NANO/MICROSCALE HEAT TRANSFER Zhuomin M. Zhang Georgia Institute of Technology Atlanta, Georgia New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto
CONTENTS Preface xiii List of Symbols xvii Chapter 1. Introduction 1 l. I Limitations of the Macroscopic Formulation / 2 1.2 The Length Scales / 3 1.3 From Ancient Philosophy to Contemporary Technologies / 5 1.3.1 Microelectronics and Information Technology / 6 1.3.2 Lasers, Optoelectronics, and Nanophotonics / 8 1.3.3 Microfabrication and Nanofabrication / 10 1.3.4 Probing and Manipulation of Small Structures / 12 1.3.5 Energy Conversion Devices / 15 1.3.6 Biomolecule Imaging and Molecular Electronics / 17 1.4 Objectives and Organization of This Book / 19 References / 22 Chapter 2. Overview of Macroscopic Thermal Sciences 25 2.1 Fundamentals of Thermodynamics / 25 2.1.1 The First Law of Thermodynamics / 26 2.1.2 Thermodynamic Equilibrium and the Second Law / 27 2.1.3 The Third Law of Thermodynamics / 31 2.2 Thermodynamic Functions and Properties / 32 2.2.1 Thermodynamic Relations / 32 2.2.2 The Gibbs Phase Rule / 34 2.2.3 Specific Heats / 36 2.3 Ideal Gas and Ideal Incompressible Models / 38 2.3.1 The Ideal Gas / 38 2.3.2 Incompressible Solids and Liquids / 40 2.4 Heat Transfer Basics / 41 2.4.1 Conduction / 42 2.4.2 Convection / 44 2.4.3 Radiation / 46 2.5 Summary / 51 References / 51 Problems / 52 Chapter 3. Elements of Statistical Thermodynamics and Quantum Theory 57 3.1 Statistical Mechanics of Independent Particles / 58 3.1.1 Macrostates versus Microstates / 59 3.1.2 Phase Space / 59 VII
VIII CONTENTS 3.1.3 Quantum Mechanics Considerations / 60 3.1.4 Equilibrium Distributions for Different Statistics / 62 3.2 Thermodynamic Relations / 67 3.2.1 Heat and Work / 67 3.2.2 Entropy / 67 3.2.3 The Lagrangian Multipliers / 68 3.2.4 Entropy at Absolute Zero Temperature / 68 3.2.5 Macroscopic Properties in Terms of the Partition Function / 69 3.3 Ideal Molecular Gases / 71 3.3.1 Monatomic Ideal Gases / 71 3.3.2 Maxwell's Velocity Distribution / 73 3.3.3 Diatomic and Polyatomic Ideal Gases / 75 3.4 Statistical Ensembles and Fluctuations / 81 3.5 Basic Quantum Mechanics / 82 3.5.1 The Schrödinger Equation / 82 3.5.2 A Particle in a Potential Well or a Box / 84 3.5.3 A Rigid Rotor / 86 3.5.4 Atomic Emission and the Bohr Radius / 88 3.5.5 A Harmonic Oscillator / 90 3.6 Emission and Absorption of Photons by Molecules or Atoms / 92 3.7 Energy, Mass, and Momentum in Terms of Relativity / 94 3.8 Summary / 96 References / 96 Problems / 96 Chapter 4. Kinetic Theory and Micro/Nanofluidics 101 4.1 Kinetic Description of Dilute Gases / 101 4.1.1 Local Average and Flux / 102 4.1.2 The Mean Free Path / 105 4.2 Transport Equations and Properties of Ideal Gases / 108 4.2.1 Shear Force and Viscosity / 109 4.2.2 Heat Diffusion / 110 4.2.3 Mass Diffusion / 112 4.2.4 Intermolecular Forces / 115 4.3 The Boltzmann Transport Equation / 116 4.3.1 Hydrodynamic Equations / 117 4.3.2 Fourier's Law and Thermal Conductivity / 119 4.4 Micro/Nanofluidics and Heat Transfer / 121 4.4.1 The Knudsen Number and Flow Regimes / 122 4.4.2 Velocity Slip and Temperature Jump / 124 ЛЛ.З Gas Conduction From the Continuum to the Free Molecule Regime / 129 4.5 Summary / 132 References / 132 Problems / 133 Chapter 5. Thermal Properties of Solids and the Size Effect 137 5.1 Specific Heat of Solids / 137 5.1.1 Lattice Vibration in Solids: The Phonon Gas / 137 5.1.2 The Debye Specific Heat Model / 139 5.1.3 Free Electron Gas in Metals / 143 5.2 Quantum Size Effect on the Specific Heat / 148 5.2.1 Periodic Boundary Conditions / 148 5.2.2 General Expressions of Lattice Specific Heat / 149 5.2.3 Dimensionality / 149 5.2.4 Thin Films Including Quantum Wells / 151 5.2.5 Nanocrystals and Carbon Nanotubes / 153
CONTENTS IX 5.3 Electrical and Thermal Conductivities of Solids / 154 5.3.1 Electrical Conductivity / 155 5.3.2 Thermal Conductivity of Metals / 158 5.3.3 Derivation of Conductivities from the BTE / 160 5.3.4 Thermal Conductivity of Insulators / 162 5.4 Thermoelectricity / 166 5.4.1 The Seebeck Effect and Thermoelectric Power / 167 5.4.2 The Peltier Effect and the Thomson Effect / 168 5.4.3 Thermoelectric Generation and Refrigeration / 170 5.4.4 Onsager's Theorem and Irreversible Thermodynamics / 172 5.5 Classical Size Effect on Conductivities and Quantum Conductance / 174 5.5.1 Classical Size Effect Based on Geometric Consideration / 174 5.5.2 Classical Size Effect Based on the BTE / 178 5.5.3 Quantum Conductance / 182 5.6 Summary / 187 References / 187 Problems / 190 Chapter 6. Electron and Phonon Transport 193 6.1 The Hall Effect / 193 6.2 General Classifications of Solids / 195 6.2.1 Electrons in Atoms / 195 6.2.2 Insulators, Conductors, and Semiconductors / 197 6.2.3 Atomic Binding in Solids / 199 6.3 Crystal Structures / 201 6.3.1 The Bravais Lattices / 201 6.3.2 Primitive Vectors and the Primitive Unit Cell / 204 6.3.3 Basis Made of Two or More Atoms / 206 6.4 Electronic Band Structures / 209 6.4.1 Reciprocal Lattices and the First Brillouin Zone / 209 6.4.2 Bloch's Theorem / 210 6.4.3 Band Structures of Metals and Semiconductors / 214 6.5 Phonon Dispersion and Scattering / 217 6.5.1 The 1-D Diatomic Chain / 217 6.5.2 Dispersion Relations for Real Crystals / 219 6.5.3 Phonon Scattering / 221 6.6 Electron Emission and Tunneling / 226 6.6.1 Photoelectric Effect / 226 6.6.2 Thermionic Emission / 227 6.6.3 Field Emission and Electron Tunneling / 229 6.7 Electrical Transport in Semiconductor Devices / 232 6.7.1 Number Density, Mobility, and the Hall Effect / 232 6.7.2 Generation and Recombination / 236 6.7.3 The p-n Junction / 238 6.7.4 Optoelectronic Applications / 240 6.8 Summary / 242 References / 242 Problems / 244 Chapter 7. Nonequilibrium Energy Transfer in Nanostructures 247 7.1 Phenomenological Theories / 248 7.1.1 Hyperbolic Heat Equation / 250 7.1.2 Dual-Phase-Lag Model / 254 7.1.3 Two-Temperature Model / 255
X CONTENTS 7.2 Heat Conduction Across Layered Structures / 262 7.2.1 Equation of Phonon Radiative Transfer (EPRT) / 265 7.2.2 Solution of the EPRT / 266 7.2.3 Thermal Boundary Resistance (TBR) / 271 7.3 Heat Conduction Regimes / 275 7.4 Summary / 278 References / 278 Problems / 281 Chapter 8. Fundamentals of Thermal Radiation 283 8.1 Electromagnetic Waves / 285 8.1.1 Maxwell's Equations / 285 8.1.2 The Wave Equation / 2S6 8.1.3 Polarization / 288 8.1.4 Energy Flux and Density / 290 8.1.5 Dielectric Function / 291 8.1.6 Propagating and Evanescent Waves / 293 8.2 Blackbody Radiation: The Photon Gas / 294 8.2.1 Planck's Law / 294 8.2.2 Radiation Thermometry / 298 8.2.3 Entropy and Radiation Pressure / 301 8.2.4 Limitations of Planck's Law / 305 8.3 Radiative Properties of Semi-Infinite Media / 306 8.3.1 Reflection and Refraction of a Plane Wave / 306 8.3.2 Emissivity / 311 8.3.3 Bidirectional Reflectance / 312 8.4 Dielectric Function Models / 314 8.4.1 Kramers-Kronig Dispersion Relations / 314 8.4.2 The Drude Model for Free Carriers / 315 8.4.3 The Lorentz Oscillator Model for Lattice Absorption / 318 8.4.4 Semiconductors / 321 8.4.5 Superconductors / 325 8.4.6 Metamaterials with a Magnetic Response / 526 8.5 Summary / 329 References / 329 Problems / 330 Chapter 9. Radiative Properties of Nanomaterials 333 9.1 Radiative Properties of a Single Layer / 333 9.1.1 The Ray Tracing Method for a Thick Layer / 334 9.1.2 Thin Films / 335 9.1.3 Partial Coherence / 340 9.1.4 Effect of Surface Scattering / 344 9.2 Radiative Properties of Multilayer Structures / 346 9.2.1 Thin Films with Two or Three Layers / 347 9.2.2 The Matrix Formulation / 348 9.2.3 Radiative Properties of Thin Films on a Thick Substrate / 350 9.2.4 Local Energy Density and Absorption Distribution / 352 9.3 Photonic Crystals / 352 9.4 Periodic Gratings / 356 9.4.1 Rigorous Coupled-Wave Analysis (RCWA) / 358 9.4.2 Effective Medium Formulations / 360 9.5 Bidirectional Reflectance Distribution Function (BRDF) / 362 9.5.1 The Analytical Model / 363 9.5.2 The Monte Carlo Method / 364
CONTENTS XI 9.5.3 Surface Characterization / 367 9.5.4 BRDF Measurements / 368 9.5.5 Comparison of Modeling with Measurements / 370 9.6 Summary / 372 References / 373 Problems I 374 Chapter 10. Near-Field Energy Transfer 377 10.1 Total Internal Reflection, Guided Waves, and Photon Tunneling / 378 10.1.1 The Goos-Hänchen Shift / 379 10.1.2 Waveguides and Optical Fibers / 382 10.1.3 Photon Tunneling by Coupled Evanescent Waves / 386 10.1.4 Thermal Energy Transfer between Closely Spaced Dielectrics / 389 10.1.5 Resonance Tunneling through Periodic Dielectric Layers / 391 10.1.6 Photon Tunneling with Negative Index Materials / 393 10.2 Polaritons or Electromagnetic Surface Waves / 395 10.2.1 Surface Plasmon and Phonon Polaritons / 396 10.2.2 Coupled Surface Polaritons and Bulk Polaritons / 401 10.2.3 Polariton-Enhanced Transmission of Layered Structures / 405 10.2.4 Radiation Transmission through Nanostructures / 408 10.2.5 Superlens for Perfect Imaging and the Energy Streamlines / 410 10.3 Spectral and Directional Control of Thermal Radiation I 414 10.3.1 Gratings and Microcavities / 417 10.3.2 Metamaterials / 421 10.3.3 Modified Photonic Crystals for Coherent Thermal Emission / 422 10.4 Radiation Heat Transfer at Nanometer Distances / 425 10.4.1 The Fluctuational Electrodynamics / 426 10.4.2 Heat Transfer between Parallel Plates / 428 10.4.3 Asymptotic Formulation / 430 10.4.4 Nanoscale Radiation Heat Transfer between Doped Silicon / 431 10.5 Summary / 436 References / 437 Problems / 440 Appendix A. Physical Constants, Conversion Factors, and SI Prefixes 443 Physical Constants / 443 Conversion Factors / 443 SI Prefixes / 443 Appendix B. Mathematical Background 445 B.l Some Useful Formulae / 445 B.l.l Series and Integrals / 445 B.l.2 The Error Function / 446 B.1.3 Stirling's Formula / 447 B.2 The Method of Lagrange Multipliers / 447 B.3 Permutation and Combination / 448 B.4 Events and Probabilities / 450 B.5 Distribution Functions and the Probability Density Function / 451 B.6 Complex Variables / 454 B.7 The Plane Wave Solution / 455 B.8 The Sommerfeld Expansion / 459 Index 461