Coupled Spin and Valley Physics in Monolayers of ${\mathrm{MoS}}_2$ and Other Group-VI Dichalcogenides

We show that inversion symmetry breaking together with spin-orbit coupling leads to coupled spin and valley physics in monolayers of ${\mathrm{MoS}}_2$ and other group-VI dichalcogenides, making possible controls of spin and valley in these 2D materials. The spin-valley coupling at the valence-band edges suppresses spin and valley relaxation, as flip of each index alone is forbidden by the valley-contrasting spin splitting. Valley Hall and spin Hall effects coexist in both electron-doped and hole-doped systems. Optical interband transitions have frequency-dependent polarization selection rules which allow selective photoexcitation of carriers with various combination of valley and spin indices. Photoinduced spin Hall and valley Hall effects can generate long lived spin and valley accumulations on sample boundaries. The physics discussed here provides a route towards the integration of valleytronics and spintronics in multivalley materials with strong spin-orbit coupling and inversion symmetry breaking.

Figure 1

(a) The unit cell of bulk $2H$-${\mathrm{MoS}}_2$, which has the inversion center located in the middle plane. It contains two unit cells of ${\mathrm{MoS}}_2$ monolayers, which lacks an inversion center. (b) Top view of the ${\mathrm{MoS}}_2$ monolayer. ${\mathit{R}}_i$ are the vectors connecting nearest Mo atoms. (c) Schematic drawing of the band structure at the band edges located at the $K$ points.

Figure 2

Coupled spin and valley physics in monolayer group-VI dichalcogenides. The electrons and holes in valley $K$ are denoted by white ‘$-$’, ‘$+$’ symbol in dark circles and their counterparts in valley $-K$ are denoted by inverse color. (a) Valley and spin Hall effects in electron and hole-doped systems (see text). (b) Valley and spin optical transition selection rules. Solid (dashed) curves denote bands with spin-down (-up) quantized along the out-of-plane direction. The splitting in the conduction band is exaggerated. ${\omega }_u$ and ${\omega }_d$ are, respectively, the transition frequencies from the two split valence-band tops to the conduction band bottom. (c) Spin and valley Hall effects of electrons and holes excited by linearly polarized optical field with frequency ${\omega }_u$. (d) Spin and valley Hall effects of electrons and holes excited by two-color optical fields with frequencies ${\omega }_u$ and ${\omega }_d$ and opposite circular polarizations.