Multifield tunable valley splitting in two-dimensional MXene Cr$_2$COOH

Manipulation of the valley degree of freedom provides a novel paradigm in quantum information technology. Here, through first-principles calculations and model analysis, we demonstrate that monolayer Cr$_2$COOH MXene is a promising candidate material for valleytronics applications. We reveal that Cr$_2$COOH is a ferromagnetic semiconductor and harbors valley features. Due to the simultaneous breaking inversion symmetry and time-reversal symmetry, the valleys are polarized spontaneously. Moreover, the valley polarization is sizeable in both the valence and conduction bands, benefiting the observation of the anomalous valley Hall effect. More remarkably, the valley splitting can be effectively tuned by the magnetization direction, strain and ferroelectric substrate. More interestingly, the ferroelectric substrate Sc$_2$CO$_2$ can not only regulate the MAE, but also tune valley polarization state. Our findings offer a practical way for realizing highly tunable valleys by multiferroic couplings.