Coulomb impurity in graphene

We consider the problem of screening of an electrically charged impurity in a clean graphene sheet. When electron-electron interactions are neglected, the screening charge has a sign opposite to that of the impurity and is localized near the impurity. Interactions between electrons smear out the induced charge density to give a large-distance tail that follows approximately, but not exactly, an $r^{-2}$ behavior and with a sign which is the same as that of the impurity.

Figure 1

Feynman diagrams for the charge density without electron-electron interactions to order ${\alpha }^3$. The filled square is the impurity site, the wavy line is the $A_{\tau }$ propagator, the line is the fermion propagator, and the filled circle is the charge density operator.

Figure 2

(Color online) The dependence of $2\pi Z^{-1}{r}^{2}n\left(r\right)$ on the distance $r$ for suspended graphene. Note that coordinate $r$ is on a logarithmic scale.

Figure 3

(Color online) The dependence of $2\pi Z^{-1}{r}^{2}n\left(r\right)$ on the distance $r$ for graphene on a substrate with $ϵ=4.5$. Note that coordinate $r$ is on a logarithmic scale.