Sliding- and twist-tunable valley polarization in bilayer NiI2

Valley, as an emerging degree of freedom of electron, has attracted extensive attention on account of its huge potential in electronic component technology. Two-dimensional (2D) materials provide an ideal platform for the research of valleytronics. Here, we study the sliding and twist effects on valley of bilayer ${\mathrm{NiI}}_{2}$ by the first-principles calculations. For a monolayer, spatial inversion symmetry maintains the degeneracy of two valleys. In the AA stacking bilayer, which can be obtained by a vertical translation operation on a monolayer structure, the valley band splitting is absent due to the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{P}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}$ joint symmetry. The interlayer sliding of the AA stacking bilayer can not break $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{P}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}$ joint symmetry and therefore there is not valley band splitting in a sliding system with respect to AA stacking. For the ${\mathrm{AA}}^{\ensuremath{'}}$ stacking bilayer, the valley band splitting occurs while the valley polarization is still absent as the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{{M}_{Z}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}$ joint symmetry. Different from the AA stacking system, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{{M}_{Z}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}$ joint symmetry of the ${\mathrm{AA}}^{\ensuremath{'}}$ system can be broken by interlayer sliding, and the valley polarization is realized. Furthermore, valley polarization is studied and it existed in twisted moir\'e structures with twist angles of $13.{174}^{\ensuremath{\circ}}$, $21.{787}^{\ensuremath{\circ}}$, $27.{796}^{\ensuremath{\circ}}$, $32.{204}^{\ensuremath{\circ}}$, ${38.213}^{\ensuremath{\circ}}$, and ${46.826}^{\ensuremath{\circ}}$, as the twisting breaks the spatial inversion symmetry. Our results broaden the valley polarization materials by interlayer sliding and twisting of 2D bilayer structures.