Resonant finite-size impurities in graphene, unitary limit, and Friedel oscillations

A unitary limit for model point scatterers in graphene is known to reveal low-energy resonances. The same limit could be achieved from hybridization of band electrons with the localized impurity level positioned in the vicinity of the Fermi level. The finite-size defects represent an easier realization of the effective unitary limit, occurring when the Fermi wavelength induced by the potential becomes of the order of the size of the defect. We calculate the induced electron density and find two signatures of a strong impurity, independent of its specific realization. The dependence of the impurity-induced electron density on the distance changes near resonances from proportional to r(-3) to proportional to r(-2). The total number of induced particles at the resonance is equal to one per degree of spin and valley degeneracy. The effects of doping on the induced density are found.