NANO/MICROSCALE HEAT TRANSFER

Zhuomin M. Zhang Georgia Institute of Technology

Atlanta, Georgia

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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

V I I I 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