Quantum anomalous Hall effect in single-layer and bilayer graphene

The quantum anomalous Hall effect can occur in single- and few-layer graphene systems that have both exchange fields and spin-orbit coupling. In this paper, we present a study of the quantum anomalous Hall effect in single-layer and gated bilayer graphene systems with Rashba spin-orbit coupling. We compute Berry curvatures at each valley point and find that for single-layer graphene the Hall conductivity is quantized at ${\sigma }_{\mathit{xy}}=2e^2/h$, with each valley contributing a unit conductance and a corresponding chiral edge state. In bilayer graphene, we find that the quantized anomalous Hall conductivity is twice that of the single-layer case when the gate voltage $U$ is smaller than the exchange field $M$, and zero otherwise. Although the Chern number vanishes when $U>M$, the system still exhibits a quantized valley Hall effect, with the edge states in opposite valleys propagating in opposite directions. The possibility of tuning between different topological states with an external gate voltage suggests possible graphene-based spintronics applications.

Figure 1

(a)-(c) Bulk states band structure. (a). $M=0$, $\alpha =0$, (b). $M/{\varepsilon }_c=0.1$, $\alpha =0$, (c). $M/{\varepsilon }_c=0.1$, $\alpha /{\varepsilon }_c=0.05$. $k_c=2\pi /a$ ($a$ is the graphene lattice constant) and ${\varepsilon }_c$ are momentum and energy cut-off of the Dirac model, beyond which the energy dispersion deviates from linearity due to trigonal warping. (d). Berry curvature $\Omega =2({\Omega }_{+-}+{\Omega }_{--})$ (the factor of two arises from the two valleys) for $\alpha /{\varepsilon }_c=0.05$, $M/{\varepsilon }_c=0.1$. $\Omega$ peaks at the $k$ value where the upper valance band has its maximum and degeneracy between opposite spin states occurs when SO coupling is absent [see panel (b)].

Figure 2

Band structure at $\alpha /{\varepsilon }_c=5\times {10}^{-3}$ and $M/{\varepsilon }_c=0.02$ for (a) $U/{\varepsilon }_c=0.01$, (b) $U/{\varepsilon }_c=0.02$, and (c) $U/{\varepsilon }_c=0.03$. (d) Band gap as a function of $U$ at $\alpha /{\varepsilon }_c=0.005$ and $M/{\varepsilon }_c=0.02$, 0.04, and 0.06 as shown in legend.

Figure 3

Phase diagram of the Chern number $\mathcal{C}$ as a function of $U$ and $M$. The valley Chern number occupies complementary regions of the phase space with ${\mathcal{C}}_v=0$ for $U<M$ and $4\hspace{0.5em}\mathrm{sgn}(U)$ for $U>M$.

Figure 4

Edge-state band structure of a single-layer graphene ribbon with $M=0.1885$ and $t_R=0.0471$ in units of the near-neighbor hopping amplitude $t$ in the tight-binding Hamiltonian in Eq. (18). These values correspond to $M/{\varepsilon }_c=0.02$ and $\alpha /{\varepsilon }_c=7.5\times {10}^{-3}$ in the low-energy Hamiltonian in Eq. (1).

Figure 5

(a–d) Probability density across the single-layer graphene sheet for the edge-state wave functions $\left|\psi \right|{}^2$ of the edge states labeled A, B, C, and D in Fig. 4. The inset is a schematic which indicates the propagation direction of the corresponding edge modes.

Figure 6

Edge-state band structure of bilayer graphene at fixed Rashba SO coupling strength (in units of the hopping amplitude $t$) $t_R=0.0471$ for (a) $M>U$, $M=0.1885$ and $U=0.0942$; (b) $U>M$, $M=0.0942$ and $U=0.2826$. These values correspond in the continuum Hamiltonian in Eq. (14) to $\alpha /{\varepsilon }_c=7.5\times {10}^{-3}$, (a) $M/{\varepsilon }_c=0.02$ and $U/{\varepsilon }_c=0.01$, and (b) $M/{\varepsilon }_c=0.01$ and $U/{\varepsilon }_c=0.03$.

Figure 7

(a–h) Probability density of the edge-state wave function $\left|\psi \right|{}^2$ for the edge states A, B, C, D, E, F, G, and H labeled in Fig. 6 for the cases $M>U$ (left) and $U>M$ (right) as a function of atom position.

Figure 8

Schematic showing the direction of edge-mode propagation (indicated by arrows) in (a) the quantum anomalous Hall phase and (b) the quantum valley Hall phase.

Figure 9

Band structure from the envelope-function approach and tight-binding model. The Rashba spin-orbit strength and exchange field in the tight-binding model are (in units of $t$) $t_R=0.02$ and $M=0.1$ in Eq. (18), corresponding to $\alpha /{\varepsilon }_c=2.12\times {10}^{-3}$ and $M/{\varepsilon }_c=0.01$ in the low-energy Dirac Hamiltonian. The width of the graphene sheet is $L=119$ in units of the nearest-neighbor lattice constant.