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

TL;DR

This work addresses how purely radial in-plane spin-orbit fields (radial Rashba) arise in commensurate twisted TMDC homobilayers. It combines first-principles density functional theory with a two-band effective model (including valley-Zeeman and Rashba SOC) to describe twist-angle dependent spin textures near the $K$ and $\Gamma$ points. A key finding is that an in-plane $180^{\circ}$ rotation symmetry stabilizes the radial Rashba pattern, while reductions in symmetry (as in WTe$_2$) lead to non-radial textures; the interlayer coupling $w$ and the radial Rashba magnitude $|\lambda_R\sin(\Phi)|$ govern the strength of the effect and vary with twist angle and supercell size. The results provide microscopic insights for engineering spin-charge conversion in twisted TMDCs and offer design principles for spintronic applications in layered materials.

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 $Γ$ 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 ($Θ=0^\circ$ and $Θ=60^\circ$), but also around $Θ=30^\circ$. 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.

Figures (10)

• Figure 1: (a)-(d) Top view of exemplary investigated commensurate supercells. In (e) possible twisted commensurate bilayer supercell sizes $N_{at}$ (number of atoms) are indicated for different twist angles $\Theta$. For each supercell data point, we color-code where the $K/K'$ points of the layers fold back to: For red downward-pointing triangles, the $K$ points of both layers fold back to the same point in the supercell's FBZ ($K\leftrightarrow K$). For blue upward-pointing triangles, $K$ of layer 1 and $K'$ of layer 2 fold on top of each other and vice versa ($K\leftrightarrow K'$). For green dots all $K$- and $K'$-points of both layers fold to $\Gamma$. The last cases are not discussed in this paper and only listed for completeness.
• Figure 2: Band structure along high symmetry points and in-plane spin textures around $K$ and $\Gamma$ of the 38.2$^\circ$ twisted WSe$_2$ homobilayer, which shows a $K\leftrightarrow K$ backfolding. Out-of-plane spin is color coded from blue (spin down) over grey to red (spin up).
• Figure 3: Band structures along high symmetry points of exemplary NbSe$_2$ homobilayers, all of which exhibit a $K\leftrightarrow K$ backfolding. (a) shows the untwisted case with a lateral shifting position, in which a metal atom of one layer resides on top of a chalcogen atom of the other layer. The split bands are layer polarized due to the breaking of the in-plane mirror symmetry. In (b), we additionally show the in-plane spin textures around $K$ and $\Gamma$ for the -38.2$^\circ$ case. (c) and (d) show the band structures of 27.8$^\circ$ and -46.8$^\circ$ cases. Out-of-plane spin is color coded from blue (spin down) over grey to red (spin up).
• Figure 4: Schematic depiction of the bands of the twisted model Hamiltonian for different cases: backfolding of both $K$-points of the layers on top of each other ($K\leftrightarrow K$), backfolding of $K$-point of layer 1 on top of $K'$ of layer 2 ($K\leftrightarrow K'$) and when describing the $\Gamma$ bands. In the latter case the backfolding is irrelevant. We depict for all cases the parabolic (valence) bands and their color coded spin-$z$ expectation values with analytical expressions of the splittings (with assumption $\lambda_{VZ}>>w$ for the first two cases and $\lambda_{VZ}<<w$ for the third case). Finally, we also depict the in-plane spin textures of the four bands ordered by energy.
• Figure 5: DFT extracted spin-orbit fields around $K$ and $\Gamma$ for (a) twisted WSe$_2$ and (b) twisted NbSe$_2$ homobilayers. In (a) the maximal range of the rings is at a $k$-radius of $0.3^\cdot10^{-3}\frac{2\pi}{a_s}$, where $a_s$ is the supercell lattice constant. In (b) one additional ring (with a $k$ radius of $0.1\frac{2\pi}{a_s}$) extends this range significantly. At each $k$ point, two arrows are plotted for two adjacent bands, e.g. VB1 and VB2. Direction (not magnitude) of the in-plane spin textures are represented by the arrows. The out-of-plane spins are color coded from red over grey to blue. In (a) we additionally show a top view of the supercell (with the 180$^\circ$ rotation axis, if present) for each different lateral shifting. Twist angles, backfolding scenarios and lateral shifts are defining the columns. The rows are defined by the band pairs (sorted by energy) and the relevant close $k$-point ($K$ or $\Gamma$). VB1/VB2 are omitted in (a) due to problems with determining spin expectation values unambiguously, due to a near-degeneracy of the bands.