Microscopic theory of quantum anomalous Hall effect in graphene

Abstract

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 ${K}^{\ensuremath{'}}$, 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 ${\ensuremath{\lambda}}_{R}$, 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 ${K}^{\ensuremath{'}}$. 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 ${K}^{\ensuremath{'}}$ are different when ${\ensuremath{\lambda}}_{R}$, $M$, and $U$ are simultaneously considered. We also show that the QAH phase is robust against weak intrinsic spin-orbit coupling ${\ensuremath{\lambda}}_{SO}$, and it transitions to a trivial phase when ${\ensuremath{\lambda}}_{SO}>(\sqrt{{M}^{2}+{\ensuremath{\lambda}}_{R}^{2}}+M)/2$. Moreover, we use a tight-binding model to reproduce the ab initio method obtained band structures through doping magnetic atoms on $3\ifmmode\times\else\texttimes\fi{}3$ and $4\ifmmode\times\else\texttimes\fi{}4$ supercells of graphene, and explain the physical mechanisms of opening a nontrivial bulk gap to realize the QAH effect in different supercells of graphene.