scholarly journals Non-Gray Phonon Transport Using a Hybrid BTE-Fourier Solver

2009 ◽  
Author(s):  
James M. Loy ◽  
Dhruv Singh ◽  
Jayathi Y. Murthy
Keyword(s):  
Lattice Energy ◽  
Phonon Transport ◽  
Lattice Temperature ◽  
Boltzmann Transport ◽  
Knudsen Numbers ◽  
Large Spread ◽  
Sequential Solution ◽  
Fourier Model ◽  
Typical Solution ◽  
Fourier Equation

Non-gray phonon transport solvers based on the Boltzmann transport equation (BTE) are frequently employed to simulate sub-micron thermal transport. Typical solution procedures using 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 the directional BTEs results in slow convergence for sequential solution procedures. In this paper, we present a hybrid BTE-Fourier model which addresses this issue. By establishing a phonon group cutoff (say Kn = 0.1), 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 a BTE solver. Once the governing equations are solved for each phonon group, their energies are then summed to find the total lattice energy and correspondingly, the lattice temperature. An iterative procedure combining the lattice temperature determination and the solutions to the modified Fourier and BTE equations is developed. The procedure is shown to work well across a range of Knudsen numbers.

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

2012 ◽  
Vol 135 (1) ◽  
Author(s):  
James M. Loy ◽  
Jayathi Y. Murthy ◽  
Dhruv Singh
Keyword(s):  
Transport Equation ◽  
Thermal Transport ◽  
Boltzmann Transport Equation ◽  
Iterative Solution ◽  
Phonon Transport ◽  
Computational Effort ◽  
Boltzmann Transport ◽  
Knudsen Numbers ◽  
Sequential Solution ◽  
Hybrid Solver

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.

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

2015 ◽  
Vol 137 (1) ◽  
Author(s):  
James M. Loy ◽  
Sanjay R. Mathur ◽  
Jayathi Y. Murthy
Keyword(s):  
Transport Equation ◽  
Convergence Rates ◽  
Boltzmann Transport Equation ◽  
Physical Space ◽  
Solution Procedure ◽  
Computational Time ◽  
Boltzmann Transport ◽  
Knudsen Numbers ◽  
Numerical Solution Methods ◽  
Sequential Solution

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.

A Fast Coupled Solver for Phonon Transport in Composites

2013 ◽  
Author(s):  
James M. Loy ◽  
Ajay Vadakkepatt ◽  
Sanjay R. Mathur ◽  
Jayathi Y. Murthy
Keyword(s):  
Relaxation Times ◽  
Computational Procedure ◽  
Phonon Transport ◽  
Superior Performance ◽  
Wave Number ◽  
Computational Techniques ◽  
Boltzmann Transport ◽  
Spatial Coupling ◽  
Sequential Solution ◽  
Solution Techniques

In recent years, computational techniques for solving phonon transport have been developed under the framework of the semiclassical Boltzmann Transport Equation (BTE). Early work addressed gray transport, but more recent work has begun to resolve wave vector and polarization dependence, including that in relaxation times. Because the relaxation time in typical materials of interest spans several orders of magnitude, typical solution techniques must address an enormous range of Knudsen numbers in the same problem. Calculation procedures which solve the BTE in phase space sequentially work well in the ballistic limit, but are slow to converge in the thick limit. Unfortunately, both extremes may be encountered simultaneously in typical wave-number (K) -resolved phonon transport problems. In previous work, we developed the coupled ordinate method (COMET) to address this problem. COMET employs a point-coupled solution to resolve coupling in K-space, and embeds this point solver as a relaxation sweep in a geometric multigrid method to maintain spatial coupling. We have demonstrated speedups of up to 200 over conventional sequential solution procedures using this method. COMET also exhibits excellent scaling on multiprocessor platforms, far beyond those obtained by sequential solvers. In this paper, we extend COMET to address interface transport in composites. Just as scattering couples phonons of different wave vectors in the bulk, reflection and transmission couple different wave vectors together at interfaces. Again, sequential solution procedures perform poorly because of the poor algorithmic coupling in K space. A computational procedure based on COMET is developed for composites, addressing multigrid agglomeration strategies to promote stronger K-space coupling at interfaces. The technique is applied to canonical superlattice geometries and superior performance over typical sequential solvers is demonstrated. Furthermore, the method is applied to realistic particle composites employing computational meshes developed from x-ray computed tomography (CT) scans of particulate beds. It is demonstrated to yield solutions where sequential solution techniques fail to converge at all.

Thermal transport in beta-gallium oxide thin-film using non-gray Boltzmann transport equation

2021 ◽  
Author(s):  
Nitish Kumar ◽  
Matthew Barry ◽  
Satish Kumar
Keyword(s):  
Thin Films ◽  
Temperature Distribution ◽  
Energy Dissipation ◽  
Thermal Transport ◽  
Energy Exchange ◽  
Domain Size ◽  
Phonon Transport ◽  
Boltzmann Transport ◽  
Phonon Modes ◽  
Fourier Model

Abstract Phonon transport  in β-Ga2O3 thin films and metal–oxide field effect transistors (MESFETs) are investigated using non-gray Boltzmann transport equations (BTE) to decipher the effect of  ballistic-diffusive phonon transport. The effects of domain size, and  energy dissipation to various phonon modes and subsequent phonon-phonon energy exchange on the thermal transport and temperature distribution is investigated using non-gray BTE. Our analysis deciphered that domain size plays a major role in thermal transport in β-Ga2O3 but energy dissipation to various phonon modes and subsequent phonon-phonon energy exchange does not affect the temperature field significantly.   Phonon transport in β-Ga2O3 MESFETs on diamond substrate is investigated using coupled non-gray BTE and Fourier model. It is established that the ballistic effects need to be considered for devices with β-Ga2O3 layer thickness less than 1 µm. A non-gray phonon BTE model should be used near hotspot in the thin β-Ga2O3 layer as the Fourier model may not give accurate temperature distribution. The results from this work will help in understanding the mechanism of phonon transport in the β-Ga2O3 thin films and energy efficient design of its FETs.

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

2012 ◽  
Author(s):  
James M. Loy ◽  
Sanjay R. Mathur ◽  
Jayathi Y. Murthy
Keyword(s):  
Transport Equation ◽  
Convergence Rates ◽  
Control Volume ◽  
Boltzmann Transport Equation ◽  
Physical Space ◽  
Convergence Acceleration ◽  
Solution Procedure ◽  
Boltzmann Transport ◽  
Knudsen Numbers ◽  
Sequential Solution

Sequential 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 3–4 orders of magnitude at room temperature, diffuse-limit solutions are not feasible for all K vectors. Consequently, non-gray solutions of the BTE almost always experience extremely slow convergence. In this paper, we develop a coupled-ordinates method for solving the phonon BTE in the relaxation time approximation. Here, inter-equation 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. The solution procedure is benchmarked against a traditional sequential solution procedure for thermal transport in silicon. Significant acceleration, between 10 to 300 times, over the sequential procedure is found for heat conduction problems.

Solving Nongray Boltzmann Transport Equation in Gallium Nitride

2017 ◽  
Vol 139 (10) ◽  
Author(s):  
Ajit K. Vallabhaneni ◽  
Liang Chen ◽  
Man P. Gupta ◽  
Satish Kumar
Keyword(s):  
Gallium Nitride ◽  
Transport Equation ◽  
Thermal Transport ◽  
Hot Spot ◽  
Boltzmann Transport Equation ◽  
Significant Loss ◽  
Computational Time ◽  
Boltzmann Transport ◽  
Phonon Modes ◽  
Fourier Model

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.

A High Thermoelectric Performance ZT in p-Type LaSe2 Crystal

2021 ◽  
Vol 871 ◽  
pp. 203-207
Author(s):  
Jian Liu
Keyword(s):  
Thermal Conductivity ◽  
Power Factor ◽  
High Power ◽  
Density Functional ◽  
Lattice Thermal Conductivity ◽  
Phonon Transport ◽  
Thermoelectric Performance ◽  
Boltzmann Transport ◽  
High Power Factor ◽  
P Type

In this work, we use first principles DFT calculations, anharmonic phonon scatter theory and Boltzmann transport method, to predict a comprehensive study on the thermoelectric properties as electronic and phonon transport of layered LaSe2 crystal. The flat-and-dispersive type band structure of LaSe2 crystal offers a high power factor. In the other hand, low lattice thermal conductivity is revealed in LaSe2 semiconductor, combined with its high power factor, the LaSe2 crystal is considered a promising thermoelectric material. It is demonstrated that p-type LaSe2 could be optimized to exhibit outstanding thermoelectric performance with a maximum ZT value of 1.41 at 1100K. Explored by density functional theory calculations, the high ZT value is due to its high Seebeck coefficient S, high electrical conductivity, and low lattice thermal conductivity .

FREQUENCY DEPENDENT PHONON TRANSPORT IN TWO-DIMENSIONAL SILICON AND DIAMOND THIN FILMS

2012 ◽  
Vol 26 (17) ◽  
pp. 1250104 ◽  
Author(s):  
B. S. YILBAS ◽  
S. BIN MANSOOR
Keyword(s):  
Thin Films ◽  
Boltzmann Transport Equation ◽  
Diamond Films ◽  
Equilibrium Temperature ◽  
Phonon Transport ◽  
Two Dimensional ◽  
Boltzmann Transport ◽  
Frequency Dependent ◽  
Aluminum Films ◽  
The Difference

Phonon transport in two-dimensional silicon and aluminum films is investigated. The frequency dependent solution of Boltzmann transport equation is obtained numerically to account for the acoustic and optical phonon branches. The influence of film size on equivalent equilibrium temperature distribution in silicon and aluminum films is presented. It is found that increasing film width influences phonon transport in the film; in which case, the difference between the equivalent equilibrium temperature due to silicon and diamond films becomes smaller for wider films than that of the thinner films.

Modeling of Localized Heating Effect in Sub-Micron Silicon Transistors

2003 ◽  
Author(s):  
Keivan Etessam-Yazdani ◽  
Sadegh M. Sadeghipour ◽  
Mehdi Asheghi
Keyword(s):  
Transport Equation ◽  
Hot Spot ◽  
Boltzmann Transport Equation ◽  
Phonon Transport ◽  
Heat Diffusion ◽  
Normal Operation ◽  
Heating Effect ◽  
Boltzmann Transport ◽  
Heat Diffusion Equation ◽  
Localized Heating

The performance and reliability of sub-micron semiconductor transistors demands accurate modeling of electron and phonon transport at nanoscales. The continued downscaling of the critical dimensions, introduces hotspots, inside transistors, with dimensions much smaller than phonon mean free path. This phenomenon, known as localized heating effect, results in a relatively high temperature at the hotspot that cannot be predicted using heat diffusion equation. While the contribution of the localized heating effect to the total device thermal resistance is significant during the normal operation of transistors, it has even greater implications for the thermoelectrical behavior of the device during an electrostatic discharge (ESD) event. The Boltzmann transport equation (BTE) can be used to capture the ballistic phonon transport in the vicinity of a hot spot but many of the existing solutions are limited to the one-dimensional and simple geometry configurations. We report our initial progress in solving the two dimensional Boltzmann transport equation for a hot spot in an infinite media (silicon) with constant temperature boundary condition and uniform heat generation configuration.

Statistical Phonon Transport Model of Thermal Transport in Silicon

2009 ◽  
Vol 1229 ◽  
Author(s):  
Thomas W Brown ◽  
Edward Hensel
Keyword(s):  
Thermal Transport ◽  
Lattice Dynamics ◽  
Transport Model ◽  
Silicon Nanowire ◽  
Boltzmann Transport Equation ◽  
Momentum Conservation ◽  
Phonon Transport ◽  
Boltzmann Transport ◽  
Crystalline Materials ◽  
Statistical Framework

AbstractThermal transport in crystalline materials at various length scales can be modeled by the Boltzmann transport equation (BTE). A statistical phonon transport (SPT) model is presented that solves the BTE in a statistical framework that incorporates a unique state-based phonon transport methodology. Anisotropy of the first Brillouin zone (BZ) is captured by utilizing directionally-dependent dispersion curves obtained from lattice dynamics calculations. A rigorous implementation of phonon energy and pseudo-momentum conservation is implemented in the ballistic thermal transport regime for a homogeneous silicon nanowire with adiabatic specular boundary conditions.

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