Modeling STM images in graphene using the effective-mass approximation

In this paper we derive an analytical expression of free Green’s functions or propagators of graphene in real space by using the effective-mass approximation. We find that the propagator is highly spatial anisotropic and has threefold symmetry. We also present a theoretical model for explaining scanning tunnel microscope (STM) images on graphene by combining our explicit formula with the $T$-matrix approach. We derive an analytical expression of the local density of states near a single impurity. The details of the superstructure in STM images induced by a single impurity are discussed as well. The standing-wave patterns in quantum corrals on the graphene surface can be seen clearly from our simulations.