Condensed Matter Physics

Condensed Matter Physics Michael P. Marder University of Texas at Austin A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Singapore Toronto

v Preface References xix xxii I ATOMICSTRUCTURE 1 1 The Idea of Crystals 3 1.1 Introduction 3 1.1.1 Why are Solids Crystalline? 4 1.2 Two-Dimensional Lattices 6 1.2.1 Bravais Lattices 6 1.2.2 Enumeration of Two-Dimensional Bravais Lattices... 7 1.2.3 Lattices with Bases 7 1.2.4 Primitive Cells 9 1.2.5 Wigner-Seitz Cells 10 1.3 Symmetries 11 1.3.1 The Space Group 11 1.3.2 Translation and Point Groups 11 Problems 13 References 15 2 Three-Dimensional Lattices 17 2.1 Introduction 17 2.1.1 Distribution Among Elements 17 2.2 Monatomic Lattices 20 2.2.1 The Simple Cubic Lattice 20 2.2.2 The Face-Centered Cubic Lattice 20 2.2.3 The Body-Centered Cubic Lattice 21 2.2.4 The Hexagonal Lattice 22 2.2.5 The Hexagonal Close-Packed Lattice 23 2.2.6 The Diamond Lattice 24 2.3 Compounds 24 2.3.1 Rocksalt Sodium Chloride 25 2.3.2 Cesium Chloride 26 2.3.3 Fluorite Calcium Fluoride 26

2.3.4 Zincblende Zinc Sulfide 26 2.3.5 Wurtzite Zinc Oxide 28 2.3.6 Perovskite Calcium Titanate 28 2.4 Classification of Lattices by Symmetry 28 2.4.1 Fourteen Bravais Lattices and Seven Crystal Systems... 30 2.5 Symmetries of Lattices with Bases 32 2.5.1 Thirty-Two Crystallographic Point Groups 32 2.5.2 Two Hundred Thirty Distinct Lattices 36 2.6 Some Macroscopic Implications of Microscopic Symmetries... 37 2.6.1 Pyroelectricity 37 2.6.2 Piezoelectricity 37 2.6.3 Optical Activity 38 Problems 38 References 41 Experimental Determination of Crystal Structures 43 3.1 Introduction 43 3.2 Theory of Scattering from Crystals 44 3.2.1 Lattice Sums 47 3.2.2 Reciprocal Lattice 48 3.2.3 Miller Indices 51 3.2.4 Scattering from a Lattice with a Basis 52 3.3 Experimental Methods 54 3.3.1 LaueMethod 55 3.3.2 Rotating Crystal Method 56 3.3.3 Powder Method 58 3.4 Further Features of Scattering Experiments 59 3.4.1 Interaction of X-Rays with Matter 60 3.4.2 Productionof X-Rays 60 3.4.3 Neutrons 61 3.4.4 Electrons 61 3.4.5 Deciphering Complex Structures 63 3.4.6 Accuracy of Structure Determinations 64 Problems 65 References 67 Surfaces and Interfaces 69 4.1 Introduction 69 4.2 Geometry of Interfaces 69 4.2.1 Coherent and Commensurate Interfaces 70 4.2.2 Stacking Period and Interplanar Spacing 71 4.2.3 Other Topics in Surface Structure 73 4.3 Experimental Observation and Creation of Surfaces 73 4.3.1 Low-Energy Electron Diffraction (LEED) 74

vii 4.3.2 Reflection High-Energy Electron Diffraction (RHEED).. 75 4.3.3 Molecular Beam Epitaxy (MBE) 76 4.3.4 Field Ion Microscopy (FIM) 77 4.3.5 Scanning Tunneling Microscopy (STM) 77 4.3.6 Atomic Force Microscopy (AFM) 82 4.3.7 High Resolution Electron Microscopy (HREM) 82 Problems 82 References 85 5 Complex Structures 87 5.1 Introduction 87 5.2 Alloys 87 5.2.1 Equilibrium Structures 87 5.2.2 Phase Diagrams 89 5.2.3 Superlattices 90 5.2.4 Phase Separation 91 5.2.5 Nonequilibrium Structures in Alloys 94 5.2.6 Dynamics of Phase Separation 95 5.3 Simulations 97 5.3.1 Monte Carlo 97 5.3.2 Molecular Dynamics 98 5.4 Liquids 99 5.4.1 Correlation Functions 99 5.4.2 Extended X-Ray Absorption Fine Structure (EXAFS)... 101 5.4.3 Calculating Correlation Functions 103 5.5 Glasses 103 5.6 Liquid Crystals 107 5.6.1 Nematics, Cholesterics, and Smectics 108 5.6.2 Liquid Crystal Order Parameter 109 5.7 Polymers 110 5.7.1 Ideal Radius of Gyration 111 5.8 Quasicrystals 115 5.8.1 One-Dimensional Quasicrystal 116 5.8.2 Two-Dimensional Quasicrystals Penrose Tiles 121 5.8.3 Experimental Observations 124 5.8.4 Fullerenes 124 Problems 125 References 129 II ELECTRONIC STRUCTURE 133 6 The Single-Electron Model 135 6.1 Introduction 135

viii 6.2 The Basic Hamiltonian 137 6.3 Densities of States 139 6.3.1 Definition of Density of States D 140 6.3.2 Results for Free Electrons 141 6.4 Statistical Mechanics of Noninteracting Electrons 143 6.5 Sommerfeld Expansion 146 6.5.1 Specific Heat of Noninteracting Electrons at Low Temperatures 149 Problems 150 References 153 7 The Schrödinger Equation and Symmetry 155 7.1 Introduction 155 7.2 Translational Symmetry Bloch's Theorem 155 7.2.1 Van Hove Singularities 160 7.2.2 Fourier Analysis of Bloch's Theorem 163 7.2.3 Kronig-Penney Model 166 7.3 Rotational Symmetry Group Representations 169 7.3.1 Classes and Characters 175 7.3.2 Consequences of point group symmetries for Schrödinger's equation 178 Problems 181 References 184 8 Nearly Free and Tightly Bound Electrons 185 8.1 Introduction 185 8.2 Nearly Free Electrons 185 8.2.1 Degenerate Perturbation Theory 187 8.3 Brillouin Zones 189 8.3.1 Nearly Free Electron Fermi Surfaces 191 8.4 Tightly Bound Electrons 194 8.4.1 Wannier Functions 194 8.4.2 Tight Binding Model - 197 Problems 199 References 202 9 Electron-Electron Interactions 203 9.1 Introduction 203 9.2 Hartree and Hartree-Fock Equations 204 9.2.1 Variational Principle 205 9.2.2 Hartree-Fock Equations 205 9.2.3 Numerical Implementation 209 9.2.4 Hartree-Fock Equations for Jellium 212 9.3 Density Functional Theory 214

ix 9.3.1 Thomas-Fermi Theory 216 9.3.2 Kohn-Sham Equations 218 9.4 Stabilityof Matter 220 Problems 223 References 226 10 CalculationofBandStructures 229 10.1 Introduction 229 10.2 Numerical Methods 230 10.2.1 Pseudopotentials and Orthogonalized Planes Waves (OPW) 230 10.2.2 Linear Combination of Atomic Orbitals (LCAO) 235 10.2.3 Plane Waves 237 10.2.4 Linear Augmented Plane Waves (LAPW) 240 10.2.5 Linearized Muffin Tin Orbitals (LMTO) 243 10.3 Definition of Metals, Insulators, and Semiconductors 246 10.4 Brief Surveyofthe Periodic Table 248 10.4.1 Noble Gases 248 10.4.2 Nearly Free Electron Metals 250 10.4.3 Semiconductors 252 10.4.4 Transition Metals 252 10.4.5 RareEarths 252 Problems 254 References 258 III MECHANICAL PROPERTIES 261 11 CohesionofSolids 263 11.1 Introduction 263 11.1.1 Radiiof Atoms 263 11.2 Noble Gases 265 11.3 Ionic Crystals 269 11.3.1 Ewald Sums 270 11.4 Metals 272 11.4.1 Use of Pseudopotentials 275 11.5 Band Structure Energy 276 11.5.1 Peierls Distortion 277 11.5.2 Structural Phase Transitions 279 11.6 Hydrogen-Bonded Solids 280 11.7 Cohesive Energy from Band Calculations 280 11.8 Classical Potentials 282 Problems 283 References 285

x 12 Elasticity 287 12.1 Introduction 287 12.2 General Theory of Linear Elasticity 287 12.2.1 SolidsofCubicSymmetry 289 12.2.2 Isotropie Solids 290 12.3 Other Constitutive Laws 295 12.3.1 Liquid Crystals 295 12.3.2 Rubber 298 12.3.3 Composite and Granulär Materials 301 Problems 301 References 303 13 Phonons 305 13.1 Introduction 305 13.2 Vibrations of a Classical Lattice 305 13.2.1 Normal Modes 307 13.2.2 Lattice with a Basis 309 13.3 Vibrations of a Quantum-Mechanical Lattice 313 13.3.1 Phonon Specific Heat 317 13.3.2 Einstein and Debye Models 321 13.3.3 Thermal Expansion 324 13.4 Inelastic Scattering from Phonons 326 13.4.1 Neutron Scattering 327 13.4.2 Formal Theory of Neutron Scattering 329 13.4.3 Averaging Exponentials 333 13.4.4 Evaluation of Structure Factor 335 13.4.5 Kohn Anomalies 336 13.5 The Mössbauer Effect 336 Problems 339 References 340 14 Dislocations and Cracks 343 14.1 Introduction 343 14.2 Dislocations 345 14.2.1 Experimental Observations of Dislocations 347 14.2.2 Force to Move a Dislocation 350 14.2.3 One-Dimensional Dislocations: Frenkel-Kontorova Model 350 14.3 Two-Dimensional Dislocations and Hexatic Phases 353 14.3.1 Impossibility of Crystalline Order in Two Dimensions.. 353 14.3.2 Orientational Order 355 14.3.3 Kosterlitz-Thouless-Berezinskii Transition 356 14.4 Cracks 363 14.4.1 Fracture of a Strip 363 14.4.2 Stresses Around an Elliptical Hole 366

XI 14.4.3 Stress Intensity Factor 368 14.4.4 Atomic Aspects of Fracture 368 Problems 370 References 373 15 Fluid Mechanics 375 15.1 Introduction 375 15.2 Newtonian Fluids 375 15.2.1 Euler's Equation 375 15.2.2 Navier-Stokes Equation 377 15.3 Polymerie Solutions 378 15.4 Plasticity 385 15.5 Superfluid 4 He 389 15.5.1 Two-Fluid Hydrodynamics 392 15.5.2 Second Sound 393 15.5.3 Origin of Superfluidity 395 15.5.4 Lagrangian Theory of Wave Function 400 15.5.5 Superfluid 3 He 403 Problems 404 References 408 IV ELECTRON TRANSPORT 411 16 Dynamics of Bloch Electrons 413 16.1 Introduction 413 16.1.1 Drude Model 413 16.2 Semiclassical Electron Dynamics 415 16.2.1 Bloch Oscillations 416 16.2.2 PMethod 417 16.2.3 Effective Mass 419 16.3 Noninteracting Electrons in an Electric Field 419 16.3.1 Zener Tunneling 422 16.4 Semiclassical Equations from Wave Packets 425 16.4.1 Formal Dynamics of Wave Packets 425 16.5 Quantizing Semiclassical Dynamics 430 16.5.1 Wannier-Stark Ladders 432 16.5.2 de Haas-van Alphen Effect 432 16.5.3 Experimental Measurements of Fermi Surfaces 434 Problems 437 References 440

xii 17 Transport Phenomena and Fermi Liquid Theory 443 17.1 Introduction 443 17.2 Boltzmann Equation 443 17.2.1 Boltzmann Equation 445 17.2.2 Relaxation Time Approximation 446 17.2.3 Relation to Rate of Production of Entropy 448 17.3 Transport Symmetries 449 17.3.1 Onsager Relations. 450 17.4 Thermoelectric Phenomena 451 17.4.1 Electrical Current 451 17.4.2 Effective Mass and Holes 453 17.4.3 Mixed Thermal and Electrical Gradients 454 17.4.4 Wiedemann-Franz Law 455 17.4.5 Thermopower Seebeck Effect 456 17.4.6 Peltier Effect 457 17.4.7 Thomson Effect 457 17.4.8 Hall Effect 459 17.4.9 Magnetoresistance 461 17.4.10 Giant Magnetoresistance 462 17.5 Fermi Liquid Theory 462 17.5.1 Basic Ideas 462 17.5.2 Statistical Mechanics of Quasi-Particles 464 17.5.3 Effective Mass 466 17.5.4 Specific Heat 468 17.5.5 Fermi Liquid Parameters 469 17.5.6 Traveling Waves 470 17.5.7 Comparison with Experiment in 3 He 473 Problems 474 References 478 18 Microscopic Theories of Conduction 481 18.1 Introduction 481 18.2 Weak Scattering Theory of Conductivity 481 18.2.1 General Formula for Relaxation Time 481 18.2.2 Matthiessen's Rule 486 18.2.3 Fluctuations 487 18.3 Metal-Insulator Transitions 488 18.3.1 Typesoflmpurities 488 18.3.2 Impurity Scattering and Green's Functions 492 18.3.3 Green's Functions 493 18.3.4 Single Impurity 497 18.4 Coherent Potential Approximation 499 18.5 Localization 500 18.5.1 Exact Results in One Dimension 501

xiii 18.5.2 Scaling Theory of Localization 505 18.5.3 Comparison with Experiment 509 Problems 510 References 514 19 Electronics 517 19.1 Introduction 517 19.2 Metal Interfaces 518 19.2.1 Work Functions 519 19.2.2 Schottky Barrier 520 19.2.3 Contact Potentials 522 19.3 Semiconductors 524 19.3.1 Pure Semiconductors 525 19.3.2 Semiconductor in Equilibrium 528 19.3.3 Intrinsic Semiconductor 530 19.3.4 Extrinsic Semiconductor 531 19.4 Diodes and Transistors 533 19.4.1 Surface States 536 19.4.2 Semiconductor Junctions 537 19.4.3 Boltzmann Equation for Semiconductors 540 19.4.4 Detailed Theory of Rectification 542 19.4.5 Transistor 545 19.5 Inversion Layers 548 19.5.1 Heterostructures 548 19.5.2 Quantum Point Contact 550 19.5.3 Quantum Dot 553 Problems 556 References 557 V OPTICAL PROPERTIES 559 20 Phenomenological Theory 561 20.1 Introduction 561 20.2 Maxwell's Equations 563 20.2.1 Traveling Waves 565 20.2.2 Mechanical Oscillators as Dielectric Function 566 20.3 Kramers-Kronig Relations 568 20.3.1 Application to Optical Experiments 570 20.4 The Kubo-Greenwood Formula 573 20.4.1 Born Approximation 573 20.4.2 Susceptibility 577 20.4.3 Many-Body Green Functions 578 Problems 578

xiv References 581 21 Optical Properties of Semiconductors 583 21.1 Introduction 583 21.2 Cyclotron Resonance 583 21.2.1 Electron Energy Surfaces 586 21.3 Semiconductor Band Gaps 588 21.3.1 Direct Transitions 588 21.3.2 Indirect Transitions 589 21.4 Excitons 591 21.4.1 Mott-Wannier Excitons 591 21.4.2 Frenkel Excitons 594 21.4.3 Electron-Hole Liquid 595 21.5 Optoelectronics 595 21.5.1 Solar Cells 595 21.5.2 Lasers 596 Problems 602 References 606 22 Optical Properties of Insuiators 609 22.1 Introduction 609 22.2 Polarization 609 22.2.1 Ferroelectrics 609 22.2.2 Clausius-Mossotti Relation 611 22.3 Optical Modes in Ionic Crystals 613 22.3.1 Polaritons 616 22.3.2 Polarons 618 22.3.3 Experimental Observations of Polarons 623 22.4 Point Defects and Color Centers 623 22.4.1 Vacancies 624 22.4.2 F Centers 625 22.4.3 Electron Spin Resonance and Electron Nuclear Double Resonance 626 22.4.4 Other Centers 628 22.4.5 Franck-Condon Effect 628 22.4.6 UrbachTails 632 Problems 633 References 635 23 Optical Properties of Metals and Inelastic Scattering 637 23.1 Introduction 637 23.1.1 Plasma Frequency 637 23.2 Metals at Low Frequencies 640 23.2.1 Anomalous Skin Effect 642

xv 23.3 Plasmons 643 23.3.1 Experimental Observation of Plasmons 644 23.4 Interband Transitions 646 23.5 Brillouin and Raman Scattering 649 23.5.1 Brillouin Scattering 650 23.5.2 Raman Scattering 651 23.5.3 Inelastic X-Ray Scattering 651 23.6 Photoemission 651 23.6.1 Measurement of Work Functions 651 23.6.2 Angle-Resolved Photoemission 654 23.6.3 Core-Level Photoemission and Charge-Transfer Insulators 658 Problems 664 References 667 VI MAGNETISM 669 24 Classical Theories of Magnetism and Ordering 671 24.1 Introduction 671 24.2 Three Views of Magnetism 671 24.2.1 From Magnetic Moments 671 24.2.2 From Conductivity 672 24.2.3 From a Free Energy 673 24.3 Magnetic Dipole Moments 675 24.3.1 Spontaneous Magnetization of Ferromagnets 678 24.3.2 Ferrimagnets 679 24.3.3 Antiferromagnets 681 24.4 Mean Field Theory and the Ising Model 682 24.4.1 Domains 684 24.4.2 Hysteresis 687 24.5 Other Order-Disorder Transitions 688 24.5.1 Alloy Superlattices 688 24.5.2 SpinGlasses 691 24.6 Critical Phenomena 691 24.6.1 Landau Free Energy 692 24.6.2 Scaling Theory 698 Problems 702 References 705 25 Magnetism of Ions and Electrons 707 25.1 Introduction 707 25.2 Atomic Magnetism 709 25.2.1 Hund'sRules 710 25.2.2 Curie's Law 714

xvi 25.3 Magnetism of the Free-Electron Gas 717 25.3.1 Pauli Paramagnetism 718 25.3.2 Landau Diamagnetism 719 25.3.3 Aharonov-Bohm Effect 722 25.4 Tightly Bound Electrons in Magnetic Fields 724 25.5 Quantum Hall Effect 728 25.5.1 Integer Quantum Hall Effect 728 25.5.2 Fractional Quantum Hall Effect 733 Problems 739 References 742 26 Quantum Mechanics of Interacting Magnetic Moments 745 26.1 Introduction 745 26.2 Origin of Ferromagnetism 745 26.2.1 Heitler-London Calculation 745 26.2.2 Spin Hamiltonian 750 26.3 Heisenberg Model 750 26.3.1 Indirect Exchange and Superexchange 752 26.3.2 Ground State 753 26.3.3 SpinWaves 753 26.3.4 Spin Waves in Antiferromagnets 756 26.3.5 Comparison with Experiment 759 26.4 Ferromagnetism in Transition Metals 759 26.4.1 Stoner Model 759 26.4.2 Calculations Within Band Theory 761 26.5 Kondo Effect 763 26.5.1 Scaling Theory 768 26.6 Hubbard Model 772 26.6.1 Mean-Field Solution 773 Problems 776 References 779 27 Superconductivity 783 27.1 Introduction 783 27.2 Phenomenology of Superconductivity 784 27.2.1 Phenomenological Free Energy 785 27.2.2 Thermodynamics of Superconductors 787 27.2.3 Landau-Ginzburg Free Energy 788 27.2.4 Type I and Type II Superconductors 789 27.2.5 Flux Quantization 794 27.2.6 The Josephson Effect 796 27.2.7 Circuits with Josephson Junction Elements 798 27.2.8 SQUIDS 799 27.2.9 Origin of Josephson's Equations 800

xvii 27.3 Microscopic Theory of Superconductivity 802 27.3.1 Electron-Ion Interaction 803 27.3.2 Formal Derivation 806 27.3.3 Instability of the Normal State: Cooper Problem 808 27.3.4 Self-Consistent Ground State 812 27.3.5 Thermodynamics of Superconductors 817 27.3.6 Superconductor in External Magnetic Field 820 27.3.7 Derivation of Meissner Effect 824 27.3.8 Comparison with Experiment 827 27.3.9 High-Temperature Superconductors 828 Problems 833 References 837 APPENDICES 841 A Lattice Sums and Fourier Transforms 843 A.l One-Dimensional Sum 843 A.2 Area Under Peaks 843 A.3 Three-Dimensional Sum 844 A.4 Discrete Case 845 A.5 Convolution 846 A.6 Using the Fast Fourier Transform 846 References 848 B Variational Techniques 849 B.l Functionals and Functional Derivatives 849 B.2 Time-Independent Schrödinger Equation 850 B.3 Time-Dependent Schrödinger Equation 851 B.4 Method of Steepest Descent 852 References 852 C Second Quantization 853 C.l Rules 853 C.l.l States 853 C.1.2 Operators 853 C.1.3 Hamiltonians 854 C.2 Derivations 855 C.2.1 Bosons 855 C.2.2 Fermions 856 Index 859