We consider two faced ultra thin membranes separated by a nanogap subject to a temperature gradient. We calculate explicitly the interaction energy from a first principle approach based on a DFT-Tight Binding model. The calculation of this energy allows the correct coupling of the vibrational modes of both membranes taking into account the effects of surface reconstructions due to the relative orientations of the faced materials.

In this work we focus on how these interactions can be combined with an harmonic atomistic model to compute the energy transfer between the membranes. We obtain the characteristic relaxation time which can be directly related to the thermal conductance.

Our model shows that at a nanometer gap, in a non equilibrium regime, this mechanism contributes significantly to the thermal energy transport.