Monte-Carlo Study of Phonon Heat Conduction in Silicon Thin Films

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Monte-Carlo Study of Phonon Heat Conduction in Silicon

Thin Films

THESIS

Presented in Partial Fulfillment of the Requirements for the Degree

Master of Science in the Graduate School of The Ohio State University

Arpit Mittal, B.E. (Honors)

Graduate Program in Mechanical Engineering

The Ohio State University

2009

Master's Examination Committee:

Dr. Sandip Mazumder, Advisor

Dr. V.V. Subramaniam

c Copyright by

Arpit Mittal

2009 ABSTRACT

Heat conduction in crystalline semiconductor materials occurs by lattice vibrations that result in the propagation of quanta of energy called phonons. The Boltzmann

Transport Equation (BTE) for phonons is a powerful tool to model both equilibrium and non-equilibrium heat conduction in crystalline solids. Non-equilibrium heat conduction occurs either when the length scales (of the device in question) are small or at low temperatures. The BTE describes the evolution of the number density (or energy) distribution for phonons as a result of transport (or drift) and inter-phonon collisions. The Monte-Carlo (MC) method has found proliﬁc use in the solution of the Boltzmann Transport Equation (BTE) for phonons. This thesis contributes to the state-of-the-art by performing a systematic study of the role of the various phonon modes on thermal conductivity predictions−in particular, optical phonons.

A procedure to calculate three-phonon scattering time-scales with the inclusion of optical phonons is described and implemented. The roles of various phonon modes are assessed. It is found that Transverse Acoustic (TA) phonons are the primary carriers of energy at low temperatures. At high temperatures (T > 200 K), Longi- tudinal Acoustic (LA) phonons carry more energy than TA phonons. When optical phonons are included, there is a signiﬁcant change in the amount of energy carried by various phonons modes, especially at room temperature, where optical modes are found to carry about 25% of the energy at steady state in silicon thin ﬁlms. Most

ii importantly, it is found that inclusion of optical phonons results in better match with experimental observations for silicon thin-film thermal conductivity. The inclusion of optical phonons is found to decrease the thermal conductivity at intermediate temperatures (50−200 K) and increase it at high temperature (>200 K), especially when the film is thin. The effect of number of stochastic samples, the dimensionality of the computational domain (two-dimensional versus three-dimensional), and the lateral (in-plane) dimension of the film on the statistical accuracy and computational efficiency is systematically studied and elucidated for all temperatures. For thin film thermal conductivity predictions, it has been found that the dimensionality of the computational domain has no impact on the accuracy of the numerical solution. In the diffusion dominated regime, three-dimensional Monte-Carlo calculations are found to be comparable to two-dimensional Monte-Carlo calculations in terms of computational efficiency. It has also been found that irrespective of dimensionality of the computational domain, the impact of the size of the lateral boundaries can be accounted for by tuning the resistance provided by the boundaries, i.e., the degree of specularity, α. It is also shown that the time averaging of the statistically stationary data can be used to reduce statistical noise and can result in considerable computational savings.

iii Dedicated to my family for their never-ending love and support

iv ACKNOWLEDGMENTS

I would like to begin by acknowledging my advisor and mentor, Professor Sandip

Mazumder, for giving me with this wonderful opportunity to learn from him. He is as excellent educator and I have beneﬁted immensely from his clarity of thought and passion for teaching. I thank him for his patience and guidance during the course of my research. I would also like to acknowledge Professor V.V. Subramaniam for serving on my thesis defense committee and for his valuable help during my defense.

I am grateful to Professor Nandini Trivedi and Professor Joseph Heremans for their valuable help and guidance. I would also like to thank my lab mates over the past two years: Ankan, Sai, Mahesh, Tom, Alex, Derek and Robert for their support.

Financial support by the Department of Energy’s Basic Energy Science Program through Grant Number DE-FG02-06ER46330 is gratefully acknowledged. Caﬀeine support by the Mechanical Engineering Graduate student Association (MEGA) is also gratefully acknowledged.

v VITA

May 2005 ...... (B.E.) with Honors, Punjab Engineer- ing College, Panjab University June 2005 - August 2007 ...... Assistant Manager, Maruti Suzuki In- dia Limited September 2007 - Present ...... Graduate Research Associate, Compu- tational Thermal-Fluids Lab, The Ohio State University

PUBLICATIONS FROM THIS WORK

Research Publications

“Monte Carlo Study of Phonon Heat Conduction in Silicon Thin Films Including Contributions of Optical Phonons”, Arpit Mittal and Sandip Mazumder, Journal of Heat Transfer (accepted).

Conference Publications

“Monte Carlo Study of Phonon Heat Conduction in Silicon Thin Films: Role of Optical Phonons,” Arpit Mittal and Sandip Mazumder, “Proceedings of the ASME Summer Heat Transfer Conference”, July 19-23, 2009, San Francisco, CA, Paper Number - HT2009-88008 FIELDS OF STUDY

Major Field: Mechanical Engineering

vi TABLE OF CONTENTS

Page

Abstract ...... ii

Dedication ...... iv

Acknowledgments ...... v

Vita...... vi

List of Tables ...... ix

List of Figures ...... x

List of Symbols ...... xii

Chapters:

1. Introduction ...... 1

1.1 Modeling of Non-Equilibrium Heat Transport ...... 3 1.2 Molecular Dynamics ...... 5 1.3 Methods Based on Boltzmann Transport Equation ...... 7 1.4 Background and Scope of the Thesis ...... 10 1.5 Thesis Overview ...... 16

2. Theory ...... 17

2.1 Sub-Continuum Heat Conduction in Solids ...... 18 2.1.1 Wave propagation in crystals ...... 18 2.1.2 Energy Quantization: Phonons ...... 23 2.2 Boltzmann Transport Equation ...... 24 2.2.1 BTE for Phonons ...... 25

vii 2.3 Dispersion Relationships for Silicon ...... 26 2.4 Phonon Interaction (Scattering) Processes ...... 28 2.4.1 Impurity Scattering ...... 30 2.4.2 Boundary Scattering ...... 30 2.5 Phonon-Phonon Interactions ...... 31

3. Monte-Carlo Solution Technique ...... 37

3.1 Number of Stochastic Samples ...... 37 3.2 Initialization of Phonon Samples ...... 39 3.2.1 Position ...... 39 3.2.2 Frequency ...... 40 3.2.3 Polarization ...... 41 3.2.4 Direction ...... 42 3.3 Linear Motion (Drift) of Phonons ...... 43 3.4 Boundary Interactions ...... 45 3.5 Intrinsic Scattering ...... 49

4. RESULTS AND DISCUSSION ...... 52

4.1 Veriﬁcation of the MC code ...... 52 4.2 Thermal Conductivity and Role of Various Phonon Modes . . . . . 54 4.3 Numerical Issues ...... 67

5. SUMMARY AND FUTURE WORK ...... 76

5.1 Summary ...... 76 5.2 Future work ...... 78

Bibliography ...... 81

viii LIST OF TABLES

Table Page

2.1 Curve-ﬁt parameters for silicon dispersion data [1]...... 28

2.2 Three-phonon interactions considered in this study for the calculation of scattering time-scales for each phonon polarization...... 34

4.1 Contribution (%) of various phonon modes towards thermal energy transport when both acoustic and optical modes are considered. Film thickness = 0.42 µm...... 65

4.2 Contribution (%) of various phonon modes towards thermal energy transport when only acoustic modes are considered. Film thickness = 0.42 µm...... 66

4.3 Statistical noise in the thermal conductivity data for both two-dimensional and three-dimensional calculations at 80 K as a function of sample size. 72

4.4 Statistical noise in the thermal conductivity for two-dimensional calculations at 300 K as a function of sample size...... 73

ix LIST OF FIGURES

Figure Page

1.1 Validity of Heat Transfer Models ...... 6

1.2 Representation of a Thin Film (a) 3D, and (b) 2D ...... 14

2.1 One-dimensional spring-mass system ...... 18

2.2 One-dimensional dispersion relationship...... 20

2.3 Brillouin zone in K-space ...... 21

2.4 General form of dispersion relation in a three-dimensional crystal [2]. 22

2.5 Curve ﬁt data for dispersion curves for silicon in [100] direction. . . 29

3.1 Position sampling in a three-dimensional geometry...... 40

3.2 Coordinate system showing the direction of phonon emission and the associated angles ...... 43

3.3 Computational Domain (a) 3D, and (b) 2D ...... 45

3.4 Phonon reﬂection at the boundary ...... 47

3.5 Thermal conductivity with and without boundary scattering. . . . . 48

4.1 Comparison of the results obtained by MC method (ensemble average) against analytical solution in the ballistic limit...... 53

4.2 Comparison of the results obtained by MC method (raw data) against analytical solution in the diﬀusion limit...... 55

x 4.3 Temperature proﬁle for various degree of specularity (α)...... 56

4.4 Group velocity of various phonon modes...... 57

4.5 Normalized cumulative number density function ...... 58

4.6 Phonon-phonon scattering time-scales computed using the present hybrid approach and Holland’s model at diﬀerent temperatures. . . 59

4.7 Comparison of mean free paths of all four phonon modes at 300 K computed using the present hybrid approach, and data obtained using molecular dynamics by Henry and Chen [3] ...... 61

4.8 Predicted and measured [4] through-plane thermal conductivity for a 0.42 µm silicon ﬁlm ...... 62

4.9 Predicted and measured [4] through-plane thermal conductivity for a 1.60 µm silicon ﬁlm ...... 63

4.10 Predicted and measured [4] through-plane thermal conductivity for a 3.0 µm silicon ﬁlm ...... 64

4.11 Predicted thermal conductivity using two-dimensional versus three- dimensional computational domains...... 68

4.12 CPU time taken for two-dimensional versus three-dimensional simu-

lations with Nprescribed = 50,000...... 69

4.13 Time-dependent energy ﬂux at the boundaries for a 0.42 µm thin ﬁlm at 80K ...... 71

4.14 Time averaged energy ﬂux (average of data presented in Fig. 4.13(a))

at the boundaries for a 0.42 µm thin ﬁlm computed using Nprescribed = 50,000...... 72

4.15 Time-dependent energy ﬂux at the boundaries for a 0.42 µm ﬁlm at 300K...... 75

xi LIST OF SYMBOLS

a lattice constant [m] a acceleration vector [m/s2] b reciprocal lattice vector [m−1] BL constant used in relaxation time calculation BTN constant used in relaxation time calculation BTU constant used in relaxation time calculation C heat Capacity [J/K] f distribution function F normalized cumulative number density function F˘ correction factor Fij interatomic force between atoms i and j [N] g spring constant [N/m] h¯ Dirac constant = 1.0546 x 10−34 [m2kg.s−1] K wave vector [m−1] −23 2 −2 −1 kB Boltzmann constant = 1.381 x 10 [m kg.s K ] K magnitude of wave vector [m−1] Kn Knudsen number L characteristic length [m] M, m1, m2 mass [kg] hni phonon occupation number ˆn unit surface normal Nb number of spectral (frequency) bins Nactual total number of phonons per unit volume Nprescribed number of phonons traced p phonon polarization P momentum vector [kg-m/s] Pscat probability of scattering of a phonon q heat ﬂux vector [W/m2] r position vector [m] rc eﬀective radius [m] Rscat,Rf random numbers between 0 and 1 Rpol,R1,R2 random numbers between 0 and 1

xii t time [s] ˆt1,ˆt2 unit surface tangents T thermodynamic temperature [K] T˜ pseudo-thermodynamic temperature [K] vg phonon group velocity vector [m/s] vg magnitude of phonon group velocity [m/s] vp phase velocity [m/s] W weight factor xn, yn displacements [m] α degree of specularity γ Gruneisen parameter ∆t time step [s] ∆ωi bandwidth of a spectral interval [rad/s] κ thermal conductivity [W/m-K] λ nucleic wavelength Λ mean free path θ polar angle θD Debye Temperature[K] ρ density [kg/m3] 3 ρd number of scattering sites per unit volume [1/m ] σ standard deviation in thermal conductivity [W/m-K] 2 σd scattering cross-section [m ] τ overall relaxation time [s] τb relaxation time for boundary scattering [s] τi relaxation time for impurity scattering [s] τij relaxation time for interaction between i and j [s] τN relaxation time for Normal (N) scattering [s] τNU combined relaxation time for N and U scattering [s] τU relaxation time for Umklapp (U) scattering [s] ωo,vs,c constants used for calculating phonon dispersion ω angular frequency [rad/s] ωo central frequency of the i-th spectral bin[rad/s] ψ azimuthal angle χ degeneracy of each phonon polarization

xiii CHAPTER 1

INTRODUCTION

In recent years, aggressive scale-down in the feature sizes of electronic devices, coupled with faster processing speeds, has resulted in large quantity of heat being generated per unit volume in these devices. The inability to remove heat eﬃ- ciently is currently one of the major stumbling blocks toward further miniaturization and advancement of electronic, optoelectronic, and micro-electro-mechanical system

(MEMS) devices. The efficient removal of heat from such devices is a daunting task, and overheating is one of the most common causes of device failure. In order to for- mulate better heat removal strategies and designs, it is first necessary to understand the fundamental mechanisms of heat transport in semiconductor thin films. Accord- ingly, there has been an increasing interest in modeling thermal transport in micro- and nanoscale semiconductor devices to gain a better fundamental understanding of the mechanism of heat conduction in crystalline thin films. Modeling techniques, based on first principles, can play the crucial role of filling gaps in our understanding by revealing information that experiments are incapable of.

Since the seventies, the microelectronics industry has seen a relentless drive toward smaller features and higher functionality on semiconductor chips. In 2001, high-performance processor chips such as the POWER4, contained more than 170

1 million transistors. This number is projected to exceed a four billion by 2010 [5].

As VLSI technology scales, thermal issues are becoming the dominant factor in determining performance, reliability and cost of high-performance integrated circuits.

Management of these issues is going to be one of the key factors in the development of next-generation microprocessors, integrated networks, and other highly integrated systems [6]. Transduction mechanisms involving thermal phenomena are also at the heart of the functionality of micro-electro-mechanical (MEMS) devices. Thus, the ability to accurately characterize and control thermal phenomena is also crucial for the design of next-generation MEMS [7] and NEMS [8] devices. It is now universally acknowledged that modeling and simulations will play a central role in enabling the leap from micro- to nano- technology [5, 6, 7, 9]. The successful production of reli- able nanoscale devices requires the development of multi-physics modeling tools that address coupled thermal, mechanical, chemical, electronic and other phenomena [5].

While appreciable progress has been made in the modeling of charge carrier transport in electronic devices, equivalent progress has not been made on the thermal side.

In order to design better heat removal strategies for next-generation electronic and

MEMS devices, thermal simulation technology needs to make rapid progress. While modeling tools have proliferated within the semiconductor and MEMS industries for characterizing thermal phenomena, these tools serve the purpose at the package and systems levels only [5]. These tools are based on solution of the continuum energy transport equation, and takes into account heat transport by conduction, convection and sometimes, radiation. Such analysis is possible using any standard computational

ﬂuid dynamics (CFD) tool, or by tools speciﬁcally designed for the microelectronics industry that essentially uses a CFD engine in the background with a user-friendly

2 frontend. In such modeling tools, heat transport by conduction is modeled using the

Fourier law of heat conduction, and thermal conductivity values of the bulk materials are used [5]. In most cases, the same tools are also used at the device level, and therein lay the problem. Depending on the application, the characteristic dimension of an electronic or MEMS device could range anywhere from few tens of nanometers to few tens of micrometers. Heat conduction in crystalline dielectric solids (which include most semiconductor materials) occurs by lattice vibrations. The lattice vibrations result in traveling waves of various frequencies and speeds ultimately causing energy to propagate within the solid. From a quantum mechanical perspective, these waves may be thought of as discrete energy packets, called phonons [10, 11, 12]. The average mean free path of the energy carrying phonons in silicon at room temperature is about 300 nm [13], which is much larger than the characteristic feature sizes in modern electronic devices. If the mean free path of the phonons is larger than the characteristic dimension of the device in question, thermodynamic equilibrium ceases to exist, and the following two facts are realized: (1) the Fourier law of heat conduction, which assumes local equilibrium, becomes invalid, and (2) bulk thermal conductivity values cannot be used to characterize thermal properties of the material because (a) these values are determined experimentally by inversion of the Fourier law itself, and (b) scattering at the geometric boundaries of the device, both adiabatic and non-adiabatic, becomes important.

1.1 Modeling of Non-Equilibrium Heat Transport

The Fourier law of heat conduction is given by :

q(r, t) = −κ(T )∇T (r, t) (1.1)

3 where κ is the thermal conductivity and q is the heat ﬂux vector. The transient heat conduction equation is therefore given by:

∂T C = ∇ • κ(T )∇T (1.2) ∂t where C is the heat capacity, κ is the thermal conductivity and T is the thermodynamic temperature. Eq. 1.2 is a parabolic differential equation having an infinite wave speed. However, phonons travel inside a medium with a finite group velocity vg and the information transfer is not instantaneous. Adding a hyperbolic time term to

Eq. 1.2 imparts a ﬁnite wave speed. This leads to the Cattaneo equation, given by:

∂2T ∂T τC + C = ∇ • κ∇T (1.3) ∂t2 ∂t where τ is the phonon relaxation time. In a device with characteristic dimension L, two dimensionless parameters are frequently used to describe the nature of phonon transport. These are a) acoustic thickness, deﬁned as, L/(τvg) and b) Knudsen number (Kn), which is the inverse of acoustic thickness, and is deﬁned as Λ/L, where

Λ is the phonon mean free path and is the product of the phonon group velocity and the relaxation time. When the mean free path of the phonons is larger than

L, the scattering events are rare and the phonon transport is termed as “ballistic”.

Ballistic transport is characterized by Kn > 1 (or acoustic thickness < 1). When the mean free path of phonons is smaller than the characteristic dimension, such that there are sufficient number of scattering events, the transport of phonons is described as “diffusion dominated”. Diffusion dominated regime is characterized by large acoustic thickness or Kn << 1. When the acoustic thickness is large, Eq. 1.3 models phonon behavior adequately. In the acoustically thin limit, however, Eq. 1.3 cannot capture the ballistic behavior of the phonons. Therefore, both the Fourier law

4 and the Cattaneo equation are unsuitable for modeling phonon transport, especially when the acoustic thickness is small [14].

In recent years, two main approaches have been adopted to model non-equilibrium heat transport, a) Molecular Dynamics (MD) simulations, and, b) Boltzmann Trans- port Equation (BTE) based methods. Both these approaches are valid as long as phonons can be assumed to be particles i.e. quantum wave eﬀects can be neglected.

Particle theory fails when the characteristic dimension of the system, L, becomes much smaller than both the mean free path of energy carriers (Phonons), Λ, and the De Broglie wavelength of the atomic nuclei, λ. For silicon at room temperature, the wavelength associated with atomic nuclei is about 0.2A˚ [3], which is about an order of magnitude smaller than the atomic separation (5.43A˚ for silicon). At low temperatures however, this wavelength increases and interference effects may become important. In such as case, material (matter) waves can no longer be neglected and particle theory leads to an inaccurate description of the physical phenomenon. The wave behavior of the system can be modeled using the Schrödingerwave equation. An excellent discussion on energy transfer by waves may be found in [15]. The validity of various approaches to model heat transfer at different length scales has been pic- torially represented in Fig. 1.1. A brief introduction to MD and BTE based methods is presented next.

1.2 Molecular Dynamics

Molecular Dynamics (MD) involves the integration in time of the Newtons equations of motion for a large collection of particles that interact with one another. A system of N particles of mass M and position ri is considered, and Newton’s second

5 Figure 1.1: Validity of Heat Transfer Models

law of motion is written for each particle as:

2 N d ri X M = F (1.4) dt2 i,j j,j6=i where Fij is the interatomic force between atoms i and j. The MD approach needs a good interatomic potential describing the underlying force interactions. The fidelity of the scheme depends upon the accuracy of the interaction potential. The nature of interatomic potentials depends upon the electronic properties of the material and are usually determined by either ab-initio (or first principles) calculations (Quantum MD) or obtained by assuming a functional form of the potential and then curve-fitting parameters based on comparison between calculated and experimentally obtained properties (Classical MD). For silicon, three-body potentials such as Stillinger-Weber [16],

Tersoﬀ [17] and Environmental Dependent Interatomic potential (EDIP) [18] have been used for making thermal conductivity predictions [19]. Detailed discussions on

6 MD calculations and interatomic potential may be found in [3, 19, 20]. For typical many-body potentials, the force calculation is so computationally expensive that it limits the application of MD approach to larger systems (micron-sized structures).

The MD approach, though, is quite suitable for nanostructures and thin ﬁlms. MD is a powerful tool to gain insight into the nature of phonon interactions in strongly non-equilibrium situations and has been extensively used to calculate relaxation times for these interactions [3, 21].

1.3 Methods Based on Boltzmann Transport Equation

Non-equilibrium heat conduction in thin ﬁlms can be alternatively modeled using the semi-classical Boltzmann Transport Equation (BTE) for phonons. A detailed discussion of the BTE for phonons is presented later in section 2.2. In the past, various

BTE-based analytical models were proposed to calculate bulk thermal conductivity of semiconductor materials. Callaway [22] proposed a phenomenological model to facilitate the calculation of thermal conductivity of semiconductor materials at low temperatures. Two main limiting assumptions of the Callaway model were elastic isotropy and the absence of dispersion. Callaway also did not make any distinction between the relaxation times of longitudinal or transverse phonons. The expression for the thermal conductivity was derived from the BTE under the relaxation time approximation and is given by [23]:

3 Z θD/T 4 x kB kbT x e κ = 2 τ(x) x 2 dx (1.5) 2π vg h¯ 0 (e − 1) where θD is the Debye temperature, vg (=|vg|) is the magnitude of the phonon group velocity, kB is the Boltzmann constant, T is the temperature at which the thermal conductivity needs to be calculated, and x =hω/k ¯ BT . Holland [24, 25] improved

7 upon the Callaway model by including phonon dispersion and separate frequency and

temperature dependent lifetimes for transverse and longitudinal phonons. Holland,

in his formulations, assumed a linear dispersion relationship, which gives a constant

group velocity for each phonon mode. The thermal conductivity expressions can then

be written as:

κ = κT + κL (1.6)

where Z θD,T /T 4 x 1 3 x e κL = CT T τT (x) x 2 dx 3 0 (e − 1)

Z θD,L/T 4 x 2 3 x e κT = CLT τL(x) x 2 dx (1.7) 3 0 (e − 1) subscripts L, T refer to longitudinal and transverse acoustic modes, respectively,

3 kB kB whereas, the constant CL,T = 2 . The thermal conductivity integrals were 2π vL,T ¯h evaluated by using some ﬁtting parameters for the phonon relaxation times for both

longitudinal and transverse modes. After tuning these ﬁtting parameters against ex-

perimental data, functional forms of the relaxation times were presented for both

transverse and longitudinal modes. The increasing complexity and the underlying as-

sumptions described above limit the use of these analytical expressions for predicting

thermal conductivity of thin ﬁlms. With the advent of modern computing, numerical

techniques to calculate thermal conductivity of semiconductors were developed.

In recent years, numerical solution of the BTE has been obtained by two tech-

niques: a) deterministic methods [2, 26, 27] and (b) stochastic or Monte-Carlo method

[28, 29]. Deterministic solutions, which solve the BTE as a partial diﬀerential equa-

tion, came to the forefront with the development of the Equation of Phonon Radiative

Transport (EPRT), ﬁrst proposed by Majumdar [30]. The EPRT is derived from the

8 BTE by drawing an analogy with the Radiative Transport Equation (RTE) [31].

Numerical techniques developed to solve the RTE have been extensively applied to

solve the phonon BTE. These include, the Control Angle-Control Volume Discrete

Ordinates Method (CA-CV DOM) [14, 26, 2] and the Modiﬁed Diﬀerential Approxi-

mation [32, 33]. In the CA-CV DOM method, the BTE is solved in the energy density

form. The CA-CV DOM formulation starts with the discretization of space, time,

angle and frequency space. Spatial discretization is carried out by taking convex poly-

hedra and the angular discretization is done by splitting the angular space, 4π, into

discrete non-overlapping solid angles ∆Ωi, each centered about the discrete direction

ˆsi. Each octant is discretized into Nθ × Nψ solid angles. The governing equation is integrated over the control volume, control angle, frequency interval, and time step, yielding a balance of phonon energy ﬂux in the direction i through the control vol-

ume faces [14]. CA-CV DOM has been found to be computationally quite expensive.

In order to reduce cost of simulation, while retaining the physics of ballistic phonon

transport, a ballistic-diﬀusive approximation to the BTE, which is similar to the

Modiﬁed Diﬀerential Approximation for the Radiative Transfer Equation, was pro-

posed in [32]. For solving the Ballistic-Diﬀusive Equation (BDE) the phonon energy

is decomposed into two components: the ballistic or boundary component (b), and a

component associated with the medium (m). The ballistic component is computed

using a ray-tracing procedure for which exact solutions are possible in simple geome-

tries. For the medium component, a diﬀuse approximation is made, resulting in a

hyperbolic heat conduction-like equation, with extra sources/sinks due to the ballistic

component. An extension to incorporate dispersion eﬀects has been presented in [34].

The solution obtained from the BDE has been found to be accurate for acoustically

9 thin problems. For larger acoustic thicknesses (or low Knudsen number), due to the

failure of the diﬀusion approximation in the medium component near boundaries,

the solution is not so accurate. A discrete development of the BTE, known as the

Lattice Boltzmann Method, has also recently been used to model sub-continuum heat

transfer [35, 36].

Monte-Carlo solution of the BTE, which is the primary topic of this thesis, has

been discussed in detail in Chapter 3. A brief history of the development of Monte-

Carlo solution of the BTE is discussed next, followed by the scope of this thesis.

1.4 Background and Scope of the Thesis

The ﬁrst formal Monte-Carlo procedure for phonon transport was presented by

Peterson [37]. Peterson’s study did not consider phonon dispersion, and did not

account for the various phonon polarizations. Mazumder and Majumdar [28] pre-

sented the ﬁrst comprehensive algorithm to solve the BTE for phonons by the MC

method with the inclusion of phonon dispersion and polarization. In their approach,

statistical samples (phonons) are drawn from six individual stochastic spaces: three

wave vector and three position vector components. The sampled phonons ﬁrst un-

dergo drift (ballistic motion) and then undergo scattering events. Lacroix et al. [29] presented a transient MC algorithm that modiﬁed the scattering algorithm used by

Mazumder and Majumdar [28] to enforce Kirchoﬀ’s Law during the scattering phase.

This modified algorithm has later been used by other researchers [38, 39, 40, 41], and has been found to accurately predict transient heat conduction in thin films in both ballistic and diffusive regimes.

10 Although Mazumder and Majumdar [28] accounted for the acoustic modes of

phonon propagation, the contribution of optical phonons towards energy transport

was neglected based on the presumption that they have slow group velocities, and

therefore, do not contribute to energy transport [11]. Although this contention is

intuitive, it is well known [27] that interaction between optical and acoustic phonons

alters the eﬀective relaxation rates of the acoustic phonons, and thereby, aﬀects energy

transport in an indirect manner. Narumanchi [27] suggests that optical phonons must

be considered even in steady-state thermal predictions. While the group velocity of

optical phonons is small at small and large values of the wave-vector, K, it is not

negligible for intermediate values of K. For Longitudinal Optical (LO) phonons, the group velocity can be comparable to the slower acoustic modes [2]. Therefore, further systematic studies are necessary to understand the eﬀect of optical phonons on thermal transport.

Narumanchi and co-workers [26, 27] and Wang [2] have considered optical phonons in their deterministic ﬁnite-volume formulation for solving the BTE. The model proposed by Narumanchi invoked two simplifying assumptions while taking optical phonons into consideration in his model. First, similar to the assumptions made in the past [28], the optical phonons were assumed to have zero group velocities. Sec- ondly, all the optical phonons were clubbed into a single band and no distinction was made between Longitudinal Optical (LO) and Transverse Optical (TO) modes.

Though Wang [2] removed the ﬁrst assumption made in Narumanchi’s work, the two polarizations of the optical phonons have not been considered in calculating phonon lifetimes. The result of their work, therefore, does not bring out explicitly the role of optical phonons in silicon thermal conductivity prediction and the subsequent impact

11 on energy transport in semiconductor thin ﬁlms. Recently, Kazan et al. [42] studied

the role of optical phonon decay into acoustic modes using the modiﬁed Callaway

theory for Germanium and found that the inclusion of optical phonons increases the

accuracy in predicting the thermal conductivity of semiconductors. Therefore, these

studies make a compelling case for inclusion of optical phonons in the numerical so-

lution of the BTE for phonons. The current work seeks to build upon the works of

Mazumder and Majumdar [28] and Lacroix et al. [29] to solve the BTE for phonons with full phonon dispersion including Longitudinal Optical (LO) and Transverse Op- tical (TO) phonons using the MC method. First, the BTE is solved considering only

Longitudinal Acoustic (LA) and Transverse Acoustic (TA) modes, and the thermal conductivity of silicon thin ﬁlms are computed. Subsequently, optical modes are also considered and the thermal conductivity results, thus obtained, are compared to the results obtained without considering optical phonons as well as experimental results obtained by Asheghi [4].

The biggest roadblock in incorporating optical phonons in the numerical solution of the BTE for phonons is the lack of three-phonon scattering time-scales for the optical modes. Using the expressions given by Han and Klemens [43], Naru- manchi and co-workers [26, 27] have presented a methodology for calculating the three-phonon interaction time-scales for the various phonon modes undergoing Umk- lapp (U) scattering, including optical phonons. In Narumanchi’s work, however, the contribution of Normal (N) processes towards phonon relaxation time calculation has not been considered. Narumanchi and co-workers [26, 27] considered only a subset of the possible interactions between diﬀerent phonon modes for the phonon lifetime calculations. The interactions considered embody the low temperature assumption,

12 thereby allowing simpliﬁcations to be made to the phonon conservation laws. While

removing this assumption is desirable, it would imply consideration of a very large

number of three-phonon interactions, and rigorous implementation of the energy and

momentum conservation laws to calculate the three-phonon interaction time-scales

over the whole wave-vector space [2]. This would increase the complexity of the

numerical simulation dramatically. Therefore, this has not been attempted in the

present work. As a start, the phonon lifetimes for U processes of various phonon

modes have been calculated using the expressions given by Han and Klemens [43]

and interactions considered by Narumanchi et al. [26, 27]. The contribution of the

N processes to the phonon lifetime data is calculated from the expressions given by

Holland [24]. Therefore, in order to calculate the overall phonon relaxation time a hybrid approach, derived from the works of Holland [24] and Han and Klemens [43] has been employed in the present work. The objective of this study is to shed some light, albeit preliminary, on the eﬀect of optical phonons on energy transport and thermal conductivity predictions in silicon thin ﬁlms.

Ever since the development of the first comprehensive Monte-Carlo procedure by Mazumder and Majumdar [28], significant advances have been made toward the improvement of such algorithms. These include alternative and, perhaps, better pro- cedures for the treatment of scattering [29, 44], alternative formulations for sampling phonons [45] and steady-state approaches aimed towards reduction of computational time and memory [39]. Despite these advances, a few critical questions remain unan- swered. For Monte-Carlo simulations aimed towards prediction of through-plane thermal conductivity of thin films, the lateral (or in-plane) dimensions (Fig. 1.2) of the

ﬁlm are free parameters. Ideally, if the lateral (or non-thermalizing) boundaries are

13 (a) 3D

(b) 2D

Figure 1.2: Representation of a Thin Film (a) 3D, and (b) 2D

made perfectly specular, the eﬀect of the lateral dimension is irrelevant because in that case, these boundaries would pose no resistance (i.e., they are symmetry planes).

In the actual physical scenario, the thermal conductivity at low temperature is dic-

tated by boundary scattering, and the boundary must be made partially specular to

keep the thermal conductivity ﬁnite in the ballistic limit. This is done by calibrating

the degree of specularity (α) against experimental data, as originally proposed by

Mazumder and Majumdar, and later used by almost all subsequent studies [39, 44].

Therefore, the choice of the lateral dimension would aﬀect the calibrated value of

α, and it is important to ensure that the calibration of and the choice of lateral di-

mensions has no bearing on the predicted thermal conductivity. Furthermore, it is

important to understand the eﬀect of the choice of lateral dimension on the compu-

tational eﬃciency.

14 Mazumder and Majumdar [28] used a 2D computational domain to predict thin

ﬁlm thermal conductivity, whereas later studies [29, 39, 44] have carried out their simulations on a 3D computational domain. Whether a 2D or 3D computational domain should be used is still a matter of debate, and systematic studies are required to elucidate the pros and cons of either choice. Statistical errors are inherent to any

MC simulation. In this particular case, the statistical noise in the heat ﬂux (or thermal conductivity) data depends on the number of stochastic samples used for the MC simulation. The number of stochastic samples, in turn, determines the computational cost. This work seeks to systematically explores this correlation between the above- mentioned numerical issues through careful analysis of the statistical errors, and the associated computational cost. To summarize, the thesis has the following objectives:

• To develop a general procedure to solve the Boltzmann transport equation for

phonons in 2-D and 3-D geometries.

• To validate the developed numerical procedure in the ballistic and diﬀusive

regimes against analytical results.

• To develop a methodology to calculate frequency and temperature dependent

relaxation time-scales for three-phonon interactions, in particular, interactions

involving optical phonons.

• To predict the thermal conductivity of silicon thin ﬁlms with and without the

inclusion of optical phonons and elucidate the role of optical phonons on thermal

energy transport.

• To investigate and highlight the relevant numerical issues pertaining to the

computational eﬃciency and statistical accuracy of Monte-Carlo methods.

15 1.5 Thesis Overview

The rest of the thesis is organized as follows. Chapter 2 presents the theoretical basics of heat conduction at the sub-continuum level. The governing equation (BTE) and the approximations made to it are introduced. Inputs required to solve the

BTE, namely, the phonon-scattering time-scales and the dispersion relationship are discussed in detail. Chapter 3 presents the Monte-Carlo solution (MC) technique used to solve the BTE. In Chapter 4, the MC solution technique is ﬁrst validated by carrying out simulations for the limiting scenarios that allow analytical solution.

Next, the thermal conductivity of silicon thin ﬁlms of various thicknesses is predicted.

These results are compared against experimental data. The role of various phonon modes on energy transfer are quantified and elucidated for different film thicknesses and temperatures. Some numerical issues pertaining to the implementation of the

MC method are also highlighted. Finally, a brief summary of the the work carried out is presented in Chapter 5, followed by some recommendations for future work.

16 CHAPTER 2

THEORY

The micro/nanoscale description of heat conduction is significantly different from that at macroscales. Under the continuum assumption, the constitutive laws such as the Fourier diffusion law describe thermal phenomenon fairly well [27]. When the continuum assumption breaks down, it becomes necessary to consider carrier transport in order to describe thermal phenomenon. In such cases, as the system is not in equilibrium, temperature loses its conventional meaning and can only be described as the measure of the energy of the system. As pointed out by Narumanchi [27], continuum based approaches may fail even at macroscales. The mean free path of the energy carriers increases at low temperatures and may become comparable to size of the physical system if the operating temperatures fall in the cryogenic range (< 100

K). In crystalline solids, thermal energy transport is best described by the propagation of acoustic waves. From a quantum mechanical perspective, these acoustic waves may be thought of as packets of energy, called phonons. In the next few sections, the theory underlying heat transport in crystalline solids in described.

17 2.1 Sub-Continuum Heat Conduction in Solids

2.1.1 Wave propagation in crystals

A crystal structure can be viewed as a three-dimensional arrangement of masses

and springs where masses represent the atoms and springs represent the chemical

bonds connecting the atoms in the crystal lattice. The movement of one or a group

of atoms can be transmitted as a wave through this spring-mass network. These

vibrations are responsible for energy transport in many solids. For simplicity, let us

consider a one-dimensional array of masses and springs as shown in Fig. 2.1. The

Figure 2.1: One-dimensional spring-mass system

particles with mass m1 and m2 form a unit cell and are connected to each other by springs, having a spring constant g. The equilibrium spacing between the nearest

neighbors is denoted by a. It is assumed that the springs behave elastically, obeying

Hooke’s law. If the chain of atoms is inﬁnitely long, the reference frame can be

attached to any of the particles. For simplicity, we assume the equilibrium coordinates

of particle with mass m1 are x, and those of particles m2 are y. Let the displacement

of the particles from their equilibrium positions be denoted by xn and yn. It is further

18 assumed that only the nearest neighbors interact with each other, i.e., the restoring

force on a particle of mass m1 depends only on its relative position with respect to the adjacent particle of mass m2. When the nearest neighbor assumption is adopted, the

force between two particles spaced by more than one lattice constant a is considered

to be zero. Thus the equations of motion of the two particles having masses m1 and

m2 particles are

d2x m n = g[y − 2x + y ] 1 dt2 n n n−1 d2y m n = g[x − 2y + x ] (2.1) 2 dt2 n+1 2n n

We look for solutions of Eq. 2.1 in the form of traveling waves, given by:

xn = A exp[i((Kx) − ωt)]

yn = B exp[i((Ky) − ωt)] (2.2)

where A and B are the amplitudes of the waves and K is the wave vector whose value is given by the relation |K| = 2π/λ. Substituting Eq. 2.2 into the equation of motions 2.1 we get:

2 (m1ω − 2g)A + g(1 + exp(−i|K|a)B = 0

2 g(1 + exp(i|K|a))A + (m2ω − 2g)B = 0 (2.3)

A non-trivial solution for a set of homogeneous equations given in Eq. 2.3 exists if

and only if the determinant in zero, i.e.,

2 m1ω − 2g g(1 + exp(−i|K|a)) 2 = 0 (2.4) g(1 + exp(i|K|a)) m2ω − 2g or

4 2 2 m1m2ω − 2g(m1 + m2)ω + 2g (1 − cos|K|a) = 0 (2.5)

19 Figure 2.2: One-dimensional dispersion relationship.

Solving Eq: 2.5, we get  s  2 2 1 1 1 1 4 |K|a ω = g  + ± + − sin2  (2.6) m1 m2 m1 m2 m1m2 2

Considering only positive solutions of ω2,

1   s  2 1 1 1 1 2 4 |K|a ω = g  + ± + − sin2  (2.7) m1 m2 m1 m2 m1m2 2

When ω is plotted as a function of K, because of the ± sign in Eq. 2.7, two curves

are obtained. The relation between the frequency ω and wave vector K is called the dispersion relationship. As seen in Fig. 2.2, there are two branches or polarizations corresponding to the solution of the Eq. 2.5, called the acoustic branch and the optical branch. The solution repeats itself in the region for K = ±π/a. This region is known as the ﬁrst Brillouin Zone (BZ) as shown in Fig. 2.3. The coeﬃcient matrix corresponding to Eq. 2.4 is called the dynamical matrix. The acoustic branch is obtained when the two atoms comprising the unit cell vibrate in phase with each other.

20 Figure 2.3: Brillouin zone in K-space

Acoustic waves correspond to frequencies that become small at long wavelengths, and give rise to sound in solids. Therefore, the phonons that belong to the lower frequency spectrum are called “acoustic phonons”. The optical branch is obtained when the two atoms vibrate out of phase with each other. If there is an electric dipole created by an uneven charge distribution in the chemical bond, then the optical mode corresponds to an oscillating dipole. An oscillating dipole scatters radiation and strongly inﬂuences the optical properties of the crystal [12]. Therefore, the frequencies that correspond to the out of phase motion of the atoms in the unit cell are called “optical phonons”.

Now, consider a three-dimensional crystal, with two atoms per unit cell. Each of the two atoms has three degrees of freedom, leading to six equations of motions.

There are therefore, six phonon polarizations, three optical and three acoustic. The optical phonons consist of one longitudinal and two transverse branches, as do the acoustic phonon branches. In the longitudinal mode, atoms vibrate in the direction

21 Figure 2.4: General form of dispersion relation in a three-dimensional crystal [2].

of wave propagation whereas in the transverse mode, they vibrate perpendicular to

the direction of propagation. The speciﬁc shape of the dispersion curves depends on

the direction of the wave vector K. Figure 2.4 shows the general form of dispersion relation in 3-D crystals. The group velocity vg is deﬁned as the gradient of frequency ω

with respect to wave vector K. This is the velocity with which the energy propagates

inside the medium. Thus,

vg = ∇Kω (2.8)

However, in real crystals some of the branches are degenerate because of one or more

geometric symmetries. Dispersion relationship for a real crystal (silicon) is discussed

in section 2.3.

22 2.1.2 Energy Quantization: Phonons

If the lattice vibrations were to be modeled as a classical spring-mass (harmonic

oscillator) system, then the energy of the system would be a continuous function of

displacement. Quantum mechanical description sets a restriction on the energy states

that can be occupied. Thus, the energy of the lattice vibration is quantized. The

quantum of energy is called a phonon in analogy with the photon for the electromag-

netic waves. Each phonon carries a quantum of energyhω ¯ . By conceiving phonons

as particles, it is possible to deﬁne a distribution function for phonons. Phonons, like

photons, are bosons and therefore, follow Bose-Einstein statistics. The equilibrium

distribution of phonons at a temperature T is given by the Bose-Einstein distribution:

1 hni = (2.9) exp ¯hω − 1 kB T

where kB is the Boltzmann constant, ω is the frequency andh ¯ is the Dirac constant.

In an elastic continuum, the wave vector K can change continuously. In a crystal lattice, however, the wave vector and, hence, the frequency can only take discrete values, which limits the number of distinguishable vibration modes in a lattice with

ﬁnite size made up of discrete atoms. The number of modes per unit frequency range per unit volume of the lattice is called the density of states, D(ω). From these

deﬁnitions, the total energy in a material of volume V = L3 can be written as

X 1 E.V = hni + hω¯ (2.10) K,p 2 K,p K,p where E is the energy per unit volume of the material. The subscript K stands for the wave vector and the subscript p stands for the particular branch of the phonon mode (polarization). The summation is over the whole wave-vector space in a three- dimensional Brillouin zone (BZ) and over all phonon polarizations. In the limit of a

23 large crystal, the wave-vector space becomes so dense that it can be replaced by an integral as follows: X Z 1 E.V = hni + hω¯ dK (2.11) K,p 2 K,p p K The integration of the wave-vector space is diﬃcult because it involves the direction and magnitude of the wave-vector. The integration over wave-vector space can be transformed to integration over frequency space by invoking the dispersion relation,

K = K(ω, p), and assuming that the Brillouin zone is isotropic, yielding

X Z 1 E = hni + hω¯ D(ω)dω (2.12) K,p 2 K,p p ω where, as deﬁned earlier, D(ω) is the phonon density of states. The quantity D(ω)dω represents the number of vibrational states between ω and ω + dω. The second term within parenthesis in Eq. 2.12 is independent of temperature, and can be summed up to yield the so-called zero-point energy.

The description of phonons as a distribution of particles characterized by diﬀerent polarizations and wave vectors (and correspondingly, frequencies) implies a departure from the wave description. A particle view of phonons is useful, when wave eﬀects such as wave coherence are not important; however, it is necessary to retain certain important aspects of wave description, such as the dispersion behavior, as well as the modeling of wave interactions through scattering.

2.2 Boltzmann Transport Equation

Since it was ﬁrst established by Ludwig Boltzmann, the Boltzmann equation or the Boltzmann Transport Equation (BTE) has been applied to model the behavior of particles that interact with each other by short range forces and follow a certain

24 statistical distribution (such as electrons, ions, phonons, and photons etc.). The

BTE is fundamental in the sense that it is valid even at the microscale, where local

thermodynamic equilibrium in space and time cannot be deﬁned. Furthermore, other

macroscopic equations such as Fourier law, Ohm’s law, Fick’s law and the hyperbolic

heat equation, which break down at small length and time-scales, can be derived from

the BTE in the macroscale limit. The general form of the BTE can be written as [12]:

∂f ∂f ∂f + v · ∇f + a · = (2.13) ∂t ∂v ∂t scat where f is the distribution function of particles, v is the particle velocity and a is

the acceleration of the particle under the action of an external force. The left hand

side of Eq. 2.13 contains the transient and the drift terms respectively, and the right

hand side represents the change in the distribution function due to various particle

interaction processes, or so-called scattering.

2.2.1 BTE for Phonons

Phonons follow Bose-Einstein statistics and interact with each other via scattering

processes, and therefore, can be modeled using the BTE. In the absence of any exter-

nal force, the acceleration vector a is zero, and therefore, the third term in Eq. 2.13

is absent. The BTE for phonons may then be written as [12]

∂f ∂f + vg · ∇f = (2.14) ∂t ∂t scat

where f(r, K, t) is the distribution function for phonons, and vg is the phonon group

velocity. The left side of Eq. 2.14 represents change of the distribution function due

to motion (or drift), whereas the right hand side represents change in the distribution

function due to collisions (or scattering). For a given state f(r, K, t), the interactions

25 should involve phonons from the set of (r, K) to set (r0, K0).

∂f X = [φ(K0, K)f(K0) − φ(K, K0)f(K)] (2.15) ∂t scat K0 where φ(K0, K)f(K0) and φ(K, K0)f(K) are scattering phase functions from K0 to

K and K to K0, respectively. Due to the complexity of the scattering term, simpli-

ﬁcations and approximations have to be made to the BTE before it can be solved.

The most common approximation used to simplify the BTE is the relaxation time approximation, whereby the scattering term is expressed as

∂f f − f = o (2.16) ∂t scat τ

where fo is the equilibrium distribution function (i.e., the Bose-Einstein distribution

function), and τ is the overall scattering time-scale of the phonon due to all scattering

processes in combination. This approximation essentially linearizes the scattering

term of the BTE and implies that whenever a system is not in equilibrium, the

scattering term will restore the system to equilibrium following an exponential decay

law: f − fo = exp(−t/τ). The study of transport phenomenon, essentially, requires

the calculation of distribution function from Eq. 2.14. Once the distribution function

is known, the energy ﬂux may then be calculated using:

X Z q(r, t) = vg(r, t)f(r, K, t)¯hωK,pD(ω)dω (2.17) p ω

2.3 Dispersion Relationships for Silicon

Prior to the solution of the BTE, information regarding phonon dispersion is

required. Phonon dispersion plays an important part in thermal conductivity calcu-

lations and is necessary to model polarization and frequency-dependent behavior [2].

26 It has been observed that the inclusion of phonon dispersion has a significant effect on the phonon mean free path [13]. The phonon dispersion relationships for a given material and a specified direction are determined by neutron scattering experiments [46, 47]. They may also be calculated from lattice dynamics either by solving the Schrödingerequation for lattice vibrations [10] or using semi-classical approaches under the harmonic approximation [48]. For bulk silicon, the experimental dispersion data has been well documented by Brockhouse [46] and Dolling [47]. For nanoscale

ﬁlms, use of bulk dispersion relationships is inappropriate due to the folding of the

Brillouin zone, also referred to as phonon confinement. According to Heino [41], the effect of phonon confinement becomes appreciable only in very thin structures around 10-20 nm thick. For the present study, since the film thicknesses considered are relatively thick, it was deemed appropriate to use bulk dispersion relationships.

Chung et al. [49] have discussed the impact of various curve fits for phonon dispersion data on thermal conductivity calculations and have proposed a model which uses quadratic curve fits for LA phonons and a cubic curve fit for TA phonons. In the present study, each phonon dispersion branch is treated under the isotropic Brillouin zone approximation by a quadratic curve fit given by [50]:

2 ωK = ωo + vs|K| + c|K| (2.18)

For the acoustic modes, the constants ωo, vs and c are chosen so as to capture the slope of the dispersion curve near the center of the Brillouin Zone (BZ) and the maximum frequency at the edge of the BZ. For longitudinal optical (LO) phonons, these values ensure that the LO frequency at the BZ boundary equals the maximum LA frequency.

For both TA and Transverse Optical (TO) phonons, the curves are ﬁtted such that the slope at the edge of the BZ is zero [1]. Quadratic curve ﬁts have been used as they

27 13 2 −7 2 Phonon Polarization ωo(10 rad/s) vs(10 m/s) c (10 m s LA 0.00 9.01 -2.00 TA 0.00 5.23 -2.26 LO 9.88 0.00 -1.60 TO 10.20 -2.57 1.12

Table 2.1: Curve-ﬁt parameters for silicon dispersion data [1].

give an accurate approximation to the phonon dispersion data and are easy to invert in order to get the phonon wave vector and the phonon group velocity data. The phonon group velocity is calculated using the expression given in Eq. 2.8. Table 2.1 lists the values of the curve-ﬁt parameters for silicon in the [100] direction [1]. The dispersion relationship in the [100] direction calculated from the curve-ﬁt, is shown in Fig. 2.5. This data is provided as input into the Monte-Carlo code in the form of constants from Table 2.1. In this particular study, since an isotropic Brillouin zone is assumed, the group velocity obtained using Eq. 2.8 is essentially the slope of the dispersion curves, and it is co-directional with the wave vector.

2.4 Phonon Interaction (Scattering) Processes

Phonons traveling in a solid may be scattered by a variety of mechanisms including

: lattice imperfections (i.e. vacancies, dislocations and impurities), interactions with electrons, interaction with other phonons (intrinsic scattering) and boundaries. All of these processes lead to interchange of energy between lattice waves. These scattering phenomena may be split into two broad categories: (i) elastic scattering, in which the energy (or the frequency) of the incident phonon in unchanged, and (ii) inelastic

28 Figure 2.5: Curve ﬁt data for dispersion curves for silicon in [100] direction.

scattering, in which three or more phonons are involved and the frequency of the phonon is modiﬁed in accordance with momentum and energy conservation rules.

Phonon scattering due to vacancies and dislocations can be neglected if the crystal is considered to be perfect. In the present context, the scattering of phonons by lattice imperfections is not considered. A detailed discussion on scattering by these mechanisms may be found in Ziman [10]. Here, a brief description of the various important mechanisms of scattering is provided as a lead-in to the description of the procedure adopted to calculate the phonon-phonon interaction time-scales.

29 2.4.1 Impurity Scattering

Experiments have shown that even a small increase in the impurity concentration can have a substantial eﬀect on heat conduction by phonons [12]. A defect can be thought of as a change in mass or spring constant within a crystal that can alter the local acoustic impedance or the vibrational characteristics of a crystal. Hence, when an incident wave encounters a change in acoustic properties, it tends to scatter. This is similar to scattering of an electromagnetic wave due to change in the refractive index. The lattice vibrations inside a solid may be thought of as a phonon gas [10], where phonons can be considered traveling in all directions and colliding between themselves and with ﬁxed impurities. Using kinetic theory of gases, the average time,

τi, between collisions with imperfections is a function of the defect density (or the number of scattering sites per unit volume) ρd, the group velocity of the phonon vg, and the scattering cross section σd and is given by:

1 τi = (2.19) ησdρd|vg| where η is a constant of order unity. According to Majumdar [12], the scattering cross section varies as ζ4 σ = πR2 (2.20) d ζ4 + 1 where ζ = ωR/vg,R is the radius of the lattice imperfection, and ω is the phonon frequency.

2.4.2 Boundary Scattering

The mean free path of a phonon is the average distance traveled by a phonon before it undergoes a scattering event. In a thin ﬁlm at low temperature, the phonon mean

30 free path may become comparable to the thickness of the ﬁlm. In such cases, phonon boundary scattering dominates over other scattering mechanisms. Casimir’s [51] expression for boundary scattering relaxation time, which can be written as

1 vg 2 p = ,L = l1l2 (2.21) τb LF˘ π ˘ where l1l2 is the sample cross section, F is a correction factor employed because of

finite length/thickness ratio and the edge roughness of the film, has shown to give good agreement with experimental data. Phonon-impurity and phonon-boundary scattering are elastic processes, since no energy or frequency change of incident phonon occurs in these interactions. In a numerical scheme for solution of the BTE, the effect of the geometric boundary can be treated exactly, and no model (such as Eq. 2.21) is necessary.

2.5 Phonon-Phonon Interactions

The scattering term on the right hand side of the BTE (Eq. 2.14) becomes com- plicated if all possible scattering mechanisms are considered rigorously. In the relaxation time approximation, scattering between diﬀerent wave vectors is not explicitly accounted for. Instead, the phonons relax to the equilibrium distribution [14]. Scat- tering of three or more phonons with each other occurs due to the anharmonic nature of the interatomic forces and the discrete nature of the lattice structure. Phonon- phonon scattering involving four or more phonons is important only at temperatures much higher than the Debye temperature(645 K for silicon) [14], which is much higher than the operating temperature of most electronic devices.

Three-phonon interactions can be classiﬁed as Normal (N) processes and Umklapp

(U) processes. Both N and U processes are governed by energy and momentum

31 conservation rules given by [11]

ω + ω0 ↔ ω00(Normal + Umklapp) (2.22)

K + K0 ↔ K00(Normal) (2.23)

K + K0 ↔ K00 + b(Umklapp) (2.24) where ω, ω0 and ω00 are the angular frequencies of the interacting phonons and K, K0, and K00 are their wave-vectors. During these interactions, two phonons may merge to form a third phonon or a single phonon may break up into two phonons. In an

Umklapp process, momentum is not conserved; the diﬀerence in phonon wave-vectors resulting in the reciprocal lattice vector, b.

A phonon with wave vector K does not carry physical momentum, but it does interact with other phonons as if it has a momentumh ¯K. This can be easily under- stood if a phonon is considered as a classical particle with energy E = Mv2/2 and

dE momentum P = Mv. Comparing the velocity of the particle v = dp with Eq. 2.8 givesh ¯K as the momentum of phonon.

Although three-phonon interactions are very important for energy transport in

crystals, the problem lies in determining the exact expressions for calculating the

time-scales for these interactions. The diﬃculty in determining these time-scales is

borne out of the fact that the expressions for calculating these time-scales are not

available due to the complex nature of the anharmonic interatomic forces. As a re-

sult, much of the published literature uses only approximate relaxation times, which

are typically derived either by using curve-ﬁts to bulk thermal conductivity [24] or

by making low-temperature approximations that allow simpliﬁcations to the conser-

vation rules [43]. It is critical to note that although only U processes are responsible

32 for thermal resistance, N processes cannot be neglected. At low temperatures, they

may help in creating suﬃcient numbers of high-K phonons to participate in U processes [52]. Therefore, it is essential to consider the contribution of both N and U processes for the calculation of the overall relaxation time. Holland [24], using perturbation analysis in combination with calibration to experimental data, developed frequency and temperature dependent expressions for the relaxation times for both longitudinal and transverse modes:

−1 2 3 τNU = BLω T (LA, Normal+Umklapp)

−1 4 τN = BTN ωT (TA, Normal) (2.25)

−1 0 TA, Umklapp for ω < ω1/2 τU = 2 (2.26) BTU ω /(sinh(¯hω)/(kBT )) TA, Umklapp for ω > ω1/2

where ω1/2 is the frequency corresponding to |K/Kmax| = 0.5 and BL, BTN , and BTU

are constants [24] that need to be obtained through calibration against experimental

13 data. For silicon, ω1/2 = 2.417 × 10 rad/s. Previous MC studies of the phonon

BTE [28, 29] have used these time-scales for thermal conductivity calculations as a

preliminary step as described in section 1.4 . Holland made two important assump-

tions for calculating these time-scales: (1) only high frequency TA phonons undergo

Umklapp processes, and (2) LA phonons do not undergo U processes at all. There-

fore, in eﬀect, Eq. 2.25 gives the time-scale for N processes, whereas the expression

in Eq. 2.26 gives the phonon interaction time-scales for U processes. Despite these

assumptions, the expressions provided by Holland continue to be popular because of

their simplicity and the ease of their implementation.

33 Longitudinal Acoustic (LA) Phonons Transverse Acoustic (TA) Phonons LA + TA(BZB) ↔ LA TA + TA(BZB) ↔ LA LA + TA ↔ LO(BZB) TA + LA(BZB) ↔ LA LA + TA ↔ TO(BZB) TA + LA ↔ LO(BZB) LA + LA ↔ LO(BZB) TA + LA ↔ TO(BZB) LA + LA ↔ TO(BZB) Longitudinal Optical (LO) Phonons Transverse Optical (TO) Phonons LO ↔ LA + TA(BZB) TO ↔ LA + TA(BZB) LO ↔ LA + LA(BZB) TO ↔ LA + LA(BZB) LO ↔ TA + LA(BZB) TO ↔ TA + LA(BZB)

Table 2.2: Three-phonon interactions considered in this study for the calculation of scattering time-scales for each phonon polarization.

Klemens [52], starting from perturbation theory, considered scattering processes due to anharmonicities, and derived a set of expressions for U processes at low temperatures. Based on the work of Klemens [52] and Han and Klemens [43], Narumanchi et al. [26] provided a methodology for calculating frequency and temperature dependent phonon lifetimes for both optical and acoustic modes for the U processes. The advantage of using the expressions given in Narumanchi’s work is that unlike the expressions given by Holland [24], only a single ﬁtting parameter, called the Gruneisen constant is used to calculate the phonon lifetimes. In the present study, a hybrid approach is used to calculate the phonon time-scales wherein the time-scale for N processes is calculated from Holland’s expressions as given by Eq. 2.25. The procedure for calculating the time-scale of U processes, as derived from Narumanchi et al. [26] and Han and Klemens [43], is described next.

For silicon, the three-phonon interactions that are believed to be important [26, 27] are shown in Table 2.2, wherein BZB refers to phonons corresponding to the ﬁrst

Brillouin zone boundary. The relaxation time, τij, for the interactions of the type LA

34 + TA (BZB) ↔ LA, TA + TA (BZB) ↔ LA, and TA + LA (BZB) ↔ LA , and for the optical modes is calculated from [43] using expressions similar to those proposed by Narumanchi and co-workers [26, 27]

2 1 χγ 2 1 1 ≈ 2 ωi ωtr ωjrc − (2.27) τij 3πρvph|vg| exp (¯hωtr/kBT ) − 1 exp (¯hωj/kBT ) − 1 where i refers to the phonon for which the time-scale needs to be calculated, hence-

forth referred to as the incoming mode, j refers to the resultant or the translated

mode, and tr refers to the mode by which the incoming mode is translated to the

resultant mode (referred to as the translational mode). The phase velocity of the

incoming mode is denoted by vp(= ω/|K|), and |vg| is the magnitude of the group

velocity of the resultant mode. ρ is the density of the solid (i.e., bulk silicon). ωi is the

frequency of the incoming phonon in rad/s, ωj is the frequency of the resultant mode,

while ωtr is the frequency of the translational mode at the Brillouin Zone Boundary

(BZB). χ is the degeneracy of the translated mode, γ is the Gruneisen constant, and

has been taken to be 0.59 for silicon [27], and rc denotes the eﬀective radius. For the

[100] direction, rc is calculated as [43]:

2π/a − |Ki| rc = √ (2.28) 2 2

where a is the lattice constant, and |Ki| is the wave-vector of the phonon (incoming mode) for which the time-scale needs to be calculated. For the interaction of the type

LA / TA + LA / TA ↔ LO/TO (BZB) , the time-scales are calculated using the expression

2 1 χγ 3 1 1 ≈ 2 ωi ωO ωtr − (2.29) τij 3πρvph|vg| exp (¯hωtr/kBT ) − 1 exp (¯hωO/kBT ) − 1

where ωO is the frequency of the optical mode at the BZB, ωi is the incoming phonon

frequency, and ωtr(= ωO − ωi) is the translated phonon frequency.

35 For a given incoming phonon frequency and polarization, the time-scale for all the possible interactions are calculated. If the various phonon-phonon interaction processes are assumed to be independent, their scattering probabilities can be added together. The overall scattering time-scale is then given by the Mathiessen’s rule [12]:

1 X 1 = (2.30) τ τ U,i j ij where is the relaxation time for a single scattering process [as calculated using Eqs. 2.27 and/or 2.29], and τU,i is the relaxation time for all U processes combined. The overall relaxation time τ is then calculated as:

1 1 1 = + (2.31) τ τN τU and is used as an input for solution of Eq. 2.14, with the scattering term expressed by Eq. 2.16.

36 CHAPTER 3

MONTE-CARLO SOLUTION TECHNIQUE

Statistical (or Monte-Carlo) methods are used to solve differential equations which involve a large number of independent variables, as demonstrated in the fields of radiation transport [31], electron transport [53], rarefied gas dynamics [54] and reacting turbulent flows [55]. In addition, Monte-Carlo methods are ideal for use in problems which involve complex physical phenomenon. The distribution function f(r, K, t) in the BTE (Eq. 2.14) is, in general, a function of seven independent variables, namely time, three space variables and three wave vector components. In case of phonons, the ability of the Monte-Carlo (MC) technique to treat the individual phonon scattering events in isolation from each other, rather than using a single relaxation time for all scattering processes, makes it very attractive for use in solving the BTE for phonons. The procedure used to solve the BTE for phonons using the MC technique is described next.

3.1 Number of Stochastic Samples

The ﬁrst step in the MC solution technique is to determine the number of phonons per unit volume that are present in the computational domain, Nactual. For a given temperature and material, Eq. 2.11 gives the energy per unit volume and involves

37 integration over the whole frequency space for each phonon polarization. In order to

perform the integration numerically, the frequency space is discretized into a number

of spectral or frequency bins, Nb, with ωo and ∆ωi being the central frequency and the bandwidth of the i − th spectral bin, respectively. This reduces the integral into a summation over discrete frequency intervals. The overall energy per unit volume may then be written as [28]:

Nb 2 X X 1 |K| d|K| E = hω¯ ∆ω χ (3.1) o,i 2 i p ¯hωo,i 2π dω p i=1 exp − 1 ωo,i kbT The summation is done over all the polarizations p with χ being the degeneracy factor

for each of the polarizations. Eq. 3.1 is valid only for an isotropic Brillouin Zone, with

|K| being the component of the wave vector in any direction. From this expression,

the number of phonons per unit volume to be initialized can be calculated as :

Nb 2 X X 1 |K| d|K| N = ∆ω χ (3.2) actual 2 i p ¯hωo,i 2π dω p i=1 exp − 1 ωo,i kbT It is important to note that for a given material, the total energy (and the number

of phonons) is only a function of temperature and volume. For thin ﬁlms at room

temperature, the actual number of phonons resulting from Eq. 3.2 is usually a very

large number. For example, for a silicon ﬁlm at 300 K this number is of the order of

1027 phonons per m3, whereas it reduces to about 1024 phonons per m3 at 10 K, which

translates to 106 − 109 phonons per µm3. Computation and memory requirements

restrict the number of phonons which may be simulated, thus requiring the use of

a scaling or a weight factor W . If Nprescribed gives the number of phonons actually

initialized/emitted into the system, then the weight factor is given by

N W = actual (3.3) Nprescribed 38 Therefore, the weight factor W is deﬁned as the ratio of the actual number of phonons, to the prescribed number of phonons (stochastic samples) to be traced during the

MC simulation. Then, each stochastic sample used during the simulation actually represents an ensemble of W phonons. The value of Nprescribed is provided as an input to the MC simulation. Once the weight factor has been calculated, phonon ensembles are drawn from the six individual stochastic spaces. These include three position vector components and two directions and velocity. Under the isotropic Brillouin zone assumption, the wave vector and the group velocity are co-directional.

3.2 Initialization of Phonon Samples

3.2.1 Position

The spatial sampling of the phonons is done by dividing the computational domains into a number of control volumes. For thin-ﬁlm thermal conductivity prediction, in two dimensions, the control volumes are considered to be a rectangular. In three dimensions, the geometry considered is a rectangular parallelepiped. For modeling devices, more general approaches may be required, and has been discussed in detail by Mazumder and Majumdar [28]. Once the geometry of the control volume has been decided, random numbers are used to determine the coordinates of the phonon ensembles inside the control volumes. In a 3-D geometry, the spatial sampling of each phonon ensemble requires three random numbers, whereas in a 2-D geometry only two random numbers are required. In a rectangular parallelepiped, the position vector is calculated as

ˆ r = R1(xOB − xOA)ˆi + R2(yOB − yOA)ˆj + +R3(zOB − zOA)k (3.4)

39 Figure 3.1: Position sampling in a three-dimensional geometry.

where R1,R2 and R3 are the three random numbers such that 0 < R1,R2,R3 < 1 and OA and OB are the position vectors of the two ends of the body diagonal of the rectangular parallelepiped.

3.2.2 Frequency

For sampling the frequency of the phonon ensembles, the dispersion relationship is required as an input. The calculation of dispersion relationship has been described in section 2.3. As discussed in section 3.1, the frequency spectrum has been discretized into Nb spectral bins. From Eq. 3.2, the number of phonons in the i − th spectral bin can be calculated as:

2 X 1 |K| d|K| N = ∆ω χ (3.5) i 2 i p ¯hωo,i 2π dω p exp − 1 ωo,i kbT A normalized cumulative number density function F is then constructed, Pi j=1 Nj Fi = (3.6) PNb j=1 Nj

where Fi represents the probability of ﬁnding a phonon having frequency less than

ωo,i + ∆ωi and Fi − Fi−1 is the probability of ﬁnding a phonon in the i − th frequency

40 bin. In order to sample the frequency of the phonon, a random number Rf , between

zero and unity, is drawn. If Fi−1 < Rf < Fi, the phonon lies in the i − th frequency bin. The location of the i − th bin, corresponding to the random number Rf is then

determined by implementing a binary search algorithm. Once the spectral bin has

been determined, the frequency of the phonon ω is calculated using ∆ω ω = ω + (2R − 1) i (3.7) o,i f 2 Eq. 3.7 assumes that the frequency varies linearly within each spectral bin. This can

only be assumed if the number of frequency bins Nb is large, so that the slope of the

dispersion relation is accurately captured and the phonon density of states are not

aﬀected. By trial and error it has been found that dividing the frequency space into

1000 intervals gives suﬃcient resolution of the frequency space. Increasing Nb beyond

this did not increase the accuracy of the solution, as was found during the validation

of the numerical code.

3.2.3 Polarization

Any phonon ensemble sampled during the course of the MC simulation has to be

assigned a polarization. The probability of a polarization, p, in the i − th spectral

interval is given by

Ni,p Pi,p = (3.8) Ni

where Ni,p is the number of phonons of polarization p in the i − th spectral interval

and Ni is th total number of phonons in the i − th spectral interval and is given by

Eq. 3.5. Therefore,

2 1 |K| d|K| hω¯ o,i 2π2 dω ∆ωi χp exp −1 ωo,i P = kbT (3.9) i,p 2 P 1 |K| d|K| p hω¯ o,i 2π2 dω ∆ωi χp exp −1 ωo,i kbT 41 Once the probability of the p − th polarization has been calculated, a normalized cumulative probability distribution function is constructed. A random number Rpol is called and if Pp−1 < Rpol < Pp then the phonon is assigned a polarization p. The procedure described above is quite general in its application and may be simpliﬁed in case of silicon, since acoustic and optical branches do not overlap and the transverse modes are degenerate. For example, if only acoustic modes are considered, then the probability of a LA phonon in the i − th spectral interval is given by [28]:

Ni(LA) hn(ωo,i, LA)iD((ωo,i, LA) Pi(LA) = = Ni(TA) + Ni(LA) hn(ωo,i, LA)iD((ωo,i, LA) + 2hn(ωo,i,TA)iD((ωo,i,TA) (3.10)

where hni is the equilibrium phonon occupation number given by Eq. 2.9. Now, if the

drawn random number is less than Pi(LA), the phonon belongs to the LA branch,

else it belongs to the TA branch. Once the frequency and polarization of the phonon

are known, the magnitude of its group velocity |vg| is calculated using Eq. 2.8.

3.2.4 Direction

The last step in the initialization stage is to assign a direction to the phonon

sample. If isotropy is assumed, the direction vector ˆs may be calculated in three

dimensions as (refer to Fig. 3.2)    sin θ cos ψ  ˆs = sin θ sin ψ (3.11)  cos θ 

where 0 ≤ ψ ≤ 2π and 0 ≤ θ ≤ π/2. Therefore, the angles can be represented

by random number relations [28] and written as,

p 2 p 2 ˆs = 1 − (2R1 − 1) cos(2πR2) ˆt1+ 1 − (2R1 − 1) sin(2πR2) ˆt2+(2R1−1)ˆn

(3.12)

42 Figure 3.2: Coordinate system showing the direction of phonon emission and the associated angles

For 2-D, ˆs may be calculated as,

cos ψ ˆs = (3.13) sin ψ

ˆs = cos(2πR2)ˆi + sin(2πR2)ˆj (3.14) where ψ = 2πR2, θ =(2R1 − 1), and, R1 and R2 are two random numbers between zero and unity.

3.3 Linear Motion (Drift) of Phonons

Once the phonon ensembles have been initialized, phonons are allowed to travel in linear paths from one point to another. In between, they may interact with each other phonons (scatter). However in the MC algorithm, the drift and scattering events are treated sequentially, which mathematically is equivalent to splitting of transport and scattering operators in the BTE. This decouples the BTE (given by Eq. 2.14)

43 at each time step into two equations: the collisionless transport equation (also called the free ﬂow equation - Eq. 3.15) and the spatially homogeneous Boltzmann equation

(Eq. 3.16). The solution of Eq. 3.15 serves as an initial value for the second step in which Eq. 3.16 is solved [56]. This operator splitting introduces an error that is first order in time. ∂f + v · ∇f = 0 (3.15) ∂t g ∂f ∂f f − f = = o (3.16) ∂t ∂t scatter τ The drift or free flow of phonons causes the phonon ensemble to move from one location to another. The position of the phonons is tracked using an explicit first order time integration scheme given by

r(t + ∆t) = r(t) + vg∆t (3.17) where ∆t is the time step. The drift of phonons may cause them to move from one spatial bin to another. Therefore, based on the new position vector, the new spatial bin to which the phonon belongs needs to be ascertained. This is an important step as drift results in re-distribution of energy (and temperature) in the various spatial bins within the computational domain. During the drift phase, the phonon ensembles also interact with the physical boundaries of the system. The implementation of boundary conditions is a critical part of modeling heat transport in micro/nanoscale devices and will be discussed in detail in section 3.4. At the end of the drift phase, the energy of ˜ the each spatial bin per unit volume, Ecell, is calculated by summing up the energy of all the phonon ensembles present within the cell. The pseudo-temperature T˜ of each cell may then be calculated by Newton-Raphson inversion of Eq. 2.12 given by

Nb 2 X X 1 |K| d|K| E˜ · W = hω¯ ∆ω χ (3.18) cell o,i 2 i p ¯hωo,i 2π dω p i=1 exp ˜ − 1 ωo,i kbT 44 (a) 3D (b) 2D

Figure 3.3: Computational Domain (a) 3D, and (b) 2D

The pseudo-temperature T˜ is calculated from Eq. 3.18 under the assumption that thermodynamic equilibrium exists within the spatial bin. As mentioned previously, in non-equilibrium processes, temperature loses its conventional meaning and can only be used as a measure of the energy of the system. Hence, the pseudo-temperature T˜ is an artiﬁcial quantity that has no physical meaning, but is calculated to facilitate the calculation of the scattering time-scales, as will be described later in section 3.5.

3.4 Boundary Interactions

While phonon-boundary interaction is most important at low temperatures, where the mean free paths of phonons are longest, boundary interactions may also be very significant at room temperature and above in very thin silicon layers. For MC simulations aimed at thermal conductivity prediction of thin structures (films, nanowires, nanotubes etc.), two types of boundary conditions are encountered: isothermal and adiabatic. For thermal conductivity prediction of thin films, the end faces of the thin structures are held at constant temperatures and correspond to the isothermal boundaries, as shown in Fig. 3.3. All phonons that strike the isothermal boundary during

45 the drift phase are thermalized, analogous to absorption/emission by a blackbody in

radiation. Depending on its temperature, the boundary also emits phonons into the

computational domain, the state of emitted phonons being completely independent

of the state of incident phonons. In the numerical scheme, any phonon incident on

an isothermal wall is tallied, and then deleted from the simulation. This provides the

net incident energy during a time interval ∆t. The number of phonons emitted from a boundary face is calculated from the Bose-Einstein distribution as:

N ! X Xb Nface = A · ∆t · hn(ωo,i, p)ihvg(ωo, i) · ˆniD(ωo,i, p)∆ωi (3.19) p i=1

where vg(ωo,i)·ˆn represents the component of the phonon group velocity normal to the

face. As phonons are emitted from the face in all directions, the quantity vg(ωo,i) · ˆn

has be to directionally averaged in order to obtain the average velocity normal to the

boundary face. 1/π|vg| for 2-D hvg(ωo,i) · ˆni = (3.20) 1/4|vg| for 3-D

Therefore, the actual number of phonons emitted by the boundary at each time step

is given by

N ! A X Xb N = · ∆t · hn(ω , p)i|v |(ωo, i)D(ω , p)∆ω (3.21) face π o,i g o,i i p i=1

for 2-D and

N ! A X Xb N = · ∆t · hn(ω , p)i|v |(ωo, i)D(ω , p)∆ω (3.22) face 4 o,i g o,i i p i=1

for 3-D. In the MC scheme, this number is scaled by the weight factor, W ,i.e., the

number of samples injected is Nface/W . The spatial location of the emitted phonons

are uniformly distributed over the entire face by calling random numbers and their

46 Figure 3.4: Phonon reﬂection at the boundary

frequency and polarization are sampled based on the procedure described in the previous sections.

An adiabatic boundary condition states that there can be no heat flux through the surface. It implies that any phonon hitting the surface has to be reflected back without any change in energy. Reflection from the surface can be classified into two types: specular and diffuse. Specular reflection causes the phonon to undergo a mirror like reflection (Fig. 3.4).

ˆsr = ˆsi + 2|ˆsi · ˆn|ˆn, (3.23)

where ˆsi and ˆsr are the direction vectors of the incident and reflected phonon, respectively. Phonons would undergo a mirror-like reflection only if the wavelength of the phonon is not comparable to the characteristic roughness of the surface. On the other hand, diffuse reflection results due to the interference of phonon wave packets departing from a surface that has a characteristic roughness comparable to or larger

47 Figure 3.5: Thermal conductivity with and without boundary scattering.

than the phonon wavelength. The fraction of phonons reﬂected specularly (or dif-

fusely) therefore depends strongly on the surface roughness and on the wavelength

of the phonons under consideration [57]. Real surfaces are neither fully diﬀuse nor

fully specular and behave as if they are partially specular. In the MC technique, the

fraction of phonons reﬂected specularly is prescribed by the degree of specularity, α.

The value of α may vary between zero (completely diffuse) and unity (completely specular). Specular reflection does not pose any resistance to the flow of thermal energy. At low temperature, phonon-phonon interactions are almost non-existent and boundary scattering is solely responsible for thermal resistance. If boundary scattering is neglected at low temperature, the phonons travel from the high-energy source to the low-energy sink without any mechanism to impede them, resulting in infinite thermal conductivity as shown in Fig. 3.5. Therefore, at low temperature, boundary scattering must be accounted for. This is done by treating the lateral boundary as

48 a partially specular surface. For a given geometry, the degree of specularity is calibrated to match the experimental thermal conductivity at low temperatures. When a phonon ensemble strikes an adiabatic boundary, a random number is drawn. If this random number is less than the calibrated value of degree of specularity, the phonon is specularly reflected, else, it undergoes diffuse reflection. Implementation of diffuse reflection is implemented by re-sampling the direction of the outgoing phonon ensemble using the expression:

ˆs = (sin θ cos ψ)ˆt1 + (sin θ sin ψ)ˆt2 + cos θˆn (3.24)

for 3D, where ˆn is the unit surface normal, and ˆt1 and ˆt2 are the local unit surface tangents (Fig. 3.2), which must be perpendicular to each other.

3.5 Intrinsic Scattering

In the MC simulation, the scattering and drift processes are treated independently.

As discussed in section 2.4, phonons may be scattered due to the presence of defects in the crystal lattice. Phonon scattering due to vacancies and dislocations can be neglected if the crystal is considered to be pure. In the absence of the afore-mentioned mechanisms, phonons may interchange energy with each other, i.e., undergo intrinsic scattering. As the primary focus of this study is to investigate the role of various phonon modes on energy transport and thermal conductivity prediction, only phonon- phonon (or intrinsic) scattering events have been considered in the MC solution of the BTE. The computation of phonon-phonon scattering time-scales has already been discussed in an earlier section (section 2.5). In the MC procedure, once the time- scale of scattering of phonons has been calculated, the probability of scattering of the

49 phonon between time t and t + ∆t may be calculated as [28]

−∆t P = 1 − exp (3.25) scat τ

A random number Rscat is drawn, and if the probability of scattering, Pscat, is greater than Rscat, then the phonon is scattered. Scattering of the phonon is implemented by re-sampling the frequency, polarization and direction of the phonon from the cumulative number density function. For thermal conductivity predictions, the duration of the time step, ∆t, is varied with ﬁlm thickness but kept constant during a simulation.

The time step is chosen such that it is smaller than the minimum scattering timescale for any of the phonons sampled during the simulation. For a 0.42 µm ﬁlm, the time step was chosen to be one pico-second for all temperatures (10-300 K).

In order to create phonons at the same rate as they are destroyed at thermal equilibrium, the distribution function used to sample the frequencies of the phonons after scattering Fscat has to be modulated by the probability of scattering. This enforces Kirchoﬀ’s law for phonons, i.e., the rate of formation of phonons of a certain state equals its rate of destruction, thereby retaining equilibrium if the initial state is already in equilibrium. The new distribution function is given by:

Pi N (T˜) × P ˜ j=1 j scat, j Fscat(T ) = (3.26) PNb ˜ j=1 Nj(T ) × Pscat, j The reason the outgoing phonon state(s) are sampled from an equilibrium distribution is because, by deﬁnition, collisions restore equilibrium. Thus, each time a phonon scatters, the local thermodynamic state is little bit closer to equilibrium. If the phonons were in a box and colliding over an inﬁnite time interval, the current algorithm will guarantee that the Bose-Einstein distribution is recovered. The alternative to this procedure is to pair phonons and have them engage in the selection

50 rules directly. However, this procedure will result in a scheme that will produce huge statistical errors because the number of phonons contained in a spatial bin is generally quite small and not adequate to address all collision possibilities. Attempts to use such a scheme showed that the Bose-Einstein distribution is never recovered with any practically feasible sample size.

The use of Fscat to resample phonon frequency ensures that during the MC simulation energy is conserved. The rigorous implementation of momentum conservation is, however, harder to address. Since the MC process treats phonons one by one, triadic

N or U interactions cannot be rigorously treated. Since only U processes contribute to the thermal resistance, when the phonons scatter through a U process, their directions after scattering are randomly chosen as in the initialization procedure. In the present work, similar to the assumptions made by Lacroix et al. [29], half of the phonons that undergo scattering are resampled for their directions. At the end of the scattering algorithm, the energy within each spatial bin is summed up and local equilibrium temperature is obtained for each spatial bin. The thermal gradient across the thin film is then calculated. The net flux through the boundaries is calculated, and the effective thermal conductivity is then obtained by inverting the Fourier law of heat conduction.

51 CHAPTER 4

RESULTS AND DISCUSSION

As discussed in section 1.4, previous works [28, 29] have not considered the trans-

port of optical phonons when predicting semiconductor thin ﬁlm thermal conductiv-

ity. The only way to quantitatively measure the role of optical phonons in thermal

conductivity predictions is to run MC simulations with and without the inclusion

of optical phonons. The following section presents the results of MC simulations

for silicon ﬁlms of various thicknesses with and without taking optical phonons into

consideration. The predicted results (thermal conductivity) are compared against

experimental data, and the role of various phonon modes on thermal transport is

identiﬁed and discussed.

4.1 Veriﬁcation of the MC code

Prior to computation of the through-plane thermal conductivity, the MC code was

verified against analytical solutions for limiting scenarios. In the ballistic limit, i.e., for the case when the phonon mean free path is much larger than the film thickness, the code reproduced a medium temperature equal to the average of the fourth power of the two boundary temperatures, as shown in Fig. 4.1. In the diffusion limit, i.e., for

the case when the phonon mean free path is much smaller than the ﬁlm thickness, the

52 Figure 4.1: Comparison of the results obtained by MC method (ensemble average) against analytical solution in the ballistic limit.

53 code reproduced a linear temperature proﬁle between the two boundary temperatures

as shown in Fig. 4.2. The eﬀect of boundary scattering was also veriﬁed by altering the

degree of specularity, α, of the lateral (non-thermalizing boundaries) and previously reported results [28, 29] were reproduced, as shown in Fig. 4.3.

4.2 Thermal Conductivity and Role of Various Phonon Modes

Computation of the thermal conductivity of silicon thin ﬁlms requires computation of the group velocities of the various phonon modes from the dispersion relationship.

Fig. 4.4 shows the variation of group velocity of the various phonon modes with frequency calculated using the dispersion relationship shown in Fig. 2.5 (as adapted from Dolling [47]). The important point to note is that contrary to popular belief

(and assumptions made by many past researchers), the group velocities of both LO and TO phonons are of the same order of magnitude as the acoustic phonons, although the frequency ranges for the acoustic modes and optical modes are diﬀerent.

Another important quantity that gets altered due to the inclusion of optical phonons is the cumulative phonon occupation number at equilibrium, as computed from the

Bose-Einstein distribution. This cumulative distribution function is critical for as- signing phonon frequency and polarization when phonons are initialized within the computational domain, or when phonons are added or deleted to the domain after a scattering event. Figure 4.5 illustrates the cumulative distribution function with and without optical phonons for various temperatures. Clearly, optical phonons alter the high-frequency part of the distribution function for temperatures greater than

100 K. It can also be seen from the cumulative distribution function that the number density of optical phonons is small at low temperatures (< 100 K) . Therefore,

54 Figure 4.2: Comparison of the results obtained by MC method (raw data) against analytical solution in the diﬀusion limit.

55 Figure 4.3: Temperature proﬁle for various degree of specularity (α)

56 Figure 4.4: Group velocity of various phonon modes.

optical phonons are not expected to directly influence thermal energy transport at these temperatures. At higher temperatures, though, the number density of optical phonons is quite significant(about 12%) and they are expected to contribute towards thermal energy transport, especially since optical phonons have high frequency and therefore, carry more enrgy per phonon than their acoustic counterparts. Figure 4.6 shows a comparison between the phonon time-scales for the acoustic modes computed using two different approaches: (1) using expressions provided by Holland (Eq. 2.25 and 2.26), and (2) using the hybrid approach based on the works of Holland [24] and Han and Klemens [43] (Eqs. 2.27 and 2.29 presented in section 2.5). While the computed time-scales are somewhat different, it is encouraging to see that the time- scales computed using the hybrid approach are of the same order of magnitude as

57 (a) 50 K (b) 100 K

Figure 4.5: Normalized cumulative number density function

58 (a) 100 K

(b) 300 K

Figure 4.6: Phonon-phonon scattering time-scales computed using the present hybrid approach and Holland’s model at diﬀerent temperatures.

59 the time-scales computed using Holland’s data. If that were not the case, these new time-scales would result in thermal conductivity that is significantly different from those predicted by Holland [24], and, Mazumder and Majumdar [28]. It is also seen that the time-scales predicted by the two methods match almost exactly at high temperatures (300 K). This result is consistent with the fact that Holland’s assumption of “near equilibrium” transport is more valid at 300 K than at 100 K, and therefore, the time-scales computed by Holland are expected to be more accurate at higher temperature. Following these observations, the hybrid approach was deemed suitable for computation of time-scales of all phonon modes. Fig. 4.7 shows the computed mean free paths of all four phonon modes, where the mean free path is defined as the product of the scattering time-scale and the phonon group velocity. The mean free path is more informative since it accounts for both the timescale for scattering and the group velocity of the phonon. As shown in Fig. 4.7, the computed mean free paths for various phonon polarizations are of the same order of magnitude and follow similar behavior as calculated by Henry and Chen [3] using molecular dynamics.

The thermal conductivity predicted by the MC simulations with and without the inclusion of optical phonons has been plotted in Figs. 4.8, 4.9 and 4.10 for three diﬀerent silicon ﬁlm thicknesses. In each case, the experimental thermal conductivity values obtained by Asheghi [4] have also been plotted.

It is clear that the inclusion of optical phonons does have an eﬀect on the predicted thermal conductivity of silicon thin ﬁlms. As discussed earlier, at low temperature (<

50K), thermal conductivity predictions are dominated by boundary scattering. Thus, inclusion of optical phonons has no impact on the predicted thermal conductivity. At intermediate temperatures (50-200 K), the occupation number of optical phonons is

60 Figure 4.7: Comparison of mean free paths of all four phonon modes at 300 K computed using the present hybrid approach, and data obtained using molecular dynamics by Henry and Chen [3]

61 Figure 4.8: Predicted and measured [4] through-plane thermal conductivity for a 0.42 µm silicon ﬁlm

62 Figure 4.9: Predicted and measured [4] through-plane thermal conductivity for a 1.60 µm silicon ﬁlm

63 Figure 4.10: Predicted and measured [4] through-plane thermal conductivity for a 3.0 µm silicon ﬁlm

64 T (K) LA TA LO TO 100 30.00 66.50 2.60 0.83 200 42.15 41.00 12.16 4.69 300 43.27 30.87 19.21 6.65

Table 4.1: Contribution (%) of various phonon modes towards thermal energy transport when both acoustic and optical modes are considered. Film thickness = 0.42 µm.

quite low. Few optical phonons are present, and they carry little energy themselves.

However, they decrease the scattering time-scales of the acoustic phonons via collisions

(Fig. 4.6(a)), thereby resulting in a decrease in the value of thermal conductivity. At high temperature (> 200K), optical phonons are excited, and a significant fraction of the energy is carried by optical phonons themselves. Thus, their inclusion enhances the thermal conductivity, and is evident in Fig. 4.8 for 300K. It is also noteworthy that the high-temperature effect is more pronounced in thin films (Fig. 4.8) than in thick films (Fig. 4.10). For example, for a 3 µm film at 300K, collisions between phonons is so abundant that it does not matter what phonons are considered in the simulation because transport is almost in the diffusion regime. The alteration of thermal conductivity shown by these results discounts earlier assumptions [28] that optical phonons can be neglected for silicon thin film thermal conductivity prediction.

Tables 4.1 and 4.2 show the contribution of the various phonon modes towards thermal transport. Diﬀerent temperatures are considered, and the contributions with and without the inclusion of optical phonons are noted. The results shown are for a

0.42 µm ﬁlm.

65 T (K) LA TA 100 31.14 68.86 200 50.45 49.55 300 58.08 41.92

Table 4.2: Contribution (%) of various phonon modes towards thermal energy transport when only acoustic modes are considered. Film thickness = 0.42 µm.

The relative contribution of various phonon modes towards thermal energy transport has been discussed extensively in the past. Holland [24], Mazumder and Ma- jumdar [28], and Hamilton and Parrot [58] have predicted that TA phonons are the predominant carriers of thermal energy, whereas Ju and Goodson [13], and Henry and Chen [3] have reported that LA phonons are the predominant carriers of thermal energy. In the present work, it is found that at lower temperatures, TA phonons carry more energy than LA phonons. At 100 K, acoustic modes account for over 95% of the total thermal energy transport, the contribution of LA phonons being 30%. As the temperature increases, contribution of LA phonons increases. Above 200 K, the contribution of LA phonons has been found to be more than TA phonons. At 300

K, acoustic modes contribute to about 75% of the total energy transport, of which about 30% is contributed by TA phonons. The contribution of optical modes has been found to be about 25% with the contribution of LO phonons accounting for about 19%. At stated earlier, these ﬁndings are for a 0.42 µm thick ﬁlm. For thicker

ﬁlms, the contribution of optical phonons was found to be smaller. For example, for a 3 µm ﬁlm at 300 K, the contribution of optical phonons was found to be about

10%. Recent studies by Goicochea et al. [21] also report similar ﬁndings: the contribution of optical phonons decrease with increase in ﬁlm thickness. For very thick

66 films (bulk material), Henry and Chen [3] and Broido et al. [59] found the contribution of optical phonons to be about 5% at 300 K. The findings of the present study suggest that even though optical phonons have slower group velocities than LA or TA modes, their contribution to thermal energy transport could be significant, especially at room temperature and if the film is sufficiently thin. The results obtained in this work corroborate the results obtained by Wang [2], who suggested that one-fifth of the energy transfer rate in hot spots is contributed by optical phonons.

4.3 Numerical Issues

As discussed earlier, one of the critical parameters in the calculation of the through-plane thermal conductivity of thin ﬁlms is the lateral dimension of the ﬁlm.

Ideally, the in-plane dimension of the film should be much larger than the film thickness (or through-plane dimension). Since boundary scattering dictates the thermal conductivity at low temperature (ballistic regime), it is necessary to calibrate the degree of specularity of the boundary to replicate the exact boundary resistance, as has been proposed by Mazumder and Majumdar [28], and used in subsequent studies [2, 27]. If the chosen lateral dimension is small, the phonons will strike the boundary more often, and a larger value of the degree of specularity α will be necessary, than in a case where the lateral dimension of the film is large. The same idea is applicable to thin films treated using a 2D versus 3D computational domain. In a

2D simulation, only two lateral boundaries pose resistance to the phonons, while in a

3D simulation, four boundaries pose resistance to the phonons. Thus, the calibrated degree of specularity in 2D versus 3D is expected to be signiﬁcantly diﬀerent. For

67 Figure 4.11: Predicted thermal conductivity using two-dimensional versus three- dimensional computational domains.

example, Mazumder and Majumdar [28] reported using a value of 0.6 for a 2D thin-

film calculation, while Wang [2], in his finite-volume formulation, used a value of 0.4 for the specularity parameter, α. In the present study, we have performed both 2D and 3D simulations, with the lateral dimension being one-tenth of the film thickness, and it was found that the calibrated value of (α) is 0.885 for the 2D and 0.965 for the

3D case. When the lateral dimension is made equal to the ﬁlm thickness, in 2D case,

(α) changes to 0.14 whereas in the 3D case, it changes to 0.67. Figure 4.11 shows the comparison between the thermal conductivity predicted using a 2D versus a 3D computational domain for a silicon ﬁlm 0.42 µm thick. It is evident that within the limits of statistical errors, the thermal conductivity predicted by the 2D calculation

68 Figure 4.12: CPU time taken for two-dimensional versus three-dimensional simulations with Nprescribed = 50,000.

as well as the 3D calculation is approximately the same. Also, the statistical variations in the thermal conductivity at diﬀerent temperatures are also comparable for the two cases. Thus, it can be concluded that the lateral dimension of the ﬁlm and/or the dimensionality of the computational domain are truly free parameters that, once calibrated, has no impact on the physical results or their statistical accuracy.

Figure 4.12 shows the CPU time comparisons for the 2D case versus the 3D case.

The timing study has been carried out on a 2.13 GHz Dell Optiplex machine with approximately 50,000 phonons (= Nprescribed) simulated in each case. Although the

3D MC simulations are more expensive than the 2D case at low temperatures, at high temperatures (> 100 K), the CPU time for both 2D and the 3D case have been found to be within a few percent of each other. Therefore, for the usual operating range of semiconductor devices, there is not much disadvantage in performing 3D simulations. At low temperatures, phonon transport is dominated by ballistic motion

69 and there are more phonon-boundary interactions in the 3D case than in 2D. Since line-surface intersections are more expensive to compute in 3D than in 2D, the overall

CPU is higher for 3D computations in the ballistic case. High temperature transport, on the other hand, is dominated by scattering events, which are point events, and therefore, the CPU times are not aﬀected by the extra CPU time taken for phonon- boundary intersection calculations. In summary, these studies clearly show that the statistical accuracy is comparable in 2D versus 3D simulations, and the computational eﬃciency is also comparable above 100 K (intrinsic scattering dominated regimes).

These results imply that in strong contrast with deterministic methods for solving the BTE, stochastic methods for solving the BTE do not require any additional computational time in going from 2D to 3D. While these attributes of the MC method, i.e., easy scalability to 3D, has been known in general, these studies provide hard evidence to corroborate this general notion.

Having resolved the longstanding issue of the suitability of 2D versus 3D simulations, we proceeded to carefully analyze the statistical errors as a function of the prescribed number of stochastic samples, Nprescribed. Table 4.3 shows the statistical noise or variation in the thermal conductivity for a 0.42 µm thin ﬁlm at 80 K. The

ﬂuxes at the left (cold) and the right (hot) boundaries have been plotted in Fig. 4.13 for two diﬀerent values of Nprescribed.

With increasing time, the two flux values converge to the same value, as expected for a one-dimensional film at steady state. The fluctuations in the fluxes are associated with probability driven events in the MC calculation, and is therefore, directly related to the number of stochastic samples traced. By increasing the value of Nprescribed from

50,000 to 500,000, the standard deviation in the ﬂux decreases from 22% to about 7%.

70 (a) Nprescribed = 50, 000

(b) Nprescribed = 500, 000

Figure 4.13: Time-dependent energy ﬂux at the boundaries for a 0.42 µm thin ﬁlm at 80K

71 σ σ Nprescribed 2D κ (%) 3D κ (%) 50,000 22.39824 23.50699 100,000 15.74578 16.45898 500,000 7.165036 7.487463

Table 4.3: Statistical noise in the thermal conductivity data for both two-dimensional and three-dimensional calculations at 80 K as a function of sample size.

Figure 4.14: Time averaged energy ﬂux (average of data presented in Fig. 4.13(a)) at the boundaries for a 0.42 µm thin ﬁlm computed using Nprescribed = 50,000.

72 Nprescribed κ (W/m-K) σ/κ(%) 50,000 57.80 187.0303 100,000 52.621 149.245 500,000 55.379 62.57

Table 4.4: Statistical noise in the thermal conductivity for two-dimensional calculations at 300 K as a function of sample size.

On the other hand, increasing the value of Nprescribed from 50,000 to 500,000 increases the computational cost eight times. It is worth noting that once the system reaches a statistically steady state, a time average of the data is equivalent to an ensemble average (i.e., a case where the simulation is repeated with different initial random number seeds). Thus, to compute the thermal conductivity at steady state, one can use time averaged values of the fluxes, rather than ensemble averaged values in order to save computational time. Figure 4.14 shows the time averaged flux values over 10 time steps for Nprescribed = 50,000. Time averaging over 10 time steps reduces the standard deviation in the statistical noise from 22% to about 7%. Worth noting is the similarity in the flux data shown in Figs. 4.13(b) and 4.14. One other important

ﬁnding is that the standard deviation in the ﬂux computed at the boundaries is about the same in the 2D and the 3D case − further corroborating the earlier claim that

2D and 3D computations have the same statistical accuracy as long as Nprescribed is the same. The statistical variation in the ﬂux computed at the boundaries is even more dramatic in the diﬀusive limit (300 K).

Table 4.4 shows the variation in the thermal conductivity for diﬀerent values of at 300 K. As can be seen, the noise is much higher in this case even when a very large value of Nprescribed is used. This is because at higher temperature there are

73 numerous scattering events, all of which are probability driven, and involve drawing

of several random numbers. This necessitates the time averaging of the data over a

very large number of time steps (∼1000) in order to extract the steady state values.

Fig. 4.15(a) shows the variation in the ﬂux at the boundaries for a 0.42 µm ﬁlm at

300 K with Nprescribed= 50,000. Figure 4.15(b) shows the same ﬂux data averaged over 1000 time steps. It is quite evident that time averaging reduces the noise in the

ﬂux data and allows for steady state predictions to be made using a smaller value of

Nprescribed. For transient calculations, however, time averaging cannot be used and a higher value of Nprescribed has to be used. The above studies demonstrate that for steady state calculations, time averaging of the statistically stationary solution is a means to avoid numerous simulations to obtain ensemble averaged data.

74 (a) Nprescribed = 50, 000

(b) time averaged over 1000 time steps

Figure 4.15: Time-dependent energy ﬂux at the boundaries for a 0.42 µm ﬁlm at 300 K

75 CHAPTER 5

SUMMARY AND FUTURE WORK

5.1 Summary

As described in Chapter 1, the main objectives of this thesis are: (a) to develop a general procedure for solution of the Boltzmann transport equation for phonons in

2-D and 3-D geometries, (b) to validate the developed numerical procedure in the ballistic and diﬀusive regimes against analytical results,(c) to develop a methodology to calculate frequency and temperature dependent relaxation time-scales for three- phonon interactions, including those involving optical phonons,(d) to predict the thermal conductivity of phonons with and without the inclusion of optical phonons and to elucidate the role of optical phonons on thermal energy transport, and (e) to highlight the relevant numerical issues pertaining to the computational eﬃciency and statistical accuracy of Monte-Carlo methods for solution of the BTE for phonons.

A general numerical procedure to solve the Boltzmann Transport Equation using the Monte-Carlo technique with frequency and time dependent phonon relaxation times as explicit inputs, as well as dispersion, has been discussed and implemented.

The procedure described is quite general, and may be applied to any semiconductor material or even conﬁned structures, provided appropriate modiﬁcations are made to the inputs. The described Monte-Carlo procedure has been successfully validated

76 against experimental data for silicon thin ﬁlms in the ballistic and diﬀusive regimes.

Silicon has been chosen because of its technological importance and the ease of avail- ability of the experimental thermal conductivity data. In order to elucidate the effect of various phonon polarizations towards thermal energy transport, thermal conductivity predictions were made between 20-300 K in silicon thin films, with and without the inclusion of optical phonons. In order to realize this, a methodology for calculating the lifetimes of various phonon polarizations, including optical phonons, has been described and implemented. This methodology is an extension of the ideas proposed by Holland [24] and Han and Klemens [43]. Therefore, the main contribution of this thesis is the first successful implementation of Monte-Carlo technique to solve the

BTE for phonons with the inclusion of optical phonons.

Numerical predictions with the inclusion of optical phonons give results which match experimental results better than when only acoustic modes are considered.

Below 50 K, thermal conductivity predictions are unaltered by the inclusion of optical phonons. Between 50 K and 200 K, the inclusion of optical phonons decreases the predicted thermal conductivity, while above 200 K, the inclusion of optical phonons increases the predicted thermal conductivity. This behavior was found to be stronger in thin ﬁlms than in ﬁlms that are relatively thick. It is also found that at temperatures below 200 K, TA phonons carry more energy than LA phonons, whereas at higher temperatures LA phonons are dominant carriers of thermal energy. The optical phonons alter thermal transport in two ways. At intermediate temperature (<

200 K), they slow down the acoustic phonons through additional three-phonon interactions. At high temperature (>200 K), they not only alter the scattering time-scales of the acoustic phonons, but also carry 10-25% of the energy themselves. Overall,

77 their role was found to be most significant at high temperature and for films that are thin. These results make a compelling case for the inclusion of optical phonons in predicting silicon thin film thermal conductivity.

Several numerical issues pertinent to predicting through-plane thermal conductivity of semiconductor thin films have been investigated and discussed. First, it is found that using 2D versus 3D computational domain is comparable from both accuracy and computational efficiency standpoint except in ballistic regimes (low temperature) where 2D simulations are computationally slightly more efficient. This finding has strong implications in terms of the feasibility of using Monte-Carlo calculations for practical 3D device structures. The analysis of statistical variations revealed that statistical noise is high in the diffusion limit where the number of probability driven events is large. It is shown that for steady state calculations, time averaging of the statistically stationary data can be used to reduce statistical noise and can result in tremendous computational savings.

5.2 Future work

Despite its ability to accurately model the inherent physics of the problem, Monte-

Carlo method suffers from the disadvantage of being computationally expensive, which limits its use as a commercial tool to solve real physical problems. As can be seen from figure 4.12, in the diffusive limit, because of the exponential increase in the number of scattering events, the performance of the Monte-Carlo code suffers.

Also the scatter in the data is quite high, as seen in Fig. 4.15(a), which necessitates either averaging over a large number of time steps or running multiple simulations.

As the scattering probability of each phonon ensemble is independent of the other,

78 a parallel implementation of the scattering algorithm is expected to yield significant speed ups. Parallel computing would also allow for the increase in the number of phonons initialized Nprescribed, thus improving the statistical accuracy. Parallel implementation of the Monte-Carlo algorithm is made to appear even more attractive by the advent of General-Purpose Graphical Processing Unit (GPGPU) based computing. Efforts are also underway in combining deterministic and stochastic methods to address the two disparate regimes of transport, namely ballistic and diffusive regimes.

Future research will also focus on removing the assumptions made in calculation of phonon lifetimes by the direct implementation of the energy and momentum conservation laws [2]. The current Monte-Carlo algorithm uses dispersion relations in the form of curve fits. In order to extend the Monte-Carlo procedure to model phonon confinement effects, dispersion relationship would need to be calculated from Lattice

Dynamics (LD). The integration of these time scales, along with dispersion curves obtained from lattice dynamics, would facilitate the treatment of anisotropic Brillouin zones. The numerical procedure described in this thesis can also be used to predict the thermal conductivity of thermoelectric devices, provided the dispersion relations and the phonon-phonon and electron-phonon interaction time-scales are available.

Recently, experiments have revealed that thermal conductivity in low-dimensional structures such as silicon nanowires scales linearly with temperature as opposed to the cubic relation found in thin ﬁlms [60, 61]. The Monte-Carlo simulations, using bulk dispersion relationships, have not been successful in reproducing similar results [62, 63]. As thermal conductivity in conﬁned structures in governed by boundary

79 scattering, these results clearly indicate that there is a need for improved characteri- zation of the surface eﬀects in low dimensional structures. Other experimentally observed phenomenon in low dimensional structures, such as thermal rectiﬁcation [64], also warrant the immediate attention of the numerical modeling community.

80 BIBLIOGRAPHY

[1] E. Pop, R. W. Dutton, and K. E. Goodson. Analytic band monte carlo model for electron transport in si including acoustic and optical phonon dispersion. Journal of Applied Physics, 96(9):4998–5005, 2004.

[2] T. Wang. Sub-Micron Thermal Transport in Ultra-scaled Metal-Oxide semiconductor devices. PhD thesis, School of Mechanical Engineering, Purdue University, 2007.

[3] A. S. Henry and G. Chen. Spectral phonon transport properties of silicon based on molecular dynamics simulations and lattice dynamics. Journal of Computa- tional and Theoretical Nanoscience, 5:141–152(12), February 2008.

[4] M. Asheghi. Thermal Transport Properties of Silicon Films. PhD thesis, 2000, Mechanical Engineering, Stanford University.

[5] International technology roadmap for semiconductors. I.T.R.S, 2003. available at http://public.itrs.net/.

[6] K. Banerjee, M. Pedram, and A.H. Ajami. Analysis and optimization of thermal issues in high-performance vlsi. ISPD 2001, pages 230–237, 2001. Sonoma, CA.

[7] D.L. DeVoe. Thermal issues in mems and microscale systems. IEEE Transactions on Components and Packaging Technologies, 25(4):576–583, 2003.

[8] K. L. Ekinci and M. L. Roukes. Nanoelectromechanical systems. Rev. Sci. Instrum, 76:061101–1, 2005.

[9] R. Nair. Eﬀect of increasing chip density on the evolution of computer architec- tures. IBM Journal of Research and Development, 46(2/3):223–234, 2002.

[10] J.M. Ziman. Electrons and Phonons. Oxford University Press,London, 1960.

[11] C. Kittel. Introduction to Solid State Physics. Wiley, New York, 1996.

[12] A. Majumdar, C.L. Tien, and F.M. Gerner. Microscale Energy Transport. Taylor and Francis, 1998.

81 [13] Y. S. Ju and K. E. Goodson. Phonon scattering in silicon ﬁlms with thickness of order 100 nm. Applied Physics Letters, 74(20):3005–3007, 1999.

[14] J.Y. Murthy, S.V.J. Narumanchi, J.A. Pascual-Gutierrez, T. Wang, C. Ni, and S.R. Mathur. Review of multi-scale simulation in sub-micron heat transfer. In- ternational Journal for Multiscale Computational Engineering, 3:5–32, 2005.

[15] G. Chen. Nanoscale Energy Transport and Conversion: A parallel treatment of Electrons, Molecules, Phonons and Photons. Oxford University Press, 2005.

[16] F.H. Stillinger and T.A. Weber. Computer simulation of local order in condensed phases of silicon. Physical Review B, 31(8):52625271, 1985.

[17] J. Tersoﬀ. New empirical approach for the structure and energy of covalent systems. Physical Review B, 37(2):69917000, 1988.

[18] J.F. Justo, M. Z. Bazant, E. Kaxiras, V.V. Bulatov, and S. Yip. Interatomic potential for silicon defects and disordered phases. Physical Review B, 58:2539, 1998.

[19] A. J. H. McGaughey and M. Kaviany. Phonon transport in molecular dynamics simulations: Formulation and thermal conductivity prediction, chapter 2 in. Advances in Heat Transfer, 39:169–225, 2006.

[20] M.P. Allen and D.J. Tildesley. Computer Simulation of Liquids. Oxford Univer- sity Press, 1989.

[21] J.V. Goicochea, M. Madrid, and C. Amon. Hierarchical modeling of heat transfer in silicon-based electronic devices. In Thermal and Thermomechanical Phenom- ena in Electronic Systems, 2008. ITHERM 2008. 11th Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, pages 1006–1017, May 2008. Orlando, Florida.

[22] J. Callaway. Model for lattice thermal conductivity at low temperatures. Phys. Rev., 113(4):1046–1051, Feb. 1959.

[23] Joseph Callaway and Hans C. von Baeyer. Eﬀect of point imperfections on lattice thermal conductivity. Phys. Rev., 120(4):1149–1154, Nov 1960.

[24] M. G. Holland. Analysis of lattice thermal conductivity. Phys. Rev., 132(6):2461– 2471, Dec 1963.

[25] M. G. Holland. Phonon scattering in semiconductors from thermal conductivity studies. Phys. Rev., 134(2A):A471–A480, Apr 1964.

82 [26] S. V. J. Narumanchi, J. Y. Murthy, and C. H. Amon. Submicron heat transport model in silicon accounting for phonon dispersion and polarization. Journal of Heat Transfer, 126(6):946–955, 2004.

[27] S.V.J. Narumanchi. Simulation of Heat Transport in Sub-Micron Conduction. PhD thesis, Department of Mechanical Engineering, Carnegie Mellon University, 2003.

[28] S. Mazumder and A. Majumdar. Monte carlo study of phonon transport in solid thin ﬁlms including dispersion and polarization. Journal of Heat Transfer, 123(4):749–759, 2001.

[29] D. Lacroix, K. Joulain, and D. Lemonnier. Monte carlo transient phonon transport in silicon and germanium at nanoscales. Phys. Rev. B, 72(6):064305, Aug 2005.

[30] A. Majumdar. Microscale heat conduction in dielectric thin ﬁlms. Journal of Heat Transfer, 115(7):7–16, 1993.

[31] M.F. Modest. Radiative Heat Transfer. Academic Press, Second Edition, 2003.

[32] G. Chen. Ballistic-diﬀusive equations for transient heat conduction from nano to macroscales. Journal of Heat Transfer, 124(2):320–328, 2002.

[33] R. Yang, G. Chen, M. Laroche, and Y. Taur. Simulation of nanoscale multidi- mensional transient heat conduction problems using ballistic-diﬀusive equations and phonon boltzmann equation. Journal of Heat Transfer, 127(3):298–306, 2005.

[34] J.Y. Murthy and S.R. Mathur. Ballistic-diﬀusive approximation for phonon transport accounting for phonon dispersion and polarization,paper ht2003-47491. ASME Heat Transfer Conference, July 2003.

[35] R.A. Escobar and C.H. Amon. Thin ﬁlm phonon heat conduction by the dispersion lattice boltzmann methods. Journal of Heat Transfer, 130:092402–8, 2008.

[36] R.A. Escobar, B. Smith, and C.H. Amon. Lattice boltzmann modeling of subcon- tinuum energy transport in crystalline and amorphous microelectronic devices. Journal of Electronic Packaging, 128:115–124, June 2006.

[37] R. B. Peterson. Direct simulation of phonon-mediated heat transfer in a debye crystal. Journal of Heat Transfer, 116(4):815–822, 1994.

83 [38] M.S. Jeng, R. Yang, D. Song, and G. Chen. Modeling the thermal conductivity and phonon transport in nanoparticle composites using monte carlo simulation. Journal of Heat Transfer, 130(4):042410, 2008.

[39] J. Randrianalisoa and D. Baillis. Monte carlo simulation of steady-state microscale phonon heat transport. Journal of Heat Transfer, 130(7):072404, 2008.

[40] J. Randrianalisoa and D. Baillis. Monte carlo simulation of cross-plane thermal conductivity of nanostructured porous silicon ﬁlms. Journal of Applied Physics, 103(5):053502, 2008.

[41] P. Heino. Thermal conduction simulations in the nanoscale. Journal of Compu- tational and Theoretical Nanoscience, 4:896–927, August 2007.

[42] M. Kazan, S. Pereira, J. Coutinho, M. R. Correia, and P. Masri. Role of optical phonon in ge thermal conductivity. Applied Physics Letters, 92(21):211903, 2008.

[43] Y.J. Han and P. G. Klemens. Anharmonic thermal resistivity of dielectric crystals at low temperatures. Phys. Rev. B, 48(9):6033–6042, Sep 1993.

[44] Y. Chen, D. Li, J. R. Lukes, and A. Majumdar. Monte carlo simulation of silicon nanowire thermal conductivity. Journal of Heat Transfer, 127(10):1129–1137, 2005.

[45] S. Aubry, C. Kimmer, P. Schelling, E. Piekos, and L. Phinney. Phon-mediated thermal transport in micro-systems. In Proceedings of the Materials Research Society, (EE1.4), November 28-29 2007. Boston, MA.

[46] B. N. Brockhouse. Lattice vibrations in silicon and germanium. Phys. Rev. Lett., 2(6):256–258, Mar 1959.

[47] G. Dolling. Lattice vibrations in crystals with the diamond structure. at the Symposium of Inelastic Scattering of Neutrons in Solids and Liquids, 2:37, 1963.

[48] A. K. Ghatak and L. S. Kothari. An introduction to lattice dynamics [by] A. K. Ghatak [and] L. S. Kothari. Addison-Wesley, London, Reading, Mass.,, 1972.

[49] J. D. Chung, A. J. H. McGaughey, and M. Kaviany. Role of phonon dispersion in lattice thermal conductivity modeling. Journal of Heat Transfer, 126(3):376–380, 2004.

[50] E. Pop. Self-heating and scaling of thin body transistors. PhD thesis, Department of Electrical Engineering, Stanford University, 2004.

[51] H.B.G. Casimir. Note on the conduction of heat in crystals. Physica, 5(6):495 – 500, 1938.

84 [52] P.G. Klemens. Theory of thermal conductivity of solids In Thermal Conductivity, volume 1. Academic Press London, 1969.

[53] C. Jacoboni and L. Reggiani. The monte carlo method for the solution of charge transport in semiconductors with applications to covalent materials. Rev. Mod. Phys., 55(3):645–705, Jul 1983.

[54] G.A. Bird. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, 1994.

[55] S.B. Pope. Pdf methods for turbulent reacting ﬂows. Progress in Energy and Combustion Science, 11:119–192, 1985.

[56] N. Bellomo and L Arlotti. Lecture notes on the mathematical theory of the Boltzmann equation. World Scientiﬁc, London, Singapore., 1995.

[57] M. Asheghi, Y. K. Leung, S. S. Wong, and K. E. Goodson. Phonon-boundary scattering in thin silicon layers. Applied Physics Letters, 71(13):1798–1800, 1997.

[58] R. A. H. Hamilton and J. E. Parrott. Variational calculation of the thermal conductivity of germanium. Phys. Rev., 178(3):1284–1292, Feb 1969.

[59] D. A. Broido, M. Malorny, G. Birner, N. Mingo, and D. A. Stewart. Intrinsic lattice thermal conductivity of semiconductors from ﬁrst principles. Applied Physics Letters, 91(23):231922, 2007.

[60] D. Li, Y. Wu, P. Kim, L. Shi, P. Yang, and A. Majumdar. Thermal conductivity of individual silicon nanowires. Applied Physics Letters, 83(14):2934–2936, 2003.

[61] R. Chen, A. I. Hochbaum, P. Murphy, J. Moore, P. Yang, and A. Majum- dar. Thermal conductance of thin silicon nanowires. Physical Review Letters, 101(10):105501, 2008.

[62] Y. Chen, D. Li, J.R. Lukes, and A. Majumdar. Monte carlo simulation of silicon nanowire thermal conductivity. Journal of Heat Transfer, 127(10):1129–1137, 2005.

[63] D. Lacroix, K. Joulain, D. Terris, and D. Lemonnier. Monte carlo simulation of phonon conﬁnement in silicon nanostructures: Application to the determina- tion of the thermal conductivity of silicon nanowires. Applied Physics Letters, 89(10):103104, 2006.

[64] D.G. Walker. Thermal rectiﬁcation mechanisms including noncontiuuum eﬀects. 18th National and 7th ISHMT-ASME Heat and Mass Transfer Conference, IIT Guwahati, India,, Jan. 4-6, 2006.