Ferro-Valleytricity\nwith In-Plane Spin Magnetization

Ferro-valleytricity that manifests spin–orbit\ncoupling (SOC)-induced\nspontaneous valley polarization is generally considered to occur in\ntwo-dimensional (2D) materials with out-of-plane spin magnetization.\nHere, we propose a mechanism to realize SOC-induced valley polarization\nand ferro-valleytricity in 2D materials with in-plane spin magnetization,\nwherein the physics correlates to non-collinear magnetism in triangular\nlattice. Our model analysis provides comprehensive ingredients that\nallow for ferro-valleytricity with in-plane spin magnetization, revealing\nthat mirror symmetry favors remarkable valley polarization and time-reversal-mirror\njoint symmetry should be excluded. Through modulating the in-plane\nspin magnetization offset, the SOC-induced valley polarization could\nbe reversed. Followed by first-principles, such a mechanism is demonstrated\nin a multiferroic triangular lattice of single-layer W<sub>3</sub>Cl<sub>8</sub>. We further show that the reversal of valley polarization\ncould also be driven by applying an electric field that modulates\nferroelectricity. Our findings greatly enrich valley physics research\nand significantly extend the scope for material classes of ferro-valleytricity.