Dynamics of quasi-particles in graphene with impurities and sharp edges from the kp method standpoint

Abstract

Dynamics of quasi-particles in graphene with an impurity and a sharp edge is considered with the kp perturbation theory that allows an unified approach without usage of any models. Dirac and Weyl equations are derived by the above-mentioned method. The wave function and its envelope function together with the scattering amplitude are found in the Born approximation. The wave functions are shown to be a superposition of virtual Bloch functions which exponential decay outward from the impurity and the edge. At distances much greater that the atomic spacing the wave functions are explicitly presented. Green's functions for Schr\"odinger and Dirac equations are derived as well. Boundary conditions for the Dirac equation for graphene with a sharp edge are also derived.