Phonon thermal transport between two in-plane, two-dimensional nanoribbons in the extreme near-field regime

The phonon thermal conductance of subnanometric vacuum gaps between two in-plane nanoribbons of two-dimensional materials (graphene and silicene) is analyzed using the atomistic Green's function method and by employing the Tersoff and Lennard-Jones potentials for describing the interatomic interactions. It is found that the phonon conductance decays exponentially with the size of the gap. Three exponential regimes have been identified. In the regime where the Lennard-Jones (L-J) potential is driven by the repulsive interatomic forces, caused by the overlap of electronic orbits, there is a sharp exponential decay in conductance as the gap increases (${e}^{\ensuremath{-}10.0d}$ for graphene). When both the repulsive and the attractive (van der Waals) interatomic forces contribute to the L-J potential, the decay rate of the conductance significantly reduces to ${e}^{\ensuremath{-}2.0d}$ for graphene and ${e}^{\ensuremath{-}2.5d}$ for silicene. In the regime where attractive van der Waals forces dominate the L-J potential, phonon conductance has the slowest exponential decay as ${e}^{\ensuremath{-}1.3d}$ for both silicene and graphene. It is also found that the contribution from the optical phonons to the conductance is non-negligible only for very small gaps between graphene nanoribbons ($d<1.6\phantom{\rule{0.16em}{0ex}}\AA{}$). The phonon conductance of the gap is shown to vary with the width of the nanoribbon very modestly, such that the thermal conductivity of the gap linearly increases with the nanoribbon widths. The results of this study are of significance for a fundamental understanding of heat transfer in the extreme near-field regime and for predicting the effect of interfaces and defects on heat transfer.