Research Papers: Micro/Nanoscale Heat Transfer

Anharmonic Phonon Interactions at Interfaces and Contributions to Thermal Boundary Conductance

[+] Author and Article Information
Patrick E. Hopkins1

Engineering Sciences Center, Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185-0346; Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904-4746pehopki@sandia.gov

John C. Duda

Engineering Sciences Center, Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185-0346; Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904-4746

Pamela M. Norris

Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904-4746


Corresponding author.

J. Heat Transfer 133(6), 062401 (Mar 09, 2011) (11 pages) doi:10.1115/1.4003549 History: Received June 16, 2010; Revised January 12, 2011; Published March 09, 2011; Online March 09, 2011

Continued reduction in characteristic dimensions in nanosystems has given rise to increasing importance of material interfaces on the overall system performance. With regard to thermal transport, this increases the need for a better fundamental understanding of the processes affecting interfacial thermal transport, as characterized by the thermal boundary conductance. When thermal boundary conductance is driven by phononic scattering events, accurate predictions of interfacial transport must account for anharmonic phononic coupling as this affects the thermal transmission. In this paper, a new model for phononic thermal boundary conductance is developed that takes into account anharmonic coupling, or inelastic scattering events, at the interface between two materials. Previous models for thermal boundary conductance are first reviewed, including the diffuse mismatch model, which only considers elastic phonon scattering events, and earlier attempts to account for inelastic phonon scattering, namely, the maximum transmission model and the higher harmonic inelastic model. A new model is derived, the anharmonic inelastic model, which provides a more physical consideration of the effects of inelastic scattering on thermal boundary conductance. This is accomplished by considering specific ranges of phonon frequency interactions and phonon number density conservation. Thus, this model considers the contributions of anharmonic, inelastically scattered phonons to thermal boundary conductance. This new anharmonic inelastic model shows improved agreement between the thermal boundary conductance predictions and experimental data at the Pb/diamond and Au/diamond interfaces due to its ability to account for the temperature dependent changing phonon population in diamond, which can couple anharmonically with multiple phonons in Pb and Au. We conclude by discussing phonon scattering selection rules at interfaces and the probability of occurrence of these higher order anharmonic interfacial phonon processes quantified in this work.

Copyright © 2011 by American Society of Mechanical EngineersThe United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States Government purposes.
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Figure 1

Predicted thermal flux in Pb using the Sine-type and Debye dispersion models normalized by the predicted thermal flux using the polynomial dispersion relation from phonon dispersion data (qreal). The Sine-type dispersion gives a much more realistic prediction of the thermal flux than the Debye dispersion. The use of the Debye dispersion model in these calculations would result in a nonphysical overprediction of the actual thermal flux in Pb, and therefore an overprediction of hK at the Pb/diamond interface.

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Figure 2

DMM, MTM, and HHIM calculations for hK at a Pb/diamond interface using the Sine-type dispersion along with experimental data of measured hK at the Pb/diamond interface and hK at the interface between Pb and hydrogen-terminated diamond (Pb/H/diamond) (8). The DMM not only underpredicts the experimental data but also does not capture the increasing trend in hK with temperature since it only accounts for two-phonon elastic scattering. The MTM greatly overpredicts the experimental data since all phonons of all frequencies in diamond are assumed to participate in hK, where the underprediction of the HHIM is due to only accounting for one type of inelastic scattering event (i.e., the higher harmonic process). Inset: each higher harmonic contribution to the total hK predicted by the HHIM.

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Figure 3

AIM n-phonon transmission coefficient calculations for various values of n for the Pb/diamond interface as a function of temperature calculated with Eq. 35. In general, the transmission probability of higher order processes increases with increasing temperature. As temperature is increased, the number of modes in diamond at any frequency increases, and therefore the transmission coefficient increases. Eventually, for the higher n processes at high temperatures, all of the Pb phonons become exhausted and the transmission coefficient become unity.

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Figure 4

AIM predictions of hK(n) as a function of temperature (Eq. 34) for Pb/diamond for n=2–17. Note that the change in temperature trends of the n-phonon transmission coefficient propagate through the hK(n) calculations as seen in the changing temperature trends beginning with the n=12 calculations. Also, note that the contributions from the n=12 and higher processes are over an order of magnitude smaller than the contributions from the n<12 processes, especially at higher temperatures. This is due to the fact that, in addition to much of the Pb phonon population being exhausted, energetic selection rules allow only high frequency, low group velocity modes in Pb to participate in these higher n processes, which decreases the incident flux and subsequent thermal boundary conductance.

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Figure 5

Total thermal boundary conductance for Pb/diamond and Au/diamond predicted from the AIM calculations using Eq. 23 compared with DMM predictions along with the Pb/diamond (8) and Au/diamond (24) experimental data. The AIM shows improved agreement compared with the DMM with the experimental data, and represents a significant improvement in modeling inelastic contributions to hK compared with the MTM and HHIM (cf. Fig. 2 for Pb/diamond comparisons).



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