  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.
