Valley-related multiple Hall effect in monolayer $\mathrm{V}{\mathrm{Si}}_2{\mathrm{P}}_4$

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.

Figure 1

(a) Crystal structure of SL $\mathrm{V}{\mathrm{Si}}_2{\mathrm{P}}_4$ from top and side views. The solid line indicates the primitive cell and inset in (a) shows the local structure containing two vertical Si atoms and their ligands. (b) 2D Brillouin zone with marking the high-symmetry points. (c) Electron localization function of SL $\mathrm{V}{\mathrm{Si}}_2{\mathrm{P}}_4$. (d) Phonon spectra of SL $\mathrm{V}{\mathrm{Si}}_2{\mathrm{P}}_4$. (e) The splitting of d orbitals under the trigonal prismatic crystal field.

Figure 2

Spin-polarized band structure of SL $\mathrm{V}{\mathrm{Si}}_2{\mathrm{P}}_4$ (a) without and (b) with considering SOC. (c) Orbital-resolved spin-polarized band structure of SL $\mathrm{V}{\mathrm{Si}}_2{\mathrm{P}}_4$ with considering SOC. (d) is same as (b), but with opposite magnetization. The Fermi level is set to 0 eV.

Figure 3

Berry curvature of SL $\mathrm{V}{\mathrm{Si}}_2{\mathrm{P}}_4$ (a) as a curve along the high-symmetry points and (b) as a counter map over the 2D Brillouin zone. (c,d) Diagrams of the anomalous valley Hall effect under electron doping and an in-plane electric field. (d) is the same as (c), but with opposite magnetization.

Figure 4

(a) Energy difference between different magnetic configurations (${\mathrm{\Delta }}_{\mathrm{AFM}-\mathrm{FM}}$: energy difference between AFM and FM configurations; ${\mathrm{\Delta }}_{\mathrm{NM}-\mathrm{FM}}$: energy difference between NM and FM configurations) as a function of strain. (b) Band structure and (c) edge state of SL $\mathrm{V}{\mathrm{Si}}_2{\mathrm{P}}_4$ under −2.1% strain. Inset in (b) shows the enlarged band edges around the $+K$ and $-K$ points. The Fermi level is set to 0 eV.