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Near-field radiative heat transfer and van der Waals friction between closely spaced graphene and amorphous SiO$_2$

We study the radiative heat transfer and the van der Waals friction between graphene and an amorphous SiO$_2$ substrate. We study the surface phonon-polaritons contribution to the low-field mobility as a function of temperature and of carrier density. We find that the electric current saturate at a high electric field, in agreement with experiment. The saturation current depends weakly on the temperature, which we attribute to the "quantum" friction between the graphene carriers and the substrate optical phonons. We calculate the frictional drag between two graphene sheets caused by van der Waals friction, and find that this drag can induce a high enough voltage which can be easily measured experimentally. We find that for nonsuspended graphene the near-field radiative heat transfer, and the heat transfer due to direct phononic coupling, are of the same order of magnitude at low electric field. The phononic contribution to the heat transfer dominates at high field. For large separation between graphene and the substrate the heat transfer is dominated by the near-field radiative heat transfer.