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Limitations of generalised grey phonon models for quasiballistic thermal transport in time-periodic regimes

Suitably superimposed grey-medium solutions of the Boltzmann transport equation (BTE) provide a simple yet accurate description of non-grey quasiballistic heat conduction in transient thermal grating experiments. Recent applications of similar strategies based on kinetic and McKelvey-Schockley-Landauer theory to time-periodic transport predicted notable conductivity suppression only at heating frequencies comparable to phonon scattering rates, in contrast to lengthscale criteria observed by several prior studies. Here we show that the frequency-integrated grey-medium approximation (FIGMA) is ill suited to tackle temporally periodic quasiballistic transport. Starting from first-principles phonon dispersions and scattering rates, we obtain semi-analytic 1D BTE solutions for semi-infinite structures subjected to sinusoidal surface heating and compare these to the approximate model counterparts. We find FIGMA-based approaches to overestimate the semiconductor surface temperature by up to one and characteristic heating frequencies for onset of quasiballistic effects by up to three orders of magnitude respectively. Our study reasserts that experimentally observed heating-frequency dependent apparent conductivities originate in the overlap of the characteristic length scale of the thermal gradient with phonon mean free paths.