Parallel Computation of the Phonon Boltzmann Transport Equation

Ni, Chunjian; Murthy, Jayathi Y.

doi:10.1080/10407780902864771

Cited by 17 publications

(3 citation statements)

References 6 publications

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“…The wide spread in phonon relaxation times (typically over three orders of magnitude) and the large number of equations ($10 8 for GaN HEMT) that need to be solved simultaneously makes BTE computationally expensive. Researchers have suggested different techniques to parallelize [15,16] and to accelerate the convergence mostly in the context of radiation transport equation (RTE) [9,[17][18][19][20][21] which is similar to BTE. For brevity, we discuss here only some of the studies relevant to the present work.…”

Section: Introductionmentioning

confidence: 99%

Solving Nongray Boltzmann Transport Equation in Gallium Nitride

Vallabhaneni

Chen

Gupta

et al. 2017

Journal of Heat Transfer

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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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Section: Introductionmentioning

confidence: 99%

Solving Nongray Boltzmann Transport Equation in Gallium Nitride

Vallabhaneni

Chen

Gupta

et al. 2017

Journal of Heat Transfer

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“…Such methods were brought to the limelight by Murthy and co-workers [39][40][41][42]. To bring the state-of-the-art in deterministic solution of the phonon BTE into perspective, until 2010, the best example of multi-dimensional computation of the frequency-dependent BTE was the one by Ni and Murthy [43] who performed calculations on a 2D rectangular domain with 80x80 cells, 64 angles, and 80 spectral bands, resulting in ~3x10 7 unknowns. In the past five years, however, significant progress has been made in the solution of the phonon BTE using deterministic techniques [44][45][46][47].…”

Section: Introductionmentioning

confidence: 99%

Phonon Boltzmann Transport Equation based modeling of time domain thermo-reflectance experiments

Ali

Mazumder

2017

International Journal of Heat and Mass Transfer

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Time Domain Thermo-Reflectance (TDTR) experiments have been recently identified as a viable pathway toward extracting the phonon mean free path spectrum of semiconductor materials. However, this requires an intervening model. It is now widely believed that the frequency and polarization dependent phonon Boltzmann Transport Equation (BTE) is the most suitable model for this purpose. In this article, TDTR experiments are simulated using large-scale parallel computations of the phonon BTE in a two-dimensional computational domain. Silicon is used as the candidate substrate material. Simulations are performed for multiple pulse and modulation cycles of the TDTR pump laser. This requires resolution of a picosecond laser pulse within a computational timeframe that spans several hundreds of nanoseconds. The metallic transducer layer on top of the substrate is modeled using the Fourier law and coupled to the BTE within the silicon substrate. Studies are conducted for four different laser spot sizes and two different modulation frequencies. The BTE results are fitted to the Fourier law, and effective thermal conductivities are extracted. It is demonstrated that the time delay of the probe laser could have a significant effect on the fitted (extracted) thermal conductivity value. The modulation frequency is found to have negligible effect on the thermal conductivity, while the spot size variation exhibits significant impact. Both trends are found to be in agreement with experimental observations. The thermal conductivity accumulation function is also computed, and the effect of the mean free path spectrum on the thermal conductivity suppression is delineated.

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“…shown that approximately 20 angles in each of the polar and azimuthal directions (resulting in a total of 400 solid angles) are necessary to obtain angular grid independent solutions for realistic three-dimensional geometries, particularly when the Knudsen number is large and transport is ballistic in nature. Other studies [19] have employed 64 angles for computations in two-dimensional (2D) geometries. It is the need for angular discretization-typically, at least several tens of solid angles-that renders deterministic solution of the BTE computationally challenging.…”

Section: Introductionmentioning

confidence: 99%

Hybrid ballistic–diffusive solution to the frequency-dependent phonon Boltzmann Transport Equation

Allu

Mazumder

2016

International Journal of Heat and Mass Transfer

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The phonon Boltzmann Transport Equation (BTE) is appropriate for modeling heat conduction in semiconductor materials at the nanoscale. However, the BTE is difficult to solve on account of the directional and spectral nature of the phonon intensity, which necessitates angular and spectral discretization, and ultimately results in a large number (typically few hundreds) of four-dimensional partial differential equations. In the ballistic (large Knudsen number) regime, the phonon intensity is highly anisotropic, and therefore, angular resolution is desirable. However, in the diffusive (small Knudsen number) regime, the intensity is fairly isotropic, and hence, angular discretization is wasteful. In such scenarios, the method of spherical harmonics (P N approximation) may be effectively used to reduce the large number of directional BTEs to a few partial differential equations. Since the Knudsen number is frequency dependent, the decision to preserve or eliminate angular discretization may be made frequency by frequency based on whether the spectral Knudsen number is large or small. In this article, a hybrid method is proposed in which for some frequency intervals (bands), full angular discretization is used, while for others, the P 1 approximation is invoked to reduce the number of directional BTEs. The accuracy and efficiency of the hybrid method is tested by solving several steady state and transient nanoscale heat conduction problems in two and threedimensional geometries. Silicon is used as the candidate material. It is found that hybridization is effective in significantly improving the efficiency of solution of the BTE-sometimes by a factor of three-without significant penalty on the accuracy.

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Parallel Computation of the Phonon Boltzmann Transport Equation

Cited by 17 publications

References 6 publications

Solving Nongray Boltzmann Transport Equation in Gallium Nitride

Solving Nongray Boltzmann Transport Equation in Gallium Nitride

Phonon Boltzmann Transport Equation based modeling of time domain thermo-reflectance experiments

Hybrid ballistic–diffusive solution to the frequency-dependent phonon Boltzmann Transport Equation

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