Analytical interpretation of nondiffusive phonon transport in thermoreflectance thermal conductivity measurements

Abstract

We derive an analytical solution to the Boltzmann transport equation (BTE) to relate nondiffusive thermal conductivity measurements by thermoreflectance techniques to the bulk thermal conductivity accumulation function, which quantifies cumulative contributions to thermal conductivity from different mean free path energy carriers (here, phonons). Our solution incorporates two experimentally defined length scales: thermal penetration depth and heating laser spot radius. We identify two thermal resistances based on the predicted spatial temperature and heat flux profiles. The first resistance is associated with the interaction between energy carriers and the surface of the solution domain. The second resistance accounts for transport of energy carriers through the solution domain and is affected by the experimentally defined length scales. Comparison of the BTE result with that from conventional heat diffusion theory enables a mapping of mean-free-path-specific contributions to the measured thermal conductivity based on the experimental length scales. In general, the measured thermal conductivity will be influenced by the smaller of the two length scales and the surface properties of the system. The result is used to compare nondiffusive thermal conductivity measurements of silicon with first-principles-based calculations of its thermal conductivity accumulation function.

Compact stochastic models for multidimensional quasiballistic thermal transport

“…A variaty of studies [6,7,11,12,15,16,18,21,24] have reconstructed the 'MFP spectrum' (cumulative conductivity function) κ Σ (Λ) from effective conductivities κ eff (χ) measured as a function of a controllable parameter χ. Although the κ Σ curve reveals the spatial extent of distinct transport regimes [23], it holds insufficient information for computing the actual quasiballistic heat dynamics.…”

“…The essential physics that underpin the quasiballistic heat flow observations have been theoretically explained by a variety of works [14][15][16][17][18][19][20][21][22][23][24] largely indebted to the Boltzmann transport equation (BTE). However, BTE solutions are formulated in terms of a wide spectrum of phonon modes and as such are too intricate and inflexible for direct processing of experimental data.…”

Steady state and modulated heat conduction in layered systems predicted by the analytical solution of the phonon Boltzmann transport equation

“…It is hence clear that this behavior of the temperature is the result of the ballistic phonons present within one MFP of the surface, mainly. This jump of the modulated temperature has been also found by Regner et al 21 and appears to be a general feature of the ballistic heat conduction predicted by the BTE. The relatively lower values predicted by the BTE (less than 0.5 degrees in the phase) could be associated with the continuous emission of phonons from z ¼ 0 without receiving feedback of phonons from z !…”

“…q sn , and v n ¼ 1 in Eqs. (15), (17), and (21). Below, we analyze separately the steady state and modulated heat conduction problems.…”

“…Based on this theory, these authors showed that the contribution of ballistic phonons with mean free paths larger than the penetration depth of the thermoreflectance signal has to be considered to correctly explain the behavior of the thermal conductivity of pure crystals. Furthermore, Regner et al 21 used the methods of moments and spherical harmonics to derive analytical approximate solutions for the modulated temperature of planar semi-infinite and spherical media, respectively. The solutions obtained for both geometries exhibit the same spatial behavior than the corresponding one predicted by the Fourier's law, but with a modified penetration depth, which coincides with the one predicted by the Cattaneo-Vernotte equation.…”