Microscopic theory of quantum anomalous Hall effect in graphene

We present a microscopic theory to give a physical picture of the formation of the quantum anomalous Hall (QAH) effect in magnetized graphene coupled with Rashba spin-orbit coupling. Based on a continuum model at valley $K$ or $K^{\prime }$, we show that there exist two distinct physical origins of the QAH effect at two different limits. For large exchange field $M$, the quantization of Hall conductance in the absence of Landau-level quantization can be regarded as a summation of the topological charges carried by skyrmions from real-spin textures and merons from AB sublattice pseudospin textures, while for strong Rashba spin-orbit coupling ${\lambda }_R$, the four-band low-energy model Hamiltonian is reduced to a two-band extended Haldane model, giving rise to a nonzero Chern number $\mathcal{C}=1$ at either $K$ or $K^{\prime }$. In the presence of staggered AB sublattice potential $U$, a topological phase transition occurs at $U=M$ from a QAH phase to a quantum valley Hall phase. We further find that the band gap responses at $K$ and $K^{\prime }$ are different when ${\lambda }_R$, $M$, and $U$ are simultaneously considered. We also show that the QAH phase is robust against weak intrinsic spin-orbit coupling ${\lambda }_{SO}$, and it transitions to a trivial phase when ${\lambda }_{SO}>(\sqrt{M^2+{\lambda }_R^2}+M)/2$. Moreover, we use a tight-binding model to reproduce the ab initio method obtained band structures through doping magnetic atoms on $3\times 3$ and $4\times 4$ supercells of graphene, and explain the physical mechanisms of opening a nontrivial bulk gap to realize the QAH effect in different supercells of graphene.

Figure 1

(a) Chern number of each valence band as a function of Rashba spin-orbit coupling ${\lambda }_R$ at fixed exchange field $M/t=0.3$. (b) Chern number of each valence band as a function of exchange field M at fixed Rashba spin-orbit coupling ${\lambda }_R/t=0.9$. The solid (blue) line represents the summation of the two valence bands. The cutoff of $k_x$ and $k_y$ is set to be $k_0=\pi /2a$.

Figure 2

Real-spin and pseudospin textures of the two valence bands of graphene below the band gap at valleys $K$ and $K^{\prime }$ in the limit of $M/{\lambda }_R\gg 1$ (here, we set $M=0.8$ and ${\lambda }_R=0.12$). (a)–(d): Real-spin textures of the 1st and 2nd valence bands at valleys $K$ and $K^{\prime }$. (a) and (c): Spins point toward the south pole uniformly. (b) and (d): Spins at the center point toward the south pole while those far away from the center point toward the north pole, which indicates a skyrmion. (e)–(h): Pseudospin textures of the 1st and 2nd valence bands at valleys $K$ and $K^{\prime }$. (e) and (f): In-plane pseudospin components share the similar winding pattern pointing toward the center, but out-of-plane pseudospin components point toward south and north poles, respectively. (g) and (h): In-plane pseudospin components point toward the center without any winding, while out-of-plane pseudospin components point toward north and south poles, respectively. Each pseudospin texture corresponds to a meron or half skyrmion.

Figure 3

Schematic plot of (a) $4\times 4$ and (b) $3\times 3$ supercells of graphene. On-site crystal field potential, Rashba spin-orbit coupling, and exchange field are only considered on the highlighted sites.

Figure 4

(a) Bulk band structure of the $3\times 3$ supercell of graphene in the presence of only on-site potential $V_0/t=0.50$. A band gap $\Delta$ opens at the $\Gamma$ point. (b) Bulk gap $\Delta$ quadratically increases as a function of $V_0$.

Figure 5

Bulk band structure of the $3\times 3$ supercell of graphene in the presence of on-site potential $V_0/t=0.50$ and exchange field $M$. (a) $M/t=0.05$, (b) $M/t=0.20$. $\uparrow$ and $\downarrow$ denote up and down spin polarization.

Figure 6

(a) Bulk band structure of the $3\times 3$ supercell of graphene in the presence of on-site potential $V_0/t=0.50$, exchange field $M/2=0.20$, and Rashba spin-orbit coupling $t_R/t=0.05$. (b) The magnification of the band gap selected by the dashed square.

Figure 7

Total Berry curvature distribution $\Omega$ in the momentum space of the occupied valence bands below the band gap. Only those around the $\Gamma$ point are nonzero and share the same negative sign.

Figure 8

(a) Bulk band structure of the $4\times 4$ supercell of graphene in the presence of on-site potential $V_0/t=0.50$. (b) Magnification of bands near the Dirac crossing point. No gap opens at $K$ and $K^{\prime }$.

Figure 9

(a) Bulk band structure of the $4\times 4$ supercell of graphene in the presence of on-site potential $V_0/t=0.50$ and exchange field $M/t=0.20$. (b) Magnification of bands near the Dirac crossing point. Arrows are used to denote the up and down spin polarizations.

Figure 10

(a) Bulk band structure of the $4\times 4$ supercell of graphene in the presence of on-site potential $V_0/t=0.50$, exchange field $M/t=0.20$, and Rashba spin-orbit coupling $t_R/2=0.05$. (b) Magnification of bands near the Dirac crossing point. Bulk band gaps open at $K$ and $K^{\prime }$.