Orbital Rashba–Edelstein Effect

• Orbital Rashba–Edelstein Effect (OREE) is a nonequilibrium phenomenon where an electric field generates a transverse orbital angular momentum accumulation in systems with broken inversion symmetry.
• It is mediated by orbital Rashba coupling that creates chiral orbital-momentum textures and drives efficient orbital-charge conversion across interfaces.
• OREE enables robust orbital torques and long-range orbital transport, offering promising applications in ultrafast magnetization control and low-power orbitronic devices.

The Orbital Rashba–Edelstein Effect (OREE) is a nonequilibrium phenomenon in systems with broken inversion symmetry, in which an in-plane electric field or charge current induces a transverse accumulation of orbital angular momentum (OAM). This conversion is mediated by orbital Rashba-type coupling in the electronic band structure, generating a chiral orbital-momentum texture in momentum space. The accumulated OAM can exert significant torques on local magnetization in magnetic systems and provide a highly efficient pathway for orbital-charge interconversion in nonmagnetic heterostructures and interfaces. Unlike its spin counterpart, the spin Rashba–Edelstein effect, the OREE is inherently nonrelativistic, often dominating even in materials with weak spin–orbit coupling. The effect is quantitatively described by the orbital Edelstein susceptibility tensor, which links the applied electric field to the resulting uniform orbital polarization.

1. Fundamental Theory and Hamiltonian Formalism

The minimal model for the OREE is based on an effective orbital Rashba Hamiltonian: $$H_{\text{orb}}(\mathbf{k}) = \alpha_{\text{orb}} (k_x L_y - k_y L_x)$$ where $\alpha_{\text{orb}}$ is the orbital Rashba parameter and $L_{x,y}$ are orbital angular-momentum operators. This term describes the linear coupling between in-plane momentum and the orbital degree of freedom, arising due to inversion symmetry breaking at surfaces, interfaces, or in bulk polar materials. The physical origin is typically strong hybridization between atomic orbitals of different angular momentum (e.g., Co-$d$ and Al-$p$ at a Co/Al interface), mediated by interfacial electrostatic fields or polar displacements (Nikolaev et al., 2024, Pezo et al., 20 Mar 2025, Go et al., 2020).

When an electric field $\mathbf{E}$ is applied, the Fermi sea is displaced, and the orbital texture winds out of equilibrium, generating a net transverse orbital moment $\delta \mathbf{L} = \chi^{L} \mathbf{E}$, with the orbital Edelstein response tensor $\chi^L_{ij}$ quantifying the effect. Higher-order terms, such as $k^3$-winding induced by atomic SOC, are symmetry dictated but typically provide subleading corrections one or more orders of magnitude smaller in magnitude (Nikolaev et al., 2024).

2. Linear Response, Kubo Formalism, and Susceptibility

The OREE is captured in linear response by the Kubo formula: $\delta L_i = \chi^L_{ij} E_j$

$$\chi^L_{ij} = -e\tau \sum_{n\mathbf{k}} \left(\frac{\partial f_{n\mathbf{k}}}{\partial \epsilon_{n\mathbf{k}}}\right) \langle u_{n\mathbf{k}} | L_i | u_{n\mathbf{k}} \rangle \langle u_{n\mathbf{k}} | v_j | u_{n\mathbf{k}} \rangle + i e \hbar \sum_{n \neq m, \mathbf{k}} \frac{f_{n\mathbf{k}} - f_{m\mathbf{k}}}{(\epsilon_{n\mathbf{k}} - \epsilon_{m\mathbf{k}})(\epsilon_{n\mathbf{k}} - \epsilon_{m\mathbf{k}} - i\hbar/ \tau)} \langle u_{m\mathbf{k}} | L_i | u_{n\mathbf{k}} \rangle \langle u_{n\mathbf{k}} | v_j | u_{m\mathbf{k}} \rangle$$

The first (intraband) term dominates in clean or moderately disordered systems, whereas the interband contribution provides dissipationless or damping-like corrections analogous to anomalous Hall responses. The susceptibility is further split into even (field-like) and odd (damping-like) parts under magnetization reversal in magnetic systems (Nikolaev et al., 2024, Pezo et al., 20 Mar 2025, Salemi et al., 2019).

In nonmagnetic or superconducting systems, the formalism is directly extended to account for inversion symmetry breaking, multiband structure, and hybridization effects, with the orbital Berry-curvature and modern theory of orbital magnetization providing foundational tools (M. et al., 2023, Ando et al., 2024). The OREE coefficient scales with the orbital Rashba parameter, momentum-relaxation time, band-resolved orbital-momentum expectation values, and Fermi surface properties.

3. Microscopic Mechanisms and Material Realizations

The emergence of OREE is strongly contingent on local bandstructure features and interface chemistry. At metallic Co/Al interfaces, DFT and Wannier function analyses reveal a helical threefold-symmetric orbital texture in the interfacial Co layer, stemming from robust hybridization between Co $d_{z^2}$, $d_{yz}$, $d_{zx}$ and Al surface $p_{x}, p_{y}$-states (Nikolaev et al., 2024, Pezo et al., 20 Mar 2025). Typical matrix elements are $\langle d_{z^2}|L_x|d_{yz}\rangle = \sqrt{3}i$ and $\langle d_{z^2}|L_y|d_{zx}\rangle = -\sqrt{3}i$, generating a dominant in-plane Rashba texture.

A similar mechanism operates at oxidized Cu surfaces, where resonant O-$p$/Cu-$d$ hybridization results in an orbital Rashba parameter $\alpha_{\text{orb}}\approx 2\,\text{eV}\cdot\AA$ and orbital moment textures up to $\sim 0.5\hbar$ at the Fermi surface—five times larger than in uncapped Cu (Go et al., 2020). Nonmagnetic Rashba semiconductors such as GeTe exhibit a colossal OREE due to polar displacement-induced orbital texture, with the orbital Edelstein susceptibility remaining virtually unchanged (and dominant) upon removing SOC (Leiva-Montecinos et al., 27 May 2025, Ovalle et al., 12 Nov 2025).

In multiorbital superconductors and oxide interfaces (KTaO$_3$, SrTiO$_3$), the OREE is enhanced at symmetry-lowered avoided crossings, where hybridization between orbitals of opposite mirror parity is maximal (Varotto et al., 2022, Ando et al., 2024). The magnitude and sign of OREE are highly sensitive to Fermi level tuning, interface-induced band orderings, and crystalline symmetry.

4. Quantitative Magnitude and Contrast with Spin Edelstein Effect

A central finding across systems is that the orbital (OREE) response frequently exceeds or dominates the spin (SEE) response by one or two orders of magnitude. For Co/Al interfaces,

$$\chi_{xy}^{L,\text{even}} \approx 2\times 10^{10}\,\hbar\,\text{m/V}$$

whereas the spin Edelstein effect is suppressed by approximately two orders of magnitude (Nikolaev et al., 2024, Pezo et al., 20 Mar 2025). In ferroelectric GeTe, the peak orbital Edelstein susceptibility is roughly ten times larger than the spin susceptibility over a broad doping range (Leiva-Montecinos et al., 27 May 2025, Ovalle et al., 12 Nov 2025). In CuMnAs and Mn$_2$Au antiferromagnets, OREE on Mn sites is 30–60 times larger than SEE (Salemi et al., 2019). Surface-oxidized Cu exhibits an OREE coefficient $$\chi_{y x}^{\text{orb}} \sim 3\times 10^{-12}\,\hbar/(\text{V\,m})$$, an order of magnitude above the comparable spin value (Go et al., 2020).

Physically, the OREE persists or is only weakly diminished when SOC is eliminated in theoretical models, in contrast to the SEE, which vanishes identically in the absence of relativistic effects. The OREE thus fundamentally derives from the coupling of orbital motion to the interfacial electrostatic field and can be realized in light-element systems.

5. Experimental Manifestations and Device Implications

The OREE has been directly detected via multiple experimental probes:

• Magnetoresistance: The angular dependence of magnetoresistance in Py/oxidized Cu heterostructures (OREMR) arises from interfacial OAM accumulation, with extracted orbital diffusion and dephasing lengths two to four times longer than their spin analogues, indicating efficient OAM transport over nanometric distances (Ding et al., 2021).
• Orbital torque: In Co/Al heterostructures, the OREE drives field-like torques up to several mT, quantitatively explaining torque enhancements upon Al insertion into Co/Pt stacks (Nikolaev et al., 2024, Pezo et al., 20 Mar 2025).
• Charge–orbital conversion: In GeTe and KTaO$_3$, OREE is the dominant channel for orbital-to-charge current conversion, as shown by drift-diffusion modeling and transport (Ovalle et al., 12 Nov 2025, Varotto et al., 2022).
• Ultrafast dynamics: Optical pump–probe and time-resolved simulations reveal that OREE-driven orbital polarization exceeds the spin polarization and supports faster, longer-lived precession and current responses than the spin analog (Busch et al., 4 May 2025).
• Magnon injection/detection: The OREE at Pt/CuOx|YIG interfaces provides record-high magnon spin injection and detection efficiencies, demonstrating both direct and inverse orbital-angular-momentum interconversion, with distinct efficiencies for the two processes due to differing interfacial relaxation rates (Mendoza-Rodarte et al., 2024).

A summary of representative OREE magnitudes and comparison to SEE/SMR is provided below:

System	OREE Susceptibility	SEE/OREE Ratio	Key Effect
Co/Al (interface)	$2\times10^{10}\,\hbar/mV$	OREE/SEE $\sim100$	Field-like orbital torque
GeTe (bulk polar)	peak $\sim10^{-6}\,\hbar/mV$	OREE/SEE $\sim10$	Colossal orbital-charge conversion
CuMnAs (AFM)	$6 \times 10^{-2}\,\mu_B$nm/V	OREE/SEE $\sim60$	Staggered orbital torques
Py/Cu* (OREMR)	λ^orb_F $=8.6$ nm	—	Long-range OAM transport, ORE magnetoresistance

All values as reported in (Nikolaev et al., 2024, Pezo et al., 20 Mar 2025, Leiva-Montecinos et al., 27 May 2025, Salemi et al., 2019, Ding et al., 2021).

6. Interfacial Sensitivity, Direct/Inverse Conversion, and Control

The OREE is fundamentally an interfacial or symmetry-breaking-driven effect. In Co/Al, insertion of a single Pt monolayer suppresses the OREE by a factor of four and inverts the sign of the field-like torque, highlighting its chemical and atomic-scale confinement (Pezo et al., 20 Mar 2025). In Pt/CuOx|YIG devices, the direct (charge-to-orbital) and inverse (orbital-to-charge) OREE efficiencies differ significantly (K_D^orb / K_I^orb ≈ 0.4), attributable to distinct scattering and dephasing processes at the interface (Mendoza-Rodarte et al., 2024).

Complete electrical control of OREE is achievable in ferroelectric Rashba semiconductors, where switching the polarization direction reverses the sign of the orbital magnetization and OREE coefficient (Leiva-Montecinos et al., 27 May 2025, Ovalle et al., 12 Nov 2025). Direct detection of orbital accumulation is possible via X-ray magnetic circular dichroism, magneto-optical Kerr, and angle-resolved photoemission with circularly polarized light (Go et al., 2020, Ding et al., 2021).

Design guidelines for robust OREE include maximizing bulk or interfacial inversion asymmetry, tuning Fermi energy near orbital-mixing band extrema, selecting materials with strong orbital character near the Fermi level (e.g. transition metal d-orbitals), and minimizing cancellation from symmetry or multi-orbital compensation (Nikolaev et al., 2024, Leiva-Montecinos et al., 27 May 2025, Varotto et al., 2022, Ando et al., 2024).

7. Future Directions and Theoretical Outlook

The OREE provides a platform for efficient angular-momentum conversion in orbitronics, enabling device architectures for memory, logic, and ultrafast magnetization control without reliance on heavy elements or strong spin–orbit coupling. Its superiority over spin analogs, long diffusion lengths in light metals, and high sensitivity to interface engineering enable low-power, CMOS-compatible, and terahertz applications (Pezo et al., 20 Mar 2025, Ding et al., 2021). Material families such as polar semiconductors, oxide 2DEGs, and antiferromagnets with strong orbital anisotropy are promising candidates for maximizing OREE. The direct separation of OREE from orbital Hall effects in drift–diffusion and macroscopic response measurements allows quantitative benchmarking of materials classes (Ovalle et al., 12 Nov 2025).

Further research avenues include ultrafast all-optical orbitronics enabled by the intrinsic speed and large magnitude of the OREE (Busch et al., 4 May 2025), exploiting tensorial control in multiband/multiorbital systems, and combining OREE with ferroelectric or gating control for nonvolatile and reconfigurable functionalities (Leiva-Montecinos et al., 27 May 2025, Ando et al., 2024). The detailed modeling and symmetry classification of OREE in complex heterostructures and hybrid systems remain areas of active study.