  Using first-principles calculations within density functional theory, we
investigate the intrinsic spin Hall effect in monolayers of group-VI
transition-metal dichalcogenides MX2 (M = Mo, W and X = S, Se). MX2 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- and a spin-Hall effect. The intrinsic spin Hall conductivity (ISHC) in
p-doped samples is found to be much larger than the ISHC in n-doped samples due
to the large spin-splitting at the valence band maximum. We also show that the
ISHC 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.
