Analytical modeling of electron energy loss spectroscopy of graphene: Ab initio study versus extended hydrodynamic model

Wake potential in graphene-insulator-graphene composite systems

“…At the same time, there were indications in recent experimental EELS data [58,64,66,67] that the Dirac physics of the low-energy interband transitions plays an important role at the energy losses 2 eV for intrinsic graphene. Accordingly, we have extended the above HD model by including a low-energy contribution describing the π → π * transitions near the K points in the BZ, which we call the Dirac correction, σ D (ω), having the proper zero-frequency limit Re[σ D (ω → 0)] = σ 0 .…”

“…At higher frequencies, corresponding to 1 eV hω 50 eV, which are of interest in the valence EELS, the effects of doping are not important, but the onset of high-energy interband electron transitions in graphene gives rise to two prominent peaks in the energy loss spectra near 4 eV and 14 eV, known as the π and σ + π plasmon peaks [64], respectively. It was found that several different sets of experimental EELS data on single-layer [58] and multilayer [65] graphene can be successfully modeled by resorting to the optical limit for the response function χ 0 i (Q, ω), given in Eq. (9).…”

“…(9). Moreover, it was found that it suffices to only consider the conductivity of intrinsic graphene (E F = 0) when modeling the main peaks in the EELS data [58,64,65], so we may use Eq. (10) with σ i (ω) = σ inter i (ω), describing both π → π * and σ → σ * interband electron transitions across the BZ in graphene.…”

“…Accordingly, we have extended the above HD model by including a low-energy contribution describing the π → π * transitions near the K points in the BZ, which we call the Dirac correction, σ D (ω), having the proper zero-frequency limit Re[σ D (ω → 0)] = σ 0 . As a result, we obtained the eHD model as [58]…”

“…The main ingredients to our model are the single-layer graphene response functions for noninteracting electrons, χ 0 i (Q, ω) with i = 1, 2, and the local dielectric function S (ω) of the bulk Al 2 O 3 . The response functions χ 0 1,2 (Q, ω) are obtained in this work at zero temperature using three complementary methods: (i) the random phase approximation (RPA) for doped graphene, taking into account just its π electron bands in the Dirac cone approximation [54,55] with the effects of damping included via the Mermin procedure [56], which we call the massless Dirac fermions (MDF) method; (ii) a phenomenological approach that describes high-energy interband transitions in graphene by a two-fluid model for its π and σ electrons, amended by a low-energy Dirac contribution near the K points in the Brillouin zone (BZ), which we call the extended hydrodynamic (eHD) model [57,58]; and (iii) an ab initio-RPA method that includes transitions between all valence and conduction bands in graphene, which we simply call the ab initio method. The propagator W is used to derive an effective 2D dielectric function, (Q, ω), of the composite sy 1 -Al 2 O 3 -sy 2 structure, which is further used to evaluate the wake potential and to facilitate understanding of the appearance and the role of certain modes by analyzing the peaks in the loss function of the structure, given by Im[−1/ (Q, ω)] [56,59].…”