fig. 1.6. Allowed rotations of angle φ about an axis passing through some lattice points A and B consistent with translational symmetry

¡ ¢ where p is an integer. But |B0 A0 | = a1 + 2a1 sin φ − π2 = a1 − 2a1 cos φ. Solving for cos φ gives 1−p cos φ = . (1.4) 2 Because −1 ≤ cos φ ≤ 1, we see that p must have only the integral values -1, 0, 1, 2, 3. This gives for the possible values of φ listed in Table 1.3. Table 1.3. Allowed rotations of the point group p cos φ -1 1 0 12 1 0 2 − 12 3 -1

φ n (= |2π/φ|) 0 or 2π 1 ± 2π 6 6 ± 2π 4 4 ± 2π 3 3 ± 2π 2 2

Although only rotations of 60o , 90o , 120o , 180o , and 360o are consistent with translational symmetry, rotations through other angles are obtained in quasicrystals (e.g., five fold rotations). The subject of quasicrystals, which do not have translational symmetry under the operations of the translation group, is an interesting modern topic in solid state physics which we will not discuss in this book. Bravais Lattice If there is only one atom associated with each lattice point, the lattice is called Bravais lattice. If there is more than one atom associated with each lattice point, the lattice is called a lattice with a basis. One atom can be considered to be located at the lattice point. For a lattice with a basis it is necessary to give the locations (or basis vectors) for the additional atoms associated with the lattice point.