Thin Films THESIS. Presented in Partial Fulfillment of the Requirements for the Degree

Transcription

1 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 By 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

2 c Copyright by Arpit Mittal 2009

3 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), Longitudinal 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

4 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 ( 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

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

6 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

7 VITA May (B.E.) with Honors, Punjab Engineering College, Panjab University June August Assistant Manager, Maruti Suzuki India Limited September Present Graduate Research Associate, Computational Thermal-Fluids Lab, The Ohio State University Research Publications PUBLICATIONS FROM THIS WORK 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 - HT FIELDS OF STUDY Major Field: Mechanical Engineering vi

8 TABLE OF CONTENTS Page Abstract Dedication Acknowledgments Vita List of Tables ii iv v vi ix List of Figures x List of Symbols xii Chapters: 1. Introduction Modeling of Non-Equilibrium Heat Transport Molecular Dynamics Methods Based on Boltzmann Transport Equation Background and Scope of the Thesis Thesis Overview Theory Sub-Continuum Heat Conduction in Solids Wave propagation in crystals Energy Quantization: Phonons Boltzmann Transport Equation BTE for Phonons vii

9 2.3 Dispersion Relationships for Silicon Phonon Interaction (Scattering) Processes Impurity Scattering Boundary Scattering Phonon-Phonon Interactions Monte-Carlo Solution Technique Number of Stochastic Samples Initialization of Phonon Samples Position Frequency Polarization Direction Linear Motion (Drift) of Phonons Boundary Interactions Intrinsic Scattering RESULTS AND DISCUSSION Verification of the MC code Thermal Conductivity and Role of Various Phonon Modes Numerical Issues SUMMARY AND FUTURE WORK Summary Future work Bibliography viii

10 LIST OF TABLES Table Page 2.1 Curve-fit parameters for silicon dispersion data [1] Three-phonon interactions considered in this study for the calculation of scattering time-scales for each phonon polarization Contribution (%) of various phonon modes towards thermal energy transport when both acoustic and optical modes are considered. Film thickness = 0.42 µm Contribution (%) of various phonon modes towards thermal energy transport when only acoustic modes are considered. Film thickness = 0.42 µm Statistical noise in the thermal conductivity data for both two-dimensional and three-dimensional calculations at 80 K as a function of sample size Statistical noise in the thermal conductivity for two-dimensional calculations at 300 K as a function of sample size ix

11 LIST OF FIGURES Figure Page 1.1 Validity of Heat Transfer Models Representation of a Thin Film (a) 3D, and (b) 2D One-dimensional spring-mass system One-dimensional dispersion relationship Brillouin zone in K-space General form of dispersion relation in a three-dimensional crystal [2] Curve fit data for dispersion curves for silicon in [100] direction Position sampling in a three-dimensional geometry Coordinate system showing the direction of phonon emission and the associated angles Computational Domain (a) 3D, and (b) 2D Phonon reflection at the boundary Thermal conductivity with and without boundary scattering Comparison of the results obtained by MC method (ensemble average) against analytical solution in the ballistic limit Comparison of the results obtained by MC method (raw data) against analytical solution in the diffusion limit x

12 4.3 Temperature profile for various degree of specularity (α) Group velocity of various phonon modes Normalized cumulative number density function Phonon-phonon scattering time-scales computed using the present hybrid approach and Holland s model at different temperatures 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] Predicted and measured [4] through-plane thermal conductivity for a 0.42 µm silicon film Predicted and measured [4] through-plane thermal conductivity for a 1.60 µm silicon film Predicted and measured [4] through-plane thermal conductivity for a 3.0 µm silicon film Predicted thermal conductivity using two-dimensional versus threedimensional computational domains CPU time taken for two-dimensional versus three-dimensional simulations with N prescribed = 50, Time-dependent energy flux at the boundaries for a 0.42 µm thin film at 80K 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 N prescribed = 50, Time-dependent energy flux at the boundaries for a 0.42 µm film at 300 K xi

13 LIST OF SYMBOLS a lattice constant [m] a acceleration vector [m/s 2 ] b reciprocal lattice vector [m 1 ] B L constant used in relaxation time calculation B T N constant used in relaxation time calculation B T U constant used in relaxation time calculation C heat Capacity [J/K] f distribution function F normalized cumulative number density function F correction factor F ij interatomic force between atoms i and j [N] g spring constant [N/m] h Dirac constant = x [m 2 kg.s 1 ] K wave vector [m 1 ] k B Boltzmann constant = x [m 2 kg.s 2 K 1 ] K magnitude of wave vector [m 1 ] Kn Knudsen number L characteristic length [m] M, m 1, m 2 mass [kg] n phonon occupation number ˆn unit surface normal N b number of spectral (frequency) bins N actual total number of phonons per unit volume N prescribed number of phonons traced p phonon polarization P momentum vector [kg-m/s] P scat probability of scattering of a phonon q heat flux vector [W/m 2 ] r position vector [m] r c effective radius [m] R scat, R f random numbers between 0 and 1 R pol, R 1, R 2 random numbers between 0 and 1 xii

14 t time [s] ˆt 1,ˆt 2 unit surface tangents T thermodynamic temperature [K] T pseudo-thermodynamic temperature [K] v g phonon group velocity vector [m/s] v g magnitude of phonon group velocity [m/s] v p phase velocity [m/s] W weight factor x n, y n displacements [m] α degree of specularity γ Gruneisen parameter t time step [s] ω i bandwidth of a spectral interval [rad/s] κ thermal conductivity [W/m-K] λ nucleic wavelength Λ mean free path θ polar angle θ D Debye Temperature[K] ρ density [kg/m 3 ] ρ d number of scattering sites per unit volume [1/m 3 ] σ standard deviation in thermal conductivity [W/m-K] σ d scattering cross-section [m 2 ] τ overall relaxation time [s] τ b relaxation time for boundary scattering [s] τ i relaxation time for impurity scattering [s] τ ij relaxation time for interaction between i and j [s] τ N relaxation time for Normal (N) scattering [s] τ NU combined relaxation time for N and U scattering [s] τ U relaxation time for Umklapp (U) scattering [s] ω o,v s,c constants used for calculating phonon dispersion ω angular frequency [rad/s] ω o central frequency of the i-th spectral bin[rad/s] ψ azimuthal angle χ degeneracy of each phonon polarization xiii

15 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 efficiently 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 formulate better heat removal strategies and designs, it is first necessary to understand the fundamental mechanisms of heat transport in semiconductor thin films. Accordingly, there has been an increasing interest in modeling thermal transport in microand nano- scale 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

16 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 reliable 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

17 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

18 where κ is the thermal conductivity and q is the heat flux vector. The transient heat conduction equation is therefore given by: C T t = κ(t ) T (1.2) 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 v g 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: τc 2 T t 2 + C T t = κ T (1.3) 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/(τv g ) 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

19 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 Transport 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 0.2Å [3], which is about an order of magnitude smaller than the atomic separation (5.43Å 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 Schrödinger wave 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 pictorially represented in Fig 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 r i is considered, and Newton s second 5

20 Figure 1.1: Validity of Heat Transfer Models law of motion is written for each particle as: M d2 r i dt 2 = N j,j i F i,j (1.4) where F ij 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

21 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]: κ = k B 2π 2 v g ( kb T h ) 3 θd /T x 4 e x τ(x) dx (1.5) (e x 1) 2 0 where θ D is the Debye temperature, v g (= v g ) is the magnitude of the phonon group velocity, k B is the Boltzmann constant, T is the temperature at which the thermal conductivity needs to be calculated, and x = hω/k B T. Holland [24, 25] improved 7

22 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 = κ T + κ L (1.6) where κ L = 1 3 κ T = 2 3 θd,t /T 0 θd,l /T 0 C T T 3 x 4 e x τ T (x) (e x 1) dx 2 C L T 3 x 4 e x τ L (x) dx (1.7) (e x 1) 2 subscripts L, T refer to longitudinal and transverse acoustic modes, respectively, ( ) 3. whereas, the constant C L,T = kb h The thermal conductivity integrals were k B 2π 2 v L,T evaluated by using some fitting parameters for the phonon relaxation times for both longitudinal and transverse modes. After tuning these fitting parameters against experimental data, functional forms of the relaxation times were presented for both transverse and longitudinal modes. The increasing complexity and the underlying assumptions 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 techniques: 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 equation, 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

23 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 Approximation [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 polyhedra and the angular discretization is done by splitting the angular space, 4π, into discrete non-overlapping solid angles Ω i, each centered about the discrete direction ŝ i. Each octant is discretized into N θ N ψ 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 volume 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 proposed 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 geometries. 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

24 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] presented 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 undergo 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

25 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. Secondly, 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

26 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 solution 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 Optical (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], Narumanchi and co-workers [26, 27] have presented a methodology for calculating the three-phonon interaction time-scales for the various phonon modes undergoing Umklapp (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

27 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 procedures 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 unanswered. 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

28 (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 dictated 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 dimensions has no bearing on the predicted thermal conductivity. Furthermore, it is important to understand the effect of the choice of lateral dimension on the computational efficiency. 14

29 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 abovementioned 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

30 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

31 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

32 2.1 Sub-Continuum Heat Conduction in Solids 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 The Figure 2.1: One-dimensional spring-mass system particles with mass m 1 and m 2 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 m 1 are x, and those of particles m 2 are y. Let the displacement of the particles from their equilibrium positions be denoted by x n and y n. It is further 18

33 assumed that only the nearest neighbors interact with each other, i.e., the restoring force on a particle of mass m 1 depends only on its relative position with respect to the adjacent particle of mass m 2. 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 m 1 and m 2 particles are m 1 d 2 x n dt 2 = g[y n 2x n + y n 1 ] m 2 d 2 y n dt 2 = g[x n+1 2y 2n + x n ] (2.1) We look for solutions of Eq. 2.1 in the form of traveling waves, given by: x n = A exp[i((kx) ωt)] y n = 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: (m 1 ω 2 2g)A + g(1 + exp( i K a)b = 0 g(1 + exp(i K a))a + (m 2 ω 2 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., m 1 ω 2 2g g(1 + exp( i K a)) g(1 + exp(i K a)) m 2 ω 2 2g = 0 (2.4) or m 1 m 2 ω 4 2g(m 1 + m 2 )ω 2 + 2g 2 (1 cos K a) = 0 (2.5) 19

34 Figure 2.2: One-dimensional dispersion relationship. Solving Eq: 2.5, we get ( ω 2 = g ) ( 1 ± + 1 ) 2 4 ( ) K a sin 2 (2.6) m 1 m 2 m 1 m 2 m 1 m 2 2 Considering only positive solutions of ω 2, ( ω = g ) ( 1 ± + 1 ) 2 4 ( ) K a sin 2 m 1 m 2 m 1 m 2 m 1 m (2.7) 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 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

35 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

36 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 v g 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, v g = K ω (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

37 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 electromagnetic waves. Each phonon carries a quantum of energy hω. 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: n = ( exp 1 hω k B T ) 1 (2.9) where k B is the Boltzmann constant, ω is the frequency and h 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 = L 3 can be written as E.V = ( n K,p + 1 ) hω K,p (2.10) 2 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 threedimensional Brillouin zone (BZ) and over all phonon polarizations. In the limit of a 23

38 large crystal, the wave-vector space becomes so dense that it can be replaced by an integral as follows: E.V = p K ( n K,p + 1 ) hω K,p dk (2.11) 2 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 E = p ω ( n K,p + 1 ) hω K,p D(ω)dω (2.12) 2 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 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

39 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 t + v f + a f v = [ ] f t scat (2.13) 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 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 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 external force, the acceleration vector a is zero, and therefore, the third term in Eq is absent. The BTE for phonons may then be written as [12] f t + v g f = [ ] f t scat (2.14) where f(r, K, t) is the distribution function for phonons, and v g is the phonon group velocity. The left side of Eq 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

40 should involve phonons from the set of (r, K) to set (r, K ). [ ] f = [φ(k, K)f(K ) φ(k, K )f(k)] (2.15) t scat K where φ(k, K)f(K ) and φ(k, K )f(k) are scattering phase functions from K to K and K to K, respectively. Due to the complexity of the scattering term, simplifications 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 t scat τ (2.16) where f o 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 f o = exp( t/τ). The study of transport phenomenon, essentially, requires the calculation of distribution function from Eq Once the distribution function is known, the energy flux may then be calculated using: q(r, t) = v g (r, t)f(r, K, t) hω K,p D(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 calculations and is necessary to model polarization and frequency-dependent behavior [2]. 26

41 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 Schrödinger equation 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 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]: ω K = ω o + v s K + c K 2 (2.18) For the acoustic modes, the constants ω o, v s 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

42 Phonon Polarization ω o (10 13 rad/s) v s (10 2 m/s) c (10 7 m 2 s LA TA LO TO 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 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 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

43 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

44 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, τ i, between collisions with imperfections is a function of the defect density (or the number of scattering sites per unit volume) ρ d, the group velocity of the phonon v g, and the scattering cross section σ d and is given by: τ i = 1 ησ d ρ d v g (2.19) where η is a constant of order unity. According to Majumdar [12], the scattering cross section varies as σ d = πr 2 ( ζ 4 ζ ) (2.20) where ζ = ωr/v g, R is the radius of the lattice imperfection, and ω is the phonon frequency 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

45 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 = v g τ b L F, L = 2 l1 l 2 (2.21) π where l 1 l 2 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 complicated 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]. Scattering 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. Phononphonon 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

46 conservation rules given by [11] ω + ω ω (Normal + Umklapp) (2.22) K + K K (Normal) (2.23) K + K K + b(umklapp) (2.24) where ω, ω and ω are the angular frequencies of the interacting phonons and K, K, and K 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 momentum hk. This can be easily understood if a phonon is considered as a classical particle with energy E = Mv 2 /2 and momentum P = Mv. Comparing the velocity of the particle v = de dp with Eq. 2.8 gives hk 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 result, 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 conservation rules [43]. It is critical to note that although only U processes are responsible 32

47 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: τ 1 NU = B Lω 2 T 3 (LA, Normal+Umklapp) τ 1 N = B T N ωt 4 (TA, Normal) (2.25) τ 1 U = { 0 TA, Umklapp for ω < ω1/2 B T U ω 2 /(sinh( hω)/(k B T )) TA, Umklapp for ω > ω 1/2 (2.26) where ω 1/2 is the frequency corresponding to K/K max = 0.5 and B L, B T N, and B T U are constants [24] that need to be obtained through calibration against experimental data. For silicon, ω 1/2 = 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 assumptions 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. Therefore, in effect, Eq gives the time-scale for N processes, whereas the expression in Eq 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

48 Longitudinal Acoustic (LA) Phonons LA + TA(BZB) LA LA + TA LO(BZB) LA + TA TO(BZB) LA + LA LO(BZB) LA + LA TO(BZB) Longitudinal Optical (LO) Phonons LO LA + TA(BZB) LO LA + LA(BZB) LO TA + LA(BZB) Transverse Acoustic (TA) Phonons TA + TA(BZB) LA TA + LA(BZB) LA TA + LA LO(BZB) TA + LA TO(BZB) Transverse Optical (TO) Phonons TO LA + TA(BZB) TO LA + 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 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, τ ij, for the interactions of the type LA 34

49 + 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] [ ] 1 χγ 2 τ ij 3πρvph 2 v g ω i ω tr ω j rc 2 1 exp ( hω tr /k B T ) 1 1 (2.27) exp ( hω j /k B T ) 1 where i refers to the phonon for which the time-scale needs to be calculated, henceforth 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 v p (= ω/ K ), and v g is the magnitude of the group velocity of the resultant mode. ρ is the density of the solid (i.e., bulk silicon). ω i is the frequency of the incoming phonon in rad/s, ω j is the frequency of the resultant mode, while ω tr 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 r c denotes the effective radius. For the [100] direction, r c is calculated as [43]: r c = 2π/a K i 2 2 (2.28) where a is the lattice constant, and K i 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 1 τ ij χγ 2 3πρv 2 ph v g ω i ω O ω 3 tr [ ] 1 exp ( hω tr /k B T ) 1 1 exp ( hω O /k B T ) 1 (2.29) where ω O is the frequency of the optical mode at the BZB, ω i is the incoming phonon frequency, and ω tr (= ω O ω i ) is the translated phonon frequency. 35

50 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 τ U,i = j 1 τ ij (2.30) where is the relaxation time for a single scattering process [as calculated using Eqs and/or 2.29], and τ U,i is the relaxation time for all U processes combined. The overall relaxation time τ is then calculated as: 1 τ = 1 τ N + 1 τ U (2.31) and is used as an input for solution of Eq. 2.14, with the scattering term expressed by Eq

51 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, N actual. For a given temperature and material, Eq gives the energy per unit volume and involves 37

52 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, N b, with ω o and ω i 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]: E = p N b i=1 ( hωo,i k b T 1 hω o,i exp ) 1 K 2 2π 2 d K dω ω i χ p (3.1) ωo,i 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 : N actual = p N b i=1 exp 1 ( hωo,i k b T ) 1 K 2 2π 2 d K dω ω i χ p (3.2) ωo,i 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 phonons per m 3, whereas it reduces to about phonons per m 3 at 10 K, which translates to phonons per µm 3. 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 N prescribed gives the number of phonons actually initialized/emitted into the system, then the weight factor is given by W = N actual N prescribed (3.3) 38

53 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 N prescribed 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. vector components and two directions and velocity. These include three position Under the isotropic Brillouin zone assumption, the wave vector and the group velocity are co-directional. 3.2 Initialization of Phonon Samples 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 r = R 1 (x OB x OA )î + R 2 (y OB y OA )ĵ + +R 3 (z OB z OA )ˆk (3.4) 39

54 Figure 3.1: Position sampling in a three-dimensional geometry. where R 1, R 2 and R 3 are the three random numbers such that 0 < R 1, R 2, R 3 < 1 and OA and OB are the position vectors of the two ends of the body diagonal of the rectangular parallelepiped 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 N b spectral bins. From Eq. 3.2, the number of phonons in the i th spectral bin can be calculated as: N i = p exp 1 ( hωo,i k b T ) 1 K 2 2π 2 d K dω ω i χ p (3.5) ωo,i A normalized cumulative number density function F is then constructed, i j=1 F i = N j Nb j=1 N j (3.6) where F i represents the probability of finding a phonon having frequency less than ω o,i + ω i and F i F i 1 is the probability of finding a phonon in the i th frequency 40

55 bin. In order to sample the frequency of the phonon, a random number R f, between zero and unity, is drawn. If F i 1 < R f < F i, the phonon lies in the i th frequency bin. The location of the i th bin, corresponding to the random number R f 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 o,i + (2R f 1) ω i 2 (3.7) Eq. 3.7 assumes that the frequency varies linearly within each spectral bin. This can only be assumed if the number of frequency bins N b 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 N b beyond this did not increase the accuracy of the solution, as was found during the validation of the numerical code 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 P i,p = N i,p N i (3.8) where N i,p is the number of phonons of polarization p in the i th spectral interval and N i is th total number of phonons in the i th spectral interval and is given by Eq Therefore, P i,p = exp p ( 1 hωo,i k b T exp ( 1 hωo,i k b T K ) 2 1 2π 2 K ) 2 1 2π 2 41 d K dω d K dω ω i χ p ωo,i (3.9) ω i χ p ωo,i

56 Once the probability of the p th polarization has been calculated, a normalized cumulative probability distribution function is constructed. A random number R pol is called and if P p 1 < R pol < P p 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]: P i (LA) = N i (LA) N i (T A) + N i (LA) = n(ω o,i, LA) D((ω o,i, LA) n(ω o,i, LA) D((ω o,i, LA) + 2 n(ω o,i, T A) D((ω o,i, T A) (3.10) where n is the equilibrium phonon occupation number given by Eq Now, if the drawn random number is less than P i (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 v g is calculated using Eq Direction The last step in the initialization stage is to assign a direction to the phonon sample. If isotropy is assumed, the direction vector ŝ may be calculated in three dimensions as (refer to Fig. 3.2) ŝ = sin θ cos ψ sin θ sin ψ cos θ (3.11) where 0 ψ 2π and 0 θ π/2. Therefore, the angles can be represented by random number relations [28] and written as, ŝ = ( 1 (2R1 1) 2 cos(2πr 2 ))ˆt 1 + ( 1 (2R1 1) 2 sin(2πr 2 ))ˆt 2 +(2R 1 1)ˆn 42 (3.12)

57 Figure 3.2: Coordinate system showing the direction of phonon emission and the associated angles For 2-D, ŝ may be calculated as, ŝ = { cos ψ sin ψ } (3.13) ŝ = cos(2πr 2 )î + sin(2πr 2 )ĵ (3.14) where ψ = 2πR 2, θ =(2R 1 1), and, R 1 and R 2 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

58 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 serves as an initial value for the second step in which Eq is solved [56]. This operator splitting introduces an error that is first order in time. f t + v g f = 0 (3.15) [ ] f f t = = f f o (3.16) t τ scatter 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) + v g 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, Ẽcell, 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 given by Ẽ cell W = N b 1 K hω o,i ( ) 2 d K hωo,i p exp 1 2π 2 dω ω i χ p (3.18) ωo,i i=1 k b T 44

59 (a) 3D (b) 2D Figure 3.3: Computational Domain (a) 3D, and (b) 2D The pseudo-temperature T is calculated from Eq 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 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 All phonons that strike the isothermal boundary during 45

60 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 b N face = A t n(ω o,i, p) v g (ωo, i) ˆn D(ω o,i, p) ω i p i=1 (3.19) where v g (ω o,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 v g (ω o,i ) ˆn has be to directionally averaged in order to obtain the average velocity normal to the boundary face. v g (ω o,i ) ˆn = { 1/π vg for 2-D 1/4 v g for 3-D (3.20) Therefore, the actual number of phonons emitted by the boundary at each time step is given by ( N face = A π t p for 2-D and ( N face = A 4 t p N b i=1 N b i=1 n(ω o,i, p) v g (ωo, i)d(ω o,i, p) ω i ) n(ω o,i, p) v g (ωo, i)d(ω o,i, p) ω i ) (3.21) (3.22) for 3-D. In the MC scheme, this number is scaled by the weight factor, W,i.e., the number of samples injected is N face /W. The spatial location of the emitted phonons are uniformly distributed over the entire face by calling random numbers and their 46

61 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). ŝ r = ŝ i + 2 ŝ i ˆn ˆn, (3.23) where ŝ i and ŝ r 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

62 Figure 3.5: Thermal conductivity with and without boundary scattering. than the phonon wavelength. The fraction of phonons reflected specularly (or diffusely) 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 Therefore, at low temperature, boundary scattering must be accounted for. This is done by treating the lateral boundary as 48

63 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: ŝ = (sin θ cos ψ)ˆt 1 + (sin θ sin ψ)ˆt 2 + cos θˆn (3.24) for 3D, where ˆn is the unit surface normal, and ˆt 1 and ˆt 2 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 phononphonon (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 timescale of scattering of phonons has been calculated, the probability of scattering of the 49

64 phonon between time t and t + t may be calculated as [28] ( ) t P scat = 1 exp τ (3.25) A random number R scat is drawn, and if the probability of scattering, P scat, is greater than R scat, 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 ( 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 F scat 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: F scat ( T ) = i j=1 N j( T ) P scat, j Nb j=1 N j( T (3.26) ) P scat, 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

65 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 F scat 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

66 CHAPTER 4 RESULTS AND DISCUSSION As discussed in section 1.4, previous works [28, 29] have not considered the transport of optical phonons when predicting semiconductor thin film thermal conductivity. 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 In the diffusion limit, i.e., for the case when the phonon mean free path is much smaller than the film thickness, the 52

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

68 code reproduced a linear temperature profile between the two boundary temperatures as shown in Fig 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 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 assigning 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

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

70 Figure 4.3: Temperature profile for various degree of specularity (α) 56

71 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 and 2.26), and (2) using the hybrid approach based on the works of Holland [24] and Han and Klemens [43] (Eqs and 2.29 presented in section 2.5). While the computed time-scales are somewhat different, it is encouraging to see that the timescales computed using the hybrid approach are of the same order of magnitude as 57

72 (a) 50 K (b) 100 K (c) 200 K (d) 300 K Figure 4.5: Normalized cumulative number density function 58

73 (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

74 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 ( K), the occupation number of optical phonons is 60

75 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

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

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

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

79 T (K) LA TA LO TO 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

80 T (K) LA TA 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 Majumdar [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

81 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

82 Figure 4.11: Predicted thermal conductivity using two-dimensional versus threedimensional computational domains. example, Mazumder and Majumdar [28] reported using a value of 0.6 for a 2D thinfilm 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 for the 2D and 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 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

83 Figure 4.12: CPU time taken for two-dimensional versus three-dimensional simulations with N prescribed = 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 (= N prescribed ) 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

84 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 phononboundary 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, N prescribed. 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 for two different values of N prescribed. 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 N prescribed from 50,000 to 500,000, the standard deviation in the flux decreases from 22% to about 7%. 70

85 (a) N prescribed = 50, 000 (b) N prescribed = 500, 000 Figure 4.13: Time-dependent energy flux at the boundaries for a 0.42 µm thin film at 80K 71

86 N prescribed 2D ( ( σ κ) (%) 3D σ κ) (%) 50, , , 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 N prescribed = 50,

87 N prescribed κ (W/m-K) σ/κ(%) 50, , , 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 N prescribed 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 N prescribed = 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 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 N prescribed 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 N prescribed is used. This is because at higher temperature there are 73

88 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 N prescribed = 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 N prescribed. For transient calculations, however, time averaging cannot be used and a higher value of N prescribed 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

89 (a) N prescribed = 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