Intrinsic spin Hall effect in monolayers of group-VI dichalcogenides: A first-principles study

Using first-principles calculations within density functional theory, we investigate the intrinsic spin Hall effect in monolayers of group-VI transition-metal dichalcogenides $M{X}_2$ ($M=\mathrm{Mo},\mathrm{W}$ and $X=\mathrm{S},\mathrm{Se}$). $M{X}_2$ monolayers are direct band-gap semiconductors with two degenerate valleys located at the corners of the hexagonal Brillouin zone. Because of the inversion symmetry breaking and the strong spin-orbit coupling, charge carriers in opposite valleys carry opposite Berry curvature and spin moment, giving rise to both a valley-Hall and a spin-Hall effect. We also show that the intrinsic spin Hall conductivity in inversion-symmetric bulk dichalcogenides is an order of magnitude smaller compared to monolayers. Our result demonstrates monolayer dichalcogenides as an ideal platform for the integration of valleytronics and spintronics.

Figure 1

(a) Side view of the unit cell of the 2$H$-$M{X}_2$ structure with $M=\mathrm{Mo}$, W and $X=\mathrm{S}$, Se. It contains two $M{X}_2$ monolayers separated by a van der Waals gap. (b) Top view of the $M{X}_2$ monolayer. The black lines indicate the unit cell in $ab$ plane. (c) The first Brillouin zone and high-symmetry points of the $M{X}_2$ monolayer. (d) The Wannier functions of the $M{X}_2$ monolayer, including five $d$ orbitals on $M$ atom and three $p$ orbitals on each $X$ atom.

Figure 2

The calculated electronic structure of MoS${}_2$ with the spin-orbit coupling. (a) The partial density of states for Mo-4$d$ and S-3$p$ orbitals, respectively, in the unit of states/eV/cell. (b) The band structure with the projection of spin operator ${\widehat{s}}_z$ (color map). The red and blue colors indicate the spin-up and -down states, respectively. The optical transitions between the VBM and the CBM are coupled exclusively with $\sigma +$ ($\sigma -$) circular polarizations at the inequivalent valleys $K$ ($K^{\prime }$) (Ref. 10).

Figure 3

The Berry curvatures of monolayer MoS${}_2$ along the high-symmetry lines (a) and in the 2D $k$ plane (b). The spin Berry curvatures of monolayer MoS${}_2$ along the high-symmetry lines (c) and in the 2D $k$ plane (d). All of the Berry curvatures are in the atomic unit (bohrs${}^2$).

Figure 4

(a) The intrinsic spin Hall conductivity ${\sigma }_{xy}^s$ ($e^2/\hslash$) as a function of the Fermi energy for monolayer MoS${}_2$. The energy zero point (true Fermi level) is at the VBM. Two large peaks close to the CBM and VBM are denoted by $P_1$ and $P_2$, respectively. (b) and (c) The band structure (up panel) and the spin Berry curvature (down panel) when the Fermi level shifts to the positions of the peak $P_1$ and $P_2$, respectively. There is a small peak of ${\Omega }^s$ with negative value along the $M$-$\Gamma$ line in (c). (d) The $n$- and $p$-doping charge as a function of the Fermi energy. The low doping regimes just above the CBM and below the VBM (indicated by red circles) are more relevant in experiments.

Figure 5

The intrinsic spin Hall conductivity ${\sigma }_{xy}^s$ in the low doping regimes for the monolayer (a)–(d) and bulk (e)–(h) of the MoS${}_2$, MoSe${}_2$, WS${}_2$, and WSe${}_2$, respectively. Dashed lines indicate the band edges. Note that the band gaps of bulk are smaller than the monolayer ones. The unit of ${\sigma }_{xy}^s$ is $e^2/\hslash$ ($\cong 2.43\times {10}^{-4}{\Omega }^{-1}$) for 2D monolayer system, whereas ${\Omega }^{-1}$ cm${}^{-1}$ for 3D bulk system. For quantitatively comparing the ${\sigma }_{xy}^s$ in bulk and monolayer, one needs to divide the ${\sigma }_{xy}^s$ in monolayer by its thickness.

Figure 6

The band structure of WS${}_2$ and WSe${}_2$ monolayers. The CBM of WSe${}_2$ monolayer still locates at $K$ point. The dashed line indicates the position of the Fermi level at carrier concentration of $1.0\times {10}^{13}$ cm${}^{-2}$ for both $p$- and $n$-doped samples.