Nonlocal Electrical Detection of Reciprocal Orbital Edelstein Effect

Spin-Orbitronics leverages the spin and orbital degrees of freedom in solids for information processing. The orbital Edelstein effect and orbital Hall effect, where the charge current induces a nonequilibrium orbital angular momentum, offer a promising method to manipulate nanomagnets efficiently using light elements. Despite extensive research, understanding the Onsager reciprocity of orbital transport, fundamentally rooted in the second law of thermodynamics and time-reversal symmetry, remains elusive. In this study, we experimentally demonstrate the Onsager reciprocity of orbital transport in an orbital Edelstein system by utilizing nonlocal measurements. This method enables the precise identification of the chemical potential generated by orbital accumulation, avoiding the limitations associated with local measurements. Remarkably, we observe that the direct and inverse orbital-charge conversion processes produce identical electric voltages, confirming Onsager reciprocity in orbital transport. Additionally, we find that the orbital decay length, approximately 100 nm at room temperature, is independent of Cu thickness and decreases with lowering temperature, revealing a distinct contrast to spin transport behavior. Our findings provide valuable insights into both the reciprocity of the charge-orbital interconversion and the nonlocal correlation of orbital degree of freedom, laying the ground for orbitronics devices with long-range interconnections.

💡 Research Summary

This paper presents a comprehensive experimental verification of the Onsager reciprocity between the direct and inverse orbital Edelstein effects (OEE) using a non‑local lateral transport geometry. The authors fabricate devices consisting of an Al₂O₃/ CuOₓ (≈3 nm)/Cu nanowire and a ferromagnetic (FM) detector electrode separated by a controllable distance d (0–350 nm). The CuOₓ layer, formed by natural oxidation, provides a uniform electronic environment that supports orbital Rashba states, enabling the generation and detection of nonequilibrium orbital angular momentum (OAM) without relying on heavy‑metal spin‑orbit coupling.

In the direct OEE (DOEE) configuration, a charge current I_c is driven along the y‑axis of the Cu nanowire. According to the orbital Rashba Hamiltonian Ĥ = α(L × k)·z, this current creates an OAM polarized along the x‑direction. At the Cu/ FM interface the OAM is partially converted into spin via the FM’s intrinsic spin‑orbit coupling, and the resulting spin‑to‑charge conversion (e.g., inverse spin Hall effect) produces a measurable voltage V. In the inverse OEE (IOEE) configuration, the FM is magnetized along x, a charge current is injected, and the FM’s spin‑orbit coupling generates an x‑polarized OAM that diffuses into the CuOₓ/Cu channel, where it is reconverted into a charge current, yielding a voltage V_IOEE.

Both configurations are probed by measuring the non‑local resistance R = V/I_c while sweeping an external magnetic field B_ext (±0.5 T) and rotating the magnetization angle Φ in the film plane. The key observation is that the change in resistance ΔR for DOEE and IOEE are equal in magnitude and opposite in sign (±0.22 mΩ), confirming the Onsager reciprocal relation L ↔ J for orbital‑charge interconversion. This result rules out artifacts such as stray‑field‑induced Hall voltages (estimated to be two orders of magnitude smaller) and current‑bypass effects (predicted decay length ≈ 47 nm, far shorter than the observed ≈ 100 nm).

A systematic study of the FM material dependence (Co₇₅Fe₂₅, Co₅₀Fe₅₀, Ni₈₁Fe₁₉) shows that the signal amplitude scales with the spin‑orbit correlation ⟨L·S⟩ of the ferromagnet, reinforcing the orbital origin of the effect. The distance dependence follows an exponential decay, yielding a lateral orbital diffusion length λ_o ≈ 100 nm at room temperature. This length is essentially independent of the Cu thickness (20–80 nm), indicating that the oxidized CuOₓ layer dominates lateral transport, while the unoxidized Cu contributes only to a short vertical decay (λ_r ≈ 25 nm) extracted from the thickness‑dependent amplitude A.

Angular scans of Φ reveal a cosine dependence of ΔR on the magnetization direction, confirming that the detected OAM is aligned with the FM magnetization and that the OAM generated by the OEE is perpendicular to the applied charge current, as predicted by theory. Temperature variation shows a reduction of λ_o at lower temperatures, contrasting with the typical temperature‑independent spin diffusion length and suggesting that orbital transport is more sensitive to thermal scattering, likely because OAM resides primarily in the partially filled 3d states of the oxidized Cu.

Overall, the work achieves three major advances: (1) it establishes a reliable non‑local electrical method to detect pure orbital accumulation via its chemical potential; (2) it experimentally validates Onsager reciprocity for orbital‑charge conversion, demonstrating that direct and inverse OEE produce identical voltages with opposite signs; (3) it quantifies the distinct lateral (≈ 100 nm) and vertical (≈ 25 nm) orbital decay lengths, their independence from Cu thickness, and their temperature dependence. These findings provide essential parameters for designing orbitronics devices that exploit long‑range orbital interconnections, offering a pathway toward low‑power, light‑element based information processing technologies.