We derive a formula to calculate the ballistic thermal conductance of a two-dimensional system directly from the dispersion relations of phonons and electrons. We apply the method to a graphene and investigate both the temperature and the Fermi energy dependences of the ballistic thermal conductance. The ballistic thermal conductance per unit length of a graphene becomes isotropic from the threefold rotational symmetry. In the intrinsic graphene where the Fermi energy crosses the Dirac point, the thermal conductance of electrons increases in proportion to $T^2$ with temperature, while the phonon conductance increases in proportion to $T^{1.5}$ due to the quadratic dispersion relation of the out-of-plane acoustic mode and prevails over the electron-derived conductance irrespective of temperature. As the Fermi energy is moved from the Dirac point for the gated graphenes, the thermal conductance of electrons increases monotonically and the temperature dependence changes from a $T^2$ dependence in the intrinsic graphene to a $T$-linear one at low temperatures. The electron thermal conductance of the gated graphenes dominates over the phonon contribution at low temperatures.

• Received 19 April 2007

DOI:https://doi.org/10.1103/PhysRevB.76.115409