Spin Hall effect in the monolayer Janus compound MoSSe enhanced by Rashba spin-orbit coupling

Sizable Rashba spin-orbit coupling (SOC) is of critical importance in potential applications of two-dimensional materials in spintronics devices. However, due to the presence of vertical mirror symmetry, Rashba SOC is absent in 2H transition-metal dichalcogenide monolayers and the spin Hall conductivities are attributed only to Zeeman splitting. We study theoretically the electronic structures and intrinsic spin Hall conductivity of two-dimensional monolayer Janus MoSSe by performing first-principles calculations as well as by using the Kubo formula with Wannier interpolations. We find out that monolayer Janus MoSSe possesses considerable spin Hall conductivities both in conduction and valence bands. In valence bands, the spin Hall conductivity of Janus MoSSe is comparable to that in ${\mathrm{MoS}}_2$ and ${\mathrm{MoSe}}_2$. Moreover, in the conduction bands, the spin Hall conductivities are enhanced up to two orders of magnitude because of strong Rashba SOC. The spin Hall conductivity can be tuned significantly by adjusting the Fermi level or external strains. Our results show that monolayer Janus MoSSe could be a potential candidate to realize two-dimensional flexible spintronics devices.

Figure 1

(a) Top and side views of monolayer Janus MoSSe. The unit cell is indicated by gray solid lines, and the molybdenum, sulfur, and selenium atoms are indicated by cyan, yellow, and orange spheres, respectively. (b) Calculated phonon dispersion along high-symmetry points in the Brillouin zone of the monolayer Janus MoSSe.

Figure 2

(a) The band structures with projected $d$ orbitals (left panel), where the $d_{x^2-y^2}+d_{xy}$ and $d_{z^2}$ orbitals are drawn with the red and blue colored dots, respectively. The decomposed DOS of $p$ orbitals of S and Se atoms as well as $d$ orbitals of Mo atoms of the monolayer Janus MoSSe (right panel). (b) The band structures with the projection of spin operator $s_z$ (color map). The red and blue colors indicate spin-up and spin-down, respectively. The right panel is an enlarged figure of the CBM at the inequivalent valleys $K$ ($K^{\prime }$). The Fermi surface is set at the zero energy level and is labeled as a red dashed line.

Figure 3

(a) From left to right, the band structures of the Janus MoSSe monolayer under tensile biaxial strains, and the tensile biaxial strains are measured as $\epsilon =2\%$, 4%, 6%, respectively. The inset labels are defined in Table 1. (b) The spin-splitting energy of Rashba SOC at the $\mathrm{\Gamma }$ point of the valence band vary with wave vector $k$ under tensile strains $\epsilon =0\%$ (black line), $\epsilon =2\%$ (green line), $\epsilon =4\%$ (blue line), and $\epsilon =6\%$ (red line). (c) The perpendicular build-in electric field ${\overline{E}}_{\mathrm{int}}$ varies with tensile biaxial strain from $0\%$ to $6\%$.

Figure 4

The spin Berry curvature near the $K$ valley in (a) the valence band and (b) the conduction band, respectively. The red and blue lines indicate the higher- and lower-energy spin-splitting band at the $K$ valley, respectively. The solid and dashed lines denote the valence and conduction bands at the $K$ valley, respectively.

Figure 5

The spin Berry curvature and the SHC of monolayer Janus MoSSe. Band structures of monolayer Janus MoSSe projected by spin Berry curvature on a log scale are shown in (a) the valence-band region and (e) the conduction-band region. The $k$-resolved spin Berry curvatures along the high-symmetry points at $E_1$ and $E_2$ in the monolayer Janus MoSSe (upper panel), ${\mathrm{MoS}}_2$ (middle panel), and ${\mathrm{MoSe}}_2$ (lower panel) are shown in panels (b) and (f), respectively. The $k$-resolved spin Berry curvatures on a log scale in the contour plot of the 2D BZ at (c) $E_1$ and (g) $E_2$ in the monolayer Janus MoSSe. The SHCs of the monolayer Janus MoSSe (black line), monolayer ${\mathrm{MoS}}_2$ (read line), and monolayer ${\mathrm{MoSe}}_2$ (blue line) are shown in the (d) valence-band region and (h) conduction-band region. For comparison, the VBM (CBM) is set at the zero energy level in the figures. The red (purple) dash horizontal line represents $E_1$ ($E_2$) in panel (a) [(e)], which are also indicated by arrow in panels (d) [(h)].

Figure 6

The SHCs of the monolayer Janus MoSSe as a function of the Fermi energy and under different tensile biaxial strains $\epsilon =0\%$ (black line), $\epsilon =2\%$ (red line), $\epsilon =4\%$ (blue line), $\epsilon =6\%$ (green line) (a) valence-band region, (b) conduction-band region. For comparison, their VBM and CBM are set at the zero energy level indicated by arrows in the figures.