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.
We discuss the conditions under which the predicted (but not yet observed) zero-field interlayer excitonic condensation in double layer graphene has a critical temperature high enough to allow detection. Crucially, disorder arising from charged impur
ities and corrugation in the lattice structure --- invariably present in all real samples --- affects the formation of the condensate via the induced charge inhomogeneity. In the former case, we use a numerical Thomas-Fermi-Dirac theory to describe the local fluctuations in the electronic density in double layer graphene devices and estimate the effect these realistic fluctuations have on the formation of the condensate. To make this estimate, we calculate the critical temperature for the interlayer excitonic superfluid transition within the mean-field BCS theory for both optimistic (unscreened) and conservative (statically screened) approximations for the screening of the interlayer Coulomb interaction. We also estimate the effect of allowing dynamic contributions to the interlayer screening. We then conduct similar calculations for double quadratic bilayer graphene, showing that the quadratic nature of the low-energy bands produces pairing with critical temperature of the same order of magnitude as the linear bands of double monolayer graphene.
