Lattice Boltzmann Modeling of Subcontinuum Energy Transport in Crystalline and Amorphous Microelectronic Devices

[+] Author and Article Information
Rodrigo Escobar

Departamento de Ingeniería Mecánica y Metalúrgica,  Pontificia Universidad Católica de Chile, Santiago, Chile

Brian Smith

Mechanical Engineering Department and Institute for Complex Engineered Systems,  Carnegie Mellon University, Pittsburgh, PA 15213

Cristina Amon

Mechanical Engineering Department and Institute for Complex Engineered Systems,  Carnegie Mellon University, Pittsburgh, PA 15213camon@cmu.edu

J. Electron. Packag 128(2), 115-124 (Jan 19, 2006) (10 pages) doi:10.1115/1.2188951 History: Received February 21, 2005; Revised January 19, 2006

Numerical simulations of time-dependent energy transport in semiconductor thin films are performed using the lattice Boltzmann method applied to phonon transport. The discrete lattice Boltzmann method is derived from the continuous Boltzmann transport equation assuming first gray dispersion and then nonlinear, frequency-dependent phonon dispersion for acoustic and optical phonons. Results indicate that a transition from diffusive to ballistic energy transport is found as the characteristic length of the system becomes comparable to the phonon mean free path. The methodology is used in representative microelectronics applications covering both crystalline and amorphous materials including silicon thin films and nanoporous silica dielectrics. Size-dependent thermal conductivity values are also computed based on steady-state temperature distributions obtained from the numerical models. For each case, reducing feature size into the subcontinuum regime decreases the thermal conductivity when compared to bulk values. Overall, simulations that consider phonon dispersion yield results more consistent with experimental correlations.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

Thin-film geometry describing one-dimensional energy transfer in a silicon film

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Figure 2

Dimensionless temperature distribution along a one-dimensional slab of length L for different Knudsen numbers

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Figure 3

Transient temperature evolution at Kn=1

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Figure 4

Transient temperature evolution inside a 30nm thin film

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Figure 5

Dimensionless temperature distribution along a one-dimensional slab of length L over different film thickness

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Figure 6

Qualitative depiction of structure (morphology) of aerogel at different length scales. Dark areas represent solid silica; white is air.

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Figure 7

Relationship between average characteristic aerogel pore size and solid length scales as a function of porosity p(22)

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Figure 8

Silica branch geometries modeled using LBM: (a) straight branch, (b) L shape, (c) U shape. Note that all thermal lengths (from point 1 to point 2) are equal. Shading corresponds to temperature distribution (Fourier solution) for unit heat flux at surface 1.

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Figure 9

LBM results for heat flow through the centerline of each silica branch geometry: (a) strip geometry, (b) L geometry, and (c) U geometry

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Figure 10

Lattice density study: U geometry at Kn=0.1

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Figure 11

Effect of combining the “primitive” shapes to form a larger structure. Top: two strip branches result in the same end-to-end temperature rise (ΔT) as one double-length strip model. Bottom: two L shapes have same temperature rise as composite model.

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Figure 12

Steady-state temperature profile in transitional and ballistic regimes

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Figure 13

Conductivity factor as a function of Knudsen number

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Figure 14

Size-dependent out-of-plane thermal conductivity values as function of Knudsen number, from diffusive (Kn=0.001) to ballistic (Kn=10) transport regimes



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