A numerical fitting routine for frequency-domain thermoreflectance measurements of nanoscale material systems having arbitrary geometries

Abstract

In this work, we develop a numerical fitting routine to extract multiple thermal parameters using frequency-domain thermoreflectance (FDTR) for materials having non-standard, non-semi-infinite geometries. The numerical fitting routine is predicated on either a 2D or 3D finite element analysis that permits the inclusion of non-semi-infinite boundary conditions, which cannot be considered in the analytical solution to the heat diffusion equation in the frequency domain. We validate the fitting routine by comparing it with the analytical solution to the heat diffusion equation used within the wider literature for FDTR and known values of thermal conductivity for semi-infinite substrates (SiO2, Al2O3, and Si). We then demonstrate its capacity to extract the thermal properties of Si when etched into micropillars that have radii on the order of the pump beam. Experimental measurements of Si micropillars with circular and square cross sections are provided and fit using the numerical fitting routine established as part of this work. Likewise, we show that the analytical solution is unsuitable for the extraction of thermal properties when the geometry deviates significantly from the standard semi-infinite case. This work is critical for measuring the thermal properties of materials having arbitrary geometries, including ultra-drawn glass fibers and laser gain media.