Research Papers: Micro/Nanoscale Heat Transfer

Relative Contributions of Inelastic and Elastic Diffuse Phonon Scattering to Thermal Boundary Conductance Across Solid Interfaces

[+] Author and Article Information
Patrick E. Hopkins1

Department of Mechanical and Aerospace Engineering, University of Virginia, P.O. Box 400746, Charlottesville, VA 22904-4746

Pamela M. Norris2

Department of Mechanical and Aerospace Engineering, University of Virginia, P.O. Box 400746, Charlottesville, VA 22904-4746pamela@virginia.edu


Present address: Engineering Sciences Center, Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185-0346.


Author to whom correspondence should be addressed.

J. Heat Transfer 131(2), 022402 (Jan 05, 2009) (9 pages) doi:10.1115/1.2995623 History: Received February 13, 2008; Revised August 06, 2008; Published January 05, 2009

The accuracy of predictions of phonon thermal boundary conductance using traditional models such as the diffuse mismatch model (DMM) varies depending on the types of material comprising the interface. The DMM assumes that phonons, undergoing diffuse scattering events, are elastically scattered, which drives the energy conductance across the interface. It has been shown that at relatively high temperatures (i.e., above the Debye temperature) previously ignored inelastic scattering events can contribute substantially to interfacial transport. In this case, the predictions from the DMM become highly inaccurate. In this paper, the effects of inelastic scattering on thermal boundary conductance at metal/dielectric interfaces are studied. Experimental transient thermoreflectance data showing inelastic trends are reviewed and compared to traditional models. Using the physical assumptions in the traditional models and experimental data, the relative contributions of inelastic and elastic scattering to thermal boundary conductance are inferred.

Copyright © 2009 by American Society of Mechanical Engineers
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Figure 1

DMM (solid line), PRL (dotted line), and JFDMM (dashed line) calculations compared to temperature dependent TTR data on (a) Pt∕AlN(17), (b) Pt∕Al2O3(17), (c) Au/diamond (14), (d) Bi∕H/diamond (16), and (e) Pb/diamond and Pb∕H/diamond (16). Note the similar temperature dependent trends of the DMM and PRL due to their assumption of elastic scattering.

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Figure 2

(Top) Experimental measurements of hBD for various interfaces (see Fig. 1) compared to their corresponding PRL. The PRL predicts a constant hBD at temperatures above the Debye temperature of the lower Debye temperature material. (Bottom) Inelastic phonon radiation limit for the four material systems in the top graph. The IPRL shows a linear increase over a temperature range of 100–400K, the same trend that is shown in the experimental data. Note, however, that the slope of the linear increase in the IPRL is much greater than the slope of the linear increase in the data.

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Figure 3

(Top) Experimental measurements of hBD for Pt∕Al2O3 and Pt∕AlN (see Fig. 1) compared to their corresponding PRL. (Bottom) Inelastic phonon radiation limit for the two material systems in the top graph. The increase in hBD over the temperature range of interest is greater in the IPRL than in the slope of the linear increase in the data, which is greater than the increase in the PRL.

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Figure 4

Regime map of thermal boundary conductance models that takes into account various degrees of elastic and inelastic diffuse phonon scattering. The DMM and PRL, which take into account varying degrees of elastic scattering, are paralleled with the JFDMM and IPRL, which take into account varying degrees of inelastic scattering.

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Figure 5

Measured hBD on Pb/diamond with best fit line extrapolated to determine the y-intercept at 3.26MWm−2K−1, which is assumed as the elastic contribution to hBD in the classical limit. This value is less than the prediction of the PRL, as expected, but agrees well with the predictions from the DMM, validating the transmission coefficient calculations in the DMM for a Pb/diamond interface.

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Figure 6

Comparison of α2inel(T) determined by 1−α1inel(T) where α1inel(T) is given by the model derived by Dames and Chen (9) and presented in Eq. 16; and B(T), which is derived in this work and presented in Eq. 17. B(T) decreases to a much lower constant value much more quickly than α2inel(T) calculated with Eq. 16. The assumption in Eq. 16 is that all phonons of all frequencies have equal probability in transmission, where Eq. 17 is based on experimental data and the IPRL. The smaller phonon transmission of side 2 phonons at higher temperatures could be due to the probability of side 2 phonons breaking down into lower frequency side 1 phonons decreasing as the side 2 phonon frequency increases.

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Figure 7

Inelastic and elastic scattering contributions to hBD of Pb/diamond in the classical limit. The total thermal boundary conductance, hBD(T), shows excellent agreement to the data at low temperatures in the classical limit, and approaches a constant value at higher temperatures that are above the Debye temperature of diamond (or at temperatures that are greater than the Debye temperature of both materials).

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Figure 8

Relative magnitude of inelastic scattering on hBD. This ratio compares the inelastic contribution to hBD to the elastic contribution as predicted via Eq. 14. The role of inelastic phonon scattering increases as the acoustic mismatch of the film and substrate becomes greater. The range in which hBD should increase linearly with temperature due to inelastic scattering also increases with acoustic mismatch.



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