Dirac-Equation for Graphene with an Arbitrary Potential: Exact Analytical Results and General Proof of Bloch's Theorem

Abstract

In this work, we reexamine the problem of the investigation of the electronic band structures of graphene using the Dirac-equation approach, with massless fermions (electrons). It is assumed that the charges experience a periodic external interaction potential of arbitrary form. First, we study all analytical properties of the wave-function that solves such an equation, and exactly solve the latter for normal incident wave-vectors, whatever is the potential expression. Second, we exactly determine the Dirac energy spectrum (at Dirac points). Thirdly, we give a general proof of the Bloch's theorem, usually encountered in Solid State Physics. Finally, the discussion is extended to nonzero gap monolayer-graphenes and to a finite number of parallel graphene layers.