Polarization of supported graphene by slowly moving charges
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We describe the dynamic polarization of graphenes pi electrons by means of the Vlasov equations in the relaxation-time approximation and evaluate the stopping and the image forces on slow ions moving parallel to a supported graphene under the gating conditions. While the effects of temperature and the gate potential on these forces are conveniently included by means of a single scaling parameter, the effects of variation in size of the gap between graphene and the SiO2 substrate are found to be quite large, pointing to the need for including this gap explicitly in modeling of polarization of graphene.