Thermal conductance of graphene and dimerite

We investigate the phonon thermal conductance of graphene regarding the graphene sheet as the large-width limit of graphene strips in the ballistic limit. We find that the thermal conductance depends weakly on the direction angle $\theta$ of the thermal flux periodically with period $\pi/3$. It is further shown that the nature of this directional dependence is the directional dependence of group velocities of the phonon modes in the graphene, originating from the $D_{6h}$ symmetry in the honeycomb structure. By breaking the $D_{6h}$ symmetry in graphene, we see more obvious anisotropic effect in the thermal conductance as demonstrated by dimerite.