Electronic optics in graphene in the semiclassical approximation

Abstract

We study above-barrier scattering of Dirac electrons by a smooth electrostatic potential combined with a coordinate-dependent mass in graphene. We assume that the potential and mass are sufficiently smooth, so that we can define a small dimensionless semiclassical parameter h 1. This electronic optics setup naturally leads to focusing and the formation of caustics, which are singularities in the density of trajectories. We construct a semiclassical approximation for the wavefunction in all points, placing particular emphasis on the region near the caustic, where the maximum of the intensity lies. Because of the matrix character of the Dirac equation, this wavefunction contains a nontrivial semiclassical phase, which is absent for a scalar wave equation and which influences the focusing. We carefully discuss the three steps in our semiclassical approach: the adiabatic reduction of the matrix equation to an effective scalar equation, the construction of the wavefunction using the Maslov canonical operator and the application of the uniform approximation to the integral expression for the wavefunction in the vicinity of a caustic. We consider several numerical examples and show that our semiclassical results are in very good agreement with the results of tight-binding calculations. In particular, we show that the semiclassical phase can have a pronounced effect on the position of the focus and its intensity.

Tunneling in the Brillouin Zone: Theory of Backscattering in Valley Hall Edge Channels

“…However, the physical insight that we provide, as well as some of our analytical results, can also be transferred to other smooth-envelope topological insulators where both helical edge states are localized around the Γ point [25][26][27][28][29][30][31][32]. Our work ties to other investigations that have adopted the WKB approximation to investigate the electronic band structure or density of states in graphene and other materials in the presence of smooth electromagnetic fields [33][34][35][36][37][38].…”

Generalized Dirac oscillators with position-dependent mass

“…In this scenario the mass in DO becomes position dependent and is related to the refractive index. In this context we would like to mention that the lower dimensional Dirac equation with a position dependent mass has also found applications in electronic optics in graphene [14]. In this work our objective is obtain solutions of the DO with a position dependent mass with/without an electric field.…”

Interplay between effects of barrier tilting and scatterers within a barrier on tunneling transport of Dirac electrons in graphene

“…26 Furthermore, the reported research by Reijnders, et al addressed the optical properties of graphene with an energy gap by using the semiclassical model. 27 Finally, Choubabi, et al considered a linear potential barrier which oscillates uniformly in time and obtained a solution for the energy spectrum including several modes associated with oscillations. 28 An investigation on impurity-assisted electron tunneling conductance was reported for a very specific case of undoped, or intrinsic graphene, 29 in which a resonant-type conductance enhancement was found for the case with a single impurity atom.…”