Condensed Matter Physics

Michael P. Marder, PhD, is the Associate Dean for Science and Mathematics Education and Professor in the Department of Physics at the University of Texas at Austin, where he has been involved in a wide variety of theoretical, numerical, and experimental investigations. He specializes in the mechanics of solids, particularly the fracture of brittle materials. Dr. Marder has carried out experimental studies of crack instabilities in plastics and rubber, and constructed analytical theories for how cracks move in crystals. Recently he has studied the way that membranes ripple due to changes in their geometry, and properties of frictional sliding at small length scales.

• Preface xixReferences xxiiI ATOMIC STRUCTURE 11 The Idea of Crystals 31.1 Introduction 31.1.1 Why are Solids Crystalline? 41.2 Two-Dimensional Lattices 61.2.1 Bravais Lattices 61.2.2 Enumeration of Two-Dimensional Bravais Lattices 71.2.3 Lattices with Bases 91.2.4 Primitive Cells 91.2.5 Wigner-Seitz Cells 101.3 Symmetries 111.3.1 The Space Group 111.3.2 Translation and Point Groups 121.3.3 Role of Symmetry 14Problems 14References 162 Three-Dimensional Lattices 172.1 Introduction 172.2 Monatomic Lattices 202.2.1 The Simple Cubic Lattice 202.2.2 The Face-Centered Cubic Lattice 202.2.3 The Body-Centered Cubic Lattice 222.2.4 The Hexagonal Lattice 232.2.5 The Hexagonal Close-Packed Lattice 232.2.6 The Diamond Lattice 242.3 Compounds 242.3.1 Rocksalt—Sodium Chloride 252.3.2 Cesium Chloride 262.3.3 Fluorite—Calcium Fluoride 262.3.4 Zincblende—Zinc Sulfide 272.3.5 Wurtzite—Zinc Oxide 282.3.6 Perovskite—Calcium Titanate 282.4 Classification of Lattices by Symmetry 302.4.1 Fourteen Bravais Lattices and Seven Crystal Systems 302.5 Symmetries of Lattices with Bases 332.5.1 Thirty-Two Crystallographic Point Groups 332.5.2 Two Hundred Thirty Distinct Lattices 362.6 Some Macroscopic Implications of Microscopic Symmetries 372.6.1 Pyroelectricity 372.6.2 Piezoelectricity 372.6.3 Optical Activity 38Problems 38References 413 Scattering and Structures 433.1 Introduction 433.2 Theory of Scattering from Crystals 443.2.1 Special Conditions for Scattering 443.2.2 Elastic Scattering from Single Atom 463.2.3 Wave Scattering from Many Atoms 473.2.4 Lattice Sums 483.2.5 Reciprocal Lattice 493.2.6 Miller Indices 513.2.7 Scattering from a Lattice with a Basis 533.3 Experimental Methods 543.3.1 Laue Method 563.3.2 Rotating Crystal Method 573.3.3 Powder Method 593.4 Further Features of Scattering Experiments 603.4.1 Interaction of X-Rays with Matter 603.4.2 Production of X-Rays 613.4.3 Neutrons 633.4.4 Electrons 633.4.5 Deciphering Complex Structures 643.4.6 Accuracy of Structure Determinations 653.5 Correlation Functions 663.5.1 Why Bragg Peaks Survive Atomic Motions 663.5.2 Extended X-Ray Absorption Fine Structure (EXAFS) 673.5.3 Dynamic Light Scattering 683.5.4 Application to Dilute Solutions 70Problems 71References 734 Surfaces and Interfaces 774.1 Introduction 774.2 Geometry of Interfaces 774.2.1 Coherent and Commensurate Interfaces 784.2.2 Stacking Period and Interplanar Spacing 794.2.3 Other Topics in Surface Structure 814.3 Experimental Observation and Creation of Surfaces 824.3.1 Low-Energy Electron Diffraction (LEED) 824.3.2 Reflection High-Energy Electron Diffraction (RHEED) 844.3.3 Molecular Beam Epitaxy (MBE) 844.3.4 Field Ion Microscopy (FIM) 854.3.5 Scanning Tunneling Microscopy (STM) 864.3.6 Atomic Force Microscopy (AFM) 914.3.7 High Resolution Electron Microscopy (HREM) 91Problems 91References 945 Beyond Crystals 975.1 Introduction 975.2 Diffusion and Random Variables 975.2.1 Brownian Motion and the Diffusion Equation 975.2.2 Diffusion 985.2.3 Derivation from Master Equation 995.2.4 Connection Between Diffusion and Random Walks 1005.3 Alloys 1015.3.1 Equilibrium Structures 1015.3.2 Phase Diagrams 1025.3.3 Superlattices 1035.3.4 Phase Separation 1045.3.5 Nonequilibrium Structures in Alloys 1065.3.6 Dynamics of Phase Separation 1085.4 Simulations 1105.4.1 Monte Carlo 1105.4.2 Molecular Dynamics 1125.5 Liquids 1135.5.1 Order Parameters and Long-and Short-Range Order 1135.5.2 Packing Spheres 1145.6 Glasses 1165.7 Liquid Crystals 1205.7.1 Nematics, Cholesterics, and Smectics 1205.7.2 Liquid Crystal Order Parameter 1225.8 Polymers 1235.8.1 Ideal Radius of Gyration 1235.9 Colloids and Diffusing-Wave Scattering 1285.9.1 Colloids 1285.9.2 Diffusing-Wave Spectroscopy 1285.10 Quasicrystals 1335.10.1 One-Dimensional Quasicrystal 1345.10.2 Two-Dimensional Quasicrystals—Penrose Tiles 1395.10.3 Experimental Observations 1415.11 Fullerenes and nanotubes 143Problems 143References 149II ELECTRONIC STRUCTURE 1536 The Free Fermi Gas and Single Electron Model 1556.1 Introduction 1556.2 Starting Hamiltonian 1576.3 Densities of States 1596.3.1 Definition of Density of States D 1606.3.2 Results for Free Electrons 1616.4 Statistical Mechanics of Noninteracting Electrons 1636.5 Sommerfeld Expansion 1666.5.1 Specific Heat of Noninteracting Electrons at Low Temper-atures 169Problems 171References 1737 Non-Interacting Electrons in a Periodic Potential 1757.1 Introduction 1757.2 Translational Symmetry—Bloch’s Theorem 1757.2.1 One Dimension 1767.2.2 Bloch’s Theorem in Three Dimensions 1807.2.3 Formal Demonstration of Bloch’s Theorem 1827.2.4 Additional Implications of Bloch’s Theorem 1837.2.5 Van Hove Singularities 1867.2.6 Kronig-Penney Model 1897.3 Rotational Symmetry—Group Representations 1927.3.1 Classes and Characters 1987.3.2 Consequences of point group symmetries for Schrödinger’s equation 201Problems 203References 2068 Nearly Free and Tightly Bound Electrons 2078.1 Introduction 2078.2 Nearly Free Electrons 2088.2.1 Degenerate Perturbation Theory 2108.3 Brillouin Zones 2118.3.1 Nearly Free Electron Fermi Surfaces 2148.4 Tightly Bound Electrons 2198.4.1 Linear Combinations of Atomic Orbitals 2198.4.2 Wannier Functions 2228.4.3 Geometric Phases 2238.4.4 Tight Binding Model 226Problems 227References 2329 Electron-Electron Interactions 2339.1 Introduction 2339.2 Hartree and Hartree-Fock Equations 2349.2.1 Variational Principle 2359.2.2 Hartree-Fock Equations 2359.2.3 Numerical Implementation 2399.2.4 Hartree-Fock Equations for Jellium 2429.3 Density Functional Theory 2449.3.1 Thomas-Fermi Theory 2479.3.2 Stability of Matter 2499.4 Quantum Monte Carlo 2529.4.1 Integrals by Monte Carlo 2529.4.2 Quantum Monte Carlo Methods 2539.4.3 Physical Results 2549.5 Kohn-Sham Equations 255Problems 258References 26210 Realistic Calculations in Solids 26510.1 Introduction 26510.2 Numerical Methods 26610.2.1 Pseudopotentials and Orthogonalized Planes Waves (OPW) 26610.2.2 Linear Combination of Atomic Orbitals (LCAO) 27110.2.3 Plane Waves 27110.2.4 Linear Augmented Plane Waves (LAPW) 27410.3 Definition of Metals, Insulators, and Semiconductors 27710.4 Brief Survey of the Periodic Table 27910.4.1 Nearly Free Electron Metals 28010.4.2 Noble Gases 28210.4.3 Semiconductors 28310.4.4 Transition Metals 28410.4.5 Rare Earths 286Problems 286References 291III MECHANICAL PROPERTIES 29311 Cohesion of Solids 29511.1 Introduction 29511.1.1 Radii of Atoms 29711.2 Noble Gases 29911.3 Tonic Crystals 30111.3.1 EwaldSums 30211.4 Metals 30511.4.1 Use of Pseudopotentials 30711.5 Band Structure Energy 30811.5.1 Peierls Distortion 30911.5.2 Structural Phase Transitions 31111.6 Hydrogen-Bonded Solids 31211.7 Cohesive Energy from Band Calculations 31211.8 Classical Potentials 313Problems 315References 31812 Elasticity 32112.1 Introduction 32112.2 Nonlinear Elasticity 32112.2.1 Rubber Elasticity 32212.2.2 Larger Extensions of Rubber 32412.3 Linear Elasticity 32512.3.1 Solids of Cubic Symmetry 32612.3.2 Isotropic Solids 32812.4 Other Constitutive Laws 33212.4.1 Liquid Crystals 33212.4.2 Granular Materials 335Problems 336References 33913 Phonons 34113.1 Introduction 34113.2 Vibrations of a Classical Lattice 34213.2.1 Classical Vibrations in One Dimension 34213.2.2 Classical Vibrations in Three Dimensions 34613.2.3 Normal Modes 34713.2.4 Lattice with a Basis 34813.3 Vibrations of a Quantum-Mechanical Lattice 35113.3.1 Phonon Specific Heat 35413.3.2 Einstein and Debye Models 35813.3.3 Thermal Expansion 36113.4 Inelastic Scattering from Phonons 36313.4.1 Neutron Scattering 36413.4.2 Formal Theory of Neutron Scattering 36613.4.3 Averaging Exponentials 37013.4.4 Evaluation of Structure Factor 37213.4.5 Kohn Anomalies 37313.5 The Mössbauer Effect 374Problems 376References 37714 Dislocations and Cracks 37914.1 Introduction 37914.2 Dislocations 38114.2.1 Experimental Observations of Dislocations 38314.2.2 Force to Move a Dislocation 38614.2.3 One-Dimensional Dislocations: Frehkel-Kontorova Model 38614.3 Two-Dimensional Dislocations and Hexatic Phases 38914.3.1 Impossibility of Crystalline Order in Two Dimensions 38914.3.2 Orientational Order 39114.3.3 Kosterlitz-Thouless-Berezinskii Transition 39214.4 Cracks 39914.4.1 Fracture of a Strip 39914.4.2 Stresses Around an Elliptical Hole 40214.4.3 Stress Intensity Factor 40414.4.4 Atomic Aspects of Fracture 405Problems 406References 40915 Fluid Mechanics 41315.1 Introduction 41315.2 Newtonian Fluids 41315.2.1 Euler’s Equation 41315.2.2 Navier-Stokes Equation 41515.3 Polymeric Solutions 41615.4 Plasticity 42315.5 Superfluid 4He 42715.5.1 Two-Fluid Hydrodynamics 43015.5.2 Second Sound 43115.5.3 Direct Observation of Two Fluids 43315.5.4 Origin of Superfluidity 43415.5.5 Lagrangian Theory of Wave Function 43915.5.6 Superfluid 3He 442Problems 443References 447IV ELECTRON TRANSPORT 45116 Dynamics of Bloch Electrons 45316.1 Introduction 45316.1.1 Drude Model 45316.2 Semiclassical Electron Dynamics 45516.2.1 Bloch Oscillations 45616.2.2 k-p̂ Method 45716.2.3 Effective Mass 45916.3 Noninteracting Electrons in an Electric Field 45916.3.1 Zener Tunneling 46216.4 Semiclassical Equations from Wave Packets 46516.4.1 Formal Dynamics of Wave Packets 46516.4.2 Dynamics from Lagrangian 46716.5 Quantizing Semiclassical Dynamics 47016.5.1 Wannier-Stark Ladders 47216.5.2 de Haas-van Alphen Effect 47316.5.3 Experimental Measurements of Fermi Surfaces 474Problems 477References 48017 Transport Phenomena and Fermi Liquid Theory 4S317.1 Introduction 48317.2 Boltzmann Equation 48317.2.1 Boltzmann Equation 48517.2.2 Including Anomalous Velocity 48617.2.3 Relaxation Time Approximation 48717.2.4 Relation to Rate of Production of Entropy 48917.3 Transport Symmetries 49017.3.1 Onsager Relations 49117.4 Thermoelectric Phenomena 49217.4.1 Electrical Current 49217.4.2 Effective Mass and Holes 49417.4.3 Mixed Thermal and Electrical Gradients 49517.4.4 Wiedemann-Franz Law 49617.4.5 Thermopower—Seebeck Effect 49717.4.6 Peltier Effect 49817.4.7 Thomson Effect 49817.4.8 Hall Effect 50017.4.9 Magnetoresistance 50217.4.10 Anomalous Hall Effect 50317.5 Fermi Liquid Theory 50417.5.1 Basic Ideas 50417.5.2 Statistical Mechanics of Quasi-Particles 50617.5.3 Effective Mass 50817.5.4 Specific Heat 51017.5.5 Fermi Liquid Parameters 51117.5.6 Traveling Waves 51217.5.7 Comparison with Experiment in 3He 515Problems 516References 52018 Microscopic Theories of Conduction 52318.1 Introduction 52318.2 Weak Scattering Theory of Conductivity 52318.2.1 Genera] Formula for Relaxation Time 52318.2.2 Matthiessen’s Rule 52818.2.3 Fluctuations 52918.3 Metal-Insulator Transitions in Disordered Solids 53018.3.1 Impurities and Disorder 53018.3.2 Non-Compensated Impurities and the Mott Transition . . 53118.4 Compensated Impurity Scattering and Green’s Functions 53418.4.1 Tight-Binding Models of Disordered Solids 53418.4.2 Green’s Functions 53618.4.3 Single Impurity 53918.4.4 Coherent Potential Approximation 54118.5 Localization 54218.5.1 Exact Results in One Dimension 54418.5.2 Scaling Theory of Localization 54718.5.3 Comparison with Experiment 55118.6 Luttinger Liquids 55318.6.1 Density of States 557Problems 560References 56419 Electronics 56719.1 Introduction 56719.2 Metal Interfaces 56819.2.1 Work Functions 56919.2.2 Schottky Barrier 57019.2.3 Contact Potentials 57219.3 Semiconductors 57419.3.1 Pure Semiconductors 57519.3.2 Semiconductor in Equilibrium 57819.3.3 Intrinsic Semiconductor 58019.3.4 Extrinsic Semiconductor 58119.4 Diodes and Transistors 58319.4.1 Surface States 58619.4.2 Semiconductor Junctions 58719.4.3 Boltzmann Equation for Semiconductors 59019.4.4 Detailed Theory of Rectification 59219.4.5 Transistor 59519.5 Inversion Layers 59819.5.1 Heterostructures 598f 9,5.2 Quantum Point Contact 60019.5.3 Quantum Dot 603Problems 606References 607V OPTICAL PROPERTIES 60920 Phenomenological Theory 61120.1 Introduction 61120.2 Maxwell’s Equations 61320.2.1 Traveling Waves 61520.2.2 Mechanical Oscillators as Dielectric Function 61620.3 Kramers-Kronig Relations 61820.3.1 Application to Optical Experiments 62020.4 The Kubo-Greenwood Formula 62320.4.1 Bom Approximation 62320.4.2 Susceptibility 62720.4.3 Many-Body Green Functions 628Problems 628References 63121 Optical Properties of Semiconductors 63321.1 Introduction 63321.2 Cyclotron Resonance 63321.2.1 Electron Energy Surfaces 63621.3 Semiconductor Band Gaps 63821.3.1 Direct Transitions 63821.3.2 Indirect Transitions 63921.4 Excitons 64121.4.1 Mott-Wannier Excitons 64121.4.2 Frenkel Excitons 64421.4.3 Electron-Hole Liquid 64521.5 Optoelectronics 64521.5.1 SolarCells 64521.5.2 Lasers 646Problems 652References 65622 Optical Properties of Insulators 65922.1 Introduction 65922.2 Polarization 65922.2.1 Ferroelectrics 65922.2.2 Berry phase theory of polarization 66122.2.3 Clausius-Mossotti Relation 66122.3 Optical Modes in Ionic Crystals 66422.3.1 Polaritons 66622.3.2 Polarons 66922.3.3 Experimental Observations of Polarons 67422.4 Point Defects and Color Centers 67422.4.1 Vacancies 67522.4.2 F Centers 67622.4.3 Electron Spin Resonance and Electron Nuclear Double Res-onance 67722.4.4 Other Centers 67922.4.5 Franck-Condon Effect 67922.4.6 Urbach Tails 683Problems 684References 68623 Optical Properties of Metals and Inelastic Scattering 68923.1 Introduction 68923.1.1 Plasma Frequency 68923.2 Metals at Low Frequencies 69223.2.1 Anomalous Skin Effect 69423.3 Plasmons 69523.3.1 Experimental Observation of Plasmons 69623.4 Interband Transitions 69823.5 Brillouin and Raman Scattering 70123.5.1 Brillouin Scattering 70223.5.2 Raman Scattering 70323.5.3 Inelastic X-Ray Scattering 70323.6 Photoemission 70323.6.1 Measurement of Work Functions 70323.6.2 Angle-Resolved Photoemission 70623.6.3 Core-Level Photoemission and Charge-Transfer Insulators 710Problems 716References 719VI MAGNETISM 72124 Classical Theories of Magnetism and Ordering 72324.1 Introduction 72324.2 Three Views of Magnetism 72324.2.1 From Magnetic Moments 72324.2.2 From Conductivity 72424.2.3 From a Free Energy 72524.3 Magnetic Dipole Moments 72724.3.1 Spontaneous Magnetization of Ferromagnets 73024.3.2 Ferrimagnets 73124.3.3 Antiferromagnets 73324.4 Mean Field Theory and the Ising Model 73424.4.1 Domains 73624.4.2 Hysteresis 73924.5 Other Order-Disorder Transitions 74024.5.1 Alloy Superlattices 74024.5.2 Spin Glasses 74324.6 Critical Phenomena 74324.6.1 Landau Free Energy 74424.6.2 Scaling Theory 750Problems 754References 75725 Magnetism of Ions and Electrons 75925.1 Introduction 75925.2 Atomic Magnetism 76125.2.1 Hund’s Rules 76225.2.2 Curie’s Law 76625.3 Magnetism of the Free-El ectron Gas 76925.3.1 Pauli Paramagnetism 77025.3.2 Landau Diamagnetism 77125.3.3 Aharonov-Bohm Effect 77425.4 Tightly Bound Electrons in Magnetic Fields Ill25.5 Quantum Hall Effect 78025.5.1 Integer Quantum Hall Effect 78025.5.2 Fractional Quantum Hall Effect 785Problems 791References 79426 Quantum Mechanics of Interacting Magnetic Moments 79726.1 Introduction 79726.2 Origin of Ferromagnetism 79726.2.1 Heitler-London Calculation 79726.2.2 Spin Hamiltonian 80226.3 Heisenberg Model 80226.3.1 Indirect Exchange and Superexchange 80426.3.2 Ground State 80526.3.3 Spin Waves 80526.3.4 Spin Waves in Antiferromagnets 80826.3.5 Comparison with Experiment 81126.4 Ferromagnetism in Transition Metals 81126.4.1 Stoner Model 81126.4.2 Calculations Within Band Theory 81326.5 Spintronics 81526.5.1 Giant Magnetoresistance 81526.5.2 Spin Torque 81626.6 Kondo Effect 81926.6.1 Scaling Theory 82426.7 Hubbard Model 82826.7.1 Mean-Field Solution 829Problems 832References 83527 Superconductivity 83927.1 Introduction 83927.2 Phenomenology of Superconductivity 84027.2.1 Phenomenological Free Energy 84127.2.2 Thermodynamics of Superconductors 84327.2.3 Landau-Ginzburg Free Energy 84427.2.4 Type I and Type II Superconductors 84527.2.5 Flux Quantization 85027.2.6 The Josephson Effect 85227.2.7 Circuits with Josephson Junction Elements 85427.2.8 SQUIDS 85527.2.9 Origin of Josephson’s Equations 85627.3 Microscopic Theory of Superconductivity 85827.3.1 Electron-Ion Interaction 85927.3.2 Instability of the Normal State: Cooper Problem 86327.3.3 Self-Consistent Ground State 86527.3.4 Thermodynamics of Superconductors 86927.3.5 Superconductor in External Magnetic Field 87327.3.6 Derivation of Meissner Effect 87627.3.7 Comparison with Experiment 87927.3.8 High-Temperature Superconductors 881Problems 888References 890APPENDICES 895A Lattice Sums and Fourier Transforms 897A. l One-Dimensional Sum 897A. 2 Area Under Peaks 897A. 3 Three-Dimensional Sum 898A. 4 Discrete Case 899A.5 Convolution 900A. 6 Using the Fast Fourier Transform 900References 902B Variational Techniques 903B. l Functionals and Functional Derivatives 903B. 2 Time-Independent Schrodinger Equation 904B. 3 Time-Dependent Schrodinger Equation 905B. 4 Method of Steepest Descent 906References 906C Second Quantization 907C. l Rules 907C. 1.1 States 907C. l.2 Operators 907C. l.3 Hamiltonians 908C.2 Derivations 909C.2.1 Bosons 909C.2.2 Fermions 910Index

"The text also gives more leisurely attention to the topics of primary interest to most students: electron and phonon bond structures." (Booknews, 1 February 2011) "In this text intended for a one-year graduate course, Marder (physics, U. of Texas, Austin) comments in the preface that this second edition incorporates the many thousands of updates and corrections suggested by readers of the first edition published in 1999, and he even gives credit to several individuals who found the most errors. He also points out that "the entire discipline of condensed matter is roughly ten percent older than when the first edition was written, so adding some new topics seemed appropriate." These new topics - chosen because of increasing recognition of their importance - include graphene and nanotubes, Berry phases, Luttinger liquids, diffusion, dynamic light scattering, and spin torques. The text also gives more leisurely attention to the topics of primary interest to most students: electron and phonon bond structures." (Reference and Research Book News, February 2011)