Foundations of Solid State Physics

Siegmar Roth is founding director of Sineurop Nanotech GmbH Stuttgart, Germany, a company synthesizing carbon nanotubes, graphene and related materials. He has obtained his PhD in Physics at the University of Vienna, Austria, and his Habilitation at the University of Karlsruhe, Germany. After some years at Siemens in Erlangen, Germany, he joined the Institut Laue Langevin and later on the High Field Magnet Lab in Grenoble, from where he moved to Stuttgart to become leader of the Research Group on Synthetic Nanostructures at the Max Planck Institute for Solid State Research. Between 2009 and 2012 he was visiting professor at the School of Electrical Engineering of Korea University.David L. Carroll is professor at the Wake Forest University. He is a trained materials scientist and received his PhD from Wesleyan University, Middletown, USA. After a stay as postdoctoral fellow at the department of materials science and engineering, University of Pennsylvania, Philadelphia from 1993-1995, he joined the Max-Planck-Institute for solid state research in Stuttgart, Germany. In 1997 he became Assistant Professor at Clemson University and 2001 Associate Professor. He moved with his group to Wake Forest University in 2003, where he founded the Center for Nanotechnology and Molecular Materials.

• Preface xiii1 Introduction 11.1 Dimensionality 21.2 Approaching Dimensionality from Outside and from Inside 41.3 Dimensionality of Carbon: Solids 81.3.1 Three-Dimensional Carbon: Diamond 101.3.2 Two-Dimensional Carbon: Graphite and Graphene 101.3.3 One-Dimensional Carbon: Cumulene, Polycarbyne, and Polyene 121.3.4 Zero-Dimensional Carbon: Fullerene 131.4 Something in Between: Topology 141.5 More Peculiarities of Dimension: One Dimension 161.6 Summary 19Exploring Concepts 20References 262 One-Dimensional Substances 292.1 A15 Compounds 322.2 Krogmann Salts 372.3 Alchemists’ Gold 402.4 Bechgaard Salts and Other Charge-Transfer Compounds 422.5 Polysulfurnitride 452.6 Phthalocyanines and Other Macrocycles 472.7 Transition Metal Chalcogenides and Halides 482.8 Halogen-Bridged Mixed-Valence Transition Metal Complexes 502.9 Returning to Carbon 522.9.1 Conducting Polymers 532.9.2 Carbon Nanotubes 552.10 Perovskites 592.11 Topological States 612.12 What Did We Forget? 622.12.1 Poly-deckers 622.12.2 Polycarbenes 632.12.3 Isolated, Freestanding Nanowires 632.12.4 Templates and Filled Pores 642.12.5 Asymmetric Growth Using Catalysts 652.12.6 Gated Semiconductor Quantum Wires 662.12.7 Few-Atom Metal Nanowires 662.13 A Summary of Our Materials 68Exploring Concepts 69References 693 Order and Symmetry: The Lattice 753.1 The Correlation Function 763.2 The Real Space Crystal Lattice and Its Basis 773.2.1 Using a Coordinate System 813.2.2 Surprises in Two-Dimensional Lattices 863.2.3 The One-Dimensional Lattice 913.2.4 Polymers as One-Dimensional Lattices 923.2.5 Carbon Nanotubes as One-Dimensional Lattices 933.3 Bonding and Binding 943.4 Spatial Symmetries Are Not Enough: Time Crystals 1013.5 Summary 102Exploring Concepts 103References 1104 The Reciprocal Lattice 1114.1 Describing Objects Using Momentum and Energy 1114.1.1 Constructing the Reciprocal Lattice 1124.1.2 The Unit Cell 1144.2 The Reciprocal Lattice and Scattering 1164.2.1 General Scattering 1164.2.2 Real Systems 1204.2.3 Applying This to Real One-Dimensional Systems 1234.3 A Summary of the Reciprocal Lattice 125Exploring Concepts 126References 1285 The Dynamic Lattice 1295.1 Crystal Vibrations and Phonons 1305.1.1 A Simple One-Dimensional Model 1335.1.1.1 A Model 1335.1.1.2 Long Wavelength Vibrations 1365.1.1.3 Short Wavelength Vibrations 1375.1.1.4 More Atoms in the Basis 1375.1.2 More Dimensions 1395.2 Quantum Considerations with Phonons 1435.2.1 Conservation of Crystal Momentum 1445.2.2 General Scattering 1445.3 Phonons Yield Thermal Properties 1475.3.1 Internal Energy and Phonons 1485.3.2 Models of Energy Distribution: f p(𝜔) and 𝜔K,p 1505.3.2.1 DuLong and Petit: Equipartition of Energy 1505.3.2.2 Einstein and Quantum Statistics 1515.3.2.3 Debye and the Spectral Analysis 1525.3.3 The Debye Approximation 1565.3.4 Generalizations of the Density of States 1595.3.5 Other Thermal Properties: Thermal Transport 1615.4 Anharmonic Effects 1625.5 Summary of Phonons 168Exploring Concepts 168References 1726 Electrons in Solids 173Evolving Pictures 174Superconductors 1766.1 Properties of Electrons: A Review 1766.1.1 Electrons Travel as Waves 1766.1.1.1 Delocalization 1766.1.1.2 Localization 1786.1.2 Electrons Arrive as Particles: Statistics 1786.1.3 The Fermi Surface 1806.2 On to the Models 1816.2.1 The Free-Electron Model 1816.2.2 Nearly Free Electrons, Energy Bands, Energy Gaps, Density of States 1846.2.2.1 Bloch’s Theorem 1856.2.2.2 The Nearly Free 1D Model 1856.2.2.3 Analyzing the 1D Nearly Free Solutions 1876.2.2.4 Extending Dispersion Curves to 3D 1906.2.3 Tight Binding or Linear Combination of Atomic Orbitals 1916.2.3.1 The Formalism 1936.2.3.2 The s-Band 1946.2.3.3 s Bands in One Dimension 1956.2.3.4 s Bands in Two Dimensions 1956.2.3.5 s Bands in Three Dimensions 1966.2.4 What About Orbitals Other Than s? 1976.2.4.1 Building Bands in a Polymer 1986.2.4.2 Bonding and Antibonding States 1986.2.4.3 The Polyenes 1996.2.4.4 Translating to Bloch’s Theorem 2036.2.5 Tight Binding with a Basis 2066.2.5.1 Hybridization 2096.2.5.2 Graphene: A Two-Dimensional Example 2116.2.5.3 Carbon Nanotubes 2136.3 Are We Done Yet? 2156.4 Summary 217Exploring Concepts 218References 2237 Electrons in Solids Part II: Spatial Heterogeneity 2257.1 Heterogeneity: Band-Level Diagrams and the Contact 2267.2 Heterogeneity in Semiconductors 2297.2.1 Semiconductors: Bandgaps and Doping 2307.2.1.1 Band-Level Diagrams 2307.2.1.2 Doping 2307.2.1.3 Carrier Concentrations in Intrinsic and Doped Semiconductors 2357.2.1.4 The Fermi Level vs. the Chemical Potential 2397.2.1.5 Spectroscopy of the Dopant Levels 2407.2.1.6 Carbon Does Not “Dope” Like Si 2427.2.2 Junctions with Semiconductors 2447.3 Other Types of Heterogeneity 2497.4 Summary 251Exploring Concepts 251References 2578 Electrons Moving in Solids 2598.1 Phenomenology of Electron Dynamics in a Material 2598.1.1 Free-Electron Metals 2598.1.2 The Free-Electron Metal as a Fluid 2628.1.3 Temperature and Conductivity 2648.2 The Semiclassical Approach: The Boltzmann Equation 2678.2.1 The Sources of Electron Scattering 2678.2.2 The Nonequilibrium Distribution Function 2688.2.3 The Relaxation Time 𝜏 2688.2.4 The Differential Equation for g(r; k; t) 2688.2.5 Introducing Collisions 2698.2.6 The Relaxation Time Approximation 2708.2.7 Isotropic Scattering from Stationary States 2718.2.8 A Simple Example: Ohm’s Law 2718.2.9 Parabolic Bands 2728.2.10 Another Simple Example: AC Conductivity and Linear Response 2738.2.11 An Example with Anisotropy: 𝜇 = 𝜇(r) and ∇rT ≠ 0 2738.2.12 The Seebeck Effect and Thermopower 2748.2.13 A Final Example: Static E and B Applied but 𝜇 ≠𝜇(r) and ∇rT = 0 2778.2.14 The Hall Effect and Magnetotransport 2798.2.15 The Curious Case of Al 2808.3 Coherent Transport: The Landauer–Büttiker Approach 281Contents ix8.4 General Remarks on Measurements 2838.4.1 Simple Conductivity 2838.4.2 Conductivity of Small Particles 2878.4.3 Conductivity of High Impedance Samples 2888.4.4 Conductivity Measurements Without Contacts 2898.5 Complications: Localization, Hopping, and General Bad Behavior 2908.6 Summary 293Exploring Concepts 293References 2979 Polarons, Solitons, Excitons, and Conducting Polymers 3019.1 Distortions, Instabilities, and Transitions in One Dimension 3039.1.1 Coupling Charge with the Lattice 3039.1.2 Peierls Instability 3059.1.3 Results of Peierls in Real Systems 3089.1.3.1 Phonon Softening and the Kohn Anomaly 3089.1.3.2 Fermi Surface Warping 3099.2 Conjugation and the Double Bond 3109.3 Conjugational Defects 3139.4 The Soliton 3179.4.1 Doping 3199.4.2 Quasiparticles 3209.5 Generation of Solitons 3259.6 Nondegenerate Ground-State Polymers: Polarons 3289.7 Fractional Charges 3329.8 Soliton Lifetime 3349.9 Conductivity and Solitons 3379.10 Fibril Conduction 3419.11 Hopping Conductivity: Variable Range Hopping vs.Fluctuation-Assisted Tunneling 3459.12 Highly Conducting Polymers 3539.13 Magnetoresistance 3549.14 Organic Molecular Devices 3609.14.1 Molecular Switches 3609.14.2 LB Diodes 3639.14.3 Organic Light-Emitting Diodes 3649.14.3.1 Fundamentals of OLEDs 3669.14.3.2 Materials for OLEDs 3709.14.3.3 Designs for OLEDs 3719.14.3.4 Performance of OLEDs 3729.14.4 Field-Induced Organic Emitters 3739.14.5 Organic Lasers and Organic Light-Emitting Transistors 3769.14.5.1 Current Densities 3799.14.5.2 Contacts 3799.14.5.3 Polarons and Triplets 3799.14.6 Organic Solar Cells 3809.14.7 Organic Field-Effect Transistors 3849.14.8 Organic Thermoelectrics 3859.15 Summary 387Exploring Concepts 388References 39010 Correlation and Coupling 40310.1 The Metal-to-Insulator Transition and the Mott Insulator 40310.1.1 The Hamiltonian 40610.1.2 The Lattice and Antiferromagnetic Ordering 40710.1.3 Other Considerations: The Particle-Hole Symmetry (PHS) 40710.1.4 The Hubbard Model in Lower Dimensions 40810.1.5 Real One-Dimensional Mott Systems 41010.2 The Superconductor 41110.2.1 The Basic Phenomena 41110.2.1.1 In What Compounds Has Superconductivity Been Observed? 41510.2.2 A Basic Model 41510.2.2.1 How Does an Attractive Potential Show Up Between Two Negatively Charged Particles? 41610.2.2.2 Cooper Pair Binding 41810.2.2.3 The BCS Ground State 42010.2.2.4 Supplementary Thoughts 42510.2.3 Superconductivity Measurements Are Tricky 42810.2.4 Superconductivity and Dimensionality 43010.2.5 More on Organic Superconductors 43110.2.5.1 One-Dimensional Organic Superconductors 43210.2.5.2 Two-Dimensional Organic Superconductors 43510.2.5.3 Three-Dimensional Organic Superconductors 43610.2.6 Trends 43810.3 The Charge Density Wave 44010.3.1 The Charge DensityWave and Peierls 44010.3.1.1 Modulation of the Electron and Mass Densities 44110.3.1.2 Starting with Polymers 44110.3.1.3 A Gap Is Introduced 44210.3.1.4 The Order Parameter 44210.3.1.5 Phase Dynamics, Pinning, Commensurability, and Solitons 44210.3.2 Peierls and Coulomb Interactions: Spin Interactions 44610.3.2.1 4kF Charge DensityWaves 44610.3.2.2 Spin PeierlsWaves 44810.3.2.3 Spin DensityWaves 44810.3.3 Phonon Dispersion: Phase and Amplitude in CDWs 45010.3.4 More on Peierls–Fröhlich Mechanisms 45210.3.5 Spin DensityWaves and the Quantized Hall Effect 45310.4 Plasmons 45410.4.1 The Drude Model and the Dielectric Function 45410.4.2 The Significance of the Plasma Frequency 45510.5 Composite Particles and Quasiparticles: A Summary 457Exploring Concepts 457References 458Intermission 46511 Magnetic Interactions 46711.1 Magnetism of the Atom 46911.2 The Crystal Field 47211.3 Magnetism in Condensed Systems 47411.3.1 Paramagnetism 47411.3.1.1 Curie Paramagnets 47611.3.1.2 The Weiss Correction 47711.3.1.3 Free-Electron Magnets 47811.3.2 Diamagnetism 47911.4 Dia- and Para-Foundations of Other Magnets 48111.5 Mechanisms of Interaction: Spin Models 48211.5.1 The Mean Field Model 48311.5.2 Ising, Heisenberg, XY, and Hopfield 48311.5.2.1 Ising Models 48311.5.2.2 Heisenberg Models 48511.5.2.3 XY models 48511.5.2.4 Hopfield Models 48711.5.3 SpinWave and Magnons 48811.5.3.1 SpinWaves 48811.5.3.2 Thermodynamics 49111.5.3.3 The Particle Nature of Magnons 49311.5.3.4 Stoner Excitations 49411.5.3.5 Coupling to the Electromagnetic Field: Magnon–Photon Coupling 49411.6 More Complicated Situations 49411.6.1 Double Exchange 49411.6.2 Super Exchange 49611.6.3 RKKY 49611.7 Time Reversal Symmetry 49711.8 Summary 498Exploring Concepts 499References 50112 Polarization of Materials 50312.1 Simple Atomic Models 50312.1.1 Linearity in the Response 50412.1.2 Relating the Fields 50712.2 Temperature Dependence 50912.3 Time Dependence: 𝜀(𝜔) 51012.4 A Familiar Equation in Optics 51312.5 Understanding the Context 51412.6 The Dielectric Function and Metals 51412.7 Piezoelectrics, Pyroelectrics, and More 51512.7.1 The h-BN Example 51812.8 Summary 519Exploring Concepts 519References 52313 Optical Interactions 52513.1 Maxwell and the Solid (Review) 52713.1.1 In a Vacuum 52713.1.2 In a Material 52813.1.3 A General Solution in the Solid 52913.1.3.1 A Fun Notational Fact 53113.2 Polarization Coupling: Polaritons 53213.2.1 Phonons with Electrical Polarization 53213.2.2 Phonons Meet Photons 53413.2.3 The Phonon–Polariton 53513.2.4 The Plasmon Polariton 53813.3 Optical Transitions, Excitons, and Exciton Polaritons 54313.3.1 Transitions 54313.3.2 Carbon Nanotubes: An Example 54613.3.3 Color Centers and Dopants 54613.3.4 Excitons 54813.3.5 Exciton Polaritons 54913.4 Kramers–Kronig 54913.5 Summary 551Exploring Concepts 552References 55514 The End and the Beginning 557Reference 558Index 559