Impurity-Induced Spin-Orbit Coupling in Graphene

We study the effect of impurities in inducing spin-orbit coupling in graphene. We show that the $s{p}^3$ distortion induced by an impurity can lead to a large increase in the spin-orbit coupling with a value comparable to the one found in diamond and other zinc-blende semiconductors. The spin-flip scattering produced by the impurity leads to spin scattering lengths of the order found in recent experiments. Our results indicate that the spin-orbit coupling can be controlled via the impurity coverage.

Figure 1

Top: Top view of the graphene lattice with its orbitals. The orbitals associated with the impurity and lattice distortion are shown in solid black. (a) $s{p}^3$ orbital at impurity position; (b) $s{p}^2$ orbital of the flat graphene lattice.

Figure 2

Top: Energy (in eV) of the $\pi$ (blue) and $\sigma$ (red) bands as a function of $A$ according to (3). Bottom: Relative value of the SO coupling at the impurity site relative to the atomic value in carbon as a function of $A$ according to (8).

Figure 3

Decay of the $\sigma$ band propagator which determines the effective spin-orbit coupling as function of the distance to the carbon atom with partial $s{p}^3$ coordination. The model for the $\sigma$ band is discussed in 10. The inset shows the model used to study the scattering process around an atom with partial $s{p}^3$ coordination (see text for details).

Figure 4

Cross section for a spin-flip process for a defect as described in the text. The parameters used are $R_1=1\AA$, $R_2=2\AA$, and ${\Delta }_{\mathrm{SO}}^I=1\mathrm{meV}$ (blue) and ${\Delta }_{\mathrm{SO}}^I=2\mathrm{meV}$ (red).