Intrinsic anomalous valley Hall effect in single-layer N b 3 I 8

A major obstacle to reaching the wide applications of valleytronics operations is the lack of suitable anomalous valley Hall (AVH) materials, which should be easily fabricated and exhibit spontaneous valley polarization. Here, we identify an excellent two-dimensional AVH material, namely single-layer $\mathrm{N}{\mathrm{b}}_{3}{\mathrm{I}}_{8}$, based on first principles. Single-layer $\mathrm{N}{\mathrm{b}}_{3}{\mathrm{I}}_{8}$ is a robust ferromagnetic semiconductor with a moderate band gap. More importantly, due to the intrinsic breakings of both time-reversal symmetry and inversion symmetry, it exhibits the long-sought spontaneous valley polarization, without requiring any external tuning. Spontaneous valley polarization can reach as large as 107 meV, which is beneficial for practical operations. Furthermore, the experimental exfoliation of single-layer $\mathrm{N}{\mathrm{b}}_{3}{\mathrm{I}}_{8}$ is highly feasible as the stable layered bulk phase exists. Our findings provide an ideal platform for exploring the AVH effect and future valleytronic applications.