Nanoscale Energy Transport and Conversion A Parallel Treatment of Electrons, Molecules, Phonons, and Photons Gang Chen Massachusetts Institute of Technology OXFORD UNIVERSITY PRESS 2005
Contents Foreword, vii Introduction, 3 1 There Is Plenty of Room at the Bottom, 4 2 Classical Definition of Temperature and Heat, 9 3 Macroscopic Theory of Heat Transfer, 9 1.3.1 Conduction, 9 1.3.2 Convection, 11 1.3.3 Radiation, 13 1.3.4 Energy Balance, 16 1.3.5 Local Equilibrium, 17 1.3.6 Scaling Trends under Macroscopic Theories, 17 4 Microscopic Picture of Heat Carriers and Their Transport, 18 1.4.1 Heat Carriers, 18 1.4.2 Allowable Energy Levels of Heat Carriers, 22 1.4.3 Statistical Distribution of Energy Carriers, 23 1.4.4 Simple Kinetic Theory, 25 1.4.5 Mean Free Path, 27 5 Micro- and Nanoscale Transport Phenomena, 28 1.5.1 Classical Size Effects, 28 1.5.2 Quantum Size Effects, 29 1.5.3 Fast Transport Phenomena, 30 6 Philosophy of This Book, 32 7 Nomenclature for Chapter 1, 34 8 References, 35 9 Exercises, 37 Material Waves and Energy Quantization, 43 1 Basic Wave Characteristics, 44 2 Wave Nature of Matter, 46
XVIII 2.2.1 Wave-Particle Duality of Light, 46 2.2.2 Material Waves, 48 2.2.3 The Schrödinger Equation, 49 2.3 Example Solutions of the Schrödinger Equation, 52 2.3.1 Free Particles, 52 2.3.2 Particle in a One-Dimensional Potential Well, 53 2.3.3 Electron Spin and the Pauli Exclusion Principle, 58 2.3.4 Harmonic Oscillator, 59 2.3.5 The Rigid Rotor, 63 2.3.6 Electronic Energy Levels of the Hydrogen Atom, 64 2.4 Summary of Chapter 2, 70 2.5 Nomenclature for Chapter 2, 72 2.6 References, 72 2.7 Exercises, 73 3 Energy States in Solids 77 3.1 Crystal Structure, 78 3.1.1 Description of Lattices in Real Space, 78 3.1.2 Real Crystals, 81 3.1.3 Crystal Bonding Potential, 84 3.1.4 Reciprocal Lattice, 87 3.2 Electron Energy States in Crystals, 91 3.2.1 One-Dimensional Periodic Potential (Kronig-Penney Model), 91 3.2.2 Electron Energy Bands in Real Crystals, 98 3.3 Lattice Vibration and Phonons, 100 3.3.1 One-Dimensional Monatomic Lattice Chains, 100 3.3.2 Energy Quantization and Phonons, 103 3.3.3 One-Dimensional Diatomic and Polyatomic Lattice Chains, 104 3.3.4 Phonons in Three-Dimensional Crystals, 105 3.4 Density of States, 105 3.4.1 Electron Density of States, 107 3.4.2 Phonon Density of States, 109 3.4.3 Photon Density of States, 110 3.4.4 Differential Density of States and Solid Angle, 111 3.5 Energy Levels in Artificial Structures, 111 3.5.1 Quantum Wells, Wires, Dots, and Carbon Nanotubes, 111 3.5.2 Artificial Periodic Structures, 114
3.6 Summary of Chapter 3, 117 3.7 Nomenclature for Chapter 3, 118 3.8 References, 119 3.9 Exercises, 121 xix 4 Statistical Thermodynamics and Thermal Energy Storage 123 4.1 Ensembles and Statistical Distribution Functions, 124 4.1.1 Microcanonical Ensemble and Entropy, 124 4.1.2 Canonical and Grand Canonical Ensembles, 127 4.1.3 Molecular Partition Functions, 130 4.1.4 Fermi-Dirac, Bose-Einstein, and Boltzmann Distributions, 134 4.2 Internal Energy and Specific Heat, 137 4.2.1 Gases, 138 4.2.2 Electrons in Crystals, 141 4.2.3 Phonons, 144 4.2.4 Photons, 146 4.3 Size Effects on Internal Energy and Specific Heat, 148 4.4 Summary of Chapter 4, 150 4.5 Nomenclature for Chapter 4, 153 4.6 References, 154 4.7 Exercises, 155 5 Energy Transfer by Waves, 159 5.1 Plane Waves, 160 5.1.1 Plane Electron Waves, 161 5.1.2 Plane Electromagnetic Waves, 161 5.1.3 Plane Acoustic Waves, 167 5.2 Interface Reflection and Refraction of a Plane Wave, 169 5.2.1 Electron Waves, 169 5.2.2 Electromagnetic Waves, 171 5.2.3 Acoustic Waves, 178 5.2.4 Thermal Boundary Resistance, 180 5.3 Wave Propagation in Thin Films, 185 5.3.1 Propagation of EM Waves, 186 5.3.2 Phonons and Acoustic Waves, 191 5.3.3 Electron Waves, 193
XX 5.4 Evanescent Waves and Tunneling, 194 5.4.1 Evanescent Waves 194 5.4.2 Tunneling 195 5.5 Energy Transfer in Nanostructures: Landauer Formalism, 198 5.6 Transition to Particle Description, 204 5.6.1 Wave Packets and Group velocity, 204 5.6.2 Coherence and Transition to Particle Description, 207 5.7 Summary of Chapter 5,216 5.8 Nomenclature for Chapter 5,218 5.9 References, 220 5.10 Exercises, 223 6 Particle Description of Transport Processes: Classical Laws, 227 6.1 The Liouville Equation and the Boltzmann Equation, 228 6.1.1 The Phase Space and Liouville's Equation, 228 6.1.2 The Boltzmann Equation, 230 6.1.3 Intensity for Energy Flow, 233 6.2 Carrier Scattering, 233 6.2.1 Scattering Integral and Relaxation Time Approximation, 234 6.2.2 Scattering of Phonons, 237 6.2.3 Scattering of Electrons, 240 6.2.4 Scattering of Photons, 240 6.2.5 Scattering of Molecules, 242 6.3 Classical Constitutive Laws, 242 6.3.1 Fourier Law and Phonon Thermal Conductivity, 243 6.3.2 Newton's Shear Stress Law, 247 6.3.3 Ohm's Law and the Wiedemann-Franz Law, 249 6.3.4 Thermoelectric Effects and Onsager Relations, 254 6.3.5 Hyperpolic Heat Conduction Equation and Its Validity, 258 6.3.6 Meaning of Local Equilibrium and Validity of Diffusion Theories, 260 6.4 Conservative Equations, 262 6.4.1 Navier-Stokes Equations, 263 6.4.2 Electrohydrodynamic Equation, 266 6.4.3 Phonon Hydrodynamic Equations, 268 6.5 Summary of Chapter 6, 273 6.7 Nomenclature for Chapter 6, 275 6.8 References, 277 6.9 Exercises, 279
7 Classical Size Effects, 282 7.1 Size Effects on Electron and Phonon Conduction Parallel to Boundaries, 283 7.1.1 Electrical Conduction along Thin Films, 285 7.1.2 Phonon Heat Conduction along Thin Films, 288 7.2 Transport Perpendicular to the Boundaries, 292 7.2.1 Thermal Radiation between Two Parallel Plates, 292 7.2.2 Heat Conduction across Thin Films and Superlattices, 299 7.2.3 Rarefied Gas Heat Conduction between Two Parallel Plates, 302 7.2.4 Current Flow across Heterojunctions, 307 7.3 Rarefied Poiseuille Flow and Knudsen Minimum, 308 7.4 Transport in Nonplanar Structures, 313 7.4.1 Thermal Radiation between Concentric Cylinders and Spheres, 314 7.4.2 Rarefied Gas How and Convection, 314 7.4.3 Phonon Heat Conduction, 315 7.4.4 Multidimensional Transport Problems, 316 7.5 Diffusion Approximation with Diffusion-Transmission Boundary Conditions, 317 7.5.1 Thermal Radiation between Two Parallel Plates, 319 7.5.2 Heat Conduction in Thin Films, 321 7.5.3 Electron Transport across an Interface: Thermionic Emission, 322 7.5.4 Velocity Slip for Rarefied Gas Flow, 327 7.6 Ballistic-Diffusive Treatments, 331 7.6.1 Modified Differential Approximation for Thermal Radiation, 331 7.6.2 Ballistic-Diffusive Equations for Phonon Transport, 333 7.7 Summary of Chapter 7, 336 7.8 Nomenclature for Chapter 7, 338 7.9 References, 340 7.10 Exercises, 344 xxi 8 Energy Conversion and Coupled Transport Processes, 348 8.1 Carrier Scattering, Generation, and Recombination, 349 8.1.1 Nonequilibrium Electron-Phonon Interactions, 349 8.1.2 Photon Absorption and Carrier Excitation, 358 8.1.3 Relaxation and Recombination of Excited Carriers, 363 8.1.4 Boltzmann Equation Revisited, 366
xxii 8.2 Coupled Nonequilibrium Electron-Phonon Transport without Recombination, 367 8.2.1 Hot Electron Effects in Short Pulse Laser Heating of Metals, 369 8.2.2 Hot Electron and Hot Phonon Effects in Semiconductor Devices, 370 8.2.3 Cold and Hot Phonons in Energy Conversion Devices, 373 8.3 Energy Exchange in Semiconductor Devices with Recombination, 373 8.3.1 Energy Source Formulation, 373 8.3.2 Energy Conversion in a p-n Junction, 376 8.3.3 Radiation Heating of Semiconductors, 384 8.4 Nanostructures for Energy Conversion, 386 8.4.1 Thermoelectric Devices, 386 8.4.2 Solar Cells and Thermophotovoltaic Power Conversion, 391 8.5 Summary of Chapter 8, 395 8.6 Nomenclature for Chapter 8, 396 8.7 References, 398 8.8 Exercises, 401 9 Liquids and Their Interfaces, 404 9.1 Bulk Liquids and Their Transport Properties, 405 9.1.1 Radial Distribution Function and van der Waals Equation of State, 405 9.1.2 Kinetic Theories of Liquids, 408 9.1.3 Brownian Motion and the Langevin Equation, 411 9.2 Forces and Potentials between Particles and Surfaces, 416 9.2.1 Intermolecular Potentials, 417 9.2.2 Van der Waals Potential and Force between Surfaces, 419 9.2.3 Electric Double Layer Potential and Force at Interfaces, 421 9.2.4 Surface Forces and Potentials Due to Molecular Structures, 427 9.2.5 Surface Tension, 428 9.3 Size Effects on Single-Phase Flow and Convection, 433 9.3.1 Pressure-Driven Flow and Heat Transfer in Micro- and Nanochannels, 433 9.3.2 Electrokinetic Hows, 436 9.4 Size Effects on Phase Transition, 438 9.4.1 Curvature Effect on Vapor Pressure of Droplets, 439 9.4.2 Curvature Effect on Equilibrium Phase Transition Temperature, 441 9.4.3 Extension to Solid Particles, 441 9.4.4 Curvature Effect on Surface Tension, 442 9.5 Summary of Chapter 9, 443 9.6 Nomenclature for Chaper 9, 445
XXÜi 9.7 References, 447 9.8 Exercises, 449 10 Molecular Dynamics Simulation 452 10.1 The Equations of Motion, 453 10.2 Interatomic Potential, 458 10.3 Statistical Foundation for Molecular Dynamic Simulations, 462 10.3.1 Time Average versus Ensemble Average, 462 10.3.2 Response Function and Kramers-Kronig Relations, 464 10.3.3 Linear Response Theory, 466 10.3.4 Linear Response to Internal Thermal Disturbance, 473 10.3.5 Microscopic Expressions of Thermodynamic and Transport Properties, 476 10.3.6 Thermostatted Ensembles, 479 10.4 Solving the Equations of Motion, 483 10.4.1 Numerical Integration of the Equations of Motion, 483 10.4.2 Initial Conditions, 485 10.4.3 Periodic Boundary Condition, 485 10.5 Molecular Dynamics Simulation of Thermal Transport, 486 10.5.1 Equilibrium Molecular Dynamics Simulation, 486 10.5.2 Nonequilibrium Molecular Dynamics Simulations, 490 10.5.3 Molecular Dynamics Simulation of Nanoscale Heat Transfer, 493 10.6 Summary of Chapter 10, 494 10.7 Nomenclature for Chapter 10,496 10.8 References, 498 10.9 Exercises, 502 Appendix A: Homogeneous Semiconductors, 505 Appendix B: Seconductor p-n Junctions, 509 Index, 513 Units and Their Conversions, 530 Physical Constants, 531