Electrical Control of the Valley–Layer Hall Effect in Ferromagnetic Bilayer Lattices

The layertronics based on the layer degree of freedom are of essential significance for the construction and application of new-generation electronic devices. Although the Hall layer effect has been realized theoretically and experimentally, it is mainly based on topological and antiferromagnetic lattices. On the basis of the low-energy effective k · p model, the mechanism of the controllable valley–layer Hall effect (V–LHE) in a bilayer ferromagnetic lattice through interlayer sliding has been proposed. Due to the broken time-reversal and inversion symmetries, the V–LHE based on the valley, layer degree of freedom, ferromagnetism, and ferroelectricity can be predicted. In addition, valley and layer indexes can be controlled by magnetization orientation and slipping, respectively. The mechanism can be demonstrated in the real bilayer CrSI lattice through first-principles calculations. Moreover, V–LHE can be effectively tuned by the perpendicular external electric field in configurations without out-of-plane polarization. These findings provide a new platform for the research of valleytronics and layertronics.