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Research Papers: Micro/Nanoscale Heat Transfer

Phonon Heat Conduction in Multidimensional Heterostructures: Predictions Using the Boltzmann Transport Equation

[+] Author and Article Information
Syed Ashraf Ali

Department of Mechanical and
Aerospace Engineering,
The Ohio State University,
Columbus, OH 43210

Sandip Mazumder

Fellow ASME
Department of Mechanical and
Aerospace Engineering,
The Ohio State University,
Columbus, OH 43210
e-mail: mazumder.2@osu.edu

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 6, 2015; final manuscript received April 24, 2015; published online June 2, 2015. Assoc. Editor: Laurent Pilon.

J. Heat Transfer 137(10), 102401 (Oct 01, 2015) (11 pages) Paper No: HT-15-1012; doi: 10.1115/1.4030565 History: Received January 06, 2015; Revised April 24, 2015; Online June 02, 2015

In this article, two models for phonon transmission across semiconductor interfaces are investigated and demonstrated in the context of large-scale spatially three-dimensional calculations of the phonon Boltzmann transport equation (BTE). These include two modified forms of the classical diffuse mismatch model (DMM): one, in which dispersion is accounted for and another, in which energy transfer between longitudinal acoustic (LA) and transverse acoustic (TA) phonons is disallowed. As opposed to the vast majority of the previous studies in which the interface is treated in isolation, and the thermal boundary conductance is calculated using closed-form analytical formulations, the present study also considers the interplay between the interface and intrinsic (volumetric) scattering of phonons. This is accomplished by incorporating the interface models into a parallel solver for the full seven-dimensional BTE for phonons. A verification study is conducted in which the thermal boundary resistance of a silicon/germanium interface is compared against the previously reported results of molecular dynamics (MD) calculations. The BTE solutions overpredicted the interfacial resistance, and the reasons for this discrepancy are discussed. It is found that due to the interplay between intrinsic and interface scattering, the interfacial thermal resistance across a Si(hot)/Ge(cold) bilayer is different from that of a Si(cold)/Ge(hot) bilayer. Finally, the phonon BTE is solved for a nanoscale three-dimensional heterostructure, comprised of multiple blocks of silicon and germanium, and the time evolution of the temperature distribution is predicted and compared against predictions using the Fourier law of heat conduction.

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Figures

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Fig. 1

Schematic representation of a diffuse interface

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Fig. 2

The dispersion relationships for the silicon/germanium system: only acoustic polarization branches are shown

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Fig. 3

Spectral transmittance of acoustic phonons from silicon to germanium predicted using two different variations of the diffuse mismatch model: (a) DDMM and (b) IPT model. The predicted transmittance by both models is independent of temperature.

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Fig. 4

Unstructured stencil showing geometric connectivity and a discrete direction (line of sight) of propagation of phonons

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Fig. 5

Schematic representation of a face straddled by two cells with two different materials, the intensities on the two sides of the interface, and their integral representations

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Fig. 6

Schematic representation of the geometry and boundary conditions used for the verification study

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Fig. 7

Steady state temperature distributions predicted using solution of the BTE with two different interface models and two different configurations: (a) Si(hot)/Ge(cold) and (b) Ge(hot)/Si(cold)

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Fig. 8

Computed MFPs for acoustic phonons in silicon and germanium at two different temperatures: (a) 300 K and (b) 500 K. The computations were performed using the dispersion relationships shown in Fig. 2, and the scattering time-scale expressions given by Eq. (19).

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Fig. 9

Schematic representation (top and side views) of the geometry and boundary conditions used for the three-dimensional heterostructure. The light gray areas represent a silicon substrate. The dark gray and dotted regions represent germanium blocks embedded within the silicon substrate.

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Fig. 10

Predicted temporal evolution of temperature (K) in the three-dimensional heterostructure: (a) BTE solution and (b) Fourier law solution. The top views are at the midplane (z = 0.25 μm), and the size views are at the central plane (y = 5 μm).

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