Valley-contrasting transport of the coupling between spin and orbital angular momentum in ferromagnetic transition metal dichalcogenides

Transition metal dichalcogenides exhibit intrinsic spin-orbit coupling that opens spin splittings in valleys, resulting in spin and orbital polarized bands. Energy valleys driven by electric fields can generate spin (anomalous) Hall effect and orbital Hall effect. Here we focus on ferromagnetic materials and demonstrate that the transport of spin and orbital Hall effects can be attributed to the spin-orbit coupling, which can be characterized by spin-orbital polarization. Driven by an electric field, orbital and spin angular momentum appear at both ends of the sample through corresponding Hall effect, leading to the formation of valley-contrasting spin-orbital polarization states. Based on the spin and orbital transport analysis, we propose the intrinsic magnetic moment as a method to manipulate the spin-orbital polarization states, which enables the four-state switching of the angular momentum at the edge of sample. Moreover, due to the conservation of angular momentum, circularly polarized optical pumping can be used instead of an electric field to selectively excite the orbital and spin angular momentum. Subsequently, we argue that the sign-reversal spin-orbital polarization requires not only spin-orbit coupling effect but also the simultaneous broken time-reversal and inversion symmetries.