Stable valley-layer coupling and design principle in 2D lattice

Stable valley-layer coupling, which can be against spin–orbit coupling (SOC), is of both fundamental and technological importance as it offers a design principle for 2D valleytronics; however, a reliable mechanism to achieve the goal is missing. In this Letter, a general rule to design such valley-layer coupling is mapped out from symmetry analysis. The degenerate valleys with valley-contrasted layer physics and protected valley-layer coupling can be present in bilayer lattice, when special symmetry operations between layers depending on the inversion center are satisfied. Such valley-layer coupling and its stability against SOC are further revealed in a real material of MnF4 based on first-principles. The distinctive properties, such as optical selection of valley and electric polarization of interlayer excitons, are observed in such a unique system. Our results not only provide a feasible principle to design materials with stable valley-layer coupling but also greatly enrich the physics and broaden the scientific impact of 2D valleytronics.