Ballistic Thermal Conductance of a Graphene Ribbon

Recent experiments on thermal transport in graphene suggest that the phonon mean free path may exceed 500 nm,\footnote{S. Ghosh, et al., ``Extremely high thermal conductivity of graphene: Prospects for thermal management applications in nanoelectronic circuits'' Applied Physics Letters, 2008. \textbf{92}: p. 151911.} with thermal conductivities in the range 3000 -- 5000 (W/m/K). In this scenario, it is expected that thermal transport is dominated by a ballistic rather than diffusive mechanism. We present an analytical theory to calculate the thermal conductance of a graphene ribbon in the ballistic regime. For that purpose, we analyze the vibrational modes of a continuum thin plate with isotropic elastic properties. To address the effect of nanoscale dimensions, we consider a finite width $w$ in the model. At low temperatures, our analytical theory shows a power law dependence of the thermal conductance per unit width, were the exponent $\beta $ is a function of the ribbon width, ranging from $\beta $ = 1 for thin graphene ribbons, towards $\beta $ = 1.5 in the limit of a large graphene sheet. Quantitative predictions of our theory at room temperature are in good agreement with experiments.\footnote{Ibid.}