On the role of the energy loss function in the image force on a charge moving over supported graphene

We use a dielectric response theory to describe electrodynamic forces on a charged particle moving parallel to a supported two-dimensional layer. Using a Kramers-Kronig relation, we show that the image force on the particle can be expressed in terms of the energy loss function of the target materials. This enables us to analyze the stopping and the image forces on the particle on equal footing in the frequency-momentum domain encompassing all the energy loss channels in the target. Using the example of a graphene layer on a silicon carbide substrate, we show that both the image and stopping forces can be decomposed into contributions coming from two modes arising from hybridization of the sheet plasmon in doped graphene and a transverse optical phonon in the substrate.