Electrical Control\nof the Valley–Layer Hall\nEffect in Ferromagnetic Bilayer Lattices

The layertronics based on the layer degree of freedom\nare of essential\nsignificance for the construction and application of new-generation\nelectronic devices. Although the Hall layer effect has been realized\ntheoretically and experimentally, it is mainly based on topological\nand antiferromagnetic lattices. On the basis of the low-energy effective <i>k</i>·<i>p</i> model, the mechanism of the controllable\nvalley–layer Hall effect (V–LHE) in a bilayer ferromagnetic\nlattice through interlayer sliding has been proposed. Due to the broken\ntime-reversal and inversion symmetries, the V–LHE based on\nthe valley, layer degree of freedom, ferromagnetism, and ferroelectricity\ncan be predicted. In addition, valley and layer indexes can be controlled\nby magnetization orientation and slipping, respectively. The mechanism\ncan be demonstrated in the real bilayer CrSI lattice through first-principles\ncalculations. Moreover, V–LHE can be effectively tuned by the\nperpendicular external electric field in configurations without out-of-plane\npolarization. These findings provide a new platform for the research\nof valleytronics and layertronics.