Complex WKB Approximations in Graphene Electron-Hole Waveguides in Magnetic Field

Abstract

Application of semiclassical analysis in studying quantum mechanical behaviour of electron has been demonstrated in various fields of modern physics such as nano-structures, electronic transport in mesoscopic systems, quantum chaotic dynamics of electronic resonators (1), (2), (3), (4) and many others. One of the examples of the application of semiclassical analysis are quantum electronic transport in waveguides and resonators in semiconductors, and in particularly in graphene structures (see, for example, (5), (6), (7)), (8), (9)). Here we review some theoretical aspects of semiclassical description of the Dirac electron motion inside graphene, a one-atom-thick allotrope of carbon (5). A general introduction of ray asymptotic method and boundary layer techniques of complex Gaussian beams (Gaussian wave packages) is given to construct semiclassical approximations of Green’s function inside electron-holes waveguide in graphene structures. The Dirac electrons motion can be controlled by application of electric and magnetic fields. This application could lead in some cases to a generation of a waveguide (drift) motion inside infinite graphene sheets. Constructions of semiclassical approximations of Green’s function inside electronic waveguides or resonators has been a key problem in the analysis of electronic transport problems both different types semiconductors ((10), (11), (12), (13)) and graphene structures such as graphene nano-ribbons (5), (6), (7)), (8), (9)). It is deservedmention that a semiclassical approximation for the Green’s function in graphene as well as a relationship between the semiclassical phase and the adiabatic Berry phase was discussed in the paper (14). An application of semiclassical analysis to graphene quantum electron dynamics is demonstrated for one important problem of constructions of semiclassical approximation of Green’s function in electronic waveguide inside graphene structure with linear potential and homogeneousmagnetic field. The problem can be described by the following 2D Dirac system with magnetic and electrostatic potentials in axial gauge A = 1/2B(−x2, x1, 0) (see (5)) vF ψ(x) +U(x)ψ(x) = Eψ(x), ψ(x) = (