Relativistic energy-momentum transfer and electromagnetic conservation laws in the interaction of moving charged particles with two-dimensional materials

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

“…On the other hand, referring to ( 13 ), one may infer that has a jump at , which accounts for the left-hand side of ( 12 ), written in the domain as Therefore, by inserting in ( 33 ), we finally obtain where we used ( 35 ). The above result is consistent with a fully relativistic formulation of the force, see the first two lines of Equation ( 28 ) in [ 17 ].…”

“…Then, the second term with the vector potential may be neglected by assuming , where c is the speed of light. 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 .…”

“…To further investigate conservative and dissipative interactions, we derive expression for the total force acting on the 2D material using the approach of [ 17 ] based on the Maxwell stress tensor formalism. By placing the material in the region defined by a slice where d is allowed to approach zero, we may express the force as the flux of the Maxwell stress tensor where the surface consists of two planes normal to the z -axis and infinitesimally above and below , respectively.…”

“…Using in place of in ( 36 ) along with ( 21 ) and ( 34 ), we see that the conservative force on the 2D material can be expressed as the negative of the image force acting on the moving charged particle, where its dynamic image potential is given by (recall that, without loss of generality, we assumed ). Clearly, the integral of over gives zero, i.e., the momentum transferred from the incident particle to the 2D material under the action of the conservative force vanishes [ 17 ]. The conservative nature of the dynamic image force was discussed in [ 36 ].…”

“…There has been recent interest in the interaction between fast charged particles and sheets of graphene, along with other 2D materials in the context of Electron Energy Loss Spectroscopy (EELS) in a Scanning Transmission Electron Microscope (STEM) [ 13 , 14 , 15 , 16 ]. This is a microscopy technique used in nanophotonics research to understand, among other properties, the nature of plasmon excitations in nanostructures by probing the energy and momentum transfer from a fast incident electron to the target material [ 17 ]. It is noteworthy that graphene and other 2D materials can be suspended over a holey grid in STEM-EELS and hence may be probed free from the influence of the nearby materials [ 18 ].…”