Electric-filed tuned anomalous valley Hall effect in A-type hexagonal antiferromagnetic monolayer

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 design principle for achieving AVHE in A-type hexagonal AFM monolayer. The design principle involves the introduction of layer-dependent electrostatic potential caused by 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 design principle can be verified in AFM $\mathrm{Cr_2CH_2}$. The layer-locked hidden Berry curvature can give rise to layer-Hall effect, including valley layer-spin Hall effect and layer-locked AVHE. Our works provide an experimentally feasible way to realize AVHE in AFM monolayer.