Modeling non-diffusive thermal transport in silicon with the phonon Boltzmann Transport Equation: Full Scattering Matrix vs Relaxation Time Approximation*

In the past decade, non-diffusive heat transport by phonons at room temperature at the micro/nanoscale was reported in many single crystal materials. Many of these results have been modeled using the relaxation time approximation (RTA) to the Peierls-Boltzmann phonon transport equation (BTE). While the RTA has been shown to fail for high Debye temperature materials such as diamond and graphene, it has been considered accurate for lower Debye temperature materials such as silicon. The objective of the present study is to test this assumption by comparing the results obtained under RTA with the full scattering matrix BTE. We consider the problem of the transient thermal grating, i.e., the relaxation of a spatially periodic perturbation of the phonon distribution. Using silicon as an example, we present calculations with both the RTA and full scattering matrix over a wide range of the thermal grating periods covering the transition from diffusive to ballistic regime.  We find that the RTA performs reasonably well at the onset of non-diffusive transport, i.e., for the heat transport distances in the micron range, but becomes increasingly inaccurate on the nanometer scale when the phonon population in the thermal grating is far from equilibrium. We will discuss the relationship of the pseudo-temperature of the RTA with the thermodynamic temperature of the full scattering matrix BTE. The experimental relevance of the BTE calculations in the context of recent nanoscale transient grating experiments with extreme ultraviolet excitation will also be discussed.