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

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.

💡 Research Summary

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This paper investigates the microscopic origin of Rashba spin–orbit coupling in two‑dimensional transition‑metal dichalcogenides (TMDs) using first‑principles density‑functional theory (DFT) and Wannier‑based tight‑binding analysis. The authors focus on the Γ‑point valence bands of both monolayers (ML) and homo‑bilayers (BL) in the 1H phase, where inversion (I) and mirror (m_z) symmetries can be broken either by an external out‑of‑plane electric field (E_ext) or by the intrinsic stacking geometry. In MLs, m_z is preserved in the absence of a field, so only a cubic spin splitting appears; a linear Rashba term emerges only when E_ext breaks m_z. In contrast, the R‑type stacking of BLs lacks both I and m_z even without external bias, giving rise to a built‑in “intrinsic orbital field” (E_n^0) that produces a zero‑field Rashba splitting.

The Rashba Hamiltonian is written as H_R = α_n^R (k_x σ_y – k_y σ_x) with α_n^R = λ_n^R (E_ext + E_n^0). By calculating the spin splitting ΔE = α_n^R k_x for several small k‑vectors and several values of E_ext, the authors extract both the Rashba coefficient λ_n^R (which quantifies the response to an applied field) and the intrinsic field E_n^0 (which quantifies the built‑in asymmetry). Linear fits of ΔE versus k_x give λ_n^R (E_ext + E_n^0); varying E_ext then yields λ_n^R and E_n^0 for each band n.

Key quantitative findings include:

• In MoSe₂ monolayers under E_ext = 0.2 V Å⁻¹, the Rashba splitting of the top valence band reaches ≈13 meV, corresponding to λ_A ≈ –0.126 eV Ų V⁻¹ (A‑band) and λ_B ≈ 0.934 eV Ų V⁻¹ (B‑band).
• In R‑type bilayers, even at zero external field the A‑band and B‑band display opposite spin textures with intrinsic fields E_A^0 = 0.151 V Å⁻¹ and E_B^0 = 0.042 V Å⁻¹. The corresponding Rashba coefficients are λ_A ≈ –0.177 eV Ų V⁻¹ and λ_B ≈ 1.135 eV Ų V⁻¹.
• The sign of λ_n^R determines whether the spin splitting grows or shrinks with increasing field; when E_ext = –E_n^0 the splitting vanishes, indicating a complete reversal of the spin texture.

A central conceptual advance is the introduction of an “orbital polarization imbalance” ξ_n, defined as the normalized difference between the plane‑averaged charge density above and below the transition‑metal plane (for monolayers) or the difference between the two constituent monolayers (for bilayers). ξ_n serves as an order parameter that tracks the internal orbital asymmetry responsible for the intrinsic field. The authors demonstrate that ξ_n exhibits a sharp step precisely at the external field where ΔE = 0, confirming that ξ_n directly encodes the competition between internal polarization and interlayer hybridization. Moreover, the direction of the step (sign change) correlates with the spin orientation of the upper Rashba‑split branch, establishing ξ_n as a predictor of spin ordering.

To rationalize these observations, the authors construct a perturbative tight‑binding model based on the Wannier Hamiltonian. The model treats atomic spin‑orbit coupling and inter‑orbital hopping (particularly d_{xz/yz} of the metal and p_{x/y} of the chalcogen) as first‑order perturbations. It reveals that the Rashba effect originates from a delicate balance: (i) the built‑in orbital field, which tends to polarize charge toward one side of the layer, and (ii) interlayer hybridization, which mixes bonding (B) and antibonding (A) states with opposite orbital character. In bilayers, the internal field is partially screened by the opposite polarization of the two constituent monolayers, leading to a smaller net Rashba splitting than in monolayers under comparable external fields.

Systematic trends across the TMD family are identified. Replacing S with heavier chalcogens (Se, Te) markedly increases both λ_n^R and |E_n^0|, reflecting enhanced polarizability (lower ionicity) and stronger atomic SOC. Conversely, changing the transition metal from Mo to W has a weaker effect because the increase in atomic SOC is offset by a reduction in orbital polarizability. The competition between these two factors explains why the Rashba coefficient does not scale simply with atomic number.

The paper concludes with practical implications for spintronic device engineering. To maximize Rashba control, one should select TMDs with heavy chalcogens and employ R‑type bilayer stacking, which provides a sizable intrinsic field and a strong response coefficient. By tuning the external field to opposite the intrinsic field, the spin texture can be inverted, offering a route to non‑volatile spin switching without magnetic fields. Moreover, the orbital polarization imbalance ξ_n can be accessed experimentally via layer‑resolved charge‑density measurements (e.g., STEM‑EELS or X‑ray standing wave techniques), providing a direct probe of the Rashba mechanism.

Overall, the work delivers a comprehensive, quantitative framework linking crystal symmetry, internal orbital polarization, interlayer hybridization, and Rashba spin splitting in TMDs. It supplies both the theoretical tools (λ_n^R, E_n^0, ξ_n) and material‑specific data needed for rational design of 2D spin‑orbitronic platforms.