Radial Rashba spin-orbit fields in commensurate twisted transition-metal dichalcogenide bilayers

Abstract

In commensurate twisted homobilayers, purely radial Rashba spin-orbit fields can emerge. We employ first-principles calculations to investigate the band structures and the spin-orbit fields close to the high-symmetry points $K$ and $\Gamma$ of several commensurate twisted transition-metal dichalcogenide homobilayers: WSe$_2$, NbSe$_2$, and WTe$_2$. The observed in-plane spin textures are mostly radial, and the main features are successfully reproduced using a model Hamiltonian based on two effective mass models including spin-orbit coupling, and a general (spin-conserving) interlayer coupling. Extracting the model Hamiltonian parameters through fitting of several twisted supercells, we find a twist angle dependency of the magnitude of the radial Rashba field, which is symmetric not only around the untwisted cases ($\Theta=0^\circ$ and $\Theta=60^\circ$), but also around $\Theta=30^\circ$. Furthermore, we observe that the interlayer coupling between the $K/K'$-points of the two layers decreases with the increase of the size of the commensurate supercells. Hence, peaks of high interlayer coupling can occur only for twist angles, where small commensurate supercells are possible. Exploring different lateral displacements between the layers, we confirm that the relevant symmetry protecting the radial Rashba is an in-plane 180$^\circ$ rotation axis. We additionally investigate the effects of atomic relaxation and modulation of the interlayer distance. Our calculations on WTe$_2$ bilayers show that their lack of $C_3$ symmetry results in spin textures that are neither radial nor tangential. Our results offer fundamental microscopic insights that are particularly relevant to engineering spin-charge conversion schemes based on twisted layered materials.