Foundations of Solid State Physics Dimensionality and Symmetry

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 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(𝜔) and 𝜔_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 𝜏 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: 𝜇 = 𝜇(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 𝜇 ≠𝜇(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–Büttiker 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–Fröhlich 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: 𝜀(𝜔) 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

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.