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Research Papers: Radiative Heat Transfer

Solving Nongray Boltzmann Transport Equation in Gallium Nitride

[+] Author and Article Information
Ajit K. Vallabhaneni

G. W. Woodruff School of
Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: ajitkvallabhaneni@gmail.com

Liang Chen, Man P. Gupta, Satish Kumar

G. W. Woodruff School of
Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received December 16, 2015; final manuscript received April 24, 2017; published online June 6, 2017. Assoc. Editor: Alan McGaughey.

J. Heat Transfer 139(10), 102701 (Jun 06, 2017) (8 pages) Paper No: HT-15-1796; doi: 10.1115/1.4036616 History: Received December 16, 2015; Revised April 24, 2017

Several studies have validated that diffusive Fourier model is inadequate to model thermal transport at submicron length scales. Hence, Boltzmann transport equation (BTE) is being utilized to improve thermal predictions in electronic devices, where ballistic effects dominate. In this work, we investigated the steady-state thermal transport in a gallium nitride (GaN) film using the BTE. The phonon properties of GaN for BTE simulations are calculated from first principles—density functional theory (DFT). Despite parallelization, solving the BTE is quite expensive and requires significant computational resources. Here, we propose two methods to accelerate the process of solving the BTE without significant loss of accuracy in temperature prediction. The first one is to use the Fourier model away from the hot-spot in the device where ballistic effects can be neglected and then couple it with a BTE model for the region close to hot-spot. The second method is to accelerate the BTE model itself by using an adaptive model which is faster to solve as BTE for phonon modes with low Knudsen number is replaced with a Fourier like equation. Both these methods involve choosing a cutoff parameter based on the phonon mean free path (mfp). For a GaN-based device considered in the present work, the first method decreases the computational time by about 70%, whereas the adaptive method reduces it by 60% compared to the case where full BTE is solved across the entire domain. Using both the methods together reduces the overall computational time by more than 85%. The methods proposed here are general and can be used for any material. These approaches are quite valuable for multiscale thermal modeling in solving device level problems at a faster pace without a significant loss of accuracy.

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Figures

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Fig. 1

(Left) block diagram of a GaN device with silicon substrate and AlGaN dielectric layer. (Right) schematic of the GaN buffer of size 50 μm × 2 μm divided into BTE and Fourier domain; the thick red line indicates the channel region where the energy is added. The exchange of temperature and heat flux information at the common interface is demonstrated in the bottom panel (see color figure online).

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Fig. 2

Flowchart showing the algorithm used for the adaptive BTE model

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Fig. 3

(a) Phonon dispersion of GaN from DFT calculations. The red dots shows the comparison with experiments from Ref. [32], and the blue lines show DFT results; (b) phonon relaxation time (τ) in picoseconds as a function of frequency at room temperature; (c) thermal conductivity of GaN as a function of temperature; (d) phonon mean free path of GaN as a function of frequency at room temperature (see color figure online).

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Fig. 4

Lattice temperature variation in the domain along the x-direction at the middle of the buffer (dotted line in the inset) aty = 1.9 × 10−6 μm. The Fourier model significantly under-predicts the hot-spot temperature.

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Fig. 5

Lattice temperature variation in the domain along the y-direction at the middle of the buffer (x = 1.0 × 10−6 μm). Inset shows the schematic of the domain with the GaN–Si interface along with boundary conditions. The thick red box indicates the region over which energy is added. The right-hand side panel shows the nonlinear temperature variation near the source on GaN side (see color figure online).

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Fig. 6

Error in heat flux with respect to full BTE as a function of Kn cutoff

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Fig. 7

(a) Spatial distribution of lattice temperature across the GaN Buffer layer using full BTE. Temperature distribution in the region highlighted in red from, (b) full BTE, (c) adaptive BTE, and (d) Fourier (the legend for temperature is same for the three cases (b), (c), and (d)) (see color figure online).

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Fig. 8

Average time per iteration in seconds and error in maximum temperature as a function of Knudsen number cutoff for adaptive model (BTE is considered in only middle region of the domain, i.e., Fr + Adap. BTE + Fr model).

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Fig. 9

Average time per iteration in seconds for the four methods considered

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