Valley–Layer–Polarized Anomalous Hall Effect in a Two-Dimensional Multiferroic van der Waals Material

Emerging phenomena such as the valley Hall effect and layer Hall effect, showing promise for next-generation electronics and valleytronic devices, have attracted considerable attention. However, most studies of the layer Hall effect have been restricted to antiferromagnetic or topological systems. Based on first-principles calculations, we establish a valley-layertronics framework and predict that two-dimensional ScI₂ is a multiferroic material exhibiting substantial spontaneous valley polarization in both its monolayer (93.4 meV) and bilayer (93.7 meV) forms. Furthermore, we uncover a tunable valley-layer-polarized anomalous Hall effect in a ferromagnetic bilayer stacking configuration of ScI₂ that can be jointly tuned by ferroelectric polarization and magnetization direction. Our findings provide a robust platform for advancing valley-layertronics applications in 2D multiferroic materials.