Valley-related multiple Hall effect in monolayer V Si 2 P 4

Two-dimensional materials with valley-related multiple Hall effect are both fundamentally intriguing and practically appealing for exploring novel phenomena and applications, but have been largely overlooked to date. Here, using first-principles calculations, we show that the valley-related multiple Hall effect can exist in single-layer $\mathrm{V}{\mathrm{Si}}_{2}{\mathrm{P}}_{4}$. We identify single-layer $\mathrm{V}{\mathrm{Si}}_{2}{\mathrm{P}}_{4}$ as a ferromagnetic semiconductor with out of plane magnetization and valley physics. Arising from the joint effect of inversion symmetry breaking and time-reversal symmetry breaking, the exotic spontaneous valley polarization occurs in single-layer $\mathrm{V}{\mathrm{Si}}_{2}{\mathrm{P}}_{4}$, thus facilitating the observation of anomalous valley Hall effect. Moreover, under external strain, band inversion can occur at only one of the valleys of single-layer $\mathrm{V}{\mathrm{Si}}_{2}{\mathrm{P}}_{4}$, enabling the long-sought valley-polarized quantum anomalous Hall effect, and meanwhile the anomalous valley Hall effect is well preserved. Our work not only enriches the research on valley-related multiple Hall effect, but also provides an ideal platform for exploring valley-polarized quantum anomalous Hall effect.