Electric-field-tuned anomalous valley Hall effect in A -type hexagonal antiferromagnetic monolayers

The combination of antiferromagnetic (AFM) spintronics and anomalous valley Hall effect (AVHE) is of great significance for potential applications in valleytronics. Here, we propose a way for achieving AVHE in $A$-type hexagonal AFM monolayer. The proposed way involves the introduction of layer-dependent electrostatic potential caused by an out-of-plane external electric field, which can break the combined symmetry ($PT$ symmetry) of spatial inversion ($P$) and time reversal ($T$), producing spin splitting. The spin order of spin splitting can be reversed by regulating the direction of electric field. Based on first-principles calculations, the way can be verified in AFM ${\mathrm{Cr}}_{2}{\mathrm{CH}}_{2}$. The layer-locked hidden Berry curvature can give rise to layer-Hall effect, including a valley layer--spin Hall effect and layer-locked AVHE. Moreover, we propose Janus monolayer ${\mathrm{Cr}}_{2}\mathrm{CHF}$ with internal electric polarization, which can also realize the AVHE. Our works provide an experimentally feasible way to realize AVHE in AFM monolayer.