Multimechanism quantum anomalous Hall and Chern number tunable states in germanene (silicene, stanene)/ MBi2Te4 heterostructures

By constructing germanene (silicene, stanene)/$M{\mathrm{Bi}}_{2}{\mathrm{Te}}_{4}$ ($M=3d$-transition elements) heterostructures, we discovered and designed multimechanism quantum-anomalous-Hall (QAH) systems, including $\mathrm{\ensuremath{\Gamma}}$-based QAH, $K\ensuremath{-}{K}^{\ensuremath{'}}$-connected QAH, and valley-polarized $K$- or ${K}^{\ensuremath{'}}$-based QAH states via first-principle computations. The unique systems possess a global gap and tunable Chern number. The coexisting conventional $\mathrm{\ensuremath{\Gamma}}$-based QAH state of $M{\mathrm{Bi}}_{2}{\mathrm{Te}}_{4}$ and valley-polarized $K$(${K}^{\ensuremath{'}}$)-based QAH state of germanene (silicene, stanene), with opposite chirality, can interact with each other. Adjusting magnetic configurations of $M{\mathrm{Bi}}_{2}{\mathrm{Te}}_{4}$-layers not only switch on (off) the QAH conductance, but also modulate Chern numbers exactly. For example, the germanene/bilayer-${\mathrm{NiBi}}_{2}{\mathrm{Te}}_{4}$ possesses the Chern number $\mathit{C}=+1$ in ferromagnetic couplings and $\mathit{C}=+2$ in antiferromagnetic couplings. The novel multimechanism QAH insulators, which are achievable in experiments, provide a new approach to spintronics and valleytronics based on topological states of matter.