Determining Phonon Mean Free Paths from Observations of Quasiballistic Thermal Transport

The mean free paths (MFPs) of thermal phonons are mostly unknown in many solids. Recent work indicates that MFPs may be measured using experimental observations of quasiballistic thermal transport, but the precise relationship between the measurements and the MFP distribution remains unclear. Here, we present a method that can accurately reconstruct the MFP distribution from quasiballistic thermal measurements without any assumptions regarding the phonon scattering mechanisms. Our result will enable a substantially improved understanding of thermal transport in many solids, particularly thermoelectrics.

Figure 1

(a) Deviational temperature decay curves $\Delta T(z=0,t)$ predicted by the numerical Monte Carlo solution (solid lines) and the heat equation (dashed lines) for the two transient grating wavelengths indicated. (b) Spectral heat flux (per phonon frequency) plotted as a function of MFP ${\Lambda }_{\omega }$ for $\lambda =7\mu \mathrm{m}$. The Fourier heat flux $Q_f$ (solid line), the MC heat flux $Q_m$ (rapidly varying line), the suppression function $S(q{\Lambda }_{\omega })$ (dashed line), and the kernel $K(q{\Lambda }_{\omega })$ (dash-dotted line) are shown.

Figure 2

Actual MFP distribution for silicon (line), the reconstructed MFP distribution (circles), and the effective thermal conductivities obtained from the MC simulations with artificially added noise (squares). The $x$ axis corresponds to the MFP for the two distributions (line and circles) and to the MC grating wavelength for the thermal conductivity data (squares). The reconstructed distribution agrees with the actual distribution to within 5% over nearly the full MFP range. The distribution can be reconstructed nearly exactly over the full MFP range if data spanning a wider range of length scales is used.

Figure 3

The MFP distribution in a 390 nm thick Si membrane at room temperature (circles) reconstructed from recently reported transient grating measurements (squares) [17]. The $y$ axis is normalized to the bulk thermal conductivity value of Si. An analytic calculation (solid line), which was computed from a first-principles calculation of the MFP distribution in Si and accounting for diffuse scattering from the membrane boundaries, agrees with the reconstructed distribution to within approximately 15%.