Chapter
1.3 Some Aspects of Statistical Approach for Micro/Nanoscale Transport
2 Crystal Dynamics and Lattice Waves
2.2 Elementary Crystallography
2.2.1 Structure of Crystal Lattices
2.2.2 Reciprocal Lattices
2.2.3 Crystal Planes and Directions
2.3.1 Chain of Identical Atoms
2.3.3 Chain of Two Types of Atoms
2.4.1 Phonon Scattering With Impurities and Defects
2.4.2 Phonon Scattering With the Crystal Boundaries
2.4.3 Phonon–Phonon Scattering
2.4.3.2 Umklapp Processes
2.5.1 The Density of States
2.5.3 Thermal Conductivity
3 Some Aspects of Statistical Thermodynamics
3.2 Statistical Mechanics
3.2.1 Microstates and Macrostates
3.2.2.1 Classical Probability
3.2.2.2 Statistical Probability
3.2.3 Probability Distributions
3.2.3.1 Discrete Distributions
3.2.3.2 Continuous Distributions
3.3.1 The Microcanonical Ensemble
3.3.1.1 Postulate 1: The Probability for All Microstates Are Equal
3.3.1.2 Postulate 2: Boltzmann Entropy Formula
3.3.1.3 Postulate 3: Largest Value of the Entropy Represents the Equilibrium State
3.3.2 Canonical Ensembles
3.3.3 Grand Canonical Ensemble
3.4 Statistical Distributions
3.4.1 Maxwell–Boltzmann Distribution
3.4.2 Fermi–Dirac Distribution
3.4.3 Bose–Einstein Distribution
4 Analysis of Energy Transport Equations at Micro/Nanoscales
4.2 Hyperbolic Heat Equation and Applications
4.2.1 Analysis and Solution of Hyperbolic Heat Equation
4.2.2 Perturbation Solution for Hyperbolic Heat Equation
4.2.3 Findings and Discussions
4.2.3.1 Temperature and Stress Fields
4.2.3.2 Perturbation Solution of Temperature Field
4.3 Electron Kinetic Theory Approach for Energy Transfer in Metallic Films
4.3.1 Formulation of Microscopic Energy Transport in Metallic Substrates
4.3.2 Parabolic Heating Model
4.3.3 Application of Laser Short-Pulse Heating
4.3.4 Findings of Numerical Simulations
4.4 Equation of Phonon Radiative Transfer
4.4.1 Transport Properties of a Dielectric Material
4.4.2 Heat Transfer Mechanism in a Thin Dielectric Film
4.4.3 Boltzmann Transport Equation
4.4.4 Equation of Phonon Radiative Transfer for Two-Dimensional Dielectric Thin Films
4.4.4.2 Equilibrium Intensity Calculation
4.4.4.3 Physical Significance of Each Term in EPRT
4.4.4.4 Polarization and Modes of Phonons
4.4.4.4.1 Longitudinal Acoustic
4.4.4.4.2 Transverse Acoustic
4.4.4.4.3 Longitudinal Optical
4.4.4.4.4 Transverse Optical
4.4.4.5 Equilibrium Equivalent Temperature
4.4.5 Equation of Phonon Radiative Transfer for One-Dimensional Dielectric Thin Films
4.4.5.1 Equilibrium Equivalent Temperature
4.4.5.3 Entropy Generation
4.4.6 One-Dimensional Metallic Thin Films
4.4.6.1 Modified Two-Equation Model
4.4.6.2 Electron–Phonon Coupling Resistance
4.4.6.3 Equation of Phonon Radiative Transfer for Metallic Films
4.4.6.3.1 EPRT for Lattice Subsystem
4.4.6.3.2 Longitudinal Acoustic
4.4.6.3.3 Transverse Acoustic
4.4.6.3.4 EPRT for Electron Subsystem
4.4.6.3.5 Definition of Temperature
4.4.6.4 Further Analysis of Phonon Radiative Transport in Metallic Thin Films
4.4.6.4.1 The Steady-State Case
4.4.8 Large Film Thickness
4.4.9 Large Film Thickness; Large Characteristic Time
4.4.10 Initial Conditions and Boundary Conditions
4.4.10.1 Initial Conditions
4.4.10.2 Boundary Conditions
4.4.10.2.1 Bottom Boundary
4.4.10.2.4 Right Boundary
4.4.10.2.6 Right Boundary
4.4.11 Interface Conditions
4.4.11.1 Acoustic Mismatch Model
4.4.11.2 Diffusive Mismatch Model
4.4.11.3 Cut-Off Mismatch Model
4.5 Thermal Stresses in Solids During Short Heating Durations and Applications
4.5.1 Analysis of Thermal Stress Under Irradiation Pulse
4.5.2 Findings and Discussion
4.6.1 Characteristics of Waves
4.6.3 Schrödinger Equation
4.6.4 Solution of the Schrödinger Equation
4.6.5 Examples of the Schrödinger Equation
4.6.5.1 The Infinite Square Well
5 Analytical Treatment of Phonon Transport in Thin Films
5.2 Analytical Solution of Reduced Equation for Phonon Radiative Transport
5.3 Analytical Solution of Steady-State Equation for Phonon Radiative Transport (EPRT) Due to Temperature Disturbance Acros...
5.3.1 Consideration of Dielectric Thin Film
5.3.2 Consideration of Thin Metallic Film
5.3.3 The Coupled System of Equations of Phonon Radiative Transfer in Lattice Subsystems of Aluminum Film
5.3.3.1 Left Boundary Condition
5.3.3.2 Right Boundary Condition
5.3.4.2 Electron Subsystem
5.3.4.3 Left Boundary Condition
5.3.4.4 Right Boundary Condition
5.4 Analytical Solution to Formulation of Electron Kinetic Theory
5.4.1 Consideration of Volumetric Heat Source (Case 1)
5.4.2 Consideration of Surface Heat Source (Case 2)
5.5 Findings and Discussions
5.5.1 Reduced Form of Equation for Phonon Radiative Transport
5.5.2 Analytical Solution of Equation for Phonon Radiative Transport for the Steady-State Temperature Disturbance Across th...
5.5.2.1 Dielectric Thin Film (Silicon and Diamond Films)
5.5.2.2 Metallic Thin Film (Aluminum Film)
5.5.3 Analytical Solution of Hyperbolic Equation Pertinent to Electron Kinetic Theory Approach
6 Heat Transfer Applications in One- and Two-Dimensional Thin Films
6.2 Transient Analysis and Case Studies
6.2.1 One-Dimensional Frequency-Dependent Analysis of Dielectric Films and Applications
6.2.1.1 A Case Study and Application
6.2.1.1.1 Initial Condition
6.2.1.1.2 Boundary Conditions
6.2.1.1.3 Limitations on the Phonon Wavenumber
6.2.1.1.4 Definition of Temperatures and Heat Flux
6.2.1.3 Findings of the Case Study
6.2.2 One-Dimensional Frequency-Independent Analysis of Metallic Thin Films and Applications
6.2.2.1 A Case Study and Application
6.2.2.1.1 Boundary Conditions
6.2.2.2.1 Solution Algorithm
6.2.2.3 Findings of the Case Study
6.2.3 Two-Dimensional Frequency-Dependent Dielectric Thin Films and Applications
6.2.3.1 A Case Study and Application
6.2.3.1.1 Initial and Boundary Conditions
6.2.3.1.1.1 Initial Condition
6.2.3.1.1.2 Top Boundary Condition
6.2.3.1.1.3 Bottom Boundary Condition
6.2.3.1.1.4 Left Boundary Condition
6.2.3.3 Findings of the Case Study
6.2.4 Two-Dimensional Frequency-Independent Metallic Thin Films and Applications
6.2.4.1 EPRT for Phonon Subsystem
6.2.4.2 EPRT for Electron Subsystem
6.2.4.3 Integral Energy Balance
6.2.4.4 Definition of Equivalent Equilibrium Temperature
6.2.4.6 A Case Study and Applications
6.2.4.6.1 Initial Condition
6.2.4.6.2 Boundary Conditions
6.2.4.7 Findings of the Case Study
6.3 Cross-Plane Transport and Case Studies
6.3.1 Cross-Plane Transport Across One-Dimensional Dielectric and Metallic Thin Films and Applications
6.3.1.1 Equation of Phonon Radiative Transfer
6.3.1.2 Initial and Boundary Conditions
6.3.1.2.1.1 Initial Condition
6.3.1.2.1.2 Left Boundary Condition
6.3.1.2.1.3 Right Boundary Condition
6.3.1.2.2.1 Initial Condition
6.3.1.2.2.2 Left Boundary Condition
6.3.1.2.2.3 Right Boundary Condition
6.3.1.2.3 Aluminum Film: Lattice (Phonon) Subsystem
6.3.1.2.3.1 Initial Condition
6.3.1.2.3.2 Left Boundary Condition
6.3.1.2.3.3 Right Boundary Condition
6.3.1.2.4 Aluminum Film: Electron Subsystem
6.3.1.2.4.1 Initial Condition
6.3.1.2.4.2 Left Boundary Condition
6.3.1.2.4.3 Right Boundary Condition
6.3.1.3 Interface Conditions
6.3.1.4 Definition of Temperature and Heat Flux
6.3.1.5 Transfer Matrix Approach for Multiple Thin Films
6.3.1.6 Refractive Index for a Dielectric Material
6.3.1.7 Extinction Coefficient and Absorption Coefficient
6.3.1.9 Findings of Laser Short-Pulse Heating of Three-Layer Thin Film Assembly
6.3.2 Cross-Plane Transport Across Two-Dimensional Dielectric Thin Films and Applications
6.3.2.1 Initial and Boundary Conditions
6.3.2.1.1.1 Initial Condition
6.3.2.1.1.2 Bottom Boundary Condition
6.3.2.1.1.3 Top Boundary Condition
6.3.2.1.1.4 Left Boundary Condition
6.3.2.2.2.1 Initial Condition
6.3.2.2.2.2 Bottom Boundary Condition
6.3.2.2.2.3 Top Boundary Condition
6.3.2.2.2.4 Right Boundary Condition
6.3.2.3 Interface Condition
6.3.2.5 Findings of Laser Short-Pulse Heating of Aluminum-Diamond Thin Films Pair
7 Thermal Boundary Resistance for Cross-Plane Transport and the Presence of Minute Vacuum Gap at Interface
7.2 Application of Thermal Boundary Resistance Across Thin Films
7.2.1 Initial and Boundary Conditions
7.2.1.1 Initial Conditions
7.2.1.2 Boundary Conditions
7.2.1.3 Interface Condition
7.2.1.3.1 Diffuse Mismatch Model
7.2.1.3.2 Cut-Off Mismatch Model
7.2.1.4 Thermal Boundary Resistance
7.2.3 Findings of Thermal Boundary Resistance Application
7.3 Applications in Cross-Plane Transport Across Dielectric Thin Films With Minute Gap
7.3.1 Initial, Boundary, and Interface Conditions
7.3.1.1 Section A (Fig. 7.9)
7.3.1.1.1 Initial Condition
7.3.1.1.2 Left Boundary Condition
7.3.1.2 Section B (Fig. 7.9)
7.3.1.2.1 Initial Condition
7.3.1.2.2 Right Boundary Condition
7.3.1.3 Interface Conditions
7.3.1.3.1 Energy Balance at Point 1 (Fig. 7.11)
7.3.1.3.2 Energy Balance at Point 2 (Fig. 7.11)
7.3.3 Findings of Application of Minute Vacuum Gap
7.4 Applications in Cross-Plane Transport Across Dielectric Thin Films With Minute Gap: Consideration of Near-Field Radiation
7.4.1 Initial, Boundary, and Interface Conditions
7.4.1.1 Section A (Fig. 7.17)
7.4.1.1.1 Initial Condition
7.4.1.1.2 Left Boundary Condition
7.4.1.2 Section B (Fig. 7.17)
7.4.1.2.1 Initial Condition
7.4.1.2.2 Right Boundary Condition
7.4.1.3 Interface Conditions
7.4.1.3.1 Energy Balance at Point 1 (Fig. 7.19)
7.4.1.3.2 Energy Balance at Point 2 (Fig. 7.19)
7.4.2 Findings of Dielectric-Dielectric Films Pair and Presence of Vacuum Gap at Interface
7.5 Applications in Cross-Plane Transport Across Metallic-Dielectric Thin Films Composite With Minute Gap Under Laser Irrad...
7.5.1.1 Electron Subsystem
7.5.1.2 Lattice Subsystem
7.5.3 Definition of Temperature and Heat Flux
7.5.3.1 Phonon Temperature in Aluminum Film
7.5.3.2 Phonon Temperature in Silicon Film
7.5.3.3 Electron Temperature in Aluminum Film
7.5.3.4 Phonon Heat Flux in Aluminum Film
7.5.3.5 Phonon Heat Flux Silicon Film
7.5.3.6 Electron Heat Flux in Aluminum Film
7.5.4 Initial, Boundary, and Interface Conditions
7.5.4.1 Aluminum Film (Fig. 7.24)
7.5.4.1.1 Phonon Subsystem
7.5.4.1.2 Electron Subsystem
7.5.4.2 Silicon Film (Fig. 7.24)
7.5.4.3 Interface Conditions
7.5.4.3.1 Energy Balance for Part 1 (k
7.5.4.3.2 Energy Balance for Part 2 (k>kcut-off)
7.5.4.3.3 Aluminum Interface Condition
7.5.4.3.4 Silicon Interface Condition
7.5.5 Findings of Laser Heating of Films Pair With Presence of Minute Vacuum Gap at the Interface
8 Phonon Radiative Transfer in Curvilinear Coordinate Systems
8.2 Curvilinear Coordinates
8.3 EPRT in Orthogonal Coordinate Systems
8.3.1 Formulation of the Path-Length Derivative
8.3.2 Calculation of the Derivatives ∂φ/∂xi and ∂ψ/∂xi
8.3.3 Equation of Phonon Radiative Transfer
8.3.4 EPRT in the Cylindrical Coordinate System
8.3.5 EPRT in the Spherical Coordinate System
8.4 EPRT in Nonorthogonal Coordinate Systems
8.4.1 Formulation of the Path-Length Derivative
8.4.2 Calculation of the Derivatives ∂φ/∂xi and ∂ψ/∂xi
8.4.3 Equation of Phonon Radiative Transfer
8.5 Application to Two-Dimensional Phonon Radiative Transfer
8.5.1 Boundary Condition Related to the Polar Angle φ
8.5.2 Boundary Condition Related to the Azimuthal Angle ψ
9.1 Analytical Solutions for Heat Equation and Thermal Stress Field
9.2 Numerical Solutions to Equation for Phonon Radiative Transport
9.2.1 Frequency-Dependent Solution of EPRT for Silicon Thin Film (One-Dimensional Case)
9.2.2 Frequency-Independent Solution of EPRT for Aluminum Thin Film (One-Dimensional Case)
9.2.3 Frequency-Dependent Solution of EPRT for Silicon Thin Film (Two-Dimensional Case)
9.2.4 Frequency-Independent Solution of EPRT for Aluminum Thin Film (Two-Dimensional Case)
9.2.5 Solution of EPRT for Cross-Plane Transport Across Dielectric and Metallic Thin Films Pair (One-Dimensional Case)
9.2.6 Solution of EPRT for Cross-Plane Transport Across Silicon and Diamond Thin Films Pair: Consideration of Diffusive Mis...
9.2.7 Solution of EPRT for Cross-Plane Transport Across Silicon and Diamond Thin Films Pair With Presence of Thermal Bounda...
9.2.8 Solution of EPRT for Cross-Plane Transport Across Silicon and Silicon Thin Films Pair With Presence of Minute Size Va...
9.2.9 Solution of EPRT for Cross-Plane Transport Across Silicon and Silicon Thin Films Pair With Presence of Minute Size Va...
9.2.10 Solution of EPRT for Cross-Plane Transport Across Aluminum and Silicon Thin Films Pair With Presence of Minute Size ...
Heat Transport in Micro- and Nanoscale Thin Films
( Micro and Nano Technologies )
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