Predicting phonon transport in semiconductor nanostructures ......Daniel P. Sellan Doctor of...

Predicting phonon transport in semiconductornanostructures using atomistic calculations and the

Boltzmann transport equation

by

Daniel P. Sellan

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Mechanical & Industrial EngineeringUniversity of Toronto

Copyright c© 2012 by Daniel P. Sellan

Abstract

Predicting phonon transport in semiconductor nanostructures using atomistic

calculations and the Boltzmann transport equation

Daniel P. Sellan

Doctor of Philosophy

Graduate Department of Mechanical & Industrial Engineering

University of Toronto

2012

The mechanisms of thermal transport in defect-free silicon nanostructures are ex-

amined using a combination of lattice dynamics (LD) calculations and the Boltzmann

transport equation (BTE). To begin, the thermal conductivity reduction in thin films

is examined using a hierarchical method that first predicts phonon transport properties

using LD calculations, and then solves the phonon BTE using the lattice Boltzmann

method. This approach, which considers all of the phonons in the first Brillouin-zone, is

used to assess the suitability of common assumptions used to reduce the computational

effort. Specifically, we assess the validity of: (i) neglecting the contributions of optical

modes, (ii) the isotropic approximation, (iii) assuming an averaged bulk mean-free path

(i.e., the Gray approximation), and (iv) using the Matthiessen rule to combine the effect

of different scattering mechanisms. Because the frequency-dependent contributions to

thermal conductivity change as the film thickness is reduced, assumptions that are valid

for bulk are not necessarily valid for thin films.

Using knowledge gained from this study, an analytical model for the length-dependence

of thin film thermal conductivity is presented and compared to the predictions of the LD-

based calculations. The model contains no fitting parameters and only requires the bulk

lattice constant, bulk thermal conductivity, and an acoustic phonon speed as inputs. By

including the mode-dependence of the phonon lifetimes resulting from phonon-phonon

ii

and phonon-boundary scattering, the model predictions capture the approach to the bulk

thermal conductivity better than predictions made using Gray models based on a single

lifetime.

Both the model and the LD-based method are used to assess a procedure commonly

used to extract bulk thermal conductivities from length-dependent molecular dynamics

simulation data. Because the mode-dependence of thermal conductivity is not included

in the derivation of this extrapolation procedure, using it can result in significant error.

Finally, phonon transport across a silicon/vacuum-gap/silicon structure is modelled

using lattice dynamics and Landauer theory. The phonons transmit thermal energy

across the vacuum gap via atomic interactions between the leads. Because the incident

phonons do not encounter a classically impenetrable potential barrier, this mechanism is

not a tunneling phenomenon. The heat flux due to phonon transport can be 4 orders of

magnitude larger than that due to photon transport predicted from near-field radiation

theory.

iii

Acknowledgements

I would like to thank Elizabeth, my partner and best friend, for her encouragement

and support. I am blessed to have someone who shows me constant and unconditional

love and support.

I would like to thank my supervisor Cristina Amon for providing me with incredible

guidance, both academic and personal, as well as for her undivided support in aiding

my development as a young scientist. I am grateful to all the members of the Advanced

Thermo/fluid Optimization, Modelling, and Simulations (ATOMS) Laboratory at the

University of Toronto, I cannot imagine having a more encouraging group of colleagues

and friends or a more positive working environment.

I would like to thank Alan McGaughey for his tireless efforts in helping me become a

better researcher. I would also like to thank all the members of the Nanoscale Transport

Phenomena Laboratory at Carnegie Mellon University for sharing their knowledge and

helping me to expand my research capabilities.

I would like to thank Nazir Kherani, Yu Sun, and Charles Ward at the University of

Toronto for serving on my PhD thesis committee. Their guidance throughout this process

has been outstanding. I would like to thank Kenneth Goodson (Stanford University) and

Chandra Veer Singh (University of Toronto) for serving as external examiners for my PhD

defence.

I would like to thank the National Science and Engineering Research Council (NSERC)

of Canada for funding this work through their Alexander Graham Bell Canada Graduate

Scholarship program.

Lastly, and obviously not the least, I would like to thank my parents for their un-

conditional love and support as well as my twin brother Mike for his expertise in almost

everything except medicine.

iv

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis Overview and Scope . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Thermal Transport Models 5

2.1 Fourier Heat Conduction Equation . . . . . . . . . . . . . . . . . . . . . 5

2.2 Boltzmann Transport Equation . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Solving the Boltzmann Transport Equation using the Lattice Boltzmann

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Lattice Boltzmann Methodology . . . . . . . . . . . . . . . . . . . 9

2.3.2 Two Dimensional Lattice Structures: D2Q7 vs. D2Q9 . . . . . . 12

3 Comparing Fourier-based and BTE-based Predictions 14

3.1 Validating the Gray LBM in the Bulk Limit . . . . . . . . . . . . . . . . 14

3.2 Modelling Sub-Continuum Thermal Transport in Thin Films: A Steady-

State Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Predicting Thermal Conductivity from LBM Results . . . . . . . 19

3.2.2 Modelling Boundary Scattering in Thermal Conductivity Predictions 20

3.2.3 Modelling Sub-Continuum Thermal Transport using Fourier-based

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

v

3.3 Heated Surface: Transient Comparison . . . . . . . . . . . . . . . . . . . 24

3.4 Fourier-based vs. BTE Solutions: A Frequency-domain Thermoreflectance

Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Lattice Dynamics Calculations and the BTE 33

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Computational Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.1 Overview of the Hierarchical Procedure . . . . . . . . . . . . . . . 35

4.2.2 Predicting Phonon Properties using Lattice Dynamics Calculations 36

4.2.3 Cross-plane Phonon Transport using the BTE . . . . . . . . . . . 38

4.3 Analyzing the Results and Assessing Common Assumptions . . . . . . . 41

4.3.1 Thermal Conductivity Predictions . . . . . . . . . . . . . . . . . . 41

4.3.2 Isotropic Approximation . . . . . . . . . . . . . . . . . . . . . . . 45

4.3.3 Simplified Model Based on the Gray Approximation . . . . . . . . 47

4.3.4 Boundary Scattering and the Matthiessen Rule . . . . . . . . . . 47

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5 Theoretical Relations and the BTE 50

5.1 Theoretical Relations for Phonon Properties . . . . . . . . . . . . . . . . 50

5.2 Model Derivation for Length-dependent Thermal Conductivity . . . . . . 52

5.3 Model Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3.1 Comparison to Predictions from Lattice Dynamics Calculations . 55

5.3.2 Comparison to Gray Models . . . . . . . . . . . . . . . . . . . . . 57

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6 Size Effects in MD Thermal Conductivity Predictions 59

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.2 Predicting Thermal Conductivity using Molecular Dynamics Simulation . 62

vi

6.2.1 Green-Kubo Method . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.2.2 Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.3 Predicting Thermal Conductivity using Lattice Dynamics Calculations . 70

6.3.1 Phonon Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.3.2 Assessing the Linear Extrapolation Procedure . . . . . . . . . . . 72

6.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.4 Assessing the Linear Extrapolation Procedure using a Length-dependent

Thermal Conductivity Model . . . . . . . . . . . . . . . . . . . . . . . . 76

6.5 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7 Thermal Transport Across a Vacuum-Gap 81

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.2 Vacuum-gap Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.3 Phonon Transmission Coefficients . . . . . . . . . . . . . . . . . . . . . . 84

7.4 Vacuum Thermal Resistance . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

8 Contributions and Future Research Directions 96

8.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8.2 Suggestions for Future Study . . . . . . . . . . . . . . . . . . . . . . . . . 100

Bibliography 105

A Appendix 118

A.1 Streaming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

A.2 Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

A.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

A.3.1 Periodic Boundary Condition . . . . . . . . . . . . . . . . . . . . 119

A.3.2 Constant Temperature Boundary Condition . . . . . . . . . . . . 119

vii

A.3.3 Constant Heat Flux Boundary Condition . . . . . . . . . . . . . . 120

A.3.4 Adiabatic Boundary Condition . . . . . . . . . . . . . . . . . . . 122

A.3.5 Corner Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

viii

List of Tables

3.1 Phonon properties for bulk silicon at temperature of 300 K. Reprinted

with permission from Ref. [1], Copyright 2011 Elsevier. . . . . . . . . . . 16

4.1 Thermal conductivity predictions for bulk SW silicon and thin films (cross-

plane direction) at a temperature of 300 K. . . . . . . . . . . . . . . . . . 42

6.1 Size-dependence of SW silicon thermal conductivity at a temperature of

500 K predicted using MD simulations and the Green-Kubo method. The

prediction uncertainty is the 95% confidence interval based on the results

of ten independent simulations. Reprinted with permission from Ref. [2],

Copyright 2010 American Physical Society. . . . . . . . . . . . . . . . . . 64

6.2 Bulk thermal conductivities, in W/m-K, for SW silicon and LJ argon

found using the Green-Kubo method (kGK – Sec. 6.2.1) and the direct

method (keDM – Sec. 6.2.2) in MD simulations and from lattice dynamics

calculations (kLD and keLD – Sec. 6.3). The direct method uncertainty is

estimated to be ±20% for SW Silicon and ±10% for LJ Argon based on

the prediction repeatability. The superscript e indicates that the value

was predicted using the linear extrapolation procedure. Reprinted with

permission from Ref. [2], Copyright 2010 American Physical Society. . . . 65

ix

6.3 Lattice constants, densities, and elastic constants for LJ argon and SW

silicon at a temperature of 0 K. Reprinted with permission from Ref. [2],


7.1 Vacuum thermal resistance predicted by lattice dynamics calculations and

near-field radiation theory for a 1 Å-wide vacuum-gap at a temperature of

300 K. Thermal boundary resistance for a Si/Si grain boundary [Σ29(001)]

and a Si/Ge interface predicted by molecular dynamics simulation using

the Stillinger-Weber potential at a temperature of 500 K. Reprinted with

permission from Ref. [6], Copyright 2012 American Physical Society. . . . 93

x

List of Figures

2.1 D1Q3, D2Q7, and D2Q9 lattice structures and respective labelling. Reprinted

with permission from Ref. [1], Copyright 2011 Elsevier. . . . . . . . . . . 11

2.2 D2Q7 lattice implementation and boundary lattice nodes. Reprinted with

permission from Ref. [1], Copyright 2011 Elsevier. . . . . . . . . . . . . . 12

3.1 (a) Temperature and (b) heat flux profile predicted by the analytical

Fourier-based solution (solid line) and the Gray LBM using the D2Q7

(open circles) and D2Q9 (open triangles) lattices for the diffuse regime.

Reprinted with permission from Ref. [1], Copyright 2011 Elsevier. . . . . 15

3.2 Thin film model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 (a) Steady-state temperature profile in a silicon film with constant temper-

ature boundary conditions predicted using the Gray LBM, Kn = 0.33. (b)

Steady-state temperature profile predicted using the Fourier-based heat

equation for a silicon film with the domain extended by 2Λ. . . . . . . . 18

3.4 (a) Effective thermal conductivity normalized by the bulk value (kfilm/kbulk)

as a function of the Knudsen number (Kn) predicted by the Gray LBM

and Eq. (3.8). (b) Steady-state boundary temperature jump normalized

by the bulk value (Tjump/∆Tbulk) as a function of the Kn predicted using

the Gray LBM and Eq. (3.10). . . . . . . . . . . . . . . . . . . . . . . . . 20

xi

3.5 (a) Transient ballistic temperature profile predicted by the Gray LBM

and the hyperbolic heat equation with Kn = 0.33 and t∗ = 1. (b) Time-

dependent temperature jump on the left and right boundaries of a silicon

film predicted by the Gray LBM, Kn = 0.33. (c) Jump in the right bound-

ary temperature, Tjj, when the thermal wave hits the cold right boundary

as a function of 1/Kn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Schematic diagram of the simplified frequency-domain thermoreflectance

setup. The sample is periodic in the x and y directions and infinite in the

z direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.7 Non-dimensional temperature, T ∗, as a function of non-dimensional sample

depth, z/Λ, for a system with a surface temperature oscillating at a non-

dimensional frequency of fτ = 0.025. Each grey line corresponds to a

given time step after the system reached steady state. . . . . . . . . . . . 29

3.8 Non-dimensional penetration depth, L/Λ, as a function of non-dimensional

frequency, fτ , for the BTE (squares) and analytical solution to the con-

duction equation (solid line). The dashed line indicates L = 2Λ. . . . . . 30

3.9 Required heat flux amplitude, h0, as a function of non-dimensional pene-

tration depth, L/Λ, for the BTE (squares) and analytical solution to the

conduction equation (solid line). . . . . . . . . . . . . . . . . . . . . . . . 31

4.1 Flow chart of the hierarchical procedure for predicting the phonon thermal

conductivity of bulk and thin films using lattice dynamics calculations

and the BTE. The theoretical/computational tools are in boxes and their

inputs and outputs are in ovals. Reprinted with permission from Ref. [3],

Copyright 2010 American Institute of Physics. . . . . . . . . . . . . . . . 34

xii

4.2 (a) Frequency dependence on wave vector magnitude, |κ|, for the entire

Brillouin zone of SW silicon at a temperature of 300 K. Dispersion curves

corresponding to the [001] direction are shown as solid lines. (b) Phonon

relaxation time dependence on frequency for SW silicon at a temperature of

300 K. Near the Brillouin zone center, the relaxation times are reasonably

represented by τ = A/ω2 (solid line) [4], where A is a constant calculated

for this data to be 2×1015 1/s. Reprinted with permission from Ref. [3],

Copyright 2010 American Institute of Physics. . . . . . . . . . . . . . . . 37

4.3 Sub-continuum temperature profile across a 17.4 nm SW silicon thin film.

Reprinted with permission from Ref. [3], Copyright 2010 American Insti-

tute of Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4 Cross-plane thin film thermal conductivity normalized by the bulk value.

Phonons from the full Brillouin zone (squares) and the isotropic approx-

imation (circles) are considered. The cross-plane thermal conductivity

found using the Matthiessen rule (solid line) and a simplified model (dashed

line, see Sec. 4.3.3) are also plotted. [A] corresponds to Ref. [5]. Reprinted

with permission from Ref. [3], Copyright 2010 American Institute of Physics. 43

4.5 (a) Cross-plane thermal conductivity contribution dependence on the bulk

phonon MFP. The MFPs for each mode are sorted using a histogram with

a bin width of 2 nm. The thermal conductivity contribution is normalized

by the total value. (b) Thermal conductivity contribution dependence on

frequency for bulk and for 556 and 34.8 nm thin films. The area under

each curve is proportional to the total thermal conductivity. Reprinted

with permission from Ref. [3], Copyright 2010 American Institute of Physics. 44

5.1 In-plane and cross-plane thermal conductivity models. . . . . . . . . . . . 54

5.2 Comparison of thermal conductivity models and lattice dynamics/BTE

predictions: (a) Cross-plane, (b) In-plane. . . . . . . . . . . . . . . . . . 56

xiii

6.1 Heat current autocorrelation function (body) and its integral (insert) for

SW silicon at temperatures of 500 and 1000 K. The HCACFs are nor-

malized by their initial values. The shaded region in the insert indicates

the time range over which the HCACF integral is averaged to predict the

thermal conductivity, kGK, which is represented by a dashed line for each

temperature. Reprinted with permission from Ref. [2], Copyright 2010

American Physical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2 Schematic diagram of the simulation cell geometry used in the direct

method simulations. Reprinted with permission from Ref. [2], Copyright

2010 American Physical Society. . . . . . . . . . . . . . . . . . . . . . . . 67

6.3 Length-dependent thermal conductivities for (a) SW silicon at temper-

atures of 500 and 1000 K and (b) LJ argon at a temperature of 40 K

predicted from the direct method and MD simulations. The dashed lines

are linear fits to the discrete data. Reprinted with permission from Ref.

[2], Copyright 2010 American Physical Society. . . . . . . . . . . . . . . . 69

6.4 Inverse of the normalized length-dependent thermal conductivities for (a)

SW silicon (T = 500 K) and (b) LJ argon (T = 40 K). The squares cor-

respond to the sample lengths used in the direct method simulations [see

Figs. 6.3(a) and 6.3(b)], but are calculated using phonon properties ob-

tained from lattice dynamics calculations. Note the difference in the scales

of the vertical axes. Reprinted with permission from Ref. [2], Copyright

2010 American Physical Society. . . . . . . . . . . . . . . . . . . . . . . . 71

6.5 Bulk thermal conductivity contribution dependence on MFP for SW silicon

(T = 500 K) and LJ argon (T = 40 K). The MFPs for each mode are sorted

using a histogram with a bin size of 2 nm for SW silicon and 0.1 nm for LJ

argon. Reprinted with permission from Ref. [2], Copyright 2010 American

Physical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

xiv

6.6 Inverse of the normalized length-dependent thermal conductivity for SW

silicon at T = 500 K (solid line). The squares correspond to the data in

Fig. 6.4(a) [note that this figure has a logarithmic horizontal axis]. The

diamonds correspond to sample lengths between 4 and 8 µm. Reprinted

with permission from Ref. [2], Copyright 2010 American Physical Society. 74

6.7 Length-dependence of the thermal conductivity using Eq. (6.6) and the

bulk thermal conductivity estimated using the linear extrapolation proce-

dure [Eq. (6.8)]. Reprinted with permission from Ref. [2], Copyright 2010


7.1 Schematic diagram of the three-dimensional Si/vacuum-gap/Si structure

for LG = 1.72 Å. The shaded region between the dark silicon atoms is the

volume associated with the perfect silicon crystal. Vacuum space, shown

in white, is added to form the vacuum-gap. What we call vacuum space

is in fact a region of finite electron density. The structure is periodic

in the x and y directions and semi-infinite in the negative and positive

z directions. Reprinted with permission from Ref. [6], Copyright 2012


7.2 Frequency dependence of (a) phonon transmission coefficient [αL→R(κκκ, ν,LG)]

for 10 000 randomly sampled phonon modes in the first Brillouin zone and

(b) the fraction of transmitted phonon energy that remains in its original

mode [η(κκκ, ν,LG)] for transmitted phonon modes with αL→R(κκκ, ν,LG) >

0.01. (c) Bulk phonon density of states for the silicon leads. The phonon

density of states is calculated using a histogram with a bin width of 0.25

THz. The dashed line around 12 THz separates acoustic modes from op-

tical modes. Reprinted with permission from Ref. [6], Copyright 2012


xv

7.3 (a) Vacuum thermal resistance (RNE) as a function of vacuum-gap width

(LG) at a temperature of 300 K. (b) Total lattice energy (E) and its

derivative (dE/dLG) as a function of vacuum-gap width. The dashed

line at LG = 1.89 Å corresponds to the Stillinger-Weber silicon potential

cutoff (LcutoffG ). Reprinted with permission from Ref. [6], Copyright 2012


7.4 Frequency-dependence of vacuum thermal conductance (1/RNE) at a tem-

perature of 300 K. The mode conductances sorted using a histogram with

a bin width of 0.25 THz. The dashed line around 12 THz separates acous-

tic modes from optical modes. Reprinted with permission from Ref. [6],


A.1 Unknown phonon populations for periodic boundary condition in the x

direction for D2Q7 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . 120

A.2 Unknown phonon populations for adiabatic boundary condition implemen-

tation on the top and left boundaries. . . . . . . . . . . . . . . . . . . . . 122

A.3 A node at the corner of constant temperature and adiabatic boundaries. . 124

xvi

Chapter 1

Introduction

1.1 Motivation

Many modern electronic technologies such as graphics processing units, solid-state mem-

ory, and semiconductor light-emitting diodes incorporate components with nanometer-

scale dimensions. The continued miniaturization of these components results in increased

power densities, and the effective removal of generated waste heat is critical to device

operation and reliability [7, 8, 9]. Understanding the underlying heat transport mech-

anisms in these nanostructured materials is thus of great importance. Unfortunately,

continuum-level thermal models, such as the Fourier heat equation, are invalid at such

small length scales, where boundary effects play a significant role.

By adjusting the thermal properties in continuum-level thermal models, sub-continuum

transport can be captured across sub-continuum regimes but not within them. For ex-

ample, applying the Fourier heat equation using an effective thermal conductivity can

give the correct heat flux across the film but not the correct temperature gradient within

it.

Although bulk thermal conductivity can be treated as a function of material and tem-

perature, the closely packed surfaces and interfaces found in nanostrucutres sufficiently

1

Chapter 1. Introduction 2

alter the thermal transport landscape to give thermal conductivity a geometry depen-

dence. This geometry dependence gives researchers the ability to engineer materials with

desired thermal characteristics. One example is the semiconductor superlattice (a peri-

odic nanostructure containing stacked thin films of alternating species), which has been

developed to reduce from bulk the lattice thermal conductivity while not dramatically

affecting the electrical conductivity. Such a material is beneficial in thermoelectric gen-

eration devices. Developing an understanding of the mechanisms of thermal transport in

these nanostructured materials is necessary to fully exploit their design.

Although these materials can be characterized by building the structure and mea-

suring the properties at a series of temperatures, such a process is difficult and time

consuming. For this reason, investigators have turned to numerical studies to assist in

both characterizing and designing nanostructured materials. Some examples of numerical

work include thermal conductivity prediction for silicon thin films [10, 11] and nanowires

[12], Si/Ge [13, 14, 15, 16, 17] and Si/SiGe [17, 18] superlattices, and carbon nanotubes

[19, 20].

Despite progress on the analysis of nanostructured materials, fundamental questions

remain about the nature of thermal transport at the carrier-level. For instance, how much

of the observed lattice thermal conductivity reduction in superlattices is due to carrier-

boundary scattering compared to changes in the carrier properties, such as modifications

to band structure. This limited understanding of carrier-level transport in nanostructures

is exacerbated by inadequate models of carrier transport in bulk materials, which rely

on fitting parameters and major approximations [4, 21].

1.2 Thesis Overview and Scope

The objective of this thesis is to develop, validate, and apply heat transport models that

can be used to optimize the design of nanostructured materials. To perform this re-


search, we use a combination of atomistic modelling techniques [e.g., molecular dynamics

(MD) simulation and lattice dynamics (LD) calculations] and formulations grounded in

statistical thermodynamics [e.g., the Boltzmann transport equation (BTE)]. This thesis

attempts to elucidate the mechanisms of transport in systems where classical and contin-

uum relations break down and the transport must be analyzed at the carrier level [i.e.,

phonon (quantized lattice vibration), electron, and photon]. My research complements

experiment and is a combination of mechanical engineering, condensed matter physics,

and computational physics. The outline of the remainder of the thesis is as follows:

In Chapter 2, details regarding thermal transport modelling techniques are intro-

duced. Continuum-level equations (e.g., Fourier heat equation) and sub-continuum equa-

tions (e.g., BTE) are defined and their limitations are discussed. Methods to solve the

BTE are contrasted, and the lattice Boltzmann method (LBM) is discussed in detail.

In Chapter 3, the steady state and transient results of Fourier-based solutions are

compared to predictions of the BTE. Effort is made to characterize the LBM solutions

to the BTE using algebraic equations. The Gray approximation is made in this chapter

for the sake of simplicity and clarity. Methods to modify the Fourier-based equations so

that they reproduce the LBM results are presented.

In Chapter 4, mode-dependent phonon properties (obtained from harmonic and an-

harmonic lattice dynamics calculations) are included in BTE-based methods to predict

the cross-plane phonon thermal conductivity of Stillinger-Weber silicon thin films as thin

as 17.4 nm. This approach, which considers all of the phonons in the first Brillouin

zone, is used to assess the suitability of common assumptions. Because the frequency-

dependent contributions to thermal conductivity change as the film thickness is reduced,

assumptions that are valid for bulk are not necessarily valid for thin films.

In Chapter 5, an analytical model for the size-dependence of thin film thermal con-

ductivity is presented and compared to the predictions made in Chapter 4 on silicon

nanostructures. The model contains no fitting parameters and only requires the bulk


lattice constant, bulk thermal conductivity, and an acoustic phonon speed as inputs. By

including the mode-dependence of the phonon lifetimes resulting from phonon-phonon

and phonon-boundary scattering, the model captures the approach to the bulk thermal

conductivity better than Gray models based on a single lifetime.

In Chapter 6, the models developed in Chapters 4 and 5 are used to show that be-

cause mode-dependent phonon properties are not considered in the derivation of an ex-

trapolation procedure commonly used to extract bulk thermal conductivity from length-

dependent MD data [22], its use can lead to considerable error.

In Chapter 7, phonon transport across a silicon/vacuum-gap/silicon structure is mod-

elled using lattice dynamics and Landauer theory. We show that the heat flux due to

phonon transport can be 4 orders of magnitude larger than that due to photon transport

predicted from near-field radiation theory.

In Chapter 8, the major contributions of the work presented in this thesis and sug-

gestions for future study are discussed.

Chapter 2

Thermal Transport Models

2.1 Fourier Heat Conduction Equation

Thermal conductivity, k, is in general, a second rank, symmetric tensor empirically de-

fined by the Fourier law,

q = −k∇T , (2.1)

where q is the heat flux vector and ∇T is the temperature gradient in the material.

Though formally defined by this empirical relation [Eq. (2.1)], the thermal conductivity

is related to the properties of the sub-continuum energy carriers, specifically, the electrons

and phonons in solids.

Assuming no mass transfer and constant properties, the Fourier law can be combined

with the energy equation to yield

ρCvk

∂T

∂t= ∇2T, (2.2)

where ρ is the mass density, Cv is the specific heat, and t is time. Given appropriate

initial and boundary conditions, the temperature profile within a system can be predicted

using Eq. (2.2), which is the standard basis for describing conduction heat transfer.

It is known that Eq. (2.2) predicts an infinite thermal-wave propagation-speed, which

leads to a considerable error at small time scales. A remedy for this problem (albeit an

5

Chapter 2. Thermal Transport Models 6

over simplification) is to introduce a finite thermal-wave propagation-speed, vg, to Eq.

(2.2) to account for the wave nature of the thermal transport [23]. This procedure results

in the hyperbolic heat equation (Cattaneo equation) in the following form:

1

v2g

∂2T

∂t2+ρCvk

∂T

∂t= ∇2T. (2.3)

Like the Fourier heat equation [Eq. (2.2)] from which the hyperbolic heat equation is

derived, Eq. (2.3) is unable to capture ballistic transport. In fact, Equations (2.2) and

(2.3) are only valid when:

(i) the system can be modelled as a continuum and behaves classically, and

(ii) the energy carrier transport, be it a result of electrons or phonons, is diffuse rather

than ballistic. This means that the scattering of the carriers is primarily a result

of interactions with other carriers or with lattice imperfections and impurities that

are distributed throughout the material volume

When ballistic transport is present, the study of carrier behaviour becomes crucial for

accurate predictions of thermal transport [7]. In the ballistic (sub-continuum) regime,

energy carriers travel from one side of the system to the other without scattering. In

other words, ballistic effects are present when the characteristic size of a system, L, be-

comes comparable to the phonon mean free path (MFP), which is the average distance

a phonon travels before it scatters. Ballistic carriers contribute less to thermal conduc-

tivity than they would in a bulk system because their MFPs are reduced to the system

length. Thermal conductivity is thus a length-dependent property when ballistic effects

are present. Ballistic effects manifest themselves in two different ways: (i) boundary

temperature jump and (ii) reduced heat flux (i.e., reduced thermal conductivity).

Thermal transport in insulators and semiconductors is dominated by phonons, while

in metals, electrons are the primary carriers [7]. Because insulators and semiconductors

are integral to many nano-structured devices, substantial effort has gone into developing


adequate theories of phonon transport. These efforts focused primarily on employing

the Boltzmann transport equation (BTE) to describe the statistical time evolution of

individual phonon modes. Because previous works typically made major approximations

and assumptions regarding the nature of phonon dynamics to reduce the computational

effort, much of this work is qualitative or semi-empirical. In Chapters 4 and 5, the suit-

ability of some of these approximations/assumptions is investigated using a quantitative

analysis.

2.2 Boltzmann Transport Equation

Different numerical methods have been proposed in the literature for solving the BTE to

model phonon transport. Joshi and Majumdar developed an equation for phonon radia-

tive transfer that employed average phonon properties [24]. Chen proposed a Ballistic-

Diffusive equation for phonon transport, where phonons are defined as transporting in

either ballistic or diffusive regimes [25, 26]. Narumanchi et al. employed the finite vol-

ume method to solve the BTE for thermal transport modelling in electronic devices

using mode-dependent phonon properties [27, 28]. Monte Carlo simulations have also

been applied to solve the BTE and obtain phonon populations in nano-scale devices

[29, 30, 31, 32].

Another approach for solving the BTE, which is adopted in the present work, is

using the lattice Boltzmann method (LBM) [33]. The LBM is a particle-based method

that uses a finite differencing technique to solve the BTE numerically and predicts the

distribution of particles in discrete directions at discrete points in time and space. The

LBM method is discussed in detail in Sec. 2.3, but first, the BTE for phonon transport

modelling is introduced.

The time-dependent phonon BTE is given by [34]

∂f(κ, ν)

∂t+ vg(κ, ν) · ∇f(κ, ν) =

[∂f(κ, ν)

∂t

]coll

, (2.4)


where f(κ, ν) is the phonon population. Here, each phonon mode is identified by its

wave vector, κ, and dispersion branch, ν. The phonon group velocity vector, vg(κ, ν), is

vg(κ, ν) =∂ω(κ, ν)

∂κ, (2.5)

where ω(κ, ν) is the phonon frequency. The left hand side of Eq. (2.4) describes the

diffusion of a system of non-interacting phonons. The term on the right hand side is the

collision operator and serves to reestablish equilibrium through phonon scattering. The

phonon modes are coupled through the collision operator.

The overriding challenge in solving the BTE is modelling the collision operator. Al-

though methods have been developed to evaluate it directly [35, 36, 37], here the re-

laxation time approximation is used to make the BTE more tractable [16, 38]. Using a

first-principles iterative approach to directly evaluate the collision operator and predict

the bulk thermal conductivity of silicon and germanium, Ward and Broido found that

the relaxation time approximation introduces 5 − 10% error (for temperatures between

100 and 800 K [37]).

Under the relaxation time approximation, phonon transport is described by a set of

mode-dependent relaxation times, τ(κ, ν), defined as the average time between scattering

events. Throughout this thesis the crystal is assumed to contain no defects, no free elec-

trons, and no internal interfaces so that τ(κ, ν) is equal to the relaxation time associated

with phonon-phonon scattering, τp−p(κ, ν). Under the relaxation time approximation the

collision term is modelled as[∂f(κ, ν)

∂t

]coll

=fBE(κ, ν)− f(κ, ν)

τp−p(κ, ν), (2.6)

where fBE(κ, ν) is the Bose-Einstein (equilibrium) distribution function,

fBE(κ, ν) =1

eχ − 1, (2.7)

and χ ≡ ~ω(κ, ν)/kBT , ~ is the Planck constant divided by 2π, kB is the Boltzmann

constant, and T is the absolute temperature. Combining Eqs. (2.4) and (2.6) yields the


phonon BTE under the relaxation time approximation,

∂f(κ, ν)

∂t+ vg(κ, ν) · ∇f(κ, ν) =

fBE(κ, ν)− f(κ, ν)τp−p(κ, ν)

. (2.8)

For bulk crystals, the steady state analytical solution of Eq. (2.8) can be combined with

the Fourier law to develop an expression for the i component of thermal conductivity

[34, 39],

ki =∑ν

∑κ

cph(κ, ν)v2g,i(κ, ν)τp−p(κ, ν). (2.9)

Here, the phonon specific heat, cph(κ, ν), is [34, 39]

cph(κ, ν) =~ω(κ, ν)

V

∂fBE(κ, ν)

∂T=kBχ

2

V

eχ

[eχ − 1]2, (2.10)

where V is the volume of the computational cell used to predict the phonon properties

(see Sec. 4.2.2), and vg,i(κ, ν) is the i component of the group velocity vector.

2.3 Solving the Boltzmann Transport Equation us-

ing the Lattice Boltzmann Method

2.3.1 Lattice Boltzmann Methodology

The LBM is a discrete representation of the time-dependent BTE under the relax-

ation time approximation [i.e., Eq. (2.8)], which was first introduced as a fluid flow

modelling technique through prediction of the distribution of fictitious fluid particles

[33]. Since then, the LBM has also been employed to model transport of electrons and

phonons. Guyer [40] introduced a lattice Boltzmann computational framework for mod-

elling phonon transport using a two-dimensional hexagonal (D2Q7 ) lattice. In the present

work, the lattice labelling follows conventions introduced by Qian et al. [41]; i.e., DnQm

where n is the dimensions of space and m-1 is the number of directions on each lattice

site. Jiaung and Ho employed a square (D2Q9 ) lattice to model phonon transport [42].


Ghai et al. used the LBM for coupled modelling of phonons and electrons in semicon-

ductor materials and metals [43]. The LBM for phonons was further developed in the

works of Escobar et al. [44, 45, 46] and Goicochea et al. [47, 48]. These latter works

employed the D2Q9 lattice with multiple phonon modes. Thouy et al. [49, 50] performed

single- and multi-mode phonon LBM simulations on a D2Q9 lattice; in order to have a

higher resolution for direction sampling, they further developed their model to include

24 different directions using a D2Q25 lattice. Recently, Christensen and Graham [51]

developed a coupled LBM/finite difference method to model the heat transfer in joint

micro- and macro-domains.

The LBM for phonon transport is divided into two main categories, namely: multi-

mode LBM and single-mode LBM. Multi-mode LBM considers mode-dependent phonon

properties and solves the BTE for each phonon mode. The single-mode LBM for phonon

transport considers average phonon properties and, like its name suggests, solves the

BTE to predict the phonon populations of only one representative phonon mode. In other

words, the Gray approximation is made. Multi-mode LBM can be considered as solving

a single-mode LBM (herein called Gray LBM) for multiple phonon modes and requires

modifications in numerical implementation to include energy coupling between modes.

Since the objective of this chapter is to discuss LBM methodology and implementation for

phonon transport, the Gray approximation has been adopted for the sake of simplicity

and clarity. Details regarding multi-mode LBM using phonon properties predicted by

lattice dynamics calculations are discussed in Chapter 4.

Under the Gray approximation, the phonon BTE [Eq. (2.8)] is not mode-dependent

and can be re-written as:

∂f

∂t+ vg.∇f =

fBE − fτ

. (2.11)

The first-order discretization of Eq. (2.11) in time and in the x direction results in the

discrete lattice Boltzmann equation (LBE) [33]:

fi(x+ ∆x, t+ ∆t)− fi(x, t) =∆t

τ

[fBEi (x, t)− fi(x, t)

], (2.12)


2 13

1 2 3

4 5 6

1 2

3 4

5 6

7 8

Figure 2.1: D1Q3, D2Q7, and D2Q9 lattice structures and respective labelling. Reprinted

with permission from Ref. [1], Copyright 2011 Elsevier.

where subscript i represents the discretized directions on each lattice site. fi is the

corresponding direction-wise phonon population (i.e., the left and right propagation di-

rections for one-dimensional phonon transport) and ∆t is the computational time step.

The lattice spacing is related to the time step by ∆x = vg∆t.

For one-dimensional simulations, the D1Q3 lattice geometry is straightforward to

implement and is shown in Fig. 2.1. Any lattice selection in two dimensions should be

plane-filling. Two possible lattice configurations that have been extensively employed for

the fluid flow simulation using the LBM are the D2Q7 and D2Q9 lattice structures, and

are shown in Fig. 2.1.

At each time step during the LBM simulation, the phonon populations move in the

direction of the assigned discrete velocity set (along the arrows in Fig. 2.1) towards the

next lattice site (streaming step); and based on a set of collision rules, new populations

are calculated at each lattice site for the next time step (collision step). The effect

of boundary conditions and external sources are also included in the computation of

new components of phonon population on each lattice site at each time step during the

collision step.

For all the lattice structures discussed in this work, the phonon population at the

center (represented by a dot in Fig. 2.1) is reserved to keep the summation of all the


123

654

123

654

123

654

123

654

123

654

constant j

constant i

∆x

∆y

x

y

Figure 2.2: D2Q7 lattice implementation and boundary lattice nodes. Reprinted with

permission from Ref. [1], Copyright 2011 Elsevier.

phonon populations (i.e., lattice energy density) on that lattice site,

fm(x, t) =m−1∑i=1

fi(x, t), (2.13)

which can then be used to calculate the temperature using Bose-Einstein statistics. De-

tails regarding LBM implementation are found in Appendix A.

2.3.2 Two Dimensional Lattice Structures: D2Q7 vs. D2Q9

All previous works on the phonon LBM modelling in two dimensions, with the exception

of Guyer [40], have preferred the D2Q9 over the D2Q7 lattice for its simple structure

and more straightforward boundary condition implementation [42, 46, 49, 52, 51]. The

inherent problem with the D2Q9 lattice, however, is that the particles on diagonal di-

rections (directions 5, 6, 7, and 8 in Fig. 2.1) travel a distance√

2 times longer than the

distance travelled by the particles on the main directions (directions 1, 2, 3, and 4 in Fig.

2.1) at each time step. In the LBM for fluid flow simulations, this numerical problem

has been accounted for by altering the direction-wise velocities of the fluid particles and


introducing a weight factor for particles moving in different directions. In the Gray LBM

for phonon transport, however, only one representative mode with average phonon prop-

erties is considered. All phonons therefore have the same velocity, vg, and cannot travel

faster along the diagonals, as required for implementation of the D2Q9 lattice. We find

that the D2Q9 lattice thus introduces significant error when investigating ballistic trans-

port [1] because of the faster traveling phonons along the diagonals. This issue can be

resolved by using the D2Q7 lattice when simulating two-dimensional phonon transport

(see Fig. 2.2).

On a D2Q7 lattice, the particles move on six equi-length directions separated from

each other by an angle of π3

radians. The phonons therefore travel the same distance

during each time step (i.e., the phonons travel at the same velocity). Although using

the D2Q7 lattice resolves the issue of fictitious high speed diagonal phonons when using

the D2Q9 lattice, implementation of the D2Q7 lattice is complex. On the left and right

boundaries of the D2Q9 lattice, every other lattice site does not fall on the physical

domain boundaries (see Fig. 2.2). Boundary condition implementation at these two

boundaries should be treated with extra care (see A.2 and A.3). Geometrical constraints

of the D2Q7 lattice implies that ∆y is√

32

∆x. It also should be noted that a constant i

line is a zigzag line, as shown in Fig. 2.2. Streaming rules for the D2Q7 lattice shown in

Fig. 2.2 are presented in A.1.

Chapter 3

Comparing Fourier-based and

BTE-based Predictions

3.1 Validating the Gray LBM in the Bulk Limit

Although later chapters will show that making the Gray approximation can lead to

significant error, it is made here so that: (i) the BTE-based predictions of thermal

transport can be characterized using simple algebraic formulations (this is impossible in

cases when dispersion is included) and (ii) the transport of individual phonon modes can

be described, the results of which will later be used to aid discussions when multiple

phonon modes are considered.

To validate the Gray LBM, steady state, diffuse-regime (i.e., bulk-like) heat transfer

in a square domain with side length L = 5 µm is modelled using the Gray LBM and

the D2Q7 and D2Q9 lattices. The predicted results are then compared to an analytical

solution to the two-dimensional Fourier heat equation. The model material is silicon.

The bulk phonon properties of silicon are presented in Table 3.1.

The entire domain was initialized to a constant temperature of Tc = 299.5 K, whereas

the temperature on the top boundary (y/L = 1) was maintained at a temperature of Th =

14

Chapter 3. Comparing Fourier-based and BTE-based Predictions 15

0 0.2 0.4 0.6 0.8 1.00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x*

T*

AnalyticalD2Q7D2Q9

y/L= 0.50

y/L= 0.25

y/L= 0.75

(a) Temperature

0

1

2

3

4

5

6

q y (x

10 W

/m )2

AnalyticalD2Q7D2Q9

y/L= 0.25

y/L= 0.50

y/L= 0.75

7

(b) Heat Flux

0 0.2 0.4 0.6 0.8 1.0 x*

Figure 3.1: (a) Temperature and (b) heat flux profile predicted by the analytical Fourier-

based solution (solid line) and the Gray LBM using the D2Q7 (open circles) and D2Q9

(open triangles) lattices for the diffuse regime. Reprinted with permission from Ref. [1],

Copyright 2011 Elsevier.

300.5 K during the simulation. Domain initialization and implementation of boundary

conditions for the Gray LBM are explained in A.2 and A.3. The steady state temperature

and heat flux profiles in the y-direction (for three different y/L values: 0.25, 0.50, and

0.75), calculated using the LBM on the D2Q7 and D2Q9 lattices, are compared in Fig.

3.1 to the values determined by an analytical solution of the Fourier heat equation [53].

T ∗ and x∗ are defined as (T − Tc)/(Th − Tc) and x/L, respectively. It can be seen

that for the steady state diffuse regime, both lattice structures perform well and the

predicted results are in good agreement with the analytical Fourier-based solution. The

heat flux predicted by the D2Q9 lattice is larger than those predicted from the Fourier

heat equation because of the artificially-fast phonons travelling in the diagonal direction

on the D2Q9 lattice, and suggests that some sub-continuum effects remain even for this

relatively large system. The difference between the heat flux results of the D2Q7 lattice

and Fourier heat equation are attributed to the existence of minor sub-continuum effects.

A lattice resolution study of the steady state results for the temperature profile (a


Table 3.1: Phonon properties for bulk silicon at temperature of 300 K. Reprinted with

permission from Ref. [1], Copyright 2011 Elsevier.

Phonon mean free path (Λ) 40× 10−9 m

Phonon group velocity (vg) 6733 ms−1

Phonon frequency (ω) 8.1825× 1013 rads−1

Specific heat (cv) 1.66× 106 Jm−3K−1

Density (ρ) 2328 kgm−3

Bulk thermal conductivity (k) 149 WK−1m−1

scalar field) showed that at least 300 lattice sites on each side of the domain are required

to have lattice-independent steady state simulations. It should be noted that, as a

rule of thumb, the lattice resolution required to obtain lattice-independent steady state

results for the heat flux (a vector field) or transient results for either temperature or

heat flux is about 20 times higher than that required for capturing the correct steady

state temperature profile. This lattice resolution corresponds to about 50 lattice sites

per physical MFP when using the Gray LBM. A longer simulation time was required to

obtain a steady state heat flux than a steady state temperature profile. This trend has

also been reported in the molecular dynamics simulations [54].

3.2 Modelling Sub-Continuum Thermal Transport in

Thin Films: A Steady-State Comparison

Thermal transport in micro- and nano-structured materials can be significantly different

than in bulk. As dimensions are reduced, boundary effects become significant and prop-

erties predicted for the bulk phase (e.g., thermal conductivity, phonon relaxation times)

may not be suitable for modelling thermal transport. One example is the semiconductor


z, Cross-Plane

x, In-Plane LPhonon-Phonon

Scattering

Diffuse Phonon-Boundary

Scattering

Perio

dic

Boun

dary

Perio

dic

Boun

dary

Figure 3.2: Thin film model.

thin film, which is common in microprocessors, solid-state memory, and semiconductor

light-emitting diodes [7, 8, 55]. The reduced thermal conductivity of thin films limits

their ability to effectively remove waste heat, which is critical to device operation and

reliability [7, 8, 9].

Consider a thin film oriented such that the cross-plane direction and thermal gradient

are along the z direction, as shown in Fig. 3.2. Phonon transport within this film is first

predicted using the Gray LBM to solve the BTE. The hot and cold boundaries of the thin

film are set to temperatures of 300.5 and 299.5 K, resulting in a temperature difference

of 1 K and an average temperature of 300 K. All phonon properties correspond to a

temperature of 300 K and are presented in Table 3.1. Fixed temperature boundary

conditions are imposed using the Bose-Einstein distribution to calculate the phonon

populations at the system boundaries. This boundary condition ensures that all phonons

scatter when they interact with the system boundaries (i.e., diffuse scattering). The use

of completely diffuse boundaries is based on the idea that the reconstruction of free silicon

surfaces disrupts phonons traveling in the cross-plane direction. Similarly, for thin films

bounded by an amorphous material (such as a silicon thin film bounded by amorphous

silica layers), the transition from a crystalline to an amorphous material presents a large

disruption to the phonon propagation and will diffusely scatter the majority of incident

phonons [11].

The LBM-predicted temperature profile across a silicon thin film is presented in


0.8

∆T

(a)

(b)

1.0

0.6

0.80

0.2

0.4

1.00 0.2 0.4 0.6

0.8 1.00 0.2 0.4 0.6

0.8

Non

-dim

ensi

onal

1.0

0.6

0

0.2

0.4

∆T

∆T

∆T

∆T

∆T

film

film

left

left

right

right

∆L=Λ

Non-dimensional Position Along Film, z/L

Non-dimensional Position Along Film, z/L

Tem

pera

ture

, T*

Non

-dim

ensi

onal

Tem

pera

ture

, T*

0.2- 1.2

∆L=Λ

Figure 3.3: (a) Steady-state temperature profile in a silicon film with constant temper-

ature boundary conditions predicted using the Gray LBM, Kn = 0.33. (b) Steady-state

temperature profile predicted using the Fourier-based heat equation for a silicon film with

the domain extended by 2Λ.

Fig. 3.3(a). Of particular interest are the temperature discontinuities at the boundaries

(∆Tleft and ∆Tright), which are an indication of ballistic phonon transport. Such discon-

tinuities are negligible in bulk systems (i.e., ∆Tright,∆Tleft → 0 as L→∞) and are not

predicted using continuum-based analysis techniques (e.g., the Fourier heat equation).

At steady state, ∆Tleft = ∆Tright (herein referred to as Tjump).

In the ballistic (sub-continuum) regime, energy carriers travel from one side of the film

to the other without scattering when the characteristic size of a system, L, becomes com-


parable to the phonon MFP, Λ(κ, ν) = |vg(κ, ν)|τ(κ, ν). For convenience, the phonon

MFP is compared to the system length using the Knudsen number:

Kn(κ, ν) =|vg(κ, ν)|τ(κ, ν)

L=

Λ(κ, ν)

L. (3.1)

3.2.1 Predicting Thermal Conductivity from LBM Results

From the phonon populations predicted by the LBM, the cross-plane thermal conductiv-

ity, kz(L), can be predicted using a one-dimensional form of the Fourier Law:

kz(L) =qzL

∆Tleft + ∆Tfilm + ∆Tright, (3.2)

where qz is the z component of the heat flux vector in the thin film (qx = qy = 0),

qz =1

V

∑ν

∑κ

vg,z(κ, ν)~ω(κ, ν) [fL(κ, ν)− fR(κ, ν)] , (3.3)

where fL and fR are the populations of the phonon modes travelling in the left and right

propagation directions [56]. The temperature differences ∆Tleft, ∆Tfilm, and ∆Tright, are

calculated from the steady state temperature profile.

The thermal conductivity reduction from bulk predicted using the Gray LBM is plot-

ted versus Kn (open circles) in Fig. 3.4(a). The thermal conductivity increases mono-

tonically to the bulk value as the system length increases (Kn decreases). The length-

dependence of the steady state boundary temperature jump, Tjump, normalized by the

total temperature difference across the film ∆Tbulk = ∆Tleft + ∆Tfilm + ∆Tright, is pre-

sented in Fig. 3.4(b). At small Knudsen numbers (large length scales), the temperature

jumps at the system boundaries diminish. Reducing the domain size increases the tem-

perature jump to an asymptotic value of Tjump/∆Tbulk = 0.5 at the pure ballistic regime.


k

/k

bulk

Gray LBM0.6

0.4

0.8

0.2

1.0

102- 1- 0 1Kn

Tju

mp

/∆T b

ulk

10 1010

0.5

0.4

0.3

0.2

0.1

0

(a) (b)

102- 1- 0 1Kn

10 10100

film

k

bulk

film k =

11+2Kn

Gray LBM T

bulk=

Kn1+2Kn

jumpT∆

Figure 3.4: (a) Effective thermal conductivity normalized by the bulk value (kfilm/kbulk)

as a function of the Knudsen number (Kn) predicted by the Gray LBM and Eq. (3.8). (b)

Steady-state boundary temperature jump normalized by the bulk value (Tjump/∆Tbulk)

as a function of the Kn predicted using the Gray LBM and Eq. (3.10).

3.2.2 Modelling Boundary Scattering in Thermal Conductivity

Predictions

One method to model boundary scattering is to solve the time-dependent BTE numeri-

cally using the LBM and appropriate boundary conditions (as described in the previous

section). A second method is to solve the steady state BTE analytically for the bulk

thermal conductivity [resulting in Eq. (2.9)] and then correct the bulk phonon-phonon

relaxation times to account for phonon-boundary scattering. To do this, phonon scat-

tering mechanisms (i.e., phonon-phonon and phonon-boundary scattering) are combined

by assuming that they are independent using the Matthiessen rule [11, 34, 3].

The Matthiessen rule can be used to model the length dependence of τ(κ, ν, L) such

that

1

τ(κ, ν, L)=

1

τ∞(κ, ν)+

1

τb(κ, ν, L), (3.4)

where τ∞(κ, ν) and τb(κ, ν, L) are the intrinsic scattering and boundary scattering re-

laxation times. Again, the crystal is assumed to contain no defects, no free electrons,

and no internal interfaces so that τ∞(κ, ν) is equal to the relaxation time associated with


phonon-phonon scattering τp−p(κ, ν). The boundary scattering relaxation time is taken

to be the average time between boundary scattering events in the absence of intrinsic

scattering, i.e.,

τb(κ, ν, L) =L/2

|vg,j(κ, ν)|, (3.5)

where |vg,j(κ, ν)| is the component of the velocity vector that is perpendicular to the film

surface.

Substituting Eq. (3.4) into Eq. (2.9) and applying Eq. (3.5) leads to an expression

that describes the length dependence of thermal conductivity,

ki(L) =∑ν

∑κ

cphv2g,i(κ, ν)τ∞(κ, ν)

[1 +

2|vg,j(κ, ν)|τ∞(κ, ν)L

]−1. (3.6)

For cross-plane phonon transport i = j, for in-plane phonon transport i 6= j. As L ap-

proaches infinity, the bracketed term in Eq. (3.6) approaches unity and ki(L) approaches

the bulk value, k∞. Equation (3.6) considers mode-dependent phonon properties. To

compare Eq. (3.6) to the Gray LBM results presented in Fig. 3.4(a), the results are sim-

plified by considering only one phonon mode traveling perpendicular to the film boundary

(i.e., the cross-plane direction). Under these conditions, Eq. (3.6) becomes:

k(L) = cphv2gτ∞

[1 +

2|vg|τ∞L

]−1= k∞

1

1 + 2Kn, (3.7)

or in a more convenient form,

kfilmkbulk

=1

1 + 2Kn. (3.8)

In the LBM, phonon-phonon scattering is modelled using τp−p while phonon-boundary

scattering is modelled using appropriate boundary conditions. These scattering mecha-

nisms are thus treated independently. Agreement between results of the Gray LBM and

Eq. (3.8) is thus expected (provided that the boundary scattering is consistently incor-

porated in both methods) because the underlying treatment that scattering mechanisms

are independent is consistent. In Fig. 3.4(a), the thermal conductivity reduction pre-

dicted using Eq. (3.8) is presented (solid line) with the results obtained using the Gray


LBM (open circles). The results are in excellent agreement (within 1% for all film thick-

nesses). The agreement between the two methods suggests that Eq. (3.5) is appropriate

for modelling diffuse boundaries. Mode-dependent results are presented in Chapter 4.

3.2.3 Modelling Sub-Continuum Thermal Transport using Fourier-

based Methods

The conventional methods for modelling bulk thermal transport, such as the Fourier heat

equation, fail to capture sub-continuum effects. To address this issue, we have shown

that numerical methods based on the BTE can be used to accurately model phonon

transport. The complex nature and relatively high computational costs of these methods,

however, have hindered their adoption by industry. Furthermore, the wide selection of

commercially available Fourier-based packages (e.g., ANSYS), which are more efficient

at large length scales, makes these sub-continuum techniques a hard sell to industry. A

method to modify the Fourier-based heat equation so that it is capable of capturing sub-

continuum effects is therefore desirable in many industrial applications. In this section,

the Gray LBM results are used to develop a method to modify the Fourier-based heat

equation so that it can recover sub-continuum effects: (i) reduced heat flux and (ii)

boundary temperature jump.

Modifying the Fourier-based heat equation to recover the heat flux reduction in bal-

listic transport is straightforward and can be obtained by using the reduced thermal

conductivity predicted by Eq. (3.8). A method to recover the temperature jumps at the

boundaries, however, requires more effort.

To derive an analytical form for the length-dependence of Tjump shown in Fig. 3.4(b),

we begin with the following formulation that was obtained by analyzing the Gray LBM

results:

q = kfilm∆Tbulk

∆x= kbulk

∆Tfilm∆x

. (3.9)


Combining Eqs. (3.8) and (3.9) and using ∆Tbulk = 2Tjump + ∆Tfilm at steady state

results in a relation for the non-dimensional steady state boundary temperature jump:

Tjump∆Tbulk

=Kn

1 + 2Kn. (3.10)

In Fig. 3.4(b) the length-dependence of the boundary temperature jump predicted using

Eq. (3.10) is presented (solid line) with the results obtained using the Gray LBM (open

circles). The results are in excellent agreement (within 1% for all film thicknesses).

From the Gray LBM simulation results, we know that the steady state tempera-

ture jump at each boundary is equal. One can obtain these steady state temperature

jumps at the physical boundaries of the system using the Fourier-based heat equation

(which predicts a linear temperature profile) by extending the domain beyond the physi-

cal boundaries by a fixed value, ∆L, and implementing the physical boundary conditions

at the end of these added imaginary sections. In this case, the steady state solution is a

linear temperature profile extended across the new length of the domain, L+2∆L. Using

the proposed relation for the steady state temperature jump [Eq. (3.10)] and trigonom-

etry, we obtain ∆L = Λ. Thus, by adding extensions of ∆L = Λ to either side of the

physical domain, the Fourier-based heat equation predicts the correct temperature jumps

at the physical system boundaries. This method is more practically implemented than

using a thermal boundary resistance. This result is shown in Fig. 3.3(b) for a silicon thin

film with length such that Kn = 0.33.

3.2.4 Summary

The LBM is used to numerically solve the BTE under the relaxation time approximation.

Effort was made throughout this chapter to characterize the LBM solutions using alge-

braic equations, which are presented for the first time. These relations were then used

to develop a method to modify Fourier-based heat equations so that they reproduce the

LBM data when ballistic transport is present. The steady state heat flux reduction due


to ballistic transport predicted by the Gray LBM can be captured using the Fourier heat

equation and an effective thermal conductivity. To obtain the temperature jumps at the

system boundaries predicted by the Gray LBM, the system domain should be extended

by a distance equal to the phonon MFP on either side.

3.3 Heated Surface: Transient Comparison

In the previous section, steady state thermal transport is addressed. In many applica-

tions, such as intrinsic heating in graphics processing units, understanding the transient

behaviour of thermal transport is important to developing appropriate cooling strategies.

As mentioned in Sec. 2.1, the Fourier heat equation predicts that energy is conducted

away from a heated region at an infinite propagation speed. This problem can lead to

significant error when investigating systems at small time scales where the thermal en-

ergy of a hot region has not yet reached a cold region. Traditionally, the hyperbolic heat

equation is used to resolve wave effects in these systems by forcing energy to propagate at

a defined speed. To illustrate this case, the transient data corresponding to the geometry

of the steady state discussed above is presented. The Gray LBM is used to simulate

phonon transport in a silicon film with a thickness such that Kn = 0.33 and the initial

temperature is equal to the right boundary, Tc = 299.5 K. The left boundary is main-

tained at Th = 300.5 K during the simulation. This system results in a thermal wave

propagating from the hot, left boundary to the cold, right boundary when the simulation

is allowed to proceed.

The transient ballistic temperature profile predicted by the Gray LBM is presented

as a solid line in Fig. 3.5(a) for t∗ = 1, where t∗ = t/τ is the non-dimensional time. The

sharp wave front predicted by the Gray LBM is not physical and is due to the inclusion

of only one average phonon mode. If multiple phonon modes are included in the LBM

simulations, the predicted wave front would be less sharp. Figure 3.5(b) shows the time


0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

*t

T jum

p/ ∆

T bul

k

Left BoundaryRight Boundary

T*

Modified Hyperbolic

0.4

0.2

0.2 0.40

0 0.6 0.8 1.0

0.8

1.0

0.6

*x

Gray LBM

Gray LBM

Tjj

0 2 4 6 8 10 12 141/Kn

0

0.1

0.2

0.3

0.4

0.5

T jj

(c)

(b)

(a)

T = 0.5ejj- 0.5Kn

Gray LBM

Figure 3.5: (a) Transient ballistic temperature profile predicted by the Gray LBM and the

hyperbolic heat equation with Kn = 0.33 and t∗ = 1. (b) Time-dependent temperature

jump on the left and right boundaries of a silicon film predicted by the Gray LBM, Kn

= 0.33. (c) Jump in the right boundary temperature, Tjj, when the thermal wave hits

the cold right boundary as a function of 1/Kn.


evolution of the temperature jump at the two boundaries. The temperature at the right

boundary remains unchanged until the heat wave travels the entire film thickness, which

is equal to 3Λ, and reaches the right boundary at t∗ = 3. Once the thermal wave hits

the cold end, the temperature jump at the right boundary follows the same trend as the

temperature jump on the left boundary. In other words, the curves are symmetric about

the steady state, asymptotic value, which is in excellent agreement with Eq. (3.10). The

jump in the right boundary temperature, Tjj, when the thermal wave hits the cold right

boundary is a function of the system length and is well described by the following relation

[see Fig. 3.5(c)]:

Tjj = 0.5e− 0.5L

Λ = 0.5e−0.5Kn . (3.11)

To model the transient ballistic thermal transport using the hyperbolic heat equation,

the time dependent temperature jumps predicted by the Gray LBM [see Fig. 3.5(b)]

were implemented numerically at the system boundaries. The temperature of the left

and right boundaries were set equal to Tleft(t∗) = Th − Tjump,left(t∗) and Tright(t∗) =

Tc + Tjump,right(t∗). The hyperbolic heat equation was then solved numerically using the

finite difference scheme. The predicted temperature profile of the modified hyperbolic

heat equation is presented as a dashed line in Fig. 3.5(a). It can be seen that the

modified hyperbolic equation can recover the temperature profile predicted from the

LBM simulation to a good extent when the transient boundary temperatures are known.

Providing a general relation for the transient boundary temperature, however, is difficult

because they depend strongly on the initial condition and system geometry, which offers

opportunity for future investigation.


3.4 Fourier-based vs. BTE Solutions: A Frequency-

domain Thermoreflectance Study

An increased interest in measuring the thermal transport properties of nanostructured

materials has led to the development of noncontact measurement techniques based on

photothermal phenomena. One such technique is the pump-probe-based frequency-

domain thermoreflectance (FDTR) method [57, 58]. In one form of FDTR [58], the

continuous wave (CW) pump laser is modulated and used to periodically heat the sur-

face of a sample. This periodic surface heating results in a periodic thermal wave that

propagates into the sample. The amplitude and phase of the thermal response at the

sample surface depend on the thermal properties of the sample beneath. Because the

reflectance of the sample surface is temperature dependent, the probe laser (also CW)

becomes modulated upon reflection and is used to monitor the thermal response. A

Fourier-based thermal model is then fit to phase response and/or amplitude data of the

reflected probe beam to extract the sample thermal conductivity or the thermal conduc-

tance of a buried interface.

The goal of this work is not to simulate a real FDTR experiment, but to assess the

Fourier-based model used to derive the data analysis method described in Ref. [59]. The

Fourier-based solution for a homogeneous solid is evaluated. To do so, phonon transport

is simulated using the Gray LBM and the resulting temperature profiles are compared

to those predicted by a solution to the Fourier-based conduction equation [53]. To be

consistent with the implementation of Ref. [53], thermal transport in a homogeneous,

semi-infinite solid that is heated at the surface by a periodic source is considered. In

addition to simulating bulk-like conditions, we also assess the ability for the Fourier-

based model to capture thermal transport when sub-continuum effects are present. To

do this, sinusoidal heat fluxes are chosen such that the modulation frequency of the

propagating thermal wave is comparable to the phonon relaxation time.


T = Constant

q =

q

�Penetration Depth,

Sam

ple

Surf

ace

Hot

Cold

Sample Thicknessx

zo

sin(ω

t)∞

Figure 3.6: Schematic diagram of the simplified frequency-domain thermoreflectance

setup. The sample is periodic in the x and y directions and infinite in the z direction.

Consider a bulk, homogeneous sample that is infinite in the x and y directions and

finite in the z direction, as shown in Fig. 3.6. Silicon is used as the model material. The

averaged, bulk phonon properties required as input to the BTE are provided in Table

3.1. The sample is initially set to a uniform temperature of Tinitial = 300 K. During the

simulation, a heat flux boundary condition [i.e., q = qo sin(ft)] is imposed at z = 0 such

that the surface-temperature oscillates at a frequency, f . The magnitude of the heat flux,

qo, is chosen such that the maximum surface temperature, Tmaxsurface, is constant and equal

to 305 K. The boundary opposite to the heated surface is maintained at a temperature of

T∞ = 300 K. To ensure that the sample is semi-infinite, the sample thicknesses is chosen

such that the propagating thermal wave decays to a temperature of 300 K well before

it reaches the boundary opposite of the heated surface. The system is allowed to evolve

in time, and the non-dimensional, steady state temperature profiles are reported. The

non-dimensional temperature, T ∗, is defined as: T ∗(z) = T (z)−T∞Tmaxsurface−T∞

.

The temperature profiles predicted by the BTE are shown in Fig. 3.7 for non-dimensional

frequency fτ = 0.025. The data was recorded after the system reached steady state. As

shown in the insert of Fig. 3.7, which is a zoom-out of the main plot, the temperature

data is symmetrical about T ∗ = 0. Each grey line corresponds to the temperature profile


fτ = 0.025 �/Λ = 9.2

0

0.2

0.4

0.6

0.8

1.0

Non

-dim

ensi

onal

Tem

pera

ture

, T

*

Non-dimensional Depth, z/Λ0 20 40 60 80

Analytical Fourier Solution,T = exp(-z/�)

z

1000z/Λ

−1

0

1

T*

BTE Solutions

=�

*

Figure 3.7: Non-dimensional temperature, T ∗, as a function of non-dimensional sample

depth, z/Λ, for a system with a surface temperature oscillating at a non-dimensional

frequency of fτ = 0.025. Each grey line corresponds to a given time step after the

system reached steady state.

predicted by the BTE at a given time step. The decay envelope of the temperature oscil-

lations is well described by an exponential decay (black line) predicted by an analytical

solution of the Fourier-based conduction equation [53],

T ∗ = exp (−z/L) , (3.12)

where L is the penetration depth (a fitting parameter) defined as the depth at which T ∗

= 1/e. For fτ = 0.025, the non-dimensional penetration depth is L/Λ = 9.2.

For each f investigated, an exponential curve is fit to the decay envelope of the

BTE temperature data (which are all similar with that shown in Fig. 3.7) to extract the

corresponding penetration depth. This BTE data is plotted as squares for L/Λ versus

fτ in Fig. 3.8. When fτ � 1, the oscillatory period of the propagating thermal wave

is much larger than the phonon relaxation time. In this continuum regime, the phonons

react almost instantaneously to the oscillations of the propagating thermal wave, and

the BTE-predicted penetration depths are well-described by an analytical solution of the


0.001 0.01 0.1 1 10 1001

10

100

Analytical Fourier Solution,

�= BTE Solution

�

� = ρc k 0.5[ ][ ]

pπfbulk

Figure 3.8: Non-dimensional penetration depth, L/Λ, as a function of non-dimensional

frequency, fτ , for the BTE (squares) and analytical solution to the conduction equation

(solid line). The dashed line indicates L = 2Λ.

Fourier-based conduction equation [53],

L =[kbulkρcpπf

]0.5, (3.13)

which is plotted as a solid line in Fig. 3.8. Here, the 1/3 coefficient typically included

in the kinetic theory expression for kbulk is not included because it is associated with

three-dimensional phonon transport and only phonons that propagate perpendicular to

the heated surface are considered in the LBM implementation [1]. Thus, kbulk = cpvgΛ.

As fτ increases and becomes comparable to unity, the BTE-predicted penetration

depths deviate from the predictions of Eq. (3.13). When fτ > 1, the oscillatory period

of the propagating thermal wave is smaller than the phonon relaxation time and the

phonons do not react to the oscillations of the thermal wave. In this sub-continuum

regime, L/Λ no longer depends on fτ . In the limit that fτ approaches infinity, L/Λ

approaches 2 (indicated by the dashed line in Fig. 3.8).

In a typical FDTR experiment, f < 20 MHz. Therefore only phonons with τ > 50 ns

correspond to fτ > 1 and can transport ballistically. Ward and Broido [37] found that in


0

10

20

Req

uire

d H

eat F

lux

Am

plitu

de,

h [

GW

/m K

] 2

0

Analytical Fourier Solution,

� = ρc k 0.5[ ][ ]

pπf

0h =

BTE Solution

101

2π�

Non-dimensional Penetration Depth,

bulk

ρc pkbulk

Figure 3.9: Required heat flux amplitude, h0, as a function of non-dimensional pen-

etration depth, L/Λ, for the BTE (squares) and analytical solution to the conduction

equation (solid line).

bulk silicon only a few low-frequency phonons have τ > 50 ns at a temperature of 300 K.

These findings may explain why a recent transient thermoreflectance study [60], which

was performed using silicon at T = 300 K and modulation frequencies between 3 and 12

MHz, did not observe any laser modulation frequency dependence of the measured ther-

mal conductivity. The modulation frequencies were too small to allow enough phonons

to travel ballistically to significantly affect the measured thermal conductivity.

One method to force more phonons to travel ballistically is to increase the modula-

tion frequency beyond 20 MHz. To do so, however, may be difficult due to limitations

of available optical equipment. Another method to increase fτ is to decrease tempera-

ture. Decreasing temperature increases relaxation times, thus more phonons will travel

ballistically. If enough phonons travel ballistically, the implied thermal conductivity will

depend on the modulation frequency. This trend can be seen in Fig. 3.9, where the re-

quired heat flux amplitude, h0, is plotted as a function of non-dimensional penetration

depth, L/Λ, for the BTE (squares) and an analytical solution to the conduction equa-

tion (solid line). Note that the same surface-temperature amplitude is maintained for all


modulation frequencies studied (the surface-temperature oscillates between a maximum

of 305 K and a minimum of 295 K for all modulation frequencies). As the modulation

frequency increases, the penetration depth decreases, and the system transitions from

a diffusive to a ballistic regime when L becomes comparable to the phonon mean free

path (i.e., L/Λ < 1). In the ballistic regime, the heat flux required to heat the BTE

system is much less than that for the Fourier-based system. This reduced heat flux is

what researchers perceive as a reduced thermal conductivity in an FDTR experiment.

3.4.1 Summary

The phonon BTE is used to show that models based on the Fourier heat conduction

equation do not to accurately describe thermal transport when sub-continuum effects are

present in an FDTR-like system. In this work, phonon transport is simplified by solving

the BTE for one phonon mode with average properties (i.e., one relaxation time, one

group velocity, and one specific heat). In bulk silicon, however, phonons with relaxation

times that span 4 orders of magnitude are present [37, 3]. Performing an FDTR measure-

ment on this system would result in some phonons transporting in a continuum regime

while others transporting in a sub-continuum regime.

Chapter 4

Lattice Dynamics Calculations and

the BTE

4.1 Introduction

In the previous chapters, the Gray approximation is made for the sake of simplicity and

clarity in the attempt to elucidate the underlying physics of phonon transport in silicon

nanostructured materials. Although the Gray approximation is used extensively in liter-

ature, it is shown in this chapter that it over simplifies the phonon picture and phonon

dispersion (i.e., multiple phonon modes) must be considered to accurately describe ther-

mal transport in systems with nanoscale features.

In this chapter, multi-mode LBM is used to predict (i) the bulk thermal conductivity

of Stillinger-Weber (SW) silicon and (ii) the cross-plane thermal conductivities of SW

films as thin as 17.4 nm, all at a temperature of 300 K. Turney et al. previously analyzed

the in-plane thermal conductivity [11]. The accuracy of the BTE-based predictions de-

pend on the accuracy of the phonon properties required as input. Given an interatomic

potential that describes the atomic interactions, mode-dependent specific heats and group

velocities can be predicted using harmonic lattice dynamics calculations [61, 62]. The

33

Chapter 4. Lattice Dynamics Calculations and the BTE 34

Phonon Properties

Force ConstantsInteratomic

Harmonic &Anharmonic

Specific Heats,Velocities, &

Relaxation Times

Bulk

Thin Film (cross-plane)

ThermalConductivity

FourierLaw

TemperatureProfile &Heat Flux

Geometry &Boundary Conditions

Steady-State BTE& Fourier Law

LBM

Lattice Dynamics

Steady-State

Figure 4.1: Flow chart of the hierarchical procedure for predicting the phonon thermal

conductivity of bulk and thin films using lattice dynamics calculations and the BTE. The

theoretical/computational tools are in boxes and their inputs and outputs are in ovals.

Reprinted with permission from Ref. [3], Copyright 2010 American Institute of Physics.

required relaxation times can be predicted from molecular dynamics (MD) simulation

[63, 64] or anharmonic lattice dynamics calculations [62, 65]. Molecular dynamics simu-

lation and anharmonic lattice dynamics calculations, the latter of which naturally include

quantum effects and is used here, are computationally expensive and can be challenging

to implement. As such, the suitability of two approximations commonly made to reduce

the computational effort are investigated. First, the isotropic approximation, where the

phonon properties of one crystalline direction are assumed to be representative of the

entire Brillouin zone. Second, the Gray approximation, where the entire Brillouin zone

is represented by a single phonon velocity and relaxation time. By investigating these

approximations, we will (i) determine when they can be used to predict the phonon prop-

erties and thermal conductivity of bulk systems without introducing significant error, and

(ii) understand how their validity changes as system lengths are reduced from bulk to

the nanometer scale.

To perform the required calculations, a hierarchical procedure (described in Sec. 4.2.1)

that uses the BTE (Secs. 2.2 and 4.2.3) and lattice dynamics calculations (Sec. 4.2.2) is

Chapter 4. Lattice Dynamics Calculations and the BTE 35

used to predict the bulk and cross-plane thin film thermal conductivities. In Sec. 4.3, we

present the thermal conductivity predictions and examine the role of optical phonons,

the isotropic and Gray approximations, and the suitability of the Matthiessen rule for

combining the effects of different scattering mechanisms. It is shown that because the

frequency-dependent contributions to thermal conductivity change as the film thickness

is reduced, approximations that are valid for bulk are not necessarily valid for thin films.

4.2 Computational Tools

4.2.1 Overview of the Hierarchical Procedure

At the atomic level, the thermal conductivity of a semiconductor is related to the trans-

port of phonons. Phonon transport is modelled using the hierarchical procedure shown

in Fig. 4.1 and described in detail throughout this chapter. Interatomic forces, which

are the derivatives of a system’s potential energy with respect to the positions of its con-

stituent atoms, are first calculated from an interatomic potential energy function. The

force constants are then used in harmonic and anharmonic lattice dynamics

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