The complete impurity scattering formalism in graphene

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 2X2 and 4X4 forms (for the two sublattices and the two inequivalent valleys of the first Brillouin zone). We compare this with the 4X4 formalism proposed in cond-mat/0608228 and cond-mat/0702019, 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.