Thermal boundary conductance accumulation and interfacial phonon transmission: Measurements and theory

Ramez Cheaito, John T. Gaskins, Matthew E. Caplan, Brian F. Donovan, Brian M. Foley, Ashutosh Giri, John C. Duda, Chester J. Szwejkowski, Costel Constantin, Harlan J. Brown-Shaklee, Jon F. Ihlefeld, and Patrick E. Hopkins

Phys. Rev. B 91, 035432 – Published 22 January 2015

Abstract

The advances in phonon spectroscopy in homogeneous solids have unveiled extremely useful physics regarding the contribution of phonon energies and mean-free paths to the thermal transport in solids. However, as material systems decrease to length scales less than the phonon mean-free paths, thermal transport can become much more impacted by scattering and transmission across interfaces between two materials than the intrinsic relaxation in the homogeneous solid. To elucidate the fundamental interactions driving this thermally limiting interfacial phonon scattering process, we analytically derive and experimentally measure a thermal boundary conductance accumulation function. We develop a semiclassical theory to calculate the thermal boundary conductance accumulation function across interfaces using the diffuse mismatch model, and validate this derivation by measuring the interface conductance between eight different metals on native oxide/silicon substrates and four different metals on sapphire substrates. Measurements were performed at room temperature using time-domain thermoreflectance and represent the first-reported values for interface conductance across several metal/native oxide/silicon and metal/sapphire interfaces. The various metal films provide a variable bandwidth of phonons incident on the metal/substrate interface. This method of varying phonons' cutoff frequency in the film while keeping the same substrate allows us to mimic the accumulation of thermal boundary conductance and thus provides a direct method to experimentally validate our theory. We show that the accumulation function can be written as the product of a weighted average of the interfacial phonon transmission function and the accumulation of the temperature derivative of the phonon flux incident on the interface; this provides the framework to extract an average, spectrally dependent phonon transmissivity from a series of thermal boundary conductance measurements. Our approach provides a platform for analyzing the spectral phononic contribution to interfacial thermal transport in our experimentally measured data of metal/substrate thermal boundary conductance. Based on the assumptions made in this work and the measurement results on different metals on native oxide/silicon and sapphire substrates, we demonstrate that high-frequency phonons dictate the transport across metal/Si interfaces, especially in low Debye temperature metals with low-cutoff frequencies.

Received 4 August 2014
Revised 27 December 2014

DOI:https://doi.org/10.1103/PhysRevB.91.035432

Authors & Affiliations

Ramez Cheaito^1,*, John T. Gaskins^1,†, Matthew E. Caplan¹, Brian F. Donovan¹, Brian M. Foley¹, Ashutosh Giri¹, John C. Duda^1,†, Chester J. Szwejkowski¹, Costel Constantin², Harlan J. Brown-Shaklee³, Jon F. Ihlefeld³, and Patrick E. Hopkins^1,‡

¹Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, Virginia 22904, USA
²Department of Physics and Astronomy, James Madison University, Harrisonburg, Virginia 22807, USA
³Electronic, Optical, and Nanomaterials Department, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA

^*These authors contributed equally to this work.
^†Current address: Seagate Technology, Bloomington, MN 55435, USA.
^‡phopkins@virginia.edu

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Vol. 91, Iss. 3 — 15 January 2015

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Figure 1
(a) Thermal boundary conductance accumulation function $α_{K} (ω_{α})$ , (b) un-normalized thermal boundary conductance accumulation function $h_{K} (ω_{α})$ , (c) accumulation of the temperature derivative of the phonon flux in the metal $α_{q T} (ω_{α})$ , (d) transmission coefficient $ζ (ω)$ , (e) spectral thermal boundary conductance $h_{K} (ω)$ , and (f) temperature derivative of phonon flux in the metal $\partial q_{1} / \partial T (ω)$ , for Al/Si (solid line) and Au/Si (dashed line) interfaces as a function of phonon frequency calculated using Eqs. (5, 6, 7, 8) at room temperature. The horizontal and vertical lines in (a) and (c) designate the portion of the spectrum contributing to 50% of the plotted quantity. The calculations suggest that the majority of heat is carried by high-frequency phonons in Au but is more evenly spread across the spectrum in Al. The features in curves are related to the Van Hove singularities [71] in the various phonon spectra and our assumptions in the DMM calculations. Discontinuities in the slopes of the various calculations occur at the frequencies corresponding to the Brillouin zone edge of either the metal or silicon. Note that for Al/Si accumulation in (a) and (b), there is a very slight second discontinuity in the trend of $α_{K} (ω_{α})$ and $h_{K} (ω_{α})$ at the aluminum TA cutoff frequency of 36.4 Trad $s^{- 1}$ , which can be observed more clearly in the $h_{K} (ω)$ in (e). The various modeling calculations shown in these plots and the matlab code used to generate these accumulation models are given in the Supplemental Material [72].
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Figure 2
(a) The dispersion curves used in the calculation of $α_{K, 1 \to 2}$ in Eqs. (6) and (8) for Al/Si and plotted in Fig 1. The shaded area represents the $k - ω$ space over which the integration is carried when $ω_{α} = 40$ Trad $s^{- 1}$ . Points $A^{'}$ and $B^{'}$ on the shaded region will coincide with $A$ and $B$ on the solid rectangle when $ω_{α}$ reaches the maximum frequency in Al where both points will move along the longitudinal acoustic branch (LA) in aluminum and silicon, respectively. (b) The dispersion curves used in the calculation of $h_{K} [υ_{1} (ω_{\max})]$ given by Eq. (10) for metal/Si and plotted in Fig. 4. The shaded area represents the $k - ω$ space over which the integration is carried when $ω_{\max} = 20$ Trad $s^{- 1}$ and $υ_{1} (ω_{\max})$ is given by the dispersion curves contained within the shaded region. Points $A^{'}$ and $B^{'}$ will coincide with $A$ and $B$ and the dispersion curves in the shaded region will coincide with the dispersion curves in the solid rectangle when $ω_{\max}$ = 70 Trad $s^{- 1}$ with point $B^{'}$ moving along the LA branch in silicon and point $A^{'}$ moving vertically along the left side of the solid rectangle. The blue arrows denote the movement direction of the vertices of the shaded region as the angular frequency increases.
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Figure 3
(a) Thermal conductivity of the silicon substrates coated with various metal films measured with TDTR. The dotted lines represent the range of accepted values for thermal conductivity of bulk silicon [67, 68, 94, 95, 96, 97]. (b) Thermal boundary conductance across the metal/native oxide/Si interfaces as a function of metal Debye temperature. The data reported in this work are shown as filled symbols while previously reported results are depicted as open symbols (Bi and Pb: Ref. [86]; Au, Pt, Al, and Cr: Ref. [62]; Al/(001) and Al/(111): Ref. [83]; Al: Ref. [84]). The thermal boundary conductance data are tabulated in the Supplemental Material [72].
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Figure 4
(a) Measured thermal boundary conductance as a function of metal cutoff frequency (phonon frequencies are taken from phonon dispersion curves in the literature) [75, 76, 102, 103, 104, 105, 106, 107, 108, 109, 110]. Our model for thermal boundary conductance as a function of $ω_{\max}$ agrees well with the data discussed in this work. S1 and S2 denote the two singularities in the calculation of $h_{K} [υ_{1} (ω_{\max})]$ . The inset shows the normalized accumulation of the temperature derivative of the phonon flux. The horizontal and vertical lines in the inset show that 50% of the computed value of $α_{q T}$ is dictated by the first 45% of the phonon spectrum. The nearly linear change in $α_{q T}$ with $ω_{\max}$ implies that the nonlinear features in the measurements and the model can be attributed to the phonon transmissivity across the metal/Si interfaces. (b) Three plots of the thermal boundary conductance accumulation function for metal/Si, $α_{K} [υ_{1} (ω_{\max})]$ , obtained by normalizing the solid line in (a) at 15, 39, and 74 Trad $s^{- 1}$ . Each of these plots is labeled by the percentage of the phonon spectrum contributing to 50% of the interfacial transport. The dashed line represents $β_{0.5}$ plotted as a function of the normalization frequency $ω_{\max, N}$ . The inset in (b) shows $β_{0.5}$ for which the TA branch in Si was multiplied by a factor of 2 and the corresponding $α_{K} [υ_{1} (ω_{\max})]$ . We provide the matlab code used to generate this accumulation model in the Supplemental Material [72].
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Figure 5
Measured thermal boundary conductance across various metal/native oxide/Si and metal/sapphire interfaces as a function of metal cutoff frequency. The Cr/Si data point (unfilled square) was taken from Ref. [62] while all other data points (filled symbols) are reported in this work.
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Figure 6
Average phonon transmission across various (a) metal/native oxide/Si and (b) metal/sapphire interfaces as a function of metal cutoff frequency. The experimental data are derived from Kapitza conductance measurements reported in this work using Eq. (13) and are shown as filled symbols, while those derived from previously reported results are depicted as open symbols (Bi and Pb: Ref. [86]; Cr: Ref. [62]). The solid line in (a) is $\bar{ζ_{1 \to 2} [υ_{1} (ω_{\max})]}$ modeled with the DMM. The acceptable agreement between the model and data suggest DMM-like average transmission of phonons across metal/native oxide/silicon interfaces at room temperature. The vertical lines in (b) represent the lowest two cutoff frequencies in sapphire taken from its dispersion curve in the $Γ \to Z$ direction [111] and are possible locations for S1 and S2 for the metal/sapphire system.
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