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

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Monte-Carlo Study of Phonon Heat Conduction in Silicon Thin Films 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 c Copyright by Arpit Mittal 2009 ABSTRACT Heat conduction in crystalline semiconductor materials occurs by lattice vibrations that result in the propagation of quanta of energy called phonons. The Boltzmann Transport Equation (BTE) for phonons is a powerful tool to model both equilibrium and non-equilibrium heat conduction in crystalline solids. Non-equilibrium heat con- duction occurs either when the length scales (of the device in question) are small or at low temperatures. The BTE describes the evolution of the number density (or energy) distribution for phonons as a result of transport (or drift) and inter-phonon collisions. The Monte-Carlo (MC) method has found prolific use in the solution of the Boltzmann Transport Equation (BTE) for phonons. This thesis contributes to the state-of-the-art by performing a systematic study of the role of the various phonon modes on thermal conductivity predictions−in particular, optical phonons. A procedure to calculate three-phonon scattering time-scales with the inclusion of optical phonons is described and implemented. The roles of various phonon modes are assessed. It is found that Transverse Acoustic (TA) phonons are the primary carriers of energy at low temperatures. At high temperatures (T > 200 K), Longi- tudinal Acoustic (LA) phonons carry more energy than TA phonons. When optical phonons are included, there is a significant change in the amount of energy carried by various phonons modes, especially at room temperature, where optical modes are found to carry about 25% of the energy at steady state in silicon thin films. Most ii importantly, it is found that inclusion of optical phonons results in better match with experimental observations for silicon thin-film thermal conductivity. The inclu- sion of optical phonons is found to decrease the thermal conductivity at intermediate temperatures (50−200 K) and increase it at high temperature (>200 K), especially when the film is thin. The effect of number of stochastic samples, the dimension- ality of the computational domain (two-dimensional versus three-dimensional), and the lateral (in-plane) dimension of the film on the statistical accuracy and compu- tational 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 solu- tion. 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 de- gree of specularity, α. It is also shown that the time averaging of the statistically stationary data can be used to reduce statistical noise and can result in considerable computational savings. iii Dedicated to my family for their never-ending love and support iv ACKNOWLEDGMENTS I would like to begin by acknowledging my advisor and mentor, Professor Sandip Mazumder, for giving me with this wonderful opportunity to learn from him. He is as excellent educator and I have benefited immensely from his clarity of thought and passion for teaching. I thank him for his patience and guidance during the course of my research. I would also like to acknowledge Professor V.V. Subramaniam for serving on my thesis defense committee and for his valuable help during my defense. I am grateful to Professor Nandini Trivedi and Professor Joseph Heremans for their valuable help and guidance. I would also like to thank my lab mates over the past two years: Ankan, Sai, Mahesh, Tom, Alex, Derek and Robert for their support. Financial support by the Department of Energy's Basic Energy Science Program through Grant Number DE-FG02-06ER46330 is gratefully acknowledged. Caffeine support by the Mechanical Engineering Graduate student Association (MEGA) is also gratefully acknowledged. v VITA May 2005 . .(B.E.) with Honors, Punjab Engineer- ing College, Panjab University June 2005 - August 2007 . Assistant Manager, Maruti Suzuki In- dia Limited September 2007 - Present . .Graduate Research Associate, Compu- tational Thermal-Fluids Lab, The Ohio State University PUBLICATIONS FROM THIS WORK Research Publications \Monte Carlo Study of Phonon Heat Conduction in Silicon Thin Films Including Contributions of Optical Phonons", Arpit Mittal and Sandip Mazumder, Journal of Heat Transfer (accepted). Conference Publications \Monte Carlo Study of Phonon Heat Conduction in Silicon Thin Films: Role of Optical Phonons," Arpit Mittal and Sandip Mazumder, \Proceedings of the ASME Summer Heat Transfer Conference", July 19-23, 2009, San Francisco, CA, Paper Number - HT2009-88008 FIELDS OF STUDY Major Field: Mechanical Engineering vi TABLE OF CONTENTS Page Abstract . ii Dedication . iv Acknowledgments . .v Vita......................................... vi List of Tables . ix List of Figures . .x List of Symbols . xii Chapters: 1. Introduction . .1 1.1 Modeling of Non-Equilibrium Heat Transport . .3 1.2 Molecular Dynamics . .5 1.3 Methods Based on Boltzmann Transport Equation . .7 1.4 Background and Scope of the Thesis . 10 1.5 Thesis Overview . 16 2. Theory . 17 2.1 Sub-Continuum Heat Conduction in Solids . 18 2.1.1 Wave propagation in crystals . 18 2.1.2 Energy Quantization: Phonons . 23 2.2 Boltzmann Transport Equation . 24 2.2.1 BTE for Phonons . 25 vii 2.3 Dispersion Relationships for Silicon . 26 2.4 Phonon Interaction (Scattering) Processes . 28 2.4.1 Impurity Scattering . 30 2.4.2 Boundary Scattering . 30 2.5 Phonon-Phonon Interactions . 31 3. Monte-Carlo Solution Technique . 37 3.1 Number of Stochastic Samples . 37 3.2 Initialization of Phonon Samples . 39 3.2.1 Position . 39 3.2.2 Frequency . 40 3.2.3 Polarization . 41 3.2.4 Direction . 42 3.3 Linear Motion (Drift) of Phonons . 43 3.4 Boundary Interactions . 45 3.5 Intrinsic Scattering . 49 4. RESULTS AND DISCUSSION . 52 4.1 Verification of the MC code . 52 4.2 Thermal Conductivity and Role of Various Phonon Modes . 54 4.3 Numerical Issues . 67 5. SUMMARY AND FUTURE WORK . 76 5.1 Summary . 76 5.2 Future work . 78 Bibliography . 81 viii LIST OF TABLES Table Page 2.1 Curve-fit parameters for silicon dispersion data [1]. 28 2.2 Three-phonon interactions considered in this study for the calculation of scattering time-scales for each phonon polarization. 34 4.1 Contribution (%) of various phonon modes towards thermal energy transport when both acoustic and optical modes are considered. Film thickness = 0.42 µm. .......................... 65 4.2 Contribution (%) of various phonon modes towards thermal energy transport when only acoustic modes are considered. Film thickness = 0.42 µm................................. 66 4.3 Statistical noise in the thermal conductivity data for both two-dimensional and three-dimensional calculations at 80 K as a function of sample size. 72 4.4 Statistical noise in the thermal conductivity for two-dimensional cal- culations at 300 K as a function of sample size. 73 ix LIST OF FIGURES Figure Page 1.1 Validity of Heat Transfer Models . .6 1.2 Representation of a Thin Film (a) 3D, and (b) 2D . 14 2.1 One-dimensional spring-mass system . 18 2.2 One-dimensional dispersion relationship. 20 2.3 Brillouin zone in K-space . 21 2.4 General form of dispersion relation in a three-dimensional crystal [2]. 22 2.5 Curve fit data for dispersion curves for silicon in [100] direction. 29 3.1 Position sampling in a three-dimensional geometry. 40 3.2 Coordinate system showing the direction of phonon emission and the associated angles . 43 3.3 Computational Domain (a) 3D, and (b) 2D . 45 3.4 Phonon reflection at the boundary . 47 3.5 Thermal conductivity with and without boundary scattering. 48 4.1 Comparison of the results obtained by MC method (ensemble aver- age) against analytical solution in the ballistic limit. 53 4.2 Comparison of the results obtained by MC method (raw data) against analytical solution in the diffusion limit. 55 x 4.3 Temperature profile for various degree of specularity (α)....... 56 4.4 Group velocity of various phonon modes. 57 4.5 Normalized cumulative number density function . 58 4.6 Phonon-phonon scattering time-scales computed using the present hybrid approach and Holland's model at different temperatures. 59 4.7 Comparison of mean free paths of all four phonon modes at 300 K computed using the present hybrid approach, and data obtained using molecular dynamics by Henry and Chen [3] . 61 4.8 Predicted and measured [4] through-plane thermal conductivity for a 0.42 µm silicon film . 62 4.9 Predicted and measured [4] through-plane thermal conductivity for a 1.60 µm silicon film . 63 4.10 Predicted and measured [4] through-plane thermal conductivity for a 3.0 µm silicon film . 64 4.11 Predicted thermal conductivity using two-dimensional versus three- dimensional computational domains. 68 4.12 CPU time taken for two-dimensional versus three-dimensional simu- lations with Nprescribed = 50,000. 69 4.13 Time-dependent energy flux at the boundaries for a 0.42 µm thin film at 80K . 71 4.14 Time averaged energy flux (average of data presented in Fig. 4.13(a)) at the boundaries for a 0.42 µm thin film computed using Nprescribed = 50,000.
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