Energy losses and transition radiation in graphene traversed by a fast charged particle under oblique incidence

Abstract

We perform fully relativistic calculations of the energy loss channels for a charged particle traversing a single layer of graphene under oblique incidence in a setting pertinent to a Scanning Transmission Electron Microscope (STEM), where we distinguish between the energy deposited in graphene in the form of electronic excitations (Ohmic loss) and the energy emitted in the far field in the form of transition radiation (TR). Our formulation of the problem uses a definition of two in-plane, dielectric functions of graphene, which describe the longitudinal and transverse excitation processes that contribute separately to those two energy loss channels. Using several models for the electric conductivity of graphene as the input in those dielectric functions enables us to discuss the effects of oblique incidence on several processes in a broad range of frequencies, from the terahertz (THz) to the ultraviolet (UV). In particular, at the THz frequencies, we demonstrate that the nonlocal effect in the graphene?s conductivity is not important in the retarded regime, and we show that the longitudinal and transverse contributions to the emitted TR spectra exhibit strongly anisotropic angular patterns that are readily distinguishable in a cathodoluminescence measurement in a STEM. Moreover, we explore the possibility of exciting the so-called transverse mode in the optical response of graphene at the mid-infrared (MIR) range of frequencies by means of a fast charged particle under oblique incidence. Finally, we demonstrate that, besides the usual high-energy peaks in the longitudinal contribution to the Ohmic energy loss in the MIR to the UV frequency range, there may arise strongly directional features in the in-plane distribution of the transverse contribution to the Ohmic energy loss for an oblique trajectory, which could be possibly observed via momentum- and angle-resolved electron energy loss spectroscopy of graphene in STEM.