Microscopic theory of quantum anomalous Hall effect in graphene

We present a microscopic theory to give a physical picture of the formation of quantum anomalous Hall (QAH) effect in graphene due to a joint effect of Rashba spin-orbit coupling $\lambda_R$ and exchange field $M$. Based on a continuum model at valley $K$ or $K'$, we show that there exist two distinct physical origins of QAH effect at two different limits. For $M/\lambda_R\gg1$, 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 \emph{AB} sublattice pseudo-spin textures; while for $\lambda_R/M\gg1$, the four-band low-energy model Hamiltonian is reduced to a two-band extended Haldane's model, giving rise to a nonzero Chern number $\mathcal{C}=1$ at either $K$ or $K'$. In the presence of staggered \emph{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 $K'$ 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 a trivial phase when $\lambda_{SO}>(\sqrt{M^2+\lambda^2_R}+M)/2$. Moreover, we use a tight-binding model to reproduce the ab-initio method obtained band structures through doping magnetic atoms on $3\times3$ and $4\times4$ supercells of graphene, and explain the physical mechanisms of opening a nontrivial bulk gap to realize the QAH effect in different supercells of graphene.Comment: 10pages, ten figure

Two-Dimensional Topological Insulator State and Topological Phase Transition in Bilayer Graphene

We show that gated bilayer graphene hosts a strong topological insulator (TI) phase in the presence of Rashba spin-orbit (SO) coupling. We find that gated bilayer graphene under preserved time-reversal symmetry is a quantum valley Hall insulator for small Rashba SO coupling $\lambda_{\mathrm{R}}$, and transitions to a strong TI when $\lambda_{\mathrm{R}} > \sqrt{U^2+t_\bot^2}$, where $U$ and $t_\bot$ are respectively the interlayer potential and tunneling energy. Different from a conventional quantum spin Hall state, the edge modes of our strong TI phase exhibit both spin and valley filtering, and thus share the properties of both quantum spin Hall and quantum valley Hall insulators. The strong TI phase remains robust in the presence of weak graphene intrinsic SO coupling.Comment: 5 pages and 4 figure