Large spontaneous valley polarization and anomalous valley Hall effect in antiferromagnetic monolayer $\mathrm{Fe_2CF_2}$

Superior to ferromagnetic (FM) materials, antiferromagnetic (AFM) materials do not have any net magnetic moment and are robust to external magnetic perturbation with ultra-high dynamic speed. To achieve spontaneous valley polarization and anomalous valley Hall effect (AVHE) in AFM materials is of great significance for potential applications in spintronics and valleytronics. Here, we predict an A-type AFM monolayer $\mathrm{Fe_2CF_2}$ with large spontaneous valley polarization. Monolayer $\mathrm{Fe_2CF_2}$ has zero Berry curvature in momentum space but non-zero layer-locked hidden Berry curvature in real space, which provides the basic conditions for the realization of AVHE. Because $\mathrm{Fe_2CF_2}$ possesses the combined symmetry ($PT$ symmetry) of spatial inversion ($P$) and time reversal ($T$) symmetry, the spin is degenerate, which prevents the AVHE. An out-of-plane external electric field can be used to produce spin splitting due to the introduction of layer-dependent electrostatic potential, and then layer-locked AVHE can be realized in $\mathrm{Fe_2CF_2}$. Moreover, the spin order of spin splitting can be reversed, when the direction of electric field is reversed. It is proved that the AVHE can be achieved in Janus $\mathrm{Fe_2CFCl}$ without external electric field due to intrinsic built-in electric field. Our works provide an AFM monolayer with excellent properties to realize AVHE.