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We investigated the thermal conductivity K of graphene ribbons and graphiteslabs as the function of their lateral dimensions. Our theoretical modelconsidered the anharmonic three-phonon processes to the second-order andincluded the angle-dependent phonon scattering from the ribbon edges. It wasfound that the long mean free path of the long-wavelength acoustic phonons ingraphene can lead to an unusual non-monotonic dependence of the thermalconductivity on the length L of a ribbon. The effect is pronounced for theribbons with the smooth edges (specularity parameter p>0.5). Our results alsosuggest that - contrary to what was previously thought - the bulk-like 3Dphonons in graphite can make a rather substantial contribution to its in-planethermal conductivity. The Umklapp-limited thermal conductivity of graphiteslabs scales, for L below ~ 10 micrometers, as log(L) while for larger L, thethermal conductivity approaches a finite value following the dependence K_0 -A\timesL^-1/2, where K_0 and A are parameters independent of the length. Ourtheoretical results clarify the scaling of the phonon thermal conductivity withthe lateral sizes in graphene and graphite. The revealed anomalous dependenceK(L) for the micrometer-size graphene ribbons can account for some of thediscrepancy in reported experimental data for graphene.

Anomalous Size Dependence of the Thermal Conductivity of Graphene Ribbons