Microscopic origin of Rashba coupling from first principles: Layer-resolved orbital asymmetry in transition metal dichalcogenides

TL;DR

This work addresses the microscopic origin of Rashba spin splitting in two-dimensional transition metal dichalcogenides by combining first-principles density functional theory with a perturbative orbital model. It introduces state-resolved Rashba parameters λ_R^n and intrinsic orbital fields E_0^n, and defines the orbital polarization imbalance ξ^n and its derivative χ_orb^n as quantitative descriptors of spin ordering and spin-splitting strength. The study shows that monolayers lack linear Rashba due to mirror symmetry, while certain bilayer stackings intrinsically break both inversion and mirror symmetries, producing a tunable Rashba effect that is strongly influenced by chalcogen mass and orbital mediators such as d_{xz/yz} and p_{x/y} orbitals. The results provide a quantitative–qualitative framework for predicting and engineering Rashba coupling in layered van der Waals systems through stacking and gating.

Abstract

Spin-orbit coupling in two-dimensional materials gives rise to a Rashba spin splitting when inversion and mirror symmetries are broken, yet its microscopic origin and quantitative characterization in transition metal dichalcogenides remains incomplete. Both symmetries are broken in certain bilayer structures, enabling Rashba splittings in the absence of external electric fields. We determine this zero-field offset and the Rashba parameters that dictate the spin splitting in the linear regime. Surprisingly, the splitting is substantially smaller in bilayers than in monolayers at typical fields. This is clarified within a perturbative microscopic model, revealing that the spin splitting results from a competition between internal polarization and interlayer hybridization. We further introduce the orbital polarization imbalance as an order parameter that captures the asymmetry of the valence bands and determines the spin ordering of the Rashba-split states. Our results are both quantitative and qualitative, as they clarify the nature and origin of Rashba coupling in transition metal dichalcogenides.

Figures (6)

• Figure 1: Band structure of MoSe$_2$ with color-coded $\langle \hat{\textbf{S}}_{y} \rangle$ projections for ML (a) and BL (b) in the R$_\mathrm{X}^\mathrm{M}$ stacking. The insets correspond to the zoom-ins of the target bands, with equal energy (13 meV) and momentum window (from ${\mathrm{K}}/{16}$ to ${\mathrm{K}^{\prime}}/{16}$) for quantitative comparison between the states. The right inset in (a) shows the spin splitting $\Delta$E for different electric fields (yellow to blue: 0.05, 0.1 and 0.2 V/$\mathrm{\AA}$). Wave function representation of the top VB of a ML under an external electric field $E_{\mathrm{ext}}$ (c), and of AVB (d) and BVB (e) in the BL at zero field. The color of the orbitals schematically represents their phase. Red arrows represent the orbital polarization at the ML level.
• Figure 2: (a) Schematic $\langle \hat{\textbf{S}}_{y} \rangle$-arrangement between spin-split AVB and BVB. The color code defined in (a) is indicated by the background colors in (b) and (c). $\Delta \mathrm{E}/k_x$ as a function of the external field $E_{\mathrm{ext}}$ for AVB and BVB in BL MoSe$_2$ (b) and BL WSe$_2$ (c). Band-dependent Rashba parameters $\lambda_\mathrm{R}^n$ in units of eV$\mathrm{\AA}^2$/V are reported for the relevant states under investigation together with the intrinsic orbital field $E_0^n$ in units of V/$\mathrm{\AA}$.
• Figure 3: Rashba coefficient $\lambda_\mathrm{R}^n$ (a) and intrinsic orbital field $E_0^n$ (b) of the VB maximum for different TMD MLs, as well as for the AVB and BVB states in the corresponding TMD BLs.
• Figure 4: Plane-averaged $\langle \rho_{nk} \rangle_{xy}$ as a function of the out-of-plane coordinate $z$ at $k=\mathrm{K}/40$ for the MoSe$_2$ BL states AVB (a) and BVB (b), respectively. The red arrows indicate the sign (not the magnitude) of the ML $\xi^{n}_{\mathrm{ML}}$ contribution to the BL $\xi^{n}$. Order parameter $\xi^{n}$ for AVB (blue arrows) and BVB (green arrows). The arrows depict the expectation value of the spin of the upper splitted band in the $z$-$y$ plane (see coordinate system). The vertical dotted lines depict the electric field that closes the Rashba splitting in the AVB (blue) and BVB (green) in units of V/$\mathrm{\AA}$.
• Figure 5: Orbital polarizability $\chi_{\mathrm{orb}}^n$ for the VB maximum of various TMD MLs, as well as the AVB and BVB states of the TMD homo-BLs.