Preface xiii
1 Introduction 1
1.1 Dimensionality 2
1.2 Approaching Dimensionality from Outside and from Inside 4
1.3 Dimensionality of Carbon: Solids 8
1.3.1 Three-Dimensional Carbon: Diamond 10
1.3.2 Two-Dimensional Carbon: Graphite and Graphene 10
1.3.3 One-Dimensional Carbon: Cumulene, Polycarbyne, and Polyene 12
1.3.4 Zero-Dimensional Carbon: Fullerene 13
1.4 Something in Between: Topology 14
1.5 More Peculiarities of Dimension: One Dimension 16
1.6 Summary 19
Exploring Concepts 20
References 26
2 One-Dimensional Substances 29
2.1 A15 Compounds 32
2.2 Krogmann Salts 37
2.3 Alchemists’ Gold 40
2.4 Bechgaard Salts and Other Charge-Transfer Compounds 42
2.5 Polysulfurnitride 45
2.6 Phthalocyanines and Other Macrocycles 47
2.7 Transition Metal Chalcogenides and Halides 48
2.8 Halogen-Bridged Mixed-Valence Transition Metal Complexes 50
2.9 Returning to Carbon 52
2.9.1 Conducting Polymers 53
2.9.2 Carbon Nanotubes 55
2.10 Perovskites 59
2.11 Topological States 61
2.12 What Did We Forget? 62
2.12.1 Poly-deckers 62
2.12.2 Polycarbenes 63
2.12.3 Isolated, Freestanding Nanowires 63
2.12.4 Templates and Filled Pores 64
2.12.5 Asymmetric Growth Using Catalysts 65
2.12.6 Gated Semiconductor Quantum Wires 66
2.12.7 Few-Atom Metal Nanowires 66
2.13 A Summary of Our Materials 68
Exploring Concepts 69
References 69
3 Order and Symmetry: The Lattice 75
3.1 The Correlation Function 76
3.2 The Real Space Crystal Lattice and Its Basis 77
3.2.1 Using a Coordinate System 81
3.2.2 Surprises in Two-Dimensional Lattices 86
3.2.3 The One-Dimensional Lattice 91
3.2.4 Polymers as One-Dimensional Lattices 92
3.2.5 Carbon Nanotubes as One-Dimensional Lattices 93
3.3 Bonding and Binding 94
3.4 Spatial Symmetries Are Not Enough: Time Crystals 101
3.5 Summary 102
Exploring Concepts 103
References 110
4 The Reciprocal Lattice 111
4.1 Describing Objects Using Momentum and Energy 111
4.1.1 Constructing the Reciprocal Lattice 112
4.1.2 The Unit Cell 114
4.2 The Reciprocal Lattice and Scattering 116
4.2.1 General Scattering 116
4.2.2 Real Systems 120
4.2.3 Applying This to Real One-Dimensional Systems 123
4.3 A Summary of the Reciprocal Lattice 125
Exploring Concepts 126
References 128
5 The Dynamic Lattice 129
5.1 Crystal Vibrations and Phonons 130
5.1.1 A Simple One-Dimensional Model 133
5.1.1.1 A Model 133
5.1.1.2 Long Wavelength Vibrations 136
5.1.1.3 Short Wavelength Vibrations 137
5.1.1.4 More Atoms in the Basis 137
5.1.2 More Dimensions 139
5.2 Quantum Considerations with Phonons 143
5.2.1 Conservation of Crystal Momentum 144
5.2.2 General Scattering 144
5.3 Phonons Yield Thermal Properties 147
5.3.1 Internal Energy and Phonons 148
5.3.2 Models of Energy Distribution: f p(?) and ?K,p 150
5.3.2.1 DuLong and Petit: Equipartition of Energy 150
5.3.2.2 Einstein and Quantum Statistics 151
5.3.2.3 Debye and the Spectral Analysis 152
5.3.3 The Debye Approximation 156
5.3.4 Generalizations of the Density of States 159
5.3.5 Other Thermal Properties: Thermal Transport 161
5.4 Anharmonic Effects 162
5.5 Summary of Phonons 168
Exploring Concepts 168
References 172
6 Electrons in Solids 173
Evolving Pictures 174
Superconductors 176
6.1 Properties of Electrons: A Review 176
6.1.1 Electrons Travel as Waves 176
6.1.1.1 Delocalization 176
6.1.1.2 Localization 178
6.1.2 Electrons Arrive as Particles: Statistics 178
6.1.3 The Fermi Surface 180
6.2 On to the Models 181
6.2.1 The Free-Electron Model 181
6.2.2 Nearly Free Electrons, Energy Bands, Energy Gaps, Density of States 184
6.2.2.1 Bloch’s Theorem 185
6.2.2.2 The Nearly Free 1D Model 185
6.2.2.3 Analyzing the 1D Nearly Free Solutions 187
6.2.2.4 Extending Dispersion Curves to 3D 190
6.2.3 Tight Binding or Linear Combination of Atomic Orbitals 191
6.2.3.1 The Formalism 193
6.2.3.2 The s-Band 194
6.2.3.3 s Bands in One Dimension 195
6.2.3.4 s Bands in Two Dimensions 195
6.2.3.5 s Bands in Three Dimensions 196
6.2.4 What About Orbitals Other Than s? 197
6.2.4.1 Building Bands in a Polymer 198
6.2.4.2 Bonding and Antibonding States 198
6.2.4.3 The Polyenes 199
6.2.4.4 Translating to Bloch’s Theorem 203
6.2.5 Tight Binding with a Basis 206
6.2.5.1 Hybridization 209
6.2.5.2 Graphene: A Two-Dimensional Example 211
6.2.5.3 Carbon Nanotubes 213
6.3 Are We Done Yet? 215
6.4 Summary 217
Exploring Concepts 218
References 223
7 Electrons in Solids Part II: Spatial Heterogeneity 225
7.1 Heterogeneity: Band-Level Diagrams and the Contact 226
7.2 Heterogeneity in Semiconductors 229
7.2.1 Semiconductors: Bandgaps and Doping 230
7.2.1.1 Band-Level Diagrams 230
7.2.1.2 Doping 230
7.2.1.3 Carrier Concentrations in Intrinsic and Doped Semiconductors 235
7.2.1.4 The Fermi Level vs. the Chemical Potential 239
7.2.1.5 Spectroscopy of the Dopant Levels 240
7.2.1.6 Carbon Does Not “Dope” Like Si 242
7.2.2 Junctions with Semiconductors 244
7.3 Other Types of Heterogeneity 249
7.4 Summary 251
Exploring Concepts 251
References 257
8 Electrons Moving in Solids 259
8.1 Phenomenology of Electron Dynamics in a Material 259
8.1.1 Free-Electron Metals 259
8.1.2 The Free-Electron Metal as a Fluid 262
8.1.3 Temperature and Conductivity 264
8.2 The Semiclassical Approach: The Boltzmann Equation 267
8.2.1 The Sources of Electron Scattering 267
8.2.2 The Nonequilibrium Distribution Function 268
8.2.3 The Relaxation Time ? 268
8.2.4 The Differential Equation for g(r; k; t) 268
8.2.5 Introducing Collisions 269
8.2.6 The Relaxation Time Approximation 270
8.2.7 Isotropic Scattering from Stationary States 271
8.2.8 A Simple Example: Ohm’s Law 271
8.2.9 Parabolic Bands 272
8.2.10 Another Simple Example: AC Conductivity and Linear Response 273
8.2.11 An Example with Anisotropy: ? = ?(r) and ∇rT ≠ 0 273
8.2.12 The Seebeck Effect and Thermopower 274
8.2.13 A Final Example: Static E and B Applied but ? ≠?(r) and ∇rT = 0 277
8.2.14 The Hall Effect and Magnetotransport 279
8.2.15 The Curious Case of Al 280
8.3 Coherent Transport: The Landauer–Buttiker Approach 281Contents ix
8.4 General Remarks on Measurements 283
8.4.1 Simple Conductivity 283
8.4.2 Conductivity of Small Particles 287
8.4.3 Conductivity of High Impedance Samples 288
8.4.4 Conductivity Measurements Without Contacts 289
8.5 Complications: Localization, Hopping, and General Bad Behavior 290
8.6 Summary 293
Exploring Concepts 293
References 297
9 Polarons, Solitons, Excitons, and Conducting Polymers 301
9.1 Distortions, Instabilities, and Transitions in One Dimension 303
9.1.1 Coupling Charge with the Lattice 303
9.1.2 Peierls Instability 305
9.1.3 Results of Peierls in Real Systems 308
9.1.3.1 Phonon Softening and the Kohn Anomaly 308
9.1.3.2 Fermi Surface Warping 309
9.2 Conjugation and the Double Bond 310
9.3 Conjugational Defects 313
9.4 The Soliton 317
9.4.1 Doping 319
9.4.2 Quasiparticles 320
9.5 Generation of Solitons 325
9.6 Nondegenerate Ground-State Polymers: Polarons 328
9.7 Fractional Charges 332
9.8 Soliton Lifetime 334
9.9 Conductivity and Solitons 337
9.10 Fibril Conduction 341
9.11 Hopping Conductivity: Variable Range Hopping vs.
Fluctuation-Assisted Tunneling 345
9.12 Highly Conducting Polymers 353
9.13 Magnetoresistance 354
9.14 Organic Molecular Devices 360
9.14.1 Molecular Switches 360
9.14.2 LB Diodes 363
9.14.3 Organic Light-Emitting Diodes 364
9.14.3.1 Fundamentals of OLEDs 366
9.14.3.2 Materials for OLEDs 370
9.14.3.3 Designs for OLEDs 371
9.14.3.4 Performance of OLEDs 372
9.14.4 Field-Induced Organic Emitters 373
9.14.5 Organic Lasers and Organic Light-Emitting Transistors 376
9.14.5.1 Current Densities 379
9.14.5.2 Contacts 379
9.14.5.3 Polarons and Triplets 379
9.14.6 Organic Solar Cells 380
9.14.7 Organic Field-Effect Transistors 384
9.14.8 Organic Thermoelectrics 385
9.15 Summary 387
Exploring Concepts 388
References 390
10 Correlation and Coupling 403
10.1 The Metal-to-Insulator Transition and the Mott Insulator 403
10.1.1 The Hamiltonian 406
10.1.2 The Lattice and Antiferromagnetic Ordering 407
10.1.3 Other Considerations: The Particle-Hole Symmetry (PHS) 407
10.1.4 The Hubbard Model in Lower Dimensions 408
10.1.5 Real One-Dimensional Mott Systems 410
10.2 The Superconductor 411
10.2.1 The Basic Phenomena 411
10.2.1.1 In What Compounds Has Superconductivity Been Observed? 415
10.2.2 A Basic Model 415
10.2.2.1 How Does an Attractive Potential Show Up Between Two Negatively Charged Particles? 416
10.2.2.2 Cooper Pair Binding 418
10.2.2.3 The BCS Ground State 420
10.2.2.4 Supplementary Thoughts 425
10.2.3 Superconductivity Measurements Are Tricky 428
10.2.4 Superconductivity and Dimensionality 430
10.2.5 More on Organic Superconductors 431
10.2.5.1 One-Dimensional Organic Superconductors 432
10.2.5.2 Two-Dimensional Organic Superconductors 435
10.2.5.3 Three-Dimensional Organic Superconductors 436
10.2.6 Trends 438
10.3 The Charge Density Wave 440
10.3.1 The Charge DensityWave and Peierls 440
10.3.1.1 Modulation of the Electron and Mass Densities 441
10.3.1.2 Starting with Polymers 441
10.3.1.3 A Gap Is Introduced 442
10.3.1.4 The Order Parameter 442
10.3.1.5 Phase Dynamics, Pinning, Commensurability, and Solitons 442
10.3.2 Peierls and Coulomb Interactions: Spin Interactions 446
10.3.2.1 4kF Charge DensityWaves 446
10.3.2.2 Spin PeierlsWaves 448
10.3.2.3 Spin DensityWaves 448
10.3.3 Phonon Dispersion: Phase and Amplitude in CDWs 450
10.3.4 More on Peierls–Frohlich Mechanisms 452
10.3.5 Spin DensityWaves and the Quantized Hall Effect 453
10.4 Plasmons 454
10.4.1 The Drude Model and the Dielectric Function 454
10.4.2 The Significance of the Plasma Frequency 455
10.5 Composite Particles and Quasiparticles: A Summary 457
Exploring Concepts 457
References 458
Intermission 465
11 Magnetic Interactions 467
11.1 Magnetism of the Atom 469
11.2 The Crystal Field 472
11.3 Magnetism in Condensed Systems 474
11.3.1 Paramagnetism 474
11.3.1.1 Curie Paramagnets 476
11.3.1.2 The Weiss Correction 477
11.3.1.3 Free-Electron Magnets 478
11.3.2 Diamagnetism 479
11.4 Dia- and Para-Foundations of Other Magnets 481
11.5 Mechanisms of Interaction: Spin Models 482
11.5.1 The Mean Field Model 483
11.5.2 Ising, Heisenberg, XY, and Hopfield 483
11.5.2.1 Ising Models 483
11.5.2.2 Heisenberg Models 485
11.5.2.3 XY models 485
11.5.2.4 Hopfield Models 487
11.5.3 SpinWave and Magnons 488
11.5.3.1 SpinWaves 488
11.5.3.2 Thermodynamics 491
11.5.3.3 The Particle Nature of Magnons 493
11.5.3.4 Stoner Excitations 494
11.5.3.5 Coupling to the Electromagnetic Field: Magnon–Photon Coupling 494
11.6 More Complicated Situations 494
11.6.1 Double Exchange 494
11.6.2 Super Exchange 496
11.6.3 RKKY 496
11.7 Time Reversal Symmetry 497
11.8 Summary 498
Exploring Concepts 499
References 501
12 Polarization of Materials 503
12.1 Simple Atomic Models 503
12.1.1 Linearity in the Response 504
12.1.2 Relating the Fields 507
12.2 Temperature Dependence 509
12.3 Time Dependence: ?(?) 510
12.4 A Familiar Equation in Optics 513
12.5 Understanding the Context 514
12.6 The Dielectric Function and Metals 514
12.7 Piezoelectrics, Pyroelectrics, and More 515
12.7.1 The h-BN Example 518
12.8 Summary 519
Exploring Concepts 519
References 523
13 Optical Interactions 525
13.1 Maxwell and the Solid (Review) 527
13.1.1 In a Vacuum 527
13.1.2 In a Material 528
13.1.3 A General Solution in the Solid 529
13.1.3.1 A Fun Notational Fact 531
13.2 Polarization Coupling: Polaritons 532
13.2.1 Phonons with Electrical Polarization 532
13.2.2 Phonons Meet Photons 534
13.2.3 The Phonon–Polariton 535
13.2.4 The Plasmon Polariton 538
13.3 Optical Transitions, Excitons, and Exciton Polaritons 543
13.3.1 Transitions 543
13.3.2 Carbon Nanotubes: An Example 546
13.3.3 Color Centers and Dopants 546
13.3.4 Excitons 548
13.3.5 Exciton Polaritons 549
13.4 Kramers–Kronig 549
13.5 Summary 551
Exploring Concepts 552
References 555
14 The End and the Beginning 557
Reference 558
Index 559
