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Research Papers

A Finite Difference Lattice Boltzmann Method to Simulate Multidimensional Subcontinuum Heat Conduction

[+] Author and Article Information
Cheng Chen

Department of Mechanical Engineering,
Binghamton University,
4400 Vestal Parkway East,
Binghamton, NY 13902
e-mail: cchen20@binghamton.edu

James Geer

Department of Mechanical Engineering,
Binghamton University,
4400 Vestal Parkway East,
Binghamton, NY 13902
e-mail: jgeer@binghamton.edu

Bahgat Sammakia

Department of Mechanical Engineering,
Binghamton University,
4400 Vestal Parkway East,
Binghamton, NY 13902
e-mail: bahgat@binghamton.edu

Contributed by the Electronic and Photonic Packaging Division of ASME for publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received April 14, 2016; final manuscript received September 27, 2016; published online October 20, 2016. Assoc. Editor: Ashish Gupta.

J. Electron. Packag 138(4), 041008 (Oct 20, 2016) (10 pages) Paper No: EP-16-1053; doi: 10.1115/1.4034856 History: Received April 14, 2016; Revised September 27, 2016

In this paper, a lattice Boltzmann method (LBM)-based model is developed to simulate the subcontinuum behavior of multidimensional heat conduction in solids. Based on a previous study (Chen et al., 2014, “Sub-Continuum Thermal Modeling Using Diffusion in the Lattice Boltzmann Transport Equation,” Int. J. Heat Mass Transfer, 79, pp. 666–675), phonon energy transport is separated to a ballistic part and a diffusive part, with phonon equilibrium assumed at boundaries. Steady-state temperature/total energy density solutions from continuum scales to ballistic scales are considered. A refined LBM-based numerical approach is applied to a two-dimensional simplified transistor model proposed by (Sinha et al. 2006, “Non-Equilibrium Phonon Distributions in Sub-100 nm Silicon Transistors,” ASME J. Heat Transfer, 128(7), pp. 638–647), and the results are compared with the Fourier-based heat conduction model. The three-dimensional (3D) LBM model is also developed and verified at both the ballistic and continuous limits. The impact of film thickness on the cross-plane and in-plane thermal conductivities is analyzed, and a new model of the supplementary diffusion term is proposed. Predictions based on the finalized model are compared with the existing in-plane thermal conductivity measurements and cross-plane thermal conductivity molecular dynamics (MD) results.

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Figures

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Fig. 1

Schematic of a 2D heat conduction model with temperature boundary conditions

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Fig. 2

Dimensionless total energy density distribution in silicon model with (a) Knx = Kny = 0.01, (b) Knx = Kny = 0.1, (c) Knx = Kny = 1, (d) Knx = Kny = 10, and (e) Knx = Kny = 100

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Fig. 3

Schematic of an x-directional heat conduction model; the top and bottom boundaries are adiabatic, while “hot” and “cold” temperatures are located as specified on the horizontal sides

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Fig. 4

Distribution of separate phonon energies and the total energy density in the heat conduction direction (x*-axis) in a square Si domain sizing: (a) 720 nm, (b) 20 nm, and (c) 3.1 nm, with Knx = Kny. Phonon energies are averaged over the y* coordinate.

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Fig. 5

Dimensions and boundary conditions of a 2D transistor model. Symmetry is taken at the center, and adiabatic conditions are defined at the horizontal sides. A hotspot is set to be semicircular at the origin, with power density of 5 W/μm3. Heat flows out via part of the top surface and the bottom boundary. Initial and the ambient temperatures are both set to be 300 K, as described by Sinha et al. [2,3].

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Fig. 6

Contour of the equivalent temperature rise for heat conduction in a bulk 90 nm geometry, solved for by (a) phonon transport LBM, Tmax = 343.7 K, and (b) Fourier's diffusion model, Tmax = 338.6 K

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Fig. 7

Dimensionless total energy density (equivalent temperature) distribution obtained by the LBM for Knx = Kny = Knz = 1 at cross plane z* = 0.5: (a) surface plot and (b) contour plot

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Fig. 8

Comparison of Si steady-state dimensionless total energy density (equivalent temperature) solved for by the LBM model with (a) gray LBM without diffusion, Knx = Kny = Knz = 100 and (b) Fourier's diffusion model Knx = Kny = Knz = 0.01, at arc y* = z* = 0.5

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Fig. 9

Dependence on the width Ly of the effective thermal conductivity in Si, solved by 3D LBM: (1) cross-plane thermal conductivity keff-y, (2) in-plane thermal conductivity keff-x, and (3) lateral thermal conductivity keff-x with Lz = Ly

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Fig. 10

Contour of (e1* − e2*), linearly related to x-ballistic heat flux, from 2D heat conduction model, Knx = 0.1: (a) Kny = 1 and (b) Kny = 0.01

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Fig. 11

Size-dependent ballistic part of the in-plane thermal conductivity with various Gc preset in the diffusion term. Fitting the function provides an optimum value Gc = 0.47Gi.

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Fig. 12

Comparison of LBM predictions for in-plane thermal conductivity with experimental works [1,5], and cross-plane thermal conductivity with MD simulation [10] for Si at 300 K. Note that the MD results are obtained for Si at 400 K and will be higher at 300 K.

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Fig. 13

Contours of the equivalent temperature rise for heat conduction in a bulk 90 nm geometry, solved by LBM with modified diffusion formula, Tmax = 358.6 K

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