The complete impurity scattering formalism in graphene

We present the complete formalism that describes scattering in graphene at lowenergies. 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 2×2 and 4×4 forms (for the two sublattices and the two inequivalent valleys of the first Brillouin zone). We compare this with the 4 × 4 formalism proposed in Refs. [1, 2], and show that the latter is incomplete. We describe how it can be adapted to accurately take into account the effects of inter-valley scattering on spatially-varying quantities such as the local density of states.