Out-of-plane carrier spin in transition-metal dichalcogenides under electric current

Significance Although out-of-plane current-induced spin polarization (CISP) is important for perpendicular-magnetization reorientation, and it has been realized by the crystalline symmetry breaking, its material examples are relatively fewer compared with in-plane CISP in Rashba systems. With the help of intrinsic spin–orbit coupling, an intriguing out-of-plane CISP is designed in transition-metal dichalcogenides by the symmetry breaking in the spin space, which provides opportunities for out-of-plane magnetization rotation and electric control of valley splitting. Moreover, the spin polarization is associated with valley-dependent responses to electric current and adds a dimension, valley degree of freedom, to the study of CISP. The symmetry argument and non-Rashba effective model also helps to illuminate these physics and broaden the scope of the CISP. Absence of spatial inversion symmetry allows a nonequilibrium spin polarization to be induced by electric currents, which, in two-dimensional systems, is conventionally analyzed using the Rashba model, leading to in-plane spin polarization. Given that the material realizations of out-of-plane current-induced spin polarization (CISP) are relatively fewer than that of in-plane CISP, but important for perpendicular-magnetization switching and electronic structure evolution, it is highly desirable to search for new prototypical materials and mechanisms to generate the out-of-plane carrier spin and promote the study of CISP. Here, we propose that an out-of-plane CISP can emerge in ferromagnetic transition-metal dichalcogenide monolayers. Taking monolayer VSe2 and VTe2 as examples, we calculate the out-of-plane CISP based on linear-response theory and first-principles methods. We deduce a general low-energy model for easy-plane ferromagnetic transition-metal dichalcogenide monolayers and find that the out-of-plane CISP is due to an in-plane magnetization together with intrinsic spin–orbit coupling inducing an anisotropic out-of-plane spin splitting in the momentum space. The CISP paves the way for magnetization rotation and electric control of the valley quantum number.