Modelling relativistic effects in momentum-resolved electron energy loss spectroscopy of graphene

We present an analytical model for the electron energy loss through a two-dimensional (2D) layer of graphene, fully taking into account relativistic effects. Using two different models for graphene's 2D conductivity, one a two-fluid hydrodynamic model with an added correction to account for the inter-band electron transitions near the Dirac point in undoped graphene, the other derived from ab initio plane-wave time-dependent density functional theory in the frequency domain (PW-TDDFT-) calculations applied on a graphene superlattice, we derive various different expressions for the probability density of energy and momentum transfer from the incident electron to graphene. To further compare with electron energy loss spectroscopy (EELS) experiments that use setups like scanning Transmission Electron Microscopy, we integrated our energy loss functions over a range of wavenumbers, and compared how the choice of range directly affects the shape, position, and relative heights of graphene's * and sigma sigma* transition peaks. Comparisons were made with experimental EELS data under different model inputs, revealing again the strong effect that the choice of wavenumber range has on the energy loss.