Research Papers: Micro/Nanoscale Heat Transfer

A Coupled Ordinates Method for Convergence Acceleration of the Phonon Boltzmann Transport Equation

[+] Author and Article Information
James M. Loy, Sanjay R. Mathur

Department of Mechanical Engineering,
The University of Texas at Austin,
Austin, TX 78712-0209

Jayathi Y. Murthy

Department of Mechanical Engineering,
The University of Texas at Austin,
Austin, TX 78712-0209
e-mail: jmurthy@me.utexas.edu

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 18, 2013; final manuscript received August 12, 2014; published online November 11, 2014. Assoc. Editor: Zhuomin Zhang.

J. Heat Transfer 137(1), 012402 (Jan 01, 2015) (10 pages) Paper No: HT-13-1026; doi: 10.1115/1.4028806 History: Received January 18, 2013; Revised August 12, 2014; Online November 11, 2014

Sequential numerical solution methods are commonly used for solving the phonon Boltzmann transport equation (BTE) because of simplicity of implementation and low storage requirements. However, they exhibit poor convergence for low Knudsen numbers. This is because sequential solution procedures couple the phonon BTEs in physical space efficiently but the coupling is inefficient in wave vector (K) space. As the Knudsen number decreases, coupling in K space becomes dominant and convergence rates fall. Since materials like silicon have K-resolved Knudsen numbers that span two to five orders of magnitude at room temperature, diffuse-limit solutions are not feasible for all K vectors. Consequently, nongray solutions of the BTE experience extremely slow convergence. In this paper, we develop a coupled-ordinates method for numerically solving the phonon BTE in the relaxation time approximation. Here, interequation coupling is treated implicitly through a point-coupled direct solution of the K-resolved BTEs at each control volume. This implicit solution is used as a relaxation sweep in a geometric multigrid method which promotes coupling in physical space. The solution procedure is benchmarked against a traditional sequential solution procedure for thermal transport in silicon. Significant acceleration in computational time, between 10 and 300 times, over the sequential procedure is found for heat conduction problems.

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International Technology Roadmap for Semiconductors (ITRS), 2002, http://www.itrs.net/Links/2002Update/2002Update.pdf
Plumbridge, W. J., Matela, R. J., and Westwater, A., 2004, Structural Integrity and Reliability in Electronics, Kluwer, Dordecht, The Netherlands.
Rowlette, J. A., and Goodson, K. E., 2008, “Fully Coupled Nonequilibrium Electron-Phonon Transport in Nanometer-Scale Silicon FETs,” IEEE Trans. Electron Devices, 55(1), pp. 220–232. [CrossRef]
International Technology Roadmap for Semiconductors, 2009, http://www.itrs.net/Links/2009ITRS/Home2009.htm
Yu, J., Xiong, J., Cheng, B., and Liu, S., 2005, “Fabrication and Characterization of Ag–TiO2 Multiphase Nanocomposite Thin Films With Enhanced Photocatalytic Activity,” Appl. Catal., B, 60(3–4), pp. 211–221. [CrossRef]
Shahil, K. M. F., and Balandin, A., 2012, “Graphene-Multilayer Graphene Nanocomposites as Highly Efficient Thermal Interface Materials,” Nano Lett., 12(2), pp. 861–867. [CrossRef] [PubMed]
Li, D., Huxtable, S. T., Abramson, A. R., and Majumdar, A., 2005, “Thermal Transport in Nanostructured Solid-State Cooling Devices,” ASME J. Heat Transfer, 127(1), pp. 108–114. [CrossRef]
Zebarjadi, M., Joshi, G., Zhu, G., Yu, B., Minnich, A., Lan, Y., Wang, X., Dresselhaus, M., Ren, Z., and Chen, G., 2011, “Power Factor Enhancement by Modulation Doping in Bulk Nanocomposites,” Nano Lett., 11(6), pp. 2225–2230. [CrossRef] [PubMed]
Granstrom, M., Petritsch, K., Arias, A. C., Lux, A., Andersson, M. R., and Friend, R. H., 1998, “Laminated Fabrication of Polymeric Photovoltaic Diodes,” Nature, 395(6699), pp. 257–260. [CrossRef]
Goodey, A. P., Eichfeld, S. M., Lew, K.-K., Redwing, J. M., and Mallouk, T. E., 2007, “Silicon Nanowire Array Photoelectrochemical Cells,” J. Am. Chem. Soc., 129(41), pp. 12344–12345. [CrossRef] [PubMed]
Bidkar, R., Tung, R. C., Alexeenko, A., Sumali, H., and Raman, A., 2009, “Unified Theory of Gas Damping of Flexible Microcantilevers at Low Ambient Pressures,” Appl. Phys. Lett., 94(16), p. 163117. [CrossRef]
Mahapatro, A., Chee, J., and Peroulis, D., 2009, “Fully Electronic Method for Quantifying the Post-release Gapheight Uncertainty of Capacitive RF MEMS Switches,” International Microwave Symposium Digest, Boston, MA, June 7–12, pp. 1645–1648.
Singh, D., Guo, X., Alexeenko, A., Murthy, J. Y., and Fisher, T. S., 2009, “Modeling of Subcontinuum Thermal Transport Across Semiconductor-Gas Interfaces,” J. Appl. Phys., 106(2), p. 024314. [CrossRef]
Majumdar, A., 1993, “Microscale Heat Conduction in Dielectric Thin Films,” ASME J. Heat Transfer, 115(1), pp. 7–16. [CrossRef]
Kittel, C., 1996, Introduction to Solid State Physics, Wiley, New York.
Sun, L., and Murthy, J. Y., 2006, “Domain Size Effects in Molecular Dynamics Simulation of Phonon Transport in Silicon,” Appl. Phys. Lett., 89(17), p. 171919. [CrossRef]
McGaughey, A. J. H., and Kaviany, M., 2006, “Phonon Transport in Molecular Dynamics Simulations: Formulation and Thermal Conductivity Prediction,” Advances in Heat Transfer, Academic Press, New York, Vol. 39, pp. 169–255. [CrossRef]
Narumanchi, S. V. J., Murthy, J. Y., and Amon, C. H., 2005, “Boltzmann Transport Equation-Based Thermal Modeling Approaches for Hotspots in Microelectronics,” Heat Mass Transfer, 42(6), pp. 478–491. [CrossRef]
Bansal, A., Meterelliyoz, M., Singh, S., Murthy, J., and Roy, K., 2006, “Compact Thermal Models for Estimation of Temperature-Dependent Power/Performance in finFET Technology,” Asia and South Pacific Conference on Design Automation, Yokohama, Japan, Jan. 24–27, pp. 237–242.
Narumanchi, S. V. J., Murthy, J. Y., and Amon, C. H., 2004, “Submicron Heat Transport Model in Silicon Accounting for Phonon Dispersion and Polarization,” ASME J. Heat Transfer, 126(6), pp. 946–955. [CrossRef]
Murthy, J. Y., and Mathur, S. R., 2002, “Computation of Sub-Micron Thermal Transport Using an Unstructured Finite Volume Method,” ASME J. Heat Transfer, 124(6), pp. 1176–1181. [CrossRef]
Pascual-Gutiérrez, J., Murthy, J. Y., and Viskanta, R., 2009, “Thermal Conductivity and Phonon Transport Properties of Silicon Using Perturbation Theory and the Environment-Dependent Interatomic Potential,” J. Appl. Phys., 106(6), p. 063532. [CrossRef]
Henry, A. S., and Chen, G., 2008, “Spectral Phonon Transport Properties of Silicon Based on Molecular Dynamics Simulations and Lattice Dynamics,” J. Comput. Theor. Nanosci., 5(2), pp. 1–12. [CrossRef]
Narumanchi, S. V. J., Murthy, J. Y., and Amon, C. H., 2005, “Comparison of Different Phonon Transport Models for Predicting Heat Conduction in Silicon-on-Insulator Transistors,” ASME J. Heat Transfer, 127(7), pp. 713–723. [CrossRef]
Loy, J. M., Murthy, J. Y., and Singh, D., 2013, “A Fast Hybrid Fourier–Boltzmann Transport Equation Solver for Nongray Phonon Transport,” ASME J. Heat Transfer, 135(1), p. 011008. [CrossRef]
Mazumder, S., and Majumdar, A., 2001, “Monte Carlo Study of Phonon Transport in Solid Thin Films Including Dispersion and Polarization,” ASME J. Heat Transfer, 123(4), pp. 749–759. [CrossRef]
Péraud, J.-P. M., and Hadjiconstantinou, N. G., 2011, “Efficient Simulation of Multidimensional Phonon Transport Using Energy-Based Variance-Reduced Monte Carlo Formulations,” Phys. Rev. B, 84(20), p. 205331. [CrossRef]
Péraud, J.-P. M., and Hadjiconstantinou, N. G., 2012, “An Alternative Approach to Efficient Simulation of Micro/Nanoscale Phonon Transport,” Appl. Phys. Lett., 101(15), p. 153114. [CrossRef]
Raithby, G. D., and Chui, E. H., 1990, “A Finite-Volome Method for Predicting a Radiant Heat Transfer in Enclosures With Participating Media,” ASME J. Heat Transfer, 112(2), pp. 415–423. [CrossRef]
Chui, E. H., and Raithby, G. D., 1992, “Implicit Solution Scheme to Improve Convergence Rate of Radiative Transfer Problems,” Numer. Heat Transfer, Part B, 22(3), pp. 251–272. [CrossRef]
Fiveland, W. A., and Jessee, J., 1996, “Acceleration Schemes for the Discrete Ordinates Method,” J. Thermophys. Heat Transfer, 10(3), pp. 445–451. [CrossRef]
Hassanzadeh, P., Raithby, G. D., and Chui, E. H., 2008, “Efficient Calculation of Radiation Heat Transfer in Participating Media,” J. Thermophys. Heat Transfer, 22(2), pp. 129–139. [CrossRef]
Mathur, S. R., and Murthy, J. Y., 2009, “An Acceleration Technique for the Computation of Participating Radiative Heat Transfer,” IMECE, Lake Buena Vista, FL, Nov. 13–19, pp. 709–717.
Mazumder, S., 2005, “A New Numerical Procedure for Coupling Radiation in Participating Media With Other Modes of Heat Transfer,” ASME J. Heat Transfer, 127(9), pp. 1037–1045. [CrossRef]
Mathur, S. R., and Murthy, J. Y., 1999, “Coupled Ordinates Method for Multigrid Acceleration of Radiation Calculations,” J. Thermophys. Heat Transfer, 13(4), pp. 467–473. [CrossRef]
Holland, M. G., 1963, “Analysis of Lattice Thermal Conductivity,” Phys. Rev., 132(6), pp. 2461–2471. [CrossRef]
Chai, J. C., Lee, H. S., and Patankar, S. V., 1994, “Finite Volume Method for Radiation Heat Transfer,” J. Thermophy. Heat Transfer, 8(3), pp. 419–425. [CrossRef]
Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Taylor & Francis, New York.
Murthy, J. Y., and Mathur, S. R., 2003, “An Improved Computational Procedure for Sub-Micron Heat Conduction,” ASME J. Heat Transfer, 125(5), pp. 904–910. [CrossRef]
Brandt, A., and Livne, O. E., 2011, Multigrid Techniques: 1984 Guide With Applications to Fluid Dynamics, SIAM, Philadelphia, PA. [CrossRef]
Mathur, S. R., and Murthy, J. Y., 1997, “A Pressure-Based Method for Unstructured Meshes,” Numer. Heat Transfer, Part B, 31(2), pp. 195–215. [CrossRef]
Heaslet, M. A., and Warming, R. F., 1965, “Radiative Transport and Wall Temperature Slip in an Absorbing Planar Medium,” Int. J. Heat Mass Transfer, 8(7), pp. 979–994. [CrossRef]
Modest, M. F., 1993, Radiative Heat Transfer, McGraw-Hill, New York.
Gironcoli, S. De, 1992, “Phonons in Si-Ge Systems: An Ab Initio Interatomic-Force-Constant Approach,” Phys. Rev. B, 46(4), pp. 2412–2419. [CrossRef]
Bazant, M. Z., Kaxiras, E., and Justo, J. F., 1997, “Environment-Dependent Interatomic Potential for Bulk Silicon,” Phys. Rev. B, 56(14), pp. 8542–8552. [CrossRef]
Mingo, N., Yang, L., Li, D., and Majumdar, A., 2003, “Predicting the Thermal Conductivity of Si and Ge Nanowires,” Nano Lett., 3(12), pp. 1713–1716. [CrossRef]


Grahic Jump Location
Fig. 1

Discretized control volume in physical space

Grahic Jump Location
Fig. 2

Schematic of a control volume in wave vector space. The Brillouin zone shown here is for a face centered cubic lattice, adapted from Ref. [15].

Grahic Jump Location
Fig. 3

Flow chart for sequential solution procedure

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Fig. 4

Flow chart for one relaxation sweep for COMET

Grahic Jump Location
Fig. 5

Convergence rates for the angular and spatial discretization. On the abscissa, N refers to Nx for the spatial mesh, and Nθ for the angular mesh. The error is defined as the average deviation from Ref. [42].

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Fig. 6

Computational domain used for benchmarking COMET

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Fig. 7

Ratio of the iteration count for the sequential and COMET procedures. The tabulation below the graph shows the values.

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Fig. 8

Ratio of the total time taken for sequential and COMET procedures. The tabulation below the graph shows the values.

Grahic Jump Location
Fig. 9

Dispersion relation for silicon in the [100] direction at 300 K using EDIP [45]

Grahic Jump Location
Fig. 10

Discretization of the Brillouin zone for a nongray dispersion relation

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Fig. 11

Phonon mean free path as a function of polarization and frequency for bulk silicon

Grahic Jump Location
Fig. 12

Ratio of the iteration count for the sequential and COMET procedures. The spatial and angular discretizations used are Nx × Ny = 50 × 50 and Nθ × Nϕ = 2 × 2 in the octant, respectively.

Grahic Jump Location
Fig. 13

Ratio of the total time for the sequential and COMET procedures. The spatial and angular discretizations used are Nx × Ny = 50 × 50 and Nθ × Nϕ = 2 × 2 in the octant, respectively.



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