Spin Hall Effect in Monolayer and Bilayer WSe$_2$*

Transition metal dichalcogenides (TMDs) are expected to possess a large spin Hall effect thanks to the strong spin-orbit coupling in these materials. The recent progress in realization of high mobility WSe$_2$ samples with Ohmic contacts [1] can facilitate the experimental access to the rich physics of the TMDs. Using non-local resistance measurements in multi-terminal, high mobility hole-doped dual-gated WSe$_2$ samples we extract the spin Hall conductivity as a function of carrier density and temperature. We find that WSe$_2$ possesses a spin Hall conductivity that is weakly dependent on the carrier density in the range $5\times10^{12}$ cm$^{-2} - 10\times10^{12}$ cm$^{-2}$, increases with reducing the temperature down to 1.5 K, and is significantly larger than $e^2/h$. We discuss the different mechanisms, namely intrinsic, side-jump, and skew scattering that contribute to the measured spin Hall conductivity. [1] B. Fallahazad $et$ $al.$, Phys. Rev. Lett. 116, 086601 (2016).