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Research Papers: Micro/Nanoscale Heat Transfer

# Monte Carlo Simulation of Steady-State Microscale Phonon Heat Transport

[+] Author and Article Information
Jaona Randrianalisoa1

CETHIL UMR5008, CNRS, INSA-Lyon, Université Lyon 1, F-69621 Villeurbanne, France

Dominique Baillis

CETHIL UMR5008, CNRS, INSA-Lyon, Université Lyon 1, F-69621 Villeurbanne, France

1

Corresponding author.

J. Heat Transfer 130(7), 072404 (May 20, 2008) (13 pages) doi:10.1115/1.2897925 History: Received February 12, 2007; Revised June 04, 2007; Published May 20, 2008

## Abstract

Heat conduction in submicron crystalline materials can be well modeled by the Boltzmann transport equation (BTE). The Monte Carlo method is effective in computing the solution of the BTE. These past years, transient Monte Carlo simulations have been developed, but they are generally memory demanding. This paper presents an alternative Monte Carlo method for analyzing heat conduction in such materials. The numerical scheme is derived from past Monte Carlo algorithms for steady-state radiative heat transfer and enables us to understand well the steady-state nature of phonon transport. Moreover, this algorithm is not memory demanding and uses very few iteration to achieve convergence. It could be computationally more advantageous than transient Monte Carlo approaches in certain cases. Similar to the famous Mazumder and Majumdar’s transient algorithm (2001, “Monte Carlo Study of Phonon Transport in Solid Thin Films Including Dispersion and Polarization  ,” ASME J. Heat Transfer, 123, pp. 749–759), the dual polarizations of phonon propagation, the nonlinear dispersion relationships, the transition between the two polarization branches, and the nongray treatment of phonon relaxation times are accounted for. Scatterings by different mechanisms are treated individually, and the creation and/or destruction of phonons due to scattering is implicitly taken into account. The proposed method successfully predicts exact solutions of phonon transport across a gallium arsenide film in the ballistic regime and that across a silicon film in the diffusion regime. Its capability to model the phonon scattering by boundaries and impurities on the phonon transport has been verified. The current simulations agree well with the previous predictions and the measurement of thermal conductivity along silicon thin films and along silicon nanowires of widths greater than $22nm$. This study confirms that the dispersion curves and relaxation times of bulk silicon are not appropriate to model phonon propagation along silicon nanowires of $22nm$ width.

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## Figures

Figure 1

Comparison between the acoustic phonon dispersion curves measured at room temperature and in the wave vector direction [001] (22) and those obtained from the BZBC relations. TA1 and TA2 means that these polarization branches are overlapping.

Figure 2

Monte Carlo flowchart for steady-state microscale phonon transport

Figure 3

Evolution of the energies absorbed per unit volume at cells at the abscissas z∕Z=0, z∕Z=0.5, and z∕Z=1 as a function of the simulation step

Figure 4

Evolution of the normalized energies absorbed step by step at cells at the abscissas z∕Z=0, z∕Z=0.5, and z∕Z=1 as a function of the simulation step

Figure 5

Evolution of the absolute net energy fluxes per unit surface at the boundaries at z∕Z=0 and z∕Z=1 as a function of the simulation step

Figure 6

Temperature profiles across silicon films of thickness 60nm issued from the first three iterations

Figure 7

Temperature profiles across gallium arsenide films of thickness 400nm at very low temperature

Figure 8

Temperature profiles across silicon films of thickness 6μm submitted to a large temperature gradient

Figure 9

Influence of scattering by impurities on the heat conduction across silicon films of thickness 500nm at very low temperatures

Figure 10

Influence of scattering by boundaries on the heat conduction across silicon films of thickness 500nm at very low temperatures.

Figure 11

Thermal conductivity along silicon thin films (lines: prediction from Refs. 15-16; square symbols: experimental data from Refs. 15-16; circle symbols: Monte Carlo simulation)

Figure 12

Thermal conductivity along silicon nanowires (lines: Monte Carlo simulation from Ref. 12; square symbols: experimental data from Ref. 17; circle symbols: current Monte Carlo simulation)

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