A Generalized Enhanced Fourier Law

2016 ◽  
Vol 139 (3) ◽  
Author(s):  
Ashok T. Ramu ◽  
John E. Bowers
Keyword(s):  
Heat Flux ◽  
Transport Equation ◽  
Phonon Mode ◽  
Boltzmann Transport Equation ◽  
Phonon Transport ◽  
Boltzmann Transport ◽  
Fourier Law ◽  
Transport Effects

A generalized enhanced Fourier law (EFL) that accounts for quasi-ballistic phonon transport effects in a formulation entirely in terms of physical observables is derived from the Boltzmann transport equation. It generalizes the previously reported EFL from a gray phonon population to an arbitrary quasi-ballistic phonon mode population, the chief advantage being its formulation in terms of observables like the heat flux and temperature, in a manner akin to the Fourier law albeit rigorous enough to describe quasi-ballistic phonon transport.

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.

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.

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.

The first-principles and BTE investigation of phonon transport in 1T-TiSe2

2021 ◽  
Author(s):  
Zhao-Liang Wang ◽  
Guofu Chen ◽  
Xiaoliang Zhang ◽  
Dawei Tang
Keyword(s):  
Density Functional Theory ◽  
Transport Equation ◽  
First Principles ◽  
Density Functional ◽  
Boltzmann Transport Equation ◽  
Phonon Transport ◽  
Boltzmann Transport ◽  
Functional Theory

Through the first-principles density functional theory and the phonon Boltzmann transport equation, we investigated the phonon transport characteristics inside 1T-TiSe2.

Validation of a Unified Nondiffusive-Diffusive Phonon Transport Model for Nanoscale Heat Transfer Simulations

2014 ◽  
Author(s):  
Ashok T. Ramu ◽  
Yanbao Ma
Keyword(s):  
Heat Transfer ◽  
Spatial Dependence ◽  
Hot Spots ◽  
Transport Model ◽  
Boltzmann Transport Equation ◽  
Phonon Transport ◽  
High Fidelity ◽  
Boltzmann Transport ◽  
Fourier Law ◽  
Limiting Case

Heat transfer in the vicinity of nanoscale hot-spots is qualitatively different from that in the macroscale, which effect stems from the breakdown of Fourier law due to phonon nondiffusive transport. In this work, we validate a recently proposed alternative, high-fidelity phonon transport model, the unified nondiffusive-diffusive (UND) model, which takes into account the mixed ballistic-diffusive nature of heat transport, as well as reduces to the Fourier law as a limiting case. In the UND model, the nondiffusive phonons are treated using the Boltzmann transport equation, while the diffusive phonon gas is treated by the Fourier law. The numerical results of Maznev et al. for the geometry and spatial dependence of variables corresponding to the transient gratings experiments of Johnson et al. are used for validation of the model.

Improved Phonon Transport Modeling Using Boltzmann Transport Equation With Anisotropic Relaxation Times

2009 ◽  
Author(s):  
Chunjian Ni ◽  
Jayathi Y. Murthy
Keyword(s):  
Relaxation Time ◽  
Transport Equation ◽  
Relaxation Times ◽  
Boltzmann Transport Equation ◽  
Phonon Transport ◽  
Dynamics Simulation ◽  
Boltzmann Transport ◽  
Scattering Model ◽  
Anisotropic Relaxation ◽  
Thermal Conductivities

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 phonon BTE model employs a single-mode relaxation time idea, but the relaxation time is a function of wave-vector. The resulting model is significantly less expensive than the full-scattering model, but incorporates directional and dispersion behavior as well as relaxation times satisfying conservation rules. A critical issue in the model development is the accounting for the role of three-phonon N scattering processes. Direct inclusion of N processes into the anisotropic relaxation time model is not possible because such an inclusion would engender thermal resistance. 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 simulation by Henry and Chen. The anisotropic relaxation time phonon BTE model is validated by comparing the predicted bulk thermal conductivities of silicon and silicon thin-film thermal conductivities with experimental measurements.

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.

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