Spin-valley coupling in a two-dimensional $\mathrm{V}{\mathrm{Si}}_2{\mathrm{N}}_4$ monolayer

Materials that integrate magnetism, miniaturization, and valley properties hold potential for spintronic and valleytronic nanodevices. Recently, ferromagnetism was reported to be able to exist in the $\mathrm{V}{\mathrm{Si}}_2{\mathrm{N}}_4$ monolayer which is half-metallic and belongs to a new kind of two-dimensional material [Hong et al., Science 369, 670 (2020)]. Using first-principles calculations and model analysis, we find that $\mathrm{V}{\mathrm{Si}}_2{\mathrm{N}}_4$ is a ferromagnetic semiconductor harboring valley-contrasting physics and a magnetic critical temperature over room temperature. By tuning magnetization orientation from in plane to out of plane, valley polarization can be generated, resulting in the anomalous valley Hall effect in $\mathrm{V}{\mathrm{Si}}_2{\mathrm{N}}_4$. Furthermore, we obtain the formula for energy splitting of valleys and adopt a tight-binding model for $\mathrm{V}{\mathrm{Si}}_2{\mathrm{N}}_4$, which elucidates the physical mechanism of spin-valley coupling. More interestingly, under 4% tensile strain, the intrinsic magnetic anisotropy of $\mathrm{V}{\mathrm{Si}}_2{\mathrm{N}}_4$ becomes out of plane, and spontaneous valley polarization is achieved. Our results highlight that $\mathrm{V}{\mathrm{Si}}_2{\mathrm{N}}_4$ is a good candidate for spintronic and valleytronic applications.

Figure 1

(a) The top and side views of the crystal structure, (b) electron localization function, (c) phonon dispersions, and (d) magnetic anisotropy of $\mathrm{V}{\mathrm{Si}}_2{\mathrm{N}}_4$ monolayer. In (a), (b), the red, gray, and blue balls represent V, N, and Si elements, respectively.

Figure 2

The band structure of $\mathrm{V}{\mathrm{Si}}_2{\mathrm{N}}_4$ monolayer with SOC when the magnetization is along (a) $+x$ and (b) $+z$. (c) The valley polarization $\mathrm{\Delta }E$ as a function of magnetization orientation. The magnetization rotates from $-z$ to $+z$. (d) The orbital-resolved band structure of $\mathrm{V}{\mathrm{Si}}_2{\mathrm{N}}_4$ monolayer with SOC when the magnetization is along $+z$.

Figure 3

For the $\mathrm{V}{\mathrm{Si}}_2{\mathrm{N}}_4$ monolayer: the distributions of Berry curvature (a) in the whole Brillouin zone and (b) along the high-symmetry points. (c) The anomalous Hall conductivity as a function of $E_F$. (d) The schematics of the anomalous valley Hall effect under hole doping (left panel) and light irradiation (right panel). The magnetization is along $+z$. The electrons and holes in valley $+K$ are denoted by white $-,+$ in the dark circles, and the red and blue arrows represent spin-up and spin-down states.

Figure 4

(a) The magnetic anisotropy (blue line) and valley polarization (orange line) as functions of strain. For the $\mathrm{V}{\mathrm{Si}}_2{\mathrm{N}}_4$ monolayer under 6% tensile strain: the distributions of Berry curvature (b) in the whole Brillouin zone and (c) along the high-symmetry points. (d) The anomalous Hall conductivity as a function of $E_F$. (e) The schematics of the anomalous valley Hall effect under electron doping (left panel) and light irradiation (right panel). The magnetization is along $+z$. The electrons and holes in valley $-K$ are denoted by dark $-,+$ in the hollow circles, and the red and blue arrows represent spin-up and spin-down states.