  We investigated theoretically the phonon thermal conductivity of single layer
graphene. The phonon dispersion for all polarizations and crystallographic
directions in graphene lattice was obtained using the valence-force field
method. The three-phonon Umklapp processes were treated exactly using an
accurate phonon dispersion and Brillouin zone, and accouting for all phonon
relaxation channels allowed by the momentum and energy conservation laws. The
uniqueness of graphene was reflected in the two-dimensional phonon density of
states and restrictions on the phonon Umklapp scattering phase-space. The
phonon scattering on defects and graphene edges has been also included in the
model. The calculations were performed for the Gruneisen parameter, which was
determined from the ab initio theory as a function of the phonon wave vector
and polarization branch, and for a range of values from experiments. It was
found that the near room-temperature thermal conductivity of single layer
graphene, calculated with a realistic Gruneisen parameter, is in the range ~
2000 - 5000 W/mK depending on the defect concentration and roughness of the
edges. Owing to the long phonon mean free path the graphene edges produce
strong effect on thermal conductivity even at room temperature. The obtained
results are in good agreement with the recent measurements of the thermal
conductivity of suspended graphene.
