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Research Papers: Heat and Mass Transfer

Simulation of Phonon Transport in Semiconductors Using a Population-Dependent Many-Body Cellular Monte Carlo Approach

[+] Author and Article Information
Flavio F. M. Sabatti

School of Electrical, Computer and
Energy Engineering,
Arizona State University,
Tempe, AZ 85287-5706
e-mail: fsabatti@asu.edu

Stephen M. Goodnick

School of Electrical, Computer and
Energy Engineering,
Arizona State University,
Tempe, AZ 85287-5706
e-mail: stephen.goodnick@asu.edu

Marco Saraniti

School of Electrical, Computer and
Energy Engineering,
Arizona State University,
Tempe, AZ 85287-5706
e-mail: marco.saraniti@asu.edu

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received June 6, 2016; final manuscript received October 18, 2016; published online December 28, 2016. Assoc. Editor: Alan McGaughey.

J. Heat Transfer 139(3), 032002 (Dec 28, 2016) (10 pages) Paper No: HT-16-1360; doi: 10.1115/1.4035042 History: Received June 06, 2016; Revised October 18, 2016

A Monte Carlo rejection technique for numerically solving the complete, nonlinear phonon Boltzmann transport equation (BTE) is presented in this work, including three particles interactions. The technique has been developed to explicitly model population-dependent scattering within a full-band cellular Monte Carlo (CMC) framework, to simulate phonon transport in semiconductors, while ensuring conservation of energy and momentum for each scattering event within gridding error. The scattering algorithm directly solves the many-body problem accounting for the instantaneous distribution of the phonons. Our general approach is capable of simulating any nonequilibrium phase space distribution of phonons using the full phonon dispersion without the need of approximations used in previous Monte Carlo simulations. In particular, no assumptions are made on the dominant modes responsible for anharmonic decay, while normal and umklapp scattering are treated on the same footing. In this work, we discuss details of the algorithmic implementation of both the three-particle scattering for the treatment of the anharmonic interactions between phonons, as well as treating isotope and impurity scattering within the same framework. The simulation code was validated by comparison with both analytical and experimental results; in particular, the simulation results show close agreement with a wide range of experimental data such as thermal conductivity as function of the isotopic composition, the temperature, and the thin-film thickness.

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Figures

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Fig. 1

The Si scattering rate produced by simulating a 300 K carrier distribution (symbols) shows good agreement with the scattering table computed for 300 K (lines). The tables have been computed using the following Grüneisen parameters: 1.194, 1.194, 1.3134, 1.194, 1.27758, and 1.194, from the lowest to the highest energetic mode, and sound velocity 6 km/s.

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Fig. 2

The CMC algorithm initialized with an out-of-equilibrium distribution (dashed line) reaches a steady-state solution (solid line) comparable to a 216 K equilibrium distribution (dots). The inset shows the time evolution of the Jensen–Shannon divergence [51] between the reference and the simulated distribution; the dot in the inset represents the divergence between the reference and the time-averaged final distribution.

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Fig. 3

The natural silicon Sinat (92.2% Si28, 4.6% Si29, and 3.1% Si30) is compared to the Si28 enriched sample. The Si28 thermal conductivity (solid line) is up to ten times higher than Sinat thermal conductivity (dashed line). The Monte Carlo simulation without the isotope scattering (solid symbols) follows the Si28 thermal conductivity. The Monte Carlo simulation with the isotope scattering (open symbols) reproduces the Sinat thermal conductivity.

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Fig. 4

Bulk Si mean free path computed at 300 K compared with previous Monte Carlo [17] and molecular dynamics [52] simulations

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Fig. 5

Comparison of the bulk Si relaxation time at 300 K for longitudinal acoustic phonons in this work with a first-principles study [53] and molecular dynamics [52] simulation. While this work shows close agreement to the molecular dynamics simulation, the comparison with the first-principles study shows an overestimation of the relaxation time for phonons close to the BZ1 boundaries.

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Fig. 6

Schematic of the simulated Si thermal resistor, TH = 310 K and TL = 290 K, with the total length equal to 2.8 μm

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Fig. 7

Comparison between the evolution of the temperature distribution obtained analytically by Eq. (27) (dashed lines) and the Monte Carlo simulation (solid lines). Each panel represents a snapshot of the temperature distribution at a specific time indicated on the left corner. The last panel shows a comparison with the steady-state result of Lacroix et al. [13].

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Fig. 8

Comparison between the CMC results (solid circles) and experimental measurement [5861] (open symbols) of the 300 K silicon thin-film thermal conductivity for different film thicknesses

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