Research Papers

A Fast Hybrid Fourier–Boltzmann Transport Equation Solver for Nongray Phonon Transport

[+] Author and Article Information
Jayathi Y. Murthy

e-mail: jmurthy@me.utexas.edu
School of Mechanical Engineering,
The University of Texas at Austin,
Austin, TX 78712-0292

Dhruv Singh

Intel Corporation,
Hillsboro, OR 97124-5506

Manuscript received March 12, 2012; final manuscript received June 20, 2012; published online December 6, 2012. Assoc. Editor: Gerard F. Jones.

J. Heat Transfer 135(1), 011008 (Dec 06, 2012) (12 pages) Paper No: HT-12-1092; doi: 10.1115/1.4007654 History: Received March 12, 2012; Revised June 20, 2012

Nongray phonon transport solvers based on the Boltzmann transport equation (BTE) are being increasingly employed to simulate submicron thermal transport in semiconductors and dielectrics. Typical sequential solution schemes encounter numerical difficulties because of the large spread in scattering rates. For frequency bands with very low Knudsen numbers, strong coupling between other BTE bands result in slow convergence of sequential solution procedures. This is due to the explicit treatment of the scattering kernel. In this paper, we present a hybrid BTE-Fourier model which addresses this issue. By establishing a phonon group cutoff Knc, phonon bands with low Knudsen numbers are solved using a modified Fourier equation which includes a scattering term as well as corrections to account for boundary temperature slip. Phonon bands with high Knudsen numbers are solved using the BTE. A low-memory iterative solution procedure employing a block-coupled solution of the modified Fourier equations and a sequential solution of BTEs is developed. The hybrid solver is shown to produce solutions well within 1% of an all-BTE solver (using Knc = 0.1), but with far less computational effort. Speedup factors between 2 and 200 are obtained for a range of steady-state heat transfer problems. The hybrid solver enables efficient and accurate simulation of thermal transport in semiconductors and dielectrics across the range of length scales from submicron to the macroscale.

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International Technology Roadmap for Semiconductors (ITRS), 2002.
Kittle, C., 1996, Introduction to Solid State Physics, John Wiley, New York.
McGaughey, A. J. H., and Kaviany, M., 2006, “Phonon Transport in Molecular Dynamics Simulations: Formulation and Thermal Conductivity Prediction,” Adv. Heat Transfer, 39, pp. 169–255. [CrossRef]
Sun, L., and Murthy, J. Y., 2006, “Domain Size Effects in Molecular Dynamics Simulation of Phonon Transport in Silicon,” Appl. Phys. Lett., 89, p. 171919. [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, pp. 141–152.
Zhang, W., Fisher, T. S., and Mingo, Z., 2007, “The Atomistic Green's Function Method: An Efficient Simulation Approach for Nanoscale Phonon Transport,” Numer. Heat Transfer, Part B, 51, pp. 333–349. [CrossRef]
Datta, S., 2005, Quantum Transport: Atom to Transistor, Cambridge University Press, Cambridge, UK.
Murthy, J. Y., and Mathur, S. R., 2003, “An Improved Computational Procedure for Sub-Micron Heat Conduction,” ASME J. Heat Transfer, 125, pp. 904–910. [CrossRef]
Murthy, J. Y., Narumanchi, S. V. J., Pascual-Gutierrez, J., Wang, T., Ni, C., and Mathur, S. R., 2005, “Review of Multiscale Simulation in Submicron Heat Transfer,” Int. J. Multiscale Comp. Eng., 3, pp. 5–32. [CrossRef]
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, pp. 946–955. [CrossRef]
Ni, C., 2009, “Phonon Transport Models for Heat Conduction With Application to Microelectronics,” Ph.D. thesis, Purdue University, West Lafayette, IN.
Singh, D., Murthy, J. Y., and Fisher, T. S., 2008, “Thermal Transport in Finite-Sized Nanocomposites,” Proceedings of the ASME Heat Transfer Summer Conference, Jacksonville, FL, Paper No. HT2008-56385. [CrossRef]
Sverdrup, P. G., 2000, “Simulation and Thermometry of Sub-Continuum Heat Transfer in Semiconductor Devices,” Ph.D. thesis, Stanford University, Palo Alto, CA.
Ju, Y. S., and Goodson, K. E., 1999, Mircroscale Heat Conduction in Integrated Circuits and Their Constituent Films, Kluwer Academic Publishers, Norwell, MA.
Jacoboni, C., and Lugli, P., 1989, The Monte Carlo Method for Semiconductor Device Simulation, Springer Verlag, Vienna.
Case, K. M., 1960, “Elementary Solutions of the Transport Equation and Their Applications,” Ann. Phys., 9, pp. 1–23. [CrossRef]
Chui, E. H., and Raithby, G. D., 1992, “Implicit Solution Scheme to Improve Convergence Rate in Radiative Transfer Problems,” Numer. Heat Transfer., 22, pp. 251–272. [CrossRef]
Fiveland, V. A., and Jessee, J. P., 1996, “Acceleration Schemes for the Discrete Ordinates Method,” J. Thermophys. Heat Transfer., 10, pp. 445–451. [CrossRef]
Mathur, S. R., and Murthy, J. Y., 1999, “Coupled Ordinates Method for Multigrid Acceleration,” J. Thermophys. Heat Transfer., 13, pp. 467–473. [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, pp. 129–139. [CrossRef]
Mathur, S. R., and Murthy, J. Y., 2009, “An Acceleration Technique for the Computation of Participating Radiative Heat Transfer,” Proceedings of IMECE, Lake Buena Vista, FL, pp. 709–717. [CrossRef]
Wang, T. J., 2007, “Sub-Micron Thermal Transport in Ultra-Scaled Metal Oxide Semiconductor (MOS) Devices,” Ph.D. thesis, Purdue University, West Lafayette, IN.
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, p. 063532. [CrossRef]
Mazumder, S., 2005, “A New Numerical Procedure for Coupling Radiation in Participating Media With Other Modes of Heat Transfer,” ASME J. Heat Transfer., 127, pp. 1037–1045. [CrossRef]
Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Taylor & Francis, London.
Briggs, W. L., Henson, V. E., and McCormick, S. F., 2000, A Multigrid Tutorial, SIAM, Philadelphia, PA.
Hutchinson, B. R., and Raithby, G. D., 1986, “A Multigrid Method Based on the Additive Correction Strategy,” Numer. Heat Transfer, 9, pp. 511–537. [CrossRef]
Heaslet, M. A., and Warming, R. F., 1964, “Radiative Transport and Wall Temperature Slip in an Absorbing Planar Medium,” Int. J. Heat Mass Transfer, 8, pp. 979–994. [CrossRef]
Modest, M., 2003, Radiative Heat Transfer, Academic Press, New York.
Mingo, N., Yang, L., Li, D., and Majumdar, A., 2003, “Predicting the Thermal Conductivity of Si and Ge Nanowires,” Nano Lett., 3, pp. 1713–1716. [CrossRef]
Loy, J., 2010, “An Acceleration Technique for the Solution of the Phonon Boltzmann Transport Equation,” M.S. thesis, Purdue University, West Lafayette, IN.


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Fig. 1

Dispersion curve for silicon in the [100] direction as a function of dimensionless wavevector using environment dependent interatomic potential (EDIP) [23]. The dashed lines represent possible discretizations of the frequency space.

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Fig. 2

Thermalizing boundary at temperature T1 with an outward-pointing normal n

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Fig. 3

Flow chart for a typical sequential solution procedure

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Fig. 4

Solution loop for partially implicit solution procedure

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Fig. 5

Comparison of dimensionless temperature profiles of different Knudsen numbers (marked in parenthesis) obtained using a single-band BTE with those of Heaslet and Warming [28]

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Fig. 7

Percent error with respect to all-BTE solution. The MFE band Knudsen number is fixed at 0.1, while varying Knudsen number of the BTE band.

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Fig. 8

Percent error with respect to all-BTE solution. The BTE band is fixed at a Knudsen number of 2, while varying the Knudsen number of the MFE band.

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Fig. 9

Two-dimensional slab with T1 = 300 K, T2 = 310 K

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Fig. 10

Iteration results of the all-BTE solver compared to the hybrid solver

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Fig. 11

Timing results of the all-BTE solver compared to the hybrid solver. The bars represent the all-BTE solution time divided by the hybrid solution time.

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Fig. 12

Mean free paths of silicon at 300 K. The dispersion is taken from Ref. [23] while the scattering rates are taken from Ref. [30].

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Fig. 13

Lattice temperature of all-BTE and hybrid solvers along the diagonal from (x,y) = (0,0) to (x,y) = (L,L). L = 100 nm.

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Fig. 14

Longitudinal branch temperatures along the diagonal from (x,y) = (0,0) to (x,y) = (L,L) obtained from BTE and hybrid solver. L = 100 nm.

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Fig. 15

Branch-wise fractional heat flux along the left wall obtained from the all-BTE (solid lines) and hybrid solvers (dotted lines). L = 100 nm.

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Fig. 16

Dimensionless total x-direction heat flux along the left boundary obtained from the all-BTE and hybrid solvers

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Fig. 17

Band-wise temperatures from (x,y) = (0,0) to (x,y) = (L,L) obtained from the hybrid solver at L = 3000 nm

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Fig. 18

Dimensionless total x-direction heat flux on the left wall obtained from the hybrid solver. L = 3000 nm.

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Fig. 6

(a) Percentage difference in predicted heat flux using MFE. Triangles indicate difference with respect to [28] when using jump boundary conditions. (b) Percentage difference in predicted heat flux using MFE as compared to Fourier's law.




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