Mathematical modeling and computer simulation of interactions of charged particles with 2D materials

Abstract

Recent interest in the nanophotonic and nanoplasmonic devices has led researechers to a detailed study of different aspects of the interactions of the moving charged particles with two dimensional (2D) materials. When a charged particle moves above a 2D material, electromagnetic forces due to the polarization of the charges in the target material cause energy dissipation. Analyzing the energy spectra and momentum change of the reflected particles provides valuable information about the internal structure of the target material. The analyses of this kind are extensively performed in the electron energy loss spectroscopy (EELS) and the high resolution EELS (HREELS) experiments. To model such interactions, we used the classical dielectric response theory in the non-retarded approximation. We obtained closed form expressions for the forces acting on the charged particles moving parallel to doped phosphorene using analytical models for its dielectric function, which expose the strongly anisotropic character of the electronic structure and dynamic response of this 2D material. The parameters of these models, which we call the optical and the semiclassical models, are supplied by the ab initio calculations. It was found that the force on the incident charge has three components, showing strong dependence on the direction of motion of the charged particle. Furthermore, we performed an analysis of the electric potential in the plane of phosphorene, revealing a rich variety of the plasmonic wake patterns induced by the moving charged particle. Our computations showed surprising analogies of these wakes with Kelvin’s ship wakes and atmospheric wakes regarding the asymmetry of the wake, slowing down of the plasmon dispersion, and the formation of Mach-like wake due to nonlocal effects in the semiclassical dielectric response function. In a related effort, we adopted the energy loss method (ELM) to compute the phosphorene carrier mobility tensor when the carrier scattering on charged impurities in the substrate is the main limiting factor of the mobility in the DC regime at low temperatures. The ELM provides a closed form expression for the phosphorene carrier’s mobility tensor in terms of a double integral, which is superior to the traditional approach for mobility calculation based on the Boltzmann transport equation that requires numerical solution of an integral equation. Using the ELM, we examined the mobilities for different statistical distributions of the charged particles, revealing strong effects of the inter-particle correlation distance on the mobilities. Finally, we evaluated the energy loss function of doped graphene when there exist a random distribution of charged impurities in the substrate. We modeled the resulting random potential landscape in graphene via local Fermi energy, with its in-plane spatial correlation governed by a Gaussian distribution, giving rise to an expression for the loss function that includes a memory function, which is the solution of a nonlinear integral equation. The overall effect of this random potential landscape was found to be a broadening and shifting of the plasmon peak in the energy loss function of doped graphene.