Effect of inhomogeneities and substrate on the dynamics of the metal-insulator transition in VO$_2$ thin films

We study the thermal relaxation dynamics of VO$_2$ films after the ultrafast photo-induced metal-insulator transition for two VO$_2$ film samples grown on Al$_2$O$_3$ and TiO$_2$ substrates. We find two orders of magnitude difference in the recovery time (a few ns for the VO$_2$/Al$_2$O$_3$ sample vs. hundreds of ns for the VO$_2$/TiO$_2$ sample). We present a theoretical model that accurately describes the MIT thermal properties and interpret the experimental measurements. We obtain quantitative results that show how the microstructure of the VO$_2$ film and the thermal conductivity of the interface between the VO$_2$ film and the substrate affect long time-scale recovery dynamics. We also obtain a simple analytic relationship between the recovery time-scale and some of the film parameters.