Localized Magnetic States in Graphene

We examine the conditions necessary for the presence of localized magnetic moments on adatoms with inner shell electrons in graphene. We show that the low density of states at the Dirac point, and the anomalous broadening of the adatom electronic level, lead to the formation of magnetic moments for arbitrarily small local charging energy. As a result, we obtain an anomalous scaling of the boundary separating magnetic and nonmagnetic states. We show that, unlike any other material, the formation of magnetic moments can be controlled by an electric field effect.

Figure 1

(a) Honeycomb lattice with an impurity atom. Black: sublattice A; white: sublattice B. Intersection of the Dirac cone spectrum, ${ϵ}_{\pm }(\mathbf{k})=\pm v_{F}k$, with the localized level spectrum, $E_f(\mathbf{k})={ϵ}_0$: (b) ${ϵ}_0>0$; (c) ${ϵ}_0<0$.

Figure 2

Boundary between magnetic and nonmagnetic impurity states in the scaling variables $x$ and $y$ for ${ϵ}_0>0$ (a) and ${ϵ}_0<0$ (b). Circles: $|{ϵ}_0|/D=0.029$, $V/D=0.14$; squares: ${ϵ}_0/D=0.043$ and $V/D=0.14$; triangles:$|{ϵ}_0|/D=0.029$, $V/D=0.03$. The upturn close to $y=1$ and $x\to 0$ on panel (b) is not visible in this scale when $V$ is very small (triangles). See details in the text.

Figure 3

$n_{\uparrow }(\mu )$, $n_{\downarrow }(\mu )$, and $\chi (\mu )$ for $|{ϵ}_0|/D=0.029$ and $V/D=0.14$ ($D\sim 7\mathrm{eV}$). Left panels: $x=11$ (dashed curves), and $x=5$ (solid curves). The impurity magnetizes inside the bubble ($n_{\uparrow }\ne n_{\downarrow })$. The vertical line marks the position of the level, ${ϵ}_0=0.2\mathrm{eV}$. On the right: ${ϵ}_0=0.2\mathrm{eV}$ (solid curve) and ${ϵ}_0=-0.2\mathrm{eV}$ (dashed curve) at $x=0.45$.

Figure 4

Spectral function (in units of 1/eV) of the $f$ electrons at the Fermi energy $\mu$ for $|{ϵ}_0|/D=0.029$ and $V/D=0.14$ ($D\sim 7\mathrm{eV}$). (a) $x=11$ (solid curve) and $x=5$ (dashed curve) for ${ϵ}_0>0$ (see Fig. 2). (b) $x=0.45$ for ${ϵ}_0>0$ (solid curve) and ${ϵ}_0<0$ (dashed curve).