Space charge and screening of a supercritical impurity cluster in monolayer graphene

A Coulomb impurity of charge $Ze$ is known to destabilize the ground state of undoped graphene with respect to creation of screening space charge when $Z$ exceeds the critical value $Z_c=1/2\alpha$ set by the material's fine structure constant $\alpha$. Recent experimental advances have made it possible to explore this transition in a controlled manner by tuning $Z$ across the critical point. Combined with relatively large value of $\alpha$ this makes it possible to study graphene's screening response to a supercritical impurity $Z\alpha \gg 1$ when the screening charge is large using a Thomas-Fermi analysis. The character of screening in this regime is controlled by the dimensionless screening parameter $Z{\alpha }^2$. Specifically, for a circular impurity cluster in the weak-screening regime $Z{\alpha }^2\ll 1$ most of the screening charge is found to reside outside the cluster. The strong-screening regime $Z{\alpha }^2\gg 1$ provides a realization of the Thomson atom: Most of the screening charge is inside the cluster, nearly perfectly neutralizing the source charge with the exception of a transition layer near the cluster's edge where the rest of the space charge is localized.