An enhanced Fourier law derivable from the Boltzmann transport equation and a sample application in determining the mean-free path of nondiffusive phonon modes

2014 ◽  
Vol 116 (9) ◽  
pp. 093501 ◽  
Author(s):  
Ashok T. Ramu ◽  
Yanbao Ma
Keyword(s):  
Transport Equation ◽  
Free Path ◽  
Boltzmann Transport Equation ◽  
Mean Free Path ◽  
Boltzmann Transport ◽  
Fourier Law ◽  
Phonon Modes ◽  
Sample Application ◽  
The Mean

scholarly journals Variational approach to extracting the phonon mean free path distribution from the spectral Boltzmann transport equation

2016 ◽  
Vol 93 (15) ◽  
Author(s):  
Vazrik Chiloyan ◽  
Lingping Zeng ◽  
Samuel Huberman ◽  
Alexei A. Maznev ◽  
Keith A. Nelson ◽  
...  
Keyword(s):  
Transport Equation ◽  
Free Path ◽  
Variational Approach ◽  
Boltzmann Transport Equation ◽  
Mean Free Path ◽  
Boltzmann Transport

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.

Interplay Between Thermoelectric and Thermionic Effects in Heterostructures

2000 ◽  
Author(s):  
Taofang Zeng ◽  
Gang Chen
Keyword(s):  
Film Thickness ◽  
Free Path ◽  
Thermionic Emission ◽  
Approximate Solutions ◽  
Mean Free Path ◽  
Boltzmann Transport ◽  
Thermoelectric Effects ◽  
Double Heterojunction ◽  
The Mean ◽  
Electron Mean Free Path

Abstract When electrons sweep through a double-heterojunction structure, there exist thermionic effects at the junctions and thermoelectric effects in the film. While both thermoelectric and thermionic effects have been studied for refrigeration and power generation applications separately, their interplay in heterostructures is not understood. This paper establishes a unified model including both thermionic and thermoelectric processes based on the Boltzmann transport equation for electrons, and the nonequilibrium interaction between electrons and phonons. Approximate solutions are obtained, leading to the electron temperature and Fermi level distributions inside heterostructures and discontinuities at the interfaces as a consequence of the highly nonequilibrium transport when the film thickness is much smaller than the electron mean free path. It is found that when the film thickness is smaller than the mean free path of electrons, the transport of electrons is controlled by thermionic emission. The coexistence of thermoelectric and thermionic effects may increase the power factor when the electron mean free path is comparable to the film thickness.

scholarly journals An Efficient Solution Scheme for the Spherical-Harmonics Expansion of the Boltzmann Transport Equation Applied to Two-Dimensional Devices

VLSI Design ◽  
1998 ◽  
Vol 6 (1-4) ◽  
pp. 239-242
Author(s):  
Maria Cristina Vecchi ◽  
Jan Mohring ◽  
Massimo Rudan
Keyword(s):  
Transport Equation ◽  
Spherical Harmonics ◽  
Numerical Scheme ◽  
Boltzmann Transport Equation ◽  
Boltzmann Transport ◽  
Cpu Time ◽  
Low Energies ◽  
Energy Domain ◽  
The Mean ◽  
Spherical Harmonics Expansion

This paper presents a novel numerical scheme applicable to the solution of the Boltzmann transport equation by means of a spherical-harmonics expansion. This scheme improves the solution at low energies, keeping the desired accuracy in the calculation of the mean quantities while saving a significant amount of CPU time. This is important in view of the applications of the method, since the typical number of nodes to be used in the combined space-energy domain is in the range of 104–105.

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 Phonon Thermal Conductivity of Two Dimensional Nano and Micro Composites in the Longitudinal Direction

2005 ◽  
Author(s):  
Ravi Prasher
Keyword(s):  
Thermal Conductivity ◽  
Free Path ◽  
Longitudinal Direction ◽  
Mean Free Path ◽  
Two Dimensional ◽  
Boltzmann Transport ◽  
Host Medium ◽  
Alumina Composite ◽  
The Mean ◽  
Longitudinal Thermal Conductivity

The three important length scales in composites made from nano/micro wires and fibers are: 1) the ratio of inter fiber distance and mean free path of the phonons in the host medium 2) the ratio of the diameter of the fiber or the wire and the mean free path of the phonons in the host medium and 3) the ratio of the diameter of the fiber and the mean free path of phonons in the fiber. Modeling of longitudinal thermal conductivity of two-dimensional nano and micro composites has not been attempted in the literature. This paper develops analytical modeling for the longitudinal thermal conductivity of nano and micro composites by solving the Boltzmann transport equation (BTE) for phonons. The paper shows the scattering of phonons in the host medium by the fiber boundaries play a very important role in deciding the thermal conductivity of nano and micro composites. The model is in good agreement with data on thermal diffusivity of Bismuth Telluride nanowire/ Alumina composite.

Modeling Nanoscale Thermal Transport via the Boltzmann Transport Equation

2004 ◽  
Author(s):  
Cristina H. Amon ◽  
Jayathi Y. Murthy ◽  
Sreekant V. J. Narumanchi
Keyword(s):  
Relaxation Time ◽  
Transport Equation ◽  
Group Velocity ◽  
Phonon Dispersion ◽  
Dispersion Model ◽  
Relaxation Times ◽  
Boltzmann Transport Equation ◽  
Boltzmann Transport ◽  
Phonon Modes ◽  
Solution Methods

In modern microelectronics, where extreme miniaturization has led to feature sizes in the sub-micron and nanoscale range, Fourier diffusion has been found to be inadequate for the prediction of heat conduction. Over the past decade, the phonon Boltzmann transport equation (BTE) in the relaxation time approximation has been employed to make thermal predictions in dielectrics and semiconductors at micron and nanoscales. This paper presents a review of the BTE-based solution methods widely employed in the literature. Particular attention is given to the problem of self-heating (hotspot) in sub-micron transistors. First, the solution approaches based on the gray formulation of the BTE are presented. In this class of solution methods, phonons are characterized by one single group velocity and relaxation time. Phonon dispersion is not accounted for in any detail. This is the most widely employed approach in the literature. The semi-gray BTE approach, moments of the Boltzmann equation, the lattice Boltzmann approach, and the ballistic-diffusive approximation are presented. Models which incorporate greater details of phonon dispersion are also discussed. This includes a full phonon dispersion model developed recently by the authors. This full phonon dispersion model satisfies energy conservation, incorporates the different phonon modes, and well as the interactions between the different modes, and accounts for the frequency dependence for both the phonon group velocity and relaxation times. Results which illustrate the differences between some of these models reveal the importance of developing models that incorporate substantial details of phonon physics.

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