Quantum anomalous Hall effect in graphene from Rashba and exchange effects

We investigate the possibility of realizing quantum anomalous Hall effect in graphene. We show that a bulk energy gap can be opened in the presence of both Rashba spin-orbit coupling and an exchange field. We calculate the Berry curvature distribution and find a nonzero Chern number for the valence bands and demonstrate the existence of gapless edge states. Inspired by this finding, we also study, by first-principles method, a concrete example of graphene with Fe atoms adsorbed on top, obtaining the same result.

Figure 1

Evolution of band structures of bulk graphene along the profile of $k_y=0$. Arrows represent the spin directions. (a) Pristine graphene: spin-up and spin-down states are degenerate; (b) when only exchange field $\lambda =0.4$ is applied, the spin-up/spin-down bands are upward/downward lifted with the four bands crossing near K and ${\text{K}}^{\prime }$ points; (c) when only Rashba SOC $t_{\text{SO}}=0.1$ is present, the spin-up and spin-down states are mixed around the band crossing points; (d) when both exchange field $\lambda =0.4$ and Rashba SOC $t_{\text{SO}}=0.1$ are present, a bulk gap is opened and all four bands become nondegenerate.

Figure 2

(Color online) (a) Energy spectrum of zigzag-edged graphene ribbons with $t_{\text{SO}}=0.2$ and $\lambda =0.18$. The Fermi level $E=0.06$ corresponds to four different edge states $A$, $B$, $C$, and $D$. $a$ is the lattice constant. (b) Wave-function distributions ${\left|\psi \right|}^2$ across the width for the four edge states (only part of the ribbons are shown): $A,B$ states are localized at the left boundary, with noticeable values of ${\left|\psi \right|}^2$ on about 30 carbon atoms while ${\left|\psi \right|}^2$ of $C$ and $D$ states are completely zero in this region.

Figure 3

(Color online) (a) Berry curvature distribution $\Omega$ (in units of $e^2/h$) of the valence bands in the momentum space. The first Brillouin zone is outlined by the dashed lines, and two inequivalent valleys are labeled as K and ${\text{K}}^{\prime }$. (b) The profile of Berry curvature distribution along $k_y=0$. The parameters used here are $t_{\text{SO}}=0.1$ and $\lambda =0.18$.

Figure 4

(Color online) [(a) and (b)] Top view of the spin configurations of the higher valence band in the $k_x\text{−}{k}_y$ plane around K and ${\text{K}}^{\prime }$ points. Blue/red (outer/inner) regimes represent the spin-up/spin-down states. The arrows represent the spin direction at each point.

Figure 5

(Color online) (a) Typical structure geometry of a Fe atom on top of the hollow position of a $4\times 4$ supercell of graphene. (b) Bulk band structure with SOC included along the high-symmetry lines. The Fermi level is exactly located in the gap. [(c) and (d)] Zooming in of the band structure at K and ${\text{K}}^{\prime }$ points. A gap $\sim 5.5\text{meV}$ circled by the green dotted curves is opened. [(e) and (f)] Berry curvature distribution $\Omega$ of the valence bands near K and ${\text{K}}^{\prime }$ points. Note: the Berry curvature at K and ${\text{K}}^{\prime }$ shares the same sign thus giving a nonzero Chern number $\mathcal{C}$.