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.

Phonon Transport Modeling Using Boltzmann Transport Equation With Anisotropic Relaxation Times

2012 ◽  
Vol 134 (8) ◽  
Author(s):  
Chunjian Ni ◽  
Jayathi Y. Murthy
Keyword(s):  
Relaxation Time ◽  
Transport Equation ◽  
Thermal Transport ◽  
Relaxation Times ◽  
Boltzmann Transport Equation ◽  
Phonon Transport ◽  
Oxide Semiconductor ◽  
Boltzmann Transport ◽  
Scattering Model ◽  
Anisotropic Relaxation

A sub-micron thermal transport model based on the phonon Boltzmann transport equation (BTE) is developed using anisotropic relaxation times. A previously-published model, the full-scattering model, developed by Wang, directly computes three-phonon scattering interactions by enforcing energy and momentum conservation. However, it is computationally very expensive because it requires the evaluation of millions of scattering interactions during the iterative numerical solution procedure. The anisotropic relaxation time model employs a single-mode relaxation time, but the relaxation time is derived from detailed consideration of three-phonon interactions satisfying conservation rules, and is a function of wave vector. The resulting model is significantly less expensive than the full-scattering model, but incorporates directional and dispersion behavior. A critical issue in the model development is the role of three-phonon normal (N) scattering processes. Following Callaway, the overall relaxation rate is modified to include the shift in the phonon distribution function due to N processes. The relaxation times so obtained are compared with the data extracted from equilibrium molecular dynamics simulations by Henry and Chen. The anisotropic relaxation time phonon BTE model is validated by comparing the predicted thermal conductivities of bulk silicon and silicon thin films with experimental measurements. The model is then used for simulating thermal transport in a silicon metal-oxide-semiconductor field effect transistor (MOSFET) and leads to results close to the full-scattering model, but uses much less computation time.

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 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.

Study of an Iterative Solution for Boltzmann Transport Equation and Calculation of Thermal Conductivity

2018 ◽  
Vol 777 ◽  
pp. 421-425 ◽  
Author(s):  
Chhengrot Sion ◽  
Chung Hao Hsu
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Numerical Calculation ◽  
Iterative Method ◽  
Transport Equation ◽  
Linear Systems ◽  
Heat Transport ◽  
Boltzmann Transport Equation ◽  
Iterative Solution ◽  
Boltzmann Transport

Many methods have been developed to predict the thermal conductivity of the material. Heat transport is complex and it contains many unknown variables, which makes the thermal conductivity hard to define. The iterative solution of Boltzmann transport equation (BTE) can make the numerical calculation and the nanoscale study of heat transfer possible. Here, we review how to apply the iterative method to solve BTE and many linear systems. This method can compute a sequence of progressively accurate iteration to approximate the solution of BTE.

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.

Heat Conduction in Nanostructured Materials Predicted by Phonon Bulk Mean Free Path Distribution

2015 ◽  
Vol 137 (7) ◽  
Author(s):  
Giuseppe Romano ◽  
Jeffrey C. Grossman
Keyword(s):  
Free Path ◽  
Nanostructured Materials ◽  
Thermal Transport ◽  
High Efficiency ◽  
Theoretical Models ◽  
Mean Free Path ◽  
Phonon Transport ◽  
Computational Effort ◽  
Boltzmann Transport ◽  
Computational Framework

We develop a computational framework, based on the Boltzmann transport equation (BTE), with the ability to compute thermal transport in nanostructured materials of any geometry using, as the only input, the bulk cumulative thermal conductivity. The main advantage of our method is twofold. First, while the scattering times and dispersion curves are unknown for most materials, the phonon mean free path (MFP) distribution can be directly obtained by experiments. As a consequence, a wider range of materials can be simulated than with the frequency-dependent (FD) approach. Second, when the MFP distribution is available from theoretical models, our approach allows one to include easily the material dispersion in the calculations without discretizing the phonon frequencies for all polarizations thereby reducing considerably computational effort. Furthermore, after deriving the ballistic and diffusive limits of our model, we develop a multiscale method that couples phonon transport across different scales, enabling efficient simulations of materials with wide phonon MFP distributions length. After validating our model against the FD approach, we apply the method to porous silicon membranes and find good agreement with experiments on mesoscale pores. By enabling the investigation of thermal transport in unexplored nanostructured materials, our method has the potential to advance high-efficiency thermoelectric devices.

Heat Pulse Propagation in Silicon Nanostructures by Solving Phonon Transport Equation

2008 ◽  
Author(s):  
Damian Terris ◽  
Karl Joulain ◽  
Denis Lemonnier
Keyword(s):  
Transport Equation ◽  
Pulse Propagation ◽  
Boltzmann Transport Equation ◽  
Phonon Transport ◽  
Isotropic Scattering ◽  
Silicon Nanostructures ◽  
Boltzmann Transport ◽  
Discrete Ordinate ◽  
Rectangular Mesh ◽  
Angle Space

The temperature evolution prediction of silicon nanofilms and nanowires can be useful to safeguard high technology systems of its deterioration. The simulation of a level and a pulse in these nanostructures is then made with Boltzmann Transport Equation (BTE) resolution using the single time approximation. The Discrete Ordinate (DO) method helps to numerate the angle space. BTE is written in cylindrical coordinates which corresponds to wires. Therefore, the cylindrical plane is considered as an isotropic scattering to mimic a nanowire and then, as a specular reflexion (which conserve z momentum) to simulate a nanofilm. Using the axisymmetry done with a specular reflexion, the cylinder is two dimensionally discretized with a regular rectangular mesh.

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