Energy Losses and Transition Radiation in Multilayer Graphene Traversed by a Fast Charged Particle

Abstract

We present a fully relativistic formulation of the energy loss of a charged particle traversing a number of graphene layers and apply it to the case of two spatially separated layers probed by an energetic electron. We focus on the THz frequency range, using a Drude model to describe the conductivity of graphene and allowing for different doping density in each layer. We distinguish two types of contributions to the electron energy loss: the energy deposited in graphene layers in the form of electronic excitations (Ohm losses), which include the excitation of Dirac plasmon polaritons (DPP), and the energy that is emitted in the form of transition radiation. We study in detail the contribution of each layer to the ohmic losses and analyze the directional decomposition of the radiation emitted in the half-spaces defined by the graphene planes. By increasing the interlayer distance and changing the relative doping densities in graphene layers, we find surprisingly strong asymmetries in both the directional and layer-wise decompositions with respect to the direction of motion of the external electron. A modal decomposition is also performed in the limit of vanishing damping in graphene, exposing quite intricate roles of bonding and antibonding hybridization between DPPs in ohmic losses.

Hydrodynamic model approach to the formation of plasmonic wakes in graphene

“…It would be interesting to extend them for graphene on hexagonal Boron Nitride for discussing the excitation of phonon-plasmon-polaritons. Also, extending this work to multilayer graphene 35 is a natural con-tinuation of this work.…”

“…a result that has been obtained in the literature before 31 using a different method based on reflection coefficients. Equation (35) has a maximum at the frequency…”

Directional effects in plasmon excitation and transition radiation from an anisotropic 2D material induced by a fast charged particle

“…It is possible to derive analytical expressions for F ext ( k , ω ), F Ohm ( k , ω ) and F rad ( k , ω ) by following the procedure described in ref. 57 and 59, which starts from the appropriate Physical definitions for the corresponding energy loss processes and takes into account the symmetry properties of all the Fourier transformed quantities related to the causality of the conductive sheet response. After some straightforward algebra, all three joint probability densities may be expressed in compact form in terms of E ext∥ ( k ,0, ω ), and the in-plane conductivity tensor for a general anisotropic sheet.…”

“…The Ohmic energy loss is obtained from the work done by the total electric field acting on the induced current density in the conducting sheet, . While the resulting F Ohm ( k , ω ) is generally nonzero over the whole ( k , ω ) space, both inside ( k < k d ) and outside ( k > k d ) the light cone, 59 taking the limit of negligible dissipation makes F Ohm ( k , ω ) proportional to a delta-function factor δ ( ω − ω d ( k )), which is peaked across the dispersion surface of a collective mode, ω = ω d ( k ), typically lying outside the light cone (see the ESI†). Furthermore, keeping our focus on the directional effects in the excitation of such modes by the external charged particle, it is also worthwhile defining a marginal probability density for Ohmic energy loss by integrating out the frequency,…”

“…51,55 Although nonretarded calculations of EELS reproduced the experimental data for graphene quite well for energy losses ≳1 eV, 50,56 a relativistic formulation of the interaction of a fast charged particle with graphene was developed to assess the role of retardation in energy losses 54,57,58 and to investigate the TR spectra from graphene in a broad range of frequencies, from the THz to the VUV. 57,[59][60][61] At the same time, there were only few experimental EELS studies of BP, which were limited to the range of electron energy losses ≳1 eV, and were accompanied by ab initio calculations of the loss function for multi-layer BP in the nonretarded regime. [27][28][29] Free-electron interactions with 2D targets were recently studied in several other contexts besides the EELS, [62][63][64][65][66] most notably as part of an effort toward designing a graphene-based source of THz radiation.…”