Local density of states and Friedel oscillation in graphene

We investigate the local density of states and Friedel oscillation in graphene around a well localized impurity in Born approximation. In our analytical calculations Green's function technique has been used taking into account both the localized atomic wavefunctions in a tight-binding scheme and the corresponding symmetries of the lattice. As a result we obtained long wavelength oscillations in the density of electrons with long range behavior proportional to the inverse square of the distance from the impurity. These leading oscillations are out of phase on nearby lattice sites (in fact for an extended defect they cancel each other within one unit cell), therefore a probe with resolution worse than a few unit cells will experience only the next to leading inverse cube decay of density oscillations even for a short range scatterer.