Reversible switching of anomalous valley Hall effect in ferrovalley Janus $1T-\mathrm{CrO}X (X=\mathrm{F},\mathrm{Cl},\mathrm{Br},\mathrm{I})$ and the multiferroic heterostructure $\mathrm{CrO}X/{\mathrm{In}}_2{\mathrm{Se}}_3$

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 $1T-\mathrm{CrO}X (X=\mathrm{F},\mathrm{Cl},\mathrm{Br},\mathrm{I})$, 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.

Figure 1

(a) The crystal structures of $1T-\mathrm{CrO}X$ ($X=\mathrm{F}$, Cl, Br, I) from top and side views and the schematic diagram of the first Brillouin zone. (b) The electron localization function of CrOCl and CrOBr. (c) The differential charge densities of CrOCl and CrOBr.

Figure 2

(a), (b) are the band structures and the orbital-projected band structures under PBE for CrOCl with spin-orbit coupling, while (c), (d) are the cases for CrOBr.

Figure 3

(a), (b) are the VS1 and VS2 curves of CrOCl and CrOBr with different magnetocrystalline angles. (c), (d) are the MAE curves of CrOCl and CrOBr with different magnetocrystalline angles. The $E_0$ is set to zero when $\theta ={90}^{\circ }$.

Figure 4

(a), (b) are the Berry curvatures of CrOCl and CrOBr. (c), (d) are the band structures of CrOCl and CrOBr for PBE and Wannier function fitting, respectively.

Figure 5

(a), (b) are the valley splitting curves of CrOCl and CrOBr with different biaxial strain. (c) The MAE curves of CrOCl and CrOBr with different biaxial strain.

Figure 6

(a) The structures of 2D vdW multiferroic heterostructure CrOCl/${\mathrm{In}}_2{\mathrm{Se}}_3$ from side views for different configurations. $P\uparrow$ and $P\downarrow$ are the polarized states of FE material ${\mathrm{In}}_2{\mathrm{Se}}_3$. (b) The spin-projected band structures of the four stacking configurations.

Figure 7

(a)–(d) The electrostatic potentials and differential charge densities of the four stacking configurations.

Figure 8

The schematic of reversible switch of the AVHE under the action of an in-plane electric field.