Condensed Matter in a Nutshell

Gerald D. Mahan is Distinguished Professor of Physics at Pennsylvania State University. His books include "Quantum Mechanics in a Nutshell" (Princeton) and "Many-Particle Physics".

"Don't skip the introduction. It will not only re-energize those synapses which remember the history of chemistry, geology, and crystal growth, but it also poses some apparently simple questions which reveal the thrust of modern material research--all in eight pages."--Bruce L. Dietrich, Planetarian

• Preface xiii Chapter 1: Introduction 1 1.1 1900-1910 1 1.2 Crystal Growth 2 1.3 Materials by Design 4 1.4 Artificial Structures 5 Chapter 2: Crystal Structures 9 2.1 Lattice Vectors 9 2.2 Reciprocal Lattice Vectors 11 2.3 Two Dimensions 13 2.4 Three Dimensions 15 2.5 Compounds 19 2.6 Measuring Crystal Structures 21 2.6.1 X-ray Scattering 22 2.6.2 Electron Scattering 23 2.6.3 Neutron Scattering 23 2.7 Structure Factor 25 2.8 EXAFS 26 2.9 Optical Lattices 28 Chapter 3: Emergy Bands 31 3.1 Bloch's Theorem 31 3.1.1 Floquet's Theorem 32 3.2 Nearly Free Electron Bands 36 3.2.1 Periodic Potentials 36 3.3 Tight-binding Bands 38 3.3.1 s-State Bands 38 3.3.2 p-State Bands 41 3.3.3 Wannier Functions 43 3.4 Semiconductor Energy Bands 44 3.4.1 What Is a Semiconductor? 44 3.4.2 Si, Ge, GaAs 47 3.4.3 HgTe and CdTe 50 3.4.4 k * p Theory 51 3.4.5 Electron Velocity 55 3.5 Density of States 55 3.5.1 Dynamical Mean Field Theory 58 3.6 Pseudopotentials 60 3.7 Measurement of Energy Bands 62 3.7.1 Cyclotron Resonance 62 3.7.2 Synchrotron Band Mapping 63 Chapter 4: Insulators 68 4.1 Rare Gas Solids 68 4.2 Ionic Crystals 69 4.2.1 Madelung energy 71 4.2.2 Polarization Interactions 72 4.2.3 Van der Waals Interaction 75 4.2.4 Ionic Radii 75 4.2.5 Repulsive Energy 76 4.2.6 Phonons 77 4.3 Dielectric Screening 78 4.3.1 Dielectric Function 78 4.3.2 Polarizabilities 80 4.4 Ferroelectrics 82 4.4.1 Microscopic Theory 83 4.4.2 Thermodynamics 87 4.4.3 SrTiO3 89 4.4.4 BaTiO3 91 Chapter 5: Free Electron Metals 94 5.1 Introduction 94 5.2 Free Electrons 96 5.2.1 Electron Density 96 5.2.2 Density of States 97 5.2.3 Nonzero Temperatures 98 5.2.4 Two Dimensions 101 5.2.5 Fermi Surfaces 102 5.2.6 Thermionic Emission 104 5.3 Magnetic Fields 105 5.3.1 Integer Quantum Hall Effect 107 5.3.2 Fractional Quantum Hall Effect 110 5.3.3 Composite Fermions 113 5.3.4 deHaas-van Alphen Effect 113 5.4 Quantization of Orbits 117 5.4.1 Cyclotron Resonance 119 Chapter 6: Electron-Electron Interactions 127 6.1 Second Quantization 128 6.1.1 Tight-binding Models 131 6.1.2 Nearly Free Electrons 131 6.1.3 Hartree Energy: Wigner-Seitz 134 6.1.4 Exchange Energy 136 6.1.5 Compressibility 138 6.2 Density Operator 141 6.2.1 Two Theorems 142 6.2.2 Equations of Motion 143 6.2.3 Plasma Oscillations 144 6.2.4 Exchange Hole 146 6.3 Density Functional Theory 148 6.3.1 Functional Derivatives 149 6.3.2 Kinetic Energy 150 6.3.3 Kohn-Sham Equations 151 6.3.4 Exchange and Correlation 152 6.3.5 Application to Atoms 154 6.3.6 Time-dependent Local Density Approximation 155 6.3.7 TDLDA in Solids 157 6.4 Dielectric Function 158 6.4.1 Random Phase Approximation 159 6.4.2 Properties of P (q, w) 161 6.4.3 Hubbard-Singwi Dielectric Functions 164 6.5 Impurities in Metals 165 6.5.1 Friedel Analysis 166 6.5.2 RKKY Interaction 170 Chapter 7: Phonons 176 7.1 Phonon Dispersion 176 7.1.1 Spring Constants 177 7.1.2 Example: Square Lattice 179 7.1.3 Polar Crystals 181 7.1.4 Phonons 181 7.1.5 Dielectric Function 185 7.2 Phonon Operators 187 7.2.1 Simple Harmonic Oscillator 187 7.2.2 Phonons in One Dimension 189 7.2.3 Binary Chain 192 7.3 Phonon Density of States 195 7.3.1 Phonon Heat Capacity 197 7.3.2 Isotopes 199 7.4 Local Modes 203 7.5 Elasticity 205 7.5.1 Stress and Strain 205 7.5.2 Isotropic Materials 208 7.5.3 Boundary Conditions 210 7.5.4 Defect Interactions 211 7.5.5 Piezoelectricity 214 7.5.6 Phonon Focusing 215 7.6 Thermal Expansion 216 7.7 Debye-Waller Factor 217 7.8 Solitons 220 7.8.1 Solitary Waves 220 7.8.2 Cnoidal Functions 222 7.8.3 Periodic Solutions 223 Chapter 8: Boson Systems 230 8.1 Second Quantization 230 8.2 Superfluidity 232 8.2.1 Bose-Einstein Condensation 232 8.2.2 Bogoliubov Theory of Superfluidity 234 8.2.3 Off-diagonal Long-range Order 240 8.3 Spin Waves 244 8.3.1 Jordan-Wigner Transformation 245 8.3.2 Holstein-Primakoff Transformation 247 8.3.3 Heisenberg Model 248 Chapter 9: Electron-Phonon Interactions 254 9.1 Semiconductors and Insulators 254 9.1.1 Deformation Potentials 255 9.1.2 Frohlich Interaction 257 9.1.3 Piezoelectric Interaction 258 9.1.4 Tight-binding Models 259 9.1.5 Electron Self-energies 260 9.2 Electron-Phonon Interaction in Metals 263 9.2.1 ? 264 9.2.2 Phonon Frequencies 267 9.2.3 Electron-Phonon Mass Enhancement 268 9.3 Peierls Transition 272 9.4 Phonon-mediated Interactions 276 9.4.1 Fixed Electrons 276 9.4.2 Dynamical Phonon Exchange 278 9.5 Electron-Phonon Effects at Defects 281 9.5.1 F-Centers 281 9.5.2 Jahn-Teller Effect 284 Chapter 10: Extrinsic Semiconductors 287 10.1 Introduction 287 10.1.1 Impurities and Defects in Silicon 288 10.1.2 Donors 289 10.1.3 Statistical Mechanics of Defects 292 10.1.4 n-p Product 294 10.1.5 Chemical Potential 295 10.1.6 Schottky Barriers 297 10.2 Localization 301 10.2.1 Mott Localization 301 10.2.2 Anderson Localization 304 10.2.3 Weak Localization 304 10.2.4 Percolation 306 10.3 Variable Range Hopping 310 10.4 Mobility Edge 311 10.5 Band Gap Narrowing 312 Chapter 11: Transport Phenomena 320 11.1 Introduction 320 11.2 Drude Theory 321 11.3 Bloch Oscillations 322 11.4 Boltzmann Equation 324 11.5 Currents 327 11.5.1 Transport Coefficients 327 11.5.2 Metals 329 11.5.3 Semiconductors and Insulators 333 11.6 Impurity Scattering 335 11.6.1 Screened Impurity Scattering 336 11.6.2 T-matrix Description 337 11.6.3 Mooij Correlation 338 11.7 Electron-Phonon Interaction 340 11.7.1 Lifetime 341 11.7.2 Semiconductors 343 11.7.3 Saturation Velocity 344 11.7.4 Metals 347 11.7.5 Temperature Relaxation 348 11.8 Ballistic Transport 350 11.9 Carrier Drag 353 11.10 Electron Tunneling 355 11.10.1 Giaever Tunneling 356 11.10.2 Esaki Diode 358 11.10.3 Schottky Barrier Tunneling 361 11.10.4 Effective Mass Matching 362 11.11 Phonon Transport 364 11.11.1 Transport in Three Dimensions 364 11.11.2 Minimum Thermal Conductivity 365 11.11.3 Kapitza Resistance 366 11.11.4 Measuring Thermal Conductivity 368 11.12 Thermoelectric Devices 370 11.12.1 Maximum Cooling 371 11.12.2 Refrigerator 373 11.12.3 Power Generation 374 Chapter 12: Optical Properties 379 12.1 Introduction 379 12.1.1 Optical Functions 379 12.1.2 Kramers-Kronig Analysis 381 12.2 Simple Metals 383 12.2.1 Drude 383 12.3 Force-Force Correlations 385 12.3.1 Impurity Scattering 386 12.3.2 Interband Scattering 388 12.4 Optical Absorption 389 12.4.1 Interband Transitions in Insulators 389 12.4.2 Wannier Excitons 392 12.4.3 Frenkel Excitons 395 12.5 X-Ray Edge Singularity 396 12.6 Photoemission 399 12.7 Conducting Polymers 401 12.8 Polaritons 404 12.8.1 Phonon Polaritons 404 12.8.2 Plasmon Polaritons 405 12.9 Surface Polaritons 406 12.9.1 Surface Plasmons 408 12.9.2 Surface Optical Phonons 410 12.9.3 Surface Charge Density 413 Chapter 13: Magnetism 418 13.1 Introduction 418 13.2 Simple Magnets 418 13.2.1 Atomic Magnets 418 13.2.2 Hund's Rules 418 13.2.3 Curie's Law 420 13.2.4 Ferromagnetism 422 13.2.5 Antiferromagnetism 423 13.3 3d Metals 424 13.4 Theories of Magnetism 425 13.4.1 Ising and Heisenberg Models 425 13.4.2 Mean Field Theory 427 13.4.3 Landau Theory 431 13.4.4 Critical Phenomena 433 13.5 Magnetic Susceptibility 434 13.6 Ising Model 436 13.6.1 One Dimension 436 13.6.2 Two and Three Dimensions 437 13.6.3 Bethe Lattice 439 13.6.4 Order-Disorder Transitions 443 13.6.5 Lattice Gas 445 13.7 Topological Phase Transitions 446 13.7.1 Vortices 447 13.7.2 XY-Model 448 13.8 Kondo Effect 452 13.8.1 sd-Interaction 453 13.8.2 Spin-flip Scattering 454 13.8.3 Kondo Resonance 456 13.9 Hubbard Model 458 13.9.1 U = 0 Solution 459 13.9.2 Atomic Limit 460 13.9.3 U > 0 460 13.9.4 Half-filling 462 Chapter 14: Superconductivity 467 14.1 Discovery of Superconductivity 467 14.1.1 Zero resistance 467 14.1.2 Meissner Effect 468 14.1.3 Three Eras of Superconductivity 469 14.2 Theories of Superconductivity 473 14.2.1 London Equation 473 14.2.2 Ginzburg-Landau Theory 475 14.2.3 Type II 478 14.3 BCS Theory 479 14.3.1 History of Theory 479 14.3.2 Effective Hamiltonian 480 14.3.3 Pairing States 481 14.3.4 Gap Equation 483 14.3.5 d-Wave Energy Gaps 486 14.3.6 Density of States 487 14.3.7 Ultrasonic Attenuation 489 14.3.8 Meissner Effect 490 14.4 Electron Tunneling 492 14.4.1 Normal-Superconductor 494 14.4.2 Superconductor-Superconductor 497 14.4.3 Josephson Tunneling 498 14.4.4 Andreev Tunneling 501 14.4.5 Corner Junctions 502 14.5 Cuprate Superconductors 503 14.5.1 Muon Rotation 503 14.5.2 Magnetic Oscillations 506 14.6 Flux Quantization 507 Chapter 15: Nanometer Physics 511 15.1 Quantum Wells 512 15.1.1 Lattice Matching 512 15.1.2 Electron States 513 15.1.3 Excitons and Donors in Quantum Wells 515 15.1.4 Modulation Doping 518 15.1.5 Electron Mobility 520 15.2 Graphene 520 15.2.1 Structure 521 15.2.2 Electron Energy Bands 522 15.2.3 Eigenvectors 525 15.2.4 Landau Levels 525 15.2.5 Electron-Phonon Interaction 526 15.2.6 Phonons 528 15.3 Carbon Nanotubes 530 15.3.1 Chirality 530 15.3.2 Electronic States 531 15.3.3 Phonons in Carbon Nanotubes 536 15.3.4 Electrical Resistivity 537 Appendix 541 Index 553