Screening of Coulomb impurities in graphene

We calculate exactly the vacuum polarization charge density in the field of a subcritical Coulomb impurity, Z vertical bar e vertical bar/r, in graphene. Our analysis is based on the exact electron Green's function, obtained by using the operator method, and leads to results that are exact in the parameter Z alpha, where alpha is the fine-structure constant of graphene. Taking into account also electron-electron interactions in the Hartree approximation, we solve the problem self-consistently in the subcritical regime, where the impurity has an effective charge Z(eff), determined by the localized induced charge. We find that an impurity with bare charge Z=1 remains subcritical, Z(eff)alpha < 1/2, for any alpha, while impurities with Z=2, 3 and higher can become supercritical at certain values of alpha.