Preface ...................................................... xvii
Acknowledgments ............................................... xxi
1 Introduction and Preliminaries ............................. 1
1.1 Principal Carriers: Phonon, Electron, Fluid Particle,
and Photon ................................................. 3
1.1.1 Phonon .............................................. 4
1.1.2 Electron (and Hole) ................................. 7
1.1.3 Fluid Particle ...................................... 8
1.1.4 Photon .............................................. 9
1.2 Equilibrium and Nonequilibrium Energy Occupancy
Distributions ............................................. 10
1.2.1 Nonequilibrium Energy Carrier Occupancy by Energy
Conversion ......................................... 10
1.2.2 Transport Phenomena Related to Energy Occupancy
Distributions ...................................... 14
1.3 Particles, Waves, Wave Packets and Quasi-Particles, and
Density of States ......................................... 16
1.4 A History of Contributions Toward Heat Transfer Physics ... 17
1.5 Fundamental Constants and Fine-Structure Scales ........... 20
1.5.1 Boltzmann and Planck Constants ..................... 20
1.5.2 Atomic Units and Fine-Structure Scales ............. 21
1.6 Principal Carriers: Concentration, Energy, Kinetics, and
Speed ..................................................... 23
1.6.1 Principal-Energy Carriers Concentration ............ 24
1.6.2 Principal-Carrier Energy ........................... 25
1.6.3 Principal-Carrier Energy Transport/Transformation
Kinetics ........................................... 26
1.6.4 Principal-Carrier Speed ............................ 29
1.7 Periodic Table of Elements ................................ 29
1.8 Heat Transfer Physics: Atomic-Level Energy Kinetics ....... 32
1.8.1 Thermal Energy Storage ............................. 32
1.8.2 Thermal Energy Transport ........................... 33
1.8.3 Thermal Energy Transformation ...................... 35
1.9 Density of States and Carrier Density ..................... 40
1.10 Ab initio/MD/BTE/Macroscopic Treatments ................... 42
1.11 Scope ..................................................... 44
1.12 Problems .................................................. 46
2 Molecular Orbitals/Potentials/Dynamics and Quantum
Energy States ............................................. 50
2.1 Interatomic Forces and Potential Wells .................... 50
2.1.1 Interatomic Forces ................................. 52
2.1.2 Intermolecular Forces .............................. 52
2.1.3 Kinetic and Potential Energies and Potential
Wells .............................................. 53
2.2 Orbitals and Interatomic Potential Models ................. 58
2.2.1 Atomic and Molecular Electron Orbitals ............. 58
2.2.2 Ab initio Computation of Interatomic Potentials .... 61
2.2.3 Potential Models and Phases ........................ 65
2.2.4 Examples of Atomic Bond Length and Energy .......... 72
2.2.5 Radial Distribution of Atoms in Dense Phase ........ 73
2.3 Molecular Ensembles, Temperature, and Thermodynamic
Relations ................................................. 75
2.3.1 Ensembles and Computational Molecular Dynamics ..... 75
2.3.2 Energy Equipartition ............................... 75
2.3.3 Thermodynamic Relations ............................ 76
2.4 Hamiltonian Mechanics ..................................... 77
2.4.1 Classical and Quantum Hamiltonians ................. 77
2.4.2 Probability and Partition Function ................. 79
2.4.3 Ergodic Hypothesis in Theoretical Statistical
Mechanics .......................................... 81
2.5 Molecular Dynamics Simulations ............................ 81
2.5.1 Ensemble and Discretization of Governing
Equations .......................................... 81
2.5.2 A Molecular Dynamics Simulation Case Study: L-J
Ar FCC ............................................. 86
2.5.3 L-J FCC MD Scales in Classical Harmonic
Oscillator ......................................... 88
2.5.4 L-J Potential Phase Transformations ................ 92
2.5.5 Atomic Displacement in Solids and Quantum Effects .. 93
2.5.6 Specific Heat Capacity ............................. 94
2.5.7 Heat Flux Vector ................................... 95
2.6 Schrцdinger Equation and Quantum Energy States ............ 96
2.6.1 Time-Dependent Schrцdinger Equation and Wave
Vector ............................................. 97
2.6.2 Bloch Wave Form .................................... 99
2.6.2 Quantum-Mechanics Formalism, Bra-Ket and Matrix
Element ........................................... 100
2.6.4 Quantum Mechanical, Harmonic Oscillator ........... 101
2.6.5 Periodic, Free Electron (Gas) Model for Metals .... 105
2.6.6 Electron Orbitals in Hydrogenlike Atoms ........... 109
2.6.7 Perturbation and Numerical Solutions to
Schrцdinger Equation .............................. 111
2.7 Problems ................................................. 115
3 Carrier Energy Transport and Transformation Theories ..... 119
3.1 Boltzmann Transport Equation ............................. 120
3.1.1 Particle Probability Distribution (Occupancy)
Function .......................................... 120
3.1.2 A Simple Derivation of BTE ........................ 120
3.1.3 In- and Out-Scattering ............................ 123
3.1.4 Relaxation-Time Approximation of Scattering and
Transport Properties .............................. 124
3.1.5 Boltzmann Transport Scales ........................ 128
3.1.6 Momentum, Energy, and Average Relaxation Times .... 129
3.1.7 Moments of BTE .................................... 130
3.1.8 Numerical Solution to BTE ......................... 131
3.2 Energy Transition Kinetics and Fermi Golden Rule ......... 131
3.2.1 Elastic and Inelastic Scattering .................. 132
3.2.2 Phonon Interaction and Transition Rates ........... 133
3.2.3 Electron (and Hole) Interaction and Transition
Rates ............................................. 134
3.2.4 Fluid Particle Interaction and Transition Rates ... 138
3.2.5 Photon Interaction and Transition Rates ........... 138
3.3 Maxwell Equations and Electromagnetic Waves .............. 138
3.3.1 Maxwell Equations ................................. 138
3.3.2 Electromagnetic Wave Propagation Equation ......... 142
3.3.3 EM Wave and Photon Energy ......................... 144
3.3.4 Electric Dipole and Emission, Absorption, and
Scattering of EM Waves ............................ 145
3.3.5 Dielectric Function and Dielectric Heating ........ 147
3.3.6 Electrical Resistivity and Mobility and Joule
Heating ........................................... 151
3.4 Onsager Coupled Transport Coefficients ................... 152
3.5 Stochastic Particle Dynamics and Transport ............... 154
3.5.1 Langevin Particle Dynamics Equation and Brownian
Motion ............................................ 154
3.5.2 Fokker-Planck Particle Conservation Equation ...... 155
3.5.3 Mean-Field Theory ................................. 155
3.6 Equilibrium Fluctuation-Dissipation and Green-Kubo
Transport Theory ......................................... 156
3.7 Macroscopic Fluid Dynamics Equations ..................... 159
3.8 Macroscopic Elastic Mechanics Equations .................. 159
3.9 Macroscopic Scales ....................................... 161
3.10 Problems ................................................. 163
4 Phonon Energy Storage, Transport, and Transformation
Kinetics ................................................. 173
4.1 Phonon Dispersion in One-Dimensional Classical Lattice
Vibration ................................................ 174
4.2 Phonon Density of States and Debye Model ................. 181
4.2.1 Phonon DOS for One-Dimensional Lattice and van
Hove Singularities ................................ 182
4.2.2 Debye and Other Phonon DOS Models ................. 184
4.3 Reciprocal Lattice, Brillouin Zone, and Primitive Cell
and Its Basis ............................................ 186
4.3.1 Reciprocal Lattice ................................ 187
4.3.2 Brillouin Zone .................................... 189
4.3.3 Primitive Cell and Its Basis: Number of Phonon
Branches .......................................... 190
4.4 Normal Modes and Dynamical Matrix ........................ 191
4.5 Quantum Theory of Lattice Vibration ...................... 196
4.6 Examples of Phonon Dispersion and DOS .................... 198
4.7 Phonon Specific Heat Capacity and Debye Average
Acoustic Speed ........................................... 201
4.7.1 Acoustic Phonon Specific Heat Capacity ............ 201
4.7.2 Estimate of Directional Acoustic Velocity ......... 206
4.8 Atomic Displacement in Lattice Vibration ................. 208
4.9 Phonon BTE and Callaway Conductivity Model ............... 211
4.9.1 Single-Mode Relaxation Time ....................... 211
4.9.2 Callaway Phonon Conductivity Model from BTE ....... 212
4.9.3 Callaway-Holland Phonon Conductivity Model ........ 215
4.9.4 Phonon Scattering Relaxation Time Models .......... 215
4.9.5 Phonon Dispersion Models: Ge As Example ........... 223
4.9.6 Comparison of Dispersion Models ................... 226
4.9.7 Lattice Thermal Conductivity Prediction ........... 228
4.10 Einstein and Cahill-Pohl Minimum Phonon Conductivities ... 231
4.11 Material Metrics of Phonon Conductivity: Slack Relation .. 233
4.11.1 Derivation of Slack Relation ...................... 234
4.11.2 Force-Constant Combinative Rule for Arbitrary
Pair-Bond ......................................... 235
4.11.3 Evaluation of Sound Velocity and Debye
Temperature ....................................... 241
4.11.4 Prediction of Grьneisen Parameter ................. 244
4.11.5 Prediction of Thermal Conductivity ................ 249
4.12 Phonon Conductivity Decomposition: Acoustic Phonons ...... 254
4.12.1 Heat Current Autocorrelation Function ............. 255
4.12.2 Phonon Conductivity Decomposition ................. 258
4.12.3 Comparison with Experiment ........................ 260
4.12.4 Phonon Conductivity Decomposition: Optical
Phonons ........................................... 261
4.14 Quantum Corrections to MD/G-K Predictions ................ 262
4.15 Phonon Conductivity from BTE: Variational Method ......... 267
4.16 Experimental Data on Phonon Conductivity ................. 269
4.17 Phonon Boundary Resistance ............................... 271
4.18 Absorption of Ultrasound Waves in Solids ................. 275
4.19 Size Effects ............................................. 276
4.19.1 Finite-Size Effect on Phonon Conductivity ......... 276
4.19.2 Superlattice Phonon Conductivity .................. 278
4.19.3 Phonon Density of States of Nanoparticles ......... 280
4.19.4 Phonon Conductivity Rectification in Anisotropic,
One-Dimensional Systems ........................... 286
4.19.5 Heat Flow in Molecular Wire ....................... 287
4.19.6 Quantum Vibrational Energy Flow in
Nanostructures .................................... 288
4.19.7 Nanocone Conductivity ............................. 289
4.20 Problems ................................................. 289
5 Electron Energy Storage, Transport, and Transformation
Kinetics ................................................. 306
5.1 Schrцdinger Equation for Periodic-Potential Electronic
Band Structure ........................................... 309
5.2 Electronic Band Structure in One-Dimensional Ionic
Lattice .................................................. 311
5.3 Three-Dimensional Bands Using Tight-Binding
Approximation ............................................ 315
5.3.1 General LCAO ...................................... 315
5.3.2 Example of Tight-Binding Approximation: FCC,
Single s-Band ..................................... 317
5.4 Ab Initio Computation of Electron Band Structure ......... 319
5.5 Electron Band Structure for Semiconductors and Effective
Mass ..................................................... 321
5.6 Periodic Electron Gas Model for Metals ................... 325
5.7 Electron/Hole Density of Carrier and States for
Semiconductors ........................................... 327
5.8 Specific Heat Capacity of Conduction Electrons ........... 331
5.9 Electron BTE for Semiconductors: Thermoelectric Force .... 334
5.10 Electron Relaxation Time and Fermi Golden Rule ........... 335
5.11 Average Relaxation Time ‹‹τe›› for Power-Law τe
(Momentum) (Ee) .......................................... 338
5.12 Thermoelectric Transport Property Tensors for Power-Law
τe(Ee) ................................................... 343
5.13 ТЕ Transport Coefficients for Cubic Semiconductors ....... 346
5.13.1 Seebeck, Peltier, and Thomson Coefficients, and
Electrical and Thermal Conductivities ............. 346
5.13.2 Electron Mean Free Path for Metals ................ 348
5.14 Magnetic Field and Hall Factor and Coefficient ........... 349
5.15 Electron-Phonon Relaxation Times in Semiconductors ....... 350
5.15.1 Electron-Phonon Wave Function ..................... 351
5.15.2 Rate of Acoustic-Phonon Scattering of Electrons ... 353
5.15.3 Rate of Optical-Phonon Scattering of Electrons .... 354
5.15.4 Summary of Electron-Scattering Mechanisms and
Relaxation-Time Relations ......................... 359
5.16 ТЕ Transport Coefficients Data for Metals and
Semiconductors ........................................... 359
5.16.1 Structural Defects in Crystalline Solids .......... 359
5.16.2 Metals ............................................ 360
5.16.3 Semiconductors .................................... 366
5.16.4 ТЕ Figure of Merit Ze ............................. 372
5.17 Ab Initio Computation of ТЕ Transport Property Tensors ... 377
5.17.1 ТЕ Transport Tensors and Variable Chemical
Potential ......................................... 377
5.17.2 Introduction to BoltzTraP ......................... 379
5.17.3 Relaxation Times Based on Kane Band Model ......... 380
5.17.4 Predicted Seebeck Coefficient and Electrical
Conductivity ...................................... 385
5.17.5 Electric and Phonon Thermal Conductivities ........ 388
5.18 Electron and Phonon Transport Under Local Thermal
Nonequilibrium ........................................... 393
5.18.1 Derivations ....................................... 393
5.18.2 Phonon Modal Energy Equations ..................... 395
5.18.3 Summary of Conservation (Electrohydrodynamic)
Equations ......................................... 396
5.19 Cooling Length in Electron-Phonon Local Thermal
Nonequilibrium ........................................... 397
5.20 Electronic Energy States of Ions in Crystals ............. 400
5.21 Electronic Energy States of Gases ........................ 404
5.22 Size Effects ............................................. 407
5.22.1 Quantum Well for Improved ТЕ ZeT .................. 408
5.22.2 Reduced Electron-Phonon Scattering Rate in
Quantum Wells ..................................... 411
5.22.3 Electronic and Phonon Thermal Conductance of
Graphene-Flake Junctions .......................... 413
5.22.4 Heterobarrier for Converting Hot-Phonon Energy
to Electric Potential ............................. 418
5.23 Problems ................................................. 422
6 Fluid Particle Energy Storage, Transport, and
Transformation Kinetics .................................. 434
6.1 Fluid Particle Quantum Energy States and Partition
Functions ................................................ 436
6.1.1 Translational Energy and Partition Function ....... 436
6.1.2 Vibrational Energy and Partition Function ......... 438
6.1.3 Rotational Energy and Partition Function .......... 439
6.1.4 Electronic Energy and Partition Function .......... 440
6.1.5 Ab Initio Computation of Vibrational and
Rotational Energy States .......................... 441
6.2 Ideal-Gas Specific Heat Capacity ......................... 443
6.3 Dense-Fluid Specific Heat Capacity: van der Waals Model .. 447
6.4 Gas BTE, ƒ0ƒ, and Thermal Velocities ...................... 451
6.4.1 Interparticle Collisions .......................... 451
6.4.2 Equilibrium Distribution Function for
Translational Energy .............................. 453
6.4.3 Inclusion of Gravitational Potential Energy ....... 456
6.5 Ideal-Gas Binary Collision Rate and Relaxation Time ...... 457
6.6 Ideal-Gas Mean Free Path and Viscosity ................... 459
6.7 Kinetic-Limit Evaporation/Condensation Heat Transfer
Rate ..................................................... 461
6.8 Ideal-Gas Thermal Conductivity from BTE .................. 462
6.8.1 Nonequilibrium BTE and Relaxation-Time
Approximation ..................................... 462
6.8.2 Thermal Conductivity .............................. 463
6.9 Liquid Thermal Conductivity from Mean Free Path and
Molecular Dynamics ....................................... 469
6.10 Effective Conductivity with Dispersed Particles in
Thermal Motion ........................................... 470
6.10.1 Langevin Derivation of Brownian Diffusion ......... 471
6.10.2 Thermal Relaxation Time and Effective Fluid
Thermal Conductivity .............................. 472
6.11 Interaction of Moving Fluid Particle and Surface ......... 474
6.11.1 Fluid Flow Regimes ................................ 474
6.11.2 Knudson-Flow-Regime Surface Accommodation and
Slip Coefficients ................................. 476
6.11.3 Slip Coefficients in Transitional-Flow Regime ..... 480
6.11.4 Solid Particle Thermophoresis in Gases ............ 481
6.11.5 Physical Adsorption and Desorption ................ 482
6.11.6 Disjoining Pressure in Ultrathin-Liquid Films ..... 486
6.12 Turbulent-Flow Structure and Boundary-Layer Transport .... 487
6.12.1 Turbulent Kinetic Energy Spectrum for
Homogeneous Turbulence ............................ 489
6.12.2 Boundary-Layer Turbulent Heat Flux ................ 491
6.12.3 Turbulent Mixing Length and Turbulent Thermal
Conductivity ...................................... 492
6.12.4 Spatial Variation of Boundary-Layer Turbulent
Mixing Length ..................................... 493
6.12.5 Turbulent Mixing Using Lagrangian Langevin
Equation .......................................... 494
6.13 Thermal Plasmas: Plasma Thermal Conductivity ............. 494
6.13.1 Free Electron Density and Plasma Thermal
Conductivity ...................................... 496
6.13.2 Thermal Nonequilibrium Plasma Energy Equation ..... 500
6.13.3 Species Concentrations for Two-Temperature
Plasmas ........................................... 501
6.13.4 Kinetics of Energy Exchange Between Electrons
and Heavier Species ............................... 501
6.14 Size Effects ............................................. 502
6.14.1 Effective Thermal Conductivity in Gas-Filled
Narrow Gaps ....................................... 502
6.14.2 Thermal Creep (Slip) Flow in Narrow Gaps .......... 508
6.15 Problems ................................................. 511
7 Photon Energy Storage, Transport, and Transformation
Kinetics ................................................. 519
7.1 Quantum-Particle Treatment: Photon Gas and Blackbody
Emission ................................................. 523
7.2 Lasers and Near-Field (EM Wave) Thermal Emission ......... 527
7.2.1 Lasers and Narrow-Band Emissions .................. 527
7.2.2 Classical EM Wave Near-Field Thermal Emission ..... 528
7.3 Quantum and Semiclassical Treatments of Photon-Matter
Interaction .............................................. 529
7.3.1 Hamiltonians of Radiation Field ................... 530
7.3.2 Photon-Matter Interactions ........................ 533
7.4 Photon Absorption and Emission in Two-Level Electronic
Systems .................................................. 534
7.4.1 Einstein Excited-State Population Rate Equation ... 535
7.4.2 Einstein Coefficients for Equilibrium Electronic
Population ........................................ 537
7.4.3 Spontaneous Versus Stimulated Emissions in
Equilibrium Thermal Cavity ƒ0ph ........................... 538
7.4.4 Spectral Absorption Coefficient and Cross-
Section Area ...................................... 539
7.5 Particle Treatment: Photon BTE with Absorption,
Emission, and Scattering ................................. 541
7.5.1 Combining Absorption and Emission ................. 543
7.5.2 Photon-Free Electron Elastic Scattering Rate
and Cross-Section Area ............................ 544
7.6 Photon Intensity: Equation of Radiative Transfer ......... 547
7.6.1 General Form of ERT ............................... 547
7.6.2 Optically Thick Limit, Mean Free Path, and
Radiant Conductivity .............................. 549
7.7 Wave Treatment: Field Enhancement and Photon
Localization ............................................. 552
7.7.1 Photon Localization in One-Dimensional
Multilayer ........................................ 552
7.7.2 Coherence and Electric Field Enhancement .......... 556
7.7.3 Comparison with Particle Treatment (ERT) .......... 558
7.8 Continuous and Band Photon Absorption .................... 562
7.8.1 Photon Absorption Coefficient for Solids .......... 562
7.8.2 Photon Absorption Coefficient for Gases ........... 567
7.9 Continuous and Band Photon Emission ...................... 571
7.9.1 Emission Mechanisms ............................... 571
7.9.2 Absorption and Emission Reciprocity (Kirchhoff
Law) .............................................. 572
7.10 Spectral Surface Emissivity .............................. 574
7.11 Radiative and Nonradiative Decays and Quantum
Efficiency ............................................... 577
7.12 Anti-Stokes Fluorescence: Photon-Electron-Phonon
Couplings ................................................ 582
7.12.1 Anti-Stokes Laser Cooling (Phonon Absorption) of
Ion-Doped Solids .................................. 582
7.12.2 Laser Cooling Efficiency .......................... 584
7.12.3 Photon-Electron-Phonon Transition Rate Using
Weak Coupling Approximation ....................... 587
7.12.4 Time Scales for Laser Cooling of Solids (Weak
Couplings) ........................................ 592
7.12.5 Optimal Host Material ............................. 596
7.12.6 Photon-Electron and Electron-Phonon Transition
Rates Using Strong Couplings (Ab Initio
Computation) ...................................... 598
7.13 Gas Lasers and Laser Cooling of Gases .................... 608
7.13.1 Molecular-Gas Lasers .............................. 608
7.13.2 Laser Doppler Cooling of Atomic Gases and Doppler
Temperature ....................................... 627
7.14 Photovoltaic Solar Cell: Reducing Phonon Emission ........ 631
7.14.1 Single-Bandgap Ideal Solar PV Efficiency .......... 634
7.14.2 Multiple-Bandgap Ideal Solar PV Efficiency ........ 636
7.14.3 Semiempirical Solar PV Efficiency ................. 639
7.15 Size Effects ............................................. 642
7.15.1 Enhanced Near-Field Radiative Heat Transfer ....... 642
7.15.2 Photon Energy Confinement by Near-Field Optical.
Microscopy ........................................ 645
7.15.3 Hot Phonon Recycling in Photonics ................. 646
7.16 Problems ................................................. 650
APPENDIX A: Tables of Properties and Universal Constants ...... 661
APPENDIX В: Derivation of Green-Kubo Relation ................. 668
APPENDIX C: Derivation of Minimum Phonon Conductivity
Relations ......................................... 676
APPENDIX D: Derivation of Phonon Boundary Resistance .......... 683
APPENDIX E: Derivation of Fermi Golden Rule ................... 689
APPENDIX F: Derivation of Equilibrium, Particle Probability
Distribution Functions ............................ 696
APPENDIX G: Phonon Contributions to the Seebeck Coefficient ... 701
APPENDIX H: Monte Carlo Method for Carrier Transport .......... 709
APPENDIX I: Ladder Operators .................................. 713
Nomenclature .................................................. 719
Abbreviations ................................................. 725
Glossary ...................................................... 727
Bibliography .................................................. 741
Index ......................................................... 765
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