Reversible switching of anomalous valley Hall effect in ferrovalley Janus 1 T − CrO X ( X = F , Cl , Br , I ) and the multiferroic heterostructure CrO X / In 2 Se 3

The central issue for practical applications of the anomalous valley Hall effect (AVHE) is the tunable and nonvolatile nature of the valley splitting. We predict a type of ferrovalley material, Janus $1T\ensuremath{-}\mathrm{CrO}X (X=\mathrm{F},\mathrm{Cl},\mathrm{Br},\mathrm{I})$, in which the switching effect of the AVHE can be achieved by adjusting the biaxial strain or building a multiferroic heterostructure $\mathrm{CrO}X/{\mathrm{In}}_{2}{\mathrm{Se}}_{3}$. Stable out of plane magnetization in $\mathrm{CrO}X$ induces the valley splitting which can reach to 112 meV in the CrOBr monolayer. Interestingly, we find that the valley splitting of CrOCl is robust against the biaxial strain both in the conduction band and the valence band. In contrast, the valley splitting of the CrOBr at the conduction band can be linearly modulated, while it has a switching response at the valence band due to the strong orbital hybridization induced by compressive strain, so a reversible switch of the AVHE can be achieved. Furthermore, the electric reversible valley splitting switch is also obtained in multiferroic van der Waals heterostructure $\mathrm{CrOCl}/{\mathrm{In}}_{2}{\mathrm{Se}}_{3}$, and the reversible switch of the AVHE can also be manipulated by controlling the polarization states of the ferroelectric layer. The AVHE in $\mathrm{CrO}X$ can be readily switched on or off by either applying biaxial strain or reversing the ferroelectric polarization of the substrate ${\mathrm{In}}_{2}{\mathrm{Se}}_{3}$, which may be a promising application in the field of valleytronics.