Theory of substrate-directed heat dissipation for single-layer graphene and other two-dimensional crystals

We present a theory of the phononic thermal (Kapitza) resistance at the interface between graphene or another single-layer two-dimensional (2D) crystal (e.g. MoS$_{2}$) and a flat substrate, based on a modified version of the cross-plane heat transfer model by Persson, Volokitin and Ueba [J. Phys.: Condens. Matter 23, 045009 (2011)]. We show how intrinsic flexural phonon damping is necessary for obtaining a finite Kapitza resistance and also generalize the theory to encased single-layer 2D crystals with a superstrate. We illustrate our model by computing the thermal boundary conductance (TBC) for bare and SiO$_{2}$-encased single-layer graphene and MoS$_{2}$ on a SiO$_{2}$ substrate, using input parameters from first-principles calculation. The estimated room temperature TBC for bare (encased) graphene and MoS$_{2}$ on SiO$_{2}$ are 34.6 (105) and 3.10 (5.07) MWK$^{-1}$m$^{-2}$, respectively. The theory predicts the existence of a phonon frequency crossover point, below which the low-frequency flexural phonons in the bare 2D crystal do not dissipate energy efficiently to the substrate. We explain within the framework of our theory how the encasement of graphene with a top SiO$_{2}$ layer introduces new low-frequency transmission channels which significantly reduce the graphene-substrate Kapitza resistance. We emphasize that the distinction between bare and encased 2D crystals must be made in the analysis of cross-plane heat dissipation to the substrate.