Stopping and image forces on a charged particle moving parallel to an anisotropic two-dimensional material

Launching Plasmons in a Two-Dimensional Material Traversed by a Fast Charged Particle

“…It was recently shown that the applicability of the Drude model of conductivity ( 43 ) can also be expanded to finite k vales by adding a correction due to high-energy interband electron transitions, which gives rise to a local form of the improved conductivity model as where accounts for static screening due to high-energy interband electron transitions in a 2D material [ 25 , 28 ]. This model was found to work particularly well for TMDs at the wavenumbers up to Å [ 25 ], while for doped graphene [ 27 , 28 ] and doped phosphorene [ 29 ], it improves the agreement with the ab initio data on plasmon dispersion for the wavenumbers . Using ( 47 ) in ( 14 ) yields an expression for the dielectric permittivity, where is introduced for simplicity.…”

“…While this assumption is necessary to validate the neglect of the magnetic fields due to the moving particle, a fully relativistic formulation of the problem showed that the magnetic field due to the induced current in a conducting sheet may only be neglected if one additionally assumes that , see Equation ( 6 ) in [ 17 ]. This inequality is automatically satisfied by assuming for a particle moving parallel to the sheet because of the resonance condition [ 28 , 29 ], but for the perpendicular trajectory in Figure 1 , a strong hybridization of the plasmon dispersion with the light line can affect the launching of plasmons with long wavelengths, even when . The proximity of the region , which was shown to give rise to a sizeable radiative energy loss in doped graphene in the THz-MIR frequency range [ 22 , 23 , 24 ], renders the issue of the nonrelativistic approximation rather nontrivial in the present context.…”

“…Finally, it would also be interesting to evaluate the spatiotemporal dependence of the collective modes excited in an anisotropic 2D material, such as phosphorene [ 28 , 29 ], which exhibits a topological transition from the elliptic to hyperbolic isofrequency plasmon dispersion curves in the plane. Yet another interesting aspect of the problem involves the effects of finite size of a 2D material target, such as plasmon launching by electron impact upon graphene ribbons [ 49 ].…”

“…This model gives rise to the characteristic square-root dispersion of the plasmon frequency in 2D materials, , for small values. However, it is also necessary to take into account a correction to the in-plane conductivity in the THz–MIR range, which comes from high-energy interband electron transitions in those materials [ 27 , 28 , 29 ]. It was shown that this correction gives rise to a saturation of the plasmon dispersion as the wavenumber k increases, resulting in slowing down of such plasmons [ 25 , 28 , 29 ].…”

“…However, it is also necessary to take into account a correction to the in-plane conductivity in the THz–MIR range, which comes from high-energy interband electron transitions in those materials [ 27 , 28 , 29 ]. It was shown that this correction gives rise to a saturation of the plasmon dispersion as the wavenumber k increases, resulting in slowing down of such plasmons [ 25 , 28 , 29 ]. Hence, it is worthwhile to explore the effects of this slowing down in the dynamics of launching such sheet plasmons by an impact of the incident charged particle.…”