Micro/Nanoscale Heat Transfer

Phonon Transport Modeling Using Boltzmann Transport Equation With Anisotropic Relaxation Times

[+] Author and Article Information
Chunjian Ni

School of Mechanical Engineering,  Purdue University, West Lafayette, IN 47907charlesni2006@gmail.com

Jayathi Y. Murthy

School of Mechanical Engineering,  Purdue University, West Lafayette, IN 47907jmurthy@ecn.purdue.edu

J. Heat Transfer 134(8), 082401 (Jun 05, 2012) (12 pages) doi:10.1115/1.4006169 History: Received September 29, 2011; Revised January 25, 2012; Published June 05, 2012; Online June 05, 2012

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.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Phonon dispersion in bulk silicon in high symmetry directions; the solid lines are obtained using the adiabatic bond charge model [23], while the solid circles are experimental data [24]

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Figure 2

(a) Control volume centered around centroid P. vg is the phonon group velocity. (b) Polar and azimuthal angle in the unit wave vector direction s. Control volume centered around centroid P. vg is the phonon group velocity.

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Figure 3

Schematic of computational domain for thermal conductivity calculation and 1D transient conduction calculation

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Figure 4

1D transient response of the anisotropic relaxation time BTE solution in the thick limit, compared with the exact Fourier conduction solution

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Figure 5

Illustration of phonon travel time computation at a location x in the computational domain

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Figure 6

1D transient response in the ballistic limit compared with the exact solution

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Figure 7

Bulk thermal conductivity of silicon. The solid circles are experimental data from Holland [32] and the solid line is the predicted bulk thermal conductivity of silicon.

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Figure 8

Scattering rates at 300 K for bands in the LA mode for two different directions (θ,φ) = (π/16, π/16) and (θ,φ) = (3π/16, 3π/16) in wave vector space

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Figure 9

Relaxation times for bands in the TA mode in the [1,0,0] direction at T = 300 K and T = 500 K

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Figure 10

In-plane thermal conductivity of silicon thin films with different thicknesses at 300 K. Experimental data are from Ju and Goodson [35].

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Figure 11

Schematic of 2D domain representing bulk silicon MOSFET

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Figure 12

Nonuniform spatial mesh used for the BTE solution

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Figure 13

Steady state lattice temperature (K) profile obtained by the anisotropic relaxation time BTE model

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Figure 14

Comparison of temperature rise along three vertical lines: x = 20 nm, x = 40 nm, and x = 120 nm, for full-scattering and anisotropic relaxation time BTE models

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Figure 15

Illustration of phonon bands and polarizations used for analysis. Dispersion curves for different phonon polarizations in all the discrete directions in the first Brillouin zone are shown. Because of the anisotropic nature of the lattice, each phonon branch, for example LA, has multiple values in different directions for a given K/Kzone .



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