Screening of Coulomb Impurities in Graphene

We calculate exactly the vacuum polarization charge density in the field of a subcritical Coulomb impurity, $Z|e|/r$, 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 $\alpha$.