Intrinsic and anisotropic Rashba spin splitting in Janus transition-metal dichalcogenide monolayers

Transition-metal dichalcogenides (TMDs) monolayers have been considered as important two-dimensional semiconductor materials for the study of fundamental physics in the field of spintronics. However, the out-of-plane mirror symmetry in TMDs may constrain electrons' degrees of freedom and it may limit spin-related applications. Recently, a newly synthesized Janus TMDs MoSSe was found to intrinsically possess both the in-plane inversion and the out-of-plane mirror-symmetry breaking. Here we performed first-principles calculations in order to systematically investigate the electronic band structures of a series of Janus monolayer TMDs with chemical formula $MXY\phantom{\rule{0.16em}{0ex}}(M=\mathrm{Mo},\phantom{\rule{0.16em}{0ex}}\mathrm{W}\phantom{\rule{0.16em}{0ex}}\mathrm{and}\phantom{\rule{0.16em}{0ex}}X,Y=\mathrm{S},\phantom{\rule{0.16em}{0ex}}\mathrm{Se},\phantom{\rule{0.16em}{0ex}}\mathrm{Te})$. It is found that they possess robust electronic properties like their parent phases. We explored also the effect of perpendicular external electric field and in-plane biaxial strain on the Rashba spin splittings. The Zeeman-type spin splitting and valley polarization at $K({K}^{\ensuremath{'}})$ point are well preserved and we observed a Rashba-type spin splitting around the $\mathrm{\ensuremath{\Gamma}}$ point for all the $MXY$ systems. We have also found that these spin splittings can be enhanced by an external electric field collinear with the local electric field derived by the polar bonds and by the compressive strain. The Rashba parameters change linearly with the external electric field, but nonlinearly with the biaxial strain. The compressive strain is found to enhance significantly the anisotropic Rashba spin splitting.