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

Abstract

We use a classical dielectric response framework to evaluate the force components acting on a point charge that moves parallel to an anisotropic two-dimensional material representing doped phosphorene. The anisotropic response of phosphorene is captured by a simple analytical conductivity model that combines contributions from its inter- and intraband electronic transitions, thereby giving rise to a topological transition between elliptic and hyperbolic isofrequency curves for its plasmon modes in the terahertz to the midinfrared frequency range. Pairing this model with a simple dielectric description of a ${\mathrm{SiO}}_{2}$ substrate further reveals the effects of plasmon hybridization with optical phonons in the substrate. We have found that the force on the incident charge has three components, which all exhibit strong velocity-dependent anisotropy when its direction of motion is varied with respect to the principal directions of the phosphorene's conductivity tensor. Besides the longitudinal stopping force and the dynamic image force, there arises a transverse stopping force, with a magnitude comparable to that of the longitudinal force, which acts parallel to the target surface and perpendicular to the trajectory of the particle, and can be therefore experimentally observable, even after its magnitude is reduced by the presence of the substrate. In addition, we have found that the dynamic image force can become repulsive when the particle moves along the armchair direction of a freestanding phosphorene, but the presence of the substrate eliminates that surprising but implausible consequence of the adopted model for anisotropic conductivity of that material.