Intrinsic and extrinsic spin-orbit coupling and spin relaxation in monolayer PtSe${}_2$

Monolayer PtSe${}_2$ is a semiconducting transition metal dichalcogenide characterized by an indirect band gap, space inversion symmetry, and high carrier mobility. Strong intrinsic spin-orbit coupling and the possibility to induce extrinsic spin-orbit fields by gating make PtSe${}_2$ attractive for fundamental spin transport studies as well as for potential spintronics applications. We perform a systematic theoretical study of the spin-orbit coupling and spin relaxation in this material. Specifically, we employ first principles methods to obtain the basic orbital and spin-orbital properties of PtSe${}_2$, also in the presence of an external transverse electric field. We calculate the spin mixing parameters $b^2$ and the spin-orbit fields $\mathrm{\Omega }$ for the Bloch states of electrons and holes. This information allows us to predict the spin lifetimes due to the Elliott-Yafet and D’yakonov-Perel mechanisms. We find that $b^2$ is rather large, on the order of ${10}^{-2}$ and ${10}^{-1}$, while $\mathrm{\Omega }$ varies strongly with doping, being about $10^{3} - 10^{4}$ns$^{-1}$for$(10-100)$ meV, carrier density in the interval ${10}^{13}-{10}^{14}$,cm${}^{-2}$ at the electric field of 1 V/nm. We estimate the spin lifetimes to be on the picosecond level.