Green’s functions and impurity scattering in graphene

Abstract

We present the complete formalism that describes scattering in graphene at low energies. We begin by analyzing the real-space free Green's function matrix and its analytical expansions at low energy, carefully incorporating the discrete lattice structure, and arbitrary forms of the atomic-orbital wave function. We then compute the real-space Green's function in the presence of an impurity. We express our results both in $2\ifmmode\times\else\texttimes\fi{}2$ and $4\ifmmode\times\else\texttimes\fi{}4$ forms (for the two sublattices and the two inequivalent valleys of the first Brillouin zone). We compare this with the $4\ifmmode\times\else\texttimes\fi{}4$ formalism proposed by and Cheianov and Fal'ko [Phys. Rev. Lett. 97, 226801 (2006)] and Mariani et al. [Phys. Rev. B 76, 165402 (2007)], and show that the latter is incomplete. We describe how it can be adapted to accurately take into account the effects of intervalley scattering on spatially varying quantities such as the local density of states.