Magnetically and electrically controllable valley splittings in MXene monolayers

The modulation of the valley structure in two-dimensional valley materials is vital in the field of valleytronics. The multiferroicity provides possibility for multiple modulations of the valley, including the magnetic and electric means. Based on the first-principle calculations, we study the valley properties and associated manipulations of multiferroic Co$_2$CF$_2$ monolayers with different stacking patterns. Our calculations show that the Co$_2$CF$_2$ monolayer in the H$^{\prime}$ phase is a ferrovalley material, with sizable valley splittings. By rotating the magnetization direction, the valley splittings can be tuned for both the magnitude and sign. The electric field, driving the reversal of the electric polarization, can also change the magnitude of the valley splittings. Besides, a metastable T$^{\prime}$ phase exhibits valley splittings as well, of which the magnitude and sign can be simultaneously controlled by applied magnetic and electric fields. These findings offer a practical way for realizing highly tunable valleys by multiferroic couplings.