We present several applications of the layered electron gas model to electron energy loss spectroscopy of freestanding films consisting of N graphene layers in a scanning transmission electron microscope. Using a two-fluid model for the single-layer polarizability, we discuss the evolution of high-energy plasmon spectra with N and find good agreement with the recent experimental data for both multilayer graphene with N<10 and thick slabs of graphite. Such applications of these analytical models help shed light on several features observed in the plasmon spectra of those structures, including the role of plasmon dispersion, dynamic interference in excitations of various plasmon eigenmodes, as well as the relevance of the bulk plasmon bands, rather than surface plasmons, in classifying the plasmon peaks.

We evaluate the stopping and image forces on a charged particle moving parallel to a doped sheet of graphene by using the dielectric-response formalism for graphene’s $\pi$-electron bands in the random phase approximation (RPA). The forces are presented as functions of the particle speed and the particle distance for a broad range of charge-carrier densities in graphene. A detailed comparison with the results from a kinetic equation model reveal the importance of interband single-particle excitations in the RPA model for high particle speeds. We also consider the effects of a finite gap between graphene and a supporting substrate, as well as the effects of a finite damping rate that is included through the use of Mermin’s procedure. The damping rate is estimated from a tentative comparison of the Mermin loss function with a high-resolution reflection electron energy loss spectroscopy experiment. In the limit of low particle speeds, several analytical results are obtained for the friction coefficient that show an intricate relationship between the charge-carrier density, the damping rate, and the particle distance, which may be relevant to surface processes and electrochemistry involving graphene.

We describe the dynamic polarization of graphene’s $\pi$ 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 $\mathrm{Si}{\mathrm{O}}_2$ substrate are found to be quite large, pointing to the need for including this gap explicitly in modeling of polarization of graphene.

The propagator of a dynamically screened Coulomb interaction $W$ in a sandwichlike structure consisting of two graphene layers separated by a slab of ${\mathrm{Al}}_2{\mathrm{O}}_3$ (or vacuum) is derived from single-layer graphene response functions and by using a local dielectric function for the bulk ${\mathrm{Al}}_2{\mathrm{O}}_3$. The response function of graphene is obtained using two approaches within the random phase approximation (RPA): an ab initio method that includes all electronic bands in graphene and a computationally less demanding method based on the massless Dirac fermion (MDF) approximation for the low-energy excitations of electrons in the $\pi$ bands. The propagator $W$ is used to derive an expression for the effective dielectric function of our sandwich structure, which is relevant for the reflection electron energy loss spectroscopy of its surface. Focusing on the range of frequencies from THz to mid-infrared, special attention is paid to finding an accurate optical limit in the ab initio method, where the response function is expressed in terms of a frequency-dependent conductivity of graphene. It was shown that the optical limit suffices for describing hybridization between the Dirac plasmons in graphene layers and the Fuchs-Kliewer phonons in both surfaces of the ${\mathrm{Al}}_2{\mathrm{O}}_3$ slab, and that the spectra obtained from both the ab initio method and the MDF approximation in the optical limit agree perfectly well for wave numbers up to about 0.1 ${\mathrm{nm}}^{-1}$. Going beyond the optical limit, the agreement between the full ab initio method and the MDF approximation was found to extend to wave numbers up to about 0.3 ${\mathrm{nm}}^{-1}$ for doped graphene layers with the Fermi energy of 0.2 eV.