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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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. 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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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. 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. 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. 10

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

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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. 16

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

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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. 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. 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. 18

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



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