Friction force on slow charges moving over supported graphene

We provide a theoretical model that describes the dielectric coupling of a two-dimensional (2D) layer of graphene, represented by a polarization function in the random phase approximation, and a semi-infinite three-dimensional (3D) substrate, represented by a surface response function in a non-local formulation. We concentrate on the role of the dynamic response of the substrate for low-frequency excitations of the combined graphene-substrate system, which give rise to the stopping force on slowly moving charges above doped graphene. A comparison of the dielectric loss function with experimental high-resolution electron energy loss spectroscopy (HREELS) data for graphene on a SiC substrate is used to estimate the effects of damping rate and the local field correction in graphene, as well as to reveal the importance of phonon excitations in an insulating substrate. While the local field correction and linearly dispersing damping rate did not yield any important effects compared to the constant damping rate in graphene, a strong signature of the hybridization between graphene's p plasmon and the substrate's phonon is found in both the HREELS spectra and the stopping force. A friction coefficient that is calculated for slow charges moving above graphene on a metallic substrate shows an interplay between the low-energy single-particle excitations in both systems.