Coulomb Impurity Problem in Graphene

We address the problem of an unscreened Coulomb charge in graphene and calculate the local density of states and displaced charge as a function of energy and distance from the impurity. This is done nonperturbatively in two different ways: (1) solving the problem exactly by studying numerically the tight-binding model on the lattice and (2) using the continuum description in terms of the 2D Dirac equation. We show that the Dirac equation, when properly regularized, provides a qualitative and quantitative low energy description of the problem. The lattice solution shows extra features that cannot be described by the Dirac equation: namely, bound state formation and strong renormalization of the van Hove singularities.

Figure 1

(a) Comparison of the LDOS (solid line) with the numerical results in the lattice (dashed line), calculated at different distances from the impurity and $g=1/6$. The DOS for $g=0$ is also included for comparison (dotted-dashed line). For clarity, curves at different $r$ have been vertically displaced. (b) LDOS at the site closest to the impurity in the lattice for $g=1/3$. (c) Quantization condition (11) with $j=1/2$, for $g=0.1$ and 0.4.

Figure 2

(a)–(c) The LDOS in the lattice (dashed line, recursion method) is compared with the first contribution in (18) (solid line) for different distances from the impurity. (d) The second contribution in (18) (solid line) and the free, linear, DOS for reference. (e) First contribution in (18); the inset is a magnification for $ϵ\simeq 0$. In all panels, $g=4/3$.

Figure 3

(a) Induced charge numerically obtained from exact diagonalization on a lattice with ${124}^2$ sites ($g<g_c$). (b) Idem ($g>g_c$). (c) Evolution of the ordered numerical spectrum with $g$, $E_n$ being the $n$th energy level. The symmetries of the lattice cause some $E_n$ to merge as $g\to 0$. (d) Analytical $\delta n(r)$ obtained using (19) and regularization discussed after (20).