Orbital Edelstein Effect: Mechanisms & Applications

• Orbital Edelstein effect is a nonequilibrium phenomenon where an in-plane electric field induces orbital magnetization in systems with momentum-dependent orbital angular momentum and broken inversion symmetry.
• It arises through diverse mechanisms such as orbital Rashba coupling, chiral intersite motion, and edge-state self-rotation, and can occur with or without spin-orbit coupling.
• The effect is characterized using modern orbital magnetization theories and probed by experimental techniques like Kerr effect, XMCD, and NV-center magnetometry in platforms ranging from Janus TMDs to ferroelectric GeTe.

The orbital Edelstein effect (OEE) is a nonequilibrium magnetoelectric response in which an in-plane electric field or a charge current induces a net orbital polarization, or equivalently an orbital magnetization, in a system whose Bloch states carry momentum-dependent orbital angular momentum and whose symmetry permits an axial response to a polar drive. It is the orbital analogue of the spin Edelstein effect, but its microscopic origin is not tied to spin alone: depending on the platform, it can arise from orbital Rashba coupling, ferroelectric or structural inversion asymmetry, chiral intersite motion, edge-state self-rotation, or superconducting phase gradients. Across metals, semiconductors, interfaces, chiral crystals, and noncentrosymmetric superconductors, the central pattern is the same: the electric field shifts the distribution function on a preexisting orbital texture, so the $+\mathbf{k}$ and $-\mathbf{k}$ contributions no longer cancel, and a net orbital moment appears (Yoda et al., 2017).

1. Definition, constitutive relations, and symmetry conditions

In linear response, the OEE is written either as an induced orbital angular momentum or as an induced orbital magnetization. Common forms used across the literature are

$$\langle L_i\rangle = \alpha^L_{ij} E_j,\qquad M_i = \chi^{OE}_{ij} E_j,\qquad M_i = \beta^{Mj}_{ij} j_j.$$

The first relation emphasizes the nonequilibrium orbital polarization, the second the magnetization response, and the third the current-to-magnetization conversion. In semiclassical transport, both $M$ and $j$ scale linearly with the relaxation time $\tau$, so the ratio $M/j$ is often an intrinsic band-structure quantity within the constant-$\tau$ approximation (Sahu et al., 13 Nov 2025).

A necessary condition is broken inversion symmetry. With time-reversal symmetry, the orbital magnetic moment satisfies $m_n(\mathbf{k})=-m_n(-\mathbf{k})$; if inversion is also present, then $m_n(\mathbf{k})=m_n(-\mathbf{k})$, forcing $-\mathbf{k}$0. Accordingly, OEE is symmetry-allowed in gyrotropic noncentrosymmetric crystals and forbidden in centrosymmetric ones (Yoda et al., 2017). The precise tensor structure depends on point group. In monolayer Janus transition-metal dichalcogenides and bulk GeTe, both with $-\mathbf{k}$1 symmetry, only in-plane antisymmetric components survive, so $-\mathbf{k}$2 or $-\mathbf{k}$3, while diagonal in-plane components vanish (Sahu et al., 13 Nov 2025). In chiral tellurium-type models, by contrast, the relevant tensor can be predominantly longitudinal along the helical axis, with a nonzero $-\mathbf{k}$4 component (Göbel et al., 7 Feb 2025).

A common misconception is that OEE always requires spin-orbit coupling. The published record is more specific. In several systems the orbital texture and the orbital response persist when SOC is turned off, showing that OEE is not, in general, a relativistic effect. However, in some geometries and model realizations the effect is Rashba-enabled and disappears without the relevant SOC term. The Au(001) ultrafast study is explicit on this point: in that model and geometry, $-\mathbf{k}$5 vanishes when Rashba SOC is switched off, even though more general OEE mechanisms can exist without SOC (Busch et al., 4 May 2025).

2. Microscopic origins: orbital textures, chirality, and intersite circulation

The microscopic basis of OEE is a momentum-space orbital texture. In Janus monolayers $-\mathbf{k}$6 the top and bottom chalcogen layers differ, breaking the out-of-plane mirror symmetry and producing an internal electric field $-\mathbf{k}$7 normal to the plane. This field mixes transition-metal $-\mathbf{k}$8, $-\mathbf{k}$9, and $$\langle L_i\rangle = \alpha^L_{ij} E_j,\qquad M_i = \chi^{OE}_{ij} E_j,\qquad M_i = \beta^{Mj}_{ij} j_j.$$0 states with $$\langle L_i\rangle = \alpha^L_{ij} E_j,\qquad M_i = \chi^{OE}_{ij} E_j,\qquad M_i = \beta^{Mj}_{ij} j_j.$$1 and $$\langle L_i\rangle = \alpha^L_{ij} E_j,\qquad M_i = \chi^{OE}_{ij} E_j,\qquad M_i = \beta^{Mj}_{ij} j_j.$$2, generating in-plane orbital angular momentum near $$\langle L_i\rangle = \alpha^L_{ij} E_j,\qquad M_i = \chi^{OE}_{ij} E_j,\qquad M_i = \beta^{Mj}_{ij} j_j.$$3, $$\langle L_i\rangle = \alpha^L_{ij} E_j,\qquad M_i = \chi^{OE}_{ij} E_j,\qquad M_i = \beta^{Mj}_{ij} j_j.$$4, and $$\langle L_i\rangle = \alpha^L_{ij} E_j,\qquad M_i = \chi^{OE}_{ij} E_j,\qquad M_i = \beta^{Mj}_{ij} j_j.$$5. Near a valley $$\langle L_i\rangle = \alpha^L_{ij} E_j,\qquad M_i = \chi^{OE}_{ij} E_j,\qquad M_i = \beta^{Mj}_{ij} j_j.$$6, the texture is summarized by

$$\langle L_i\rangle = \alpha^L_{ij} E_j,\qquad M_i = \chi^{OE}_{ij} E_j,\qquad M_i = \beta^{Mj}_{ij} j_j.$$7

with the $$\langle L_i\rangle = \alpha^L_{ij} E_j,\qquad M_i = \chi^{OE}_{ij} E_j,\qquad M_i = \beta^{Mj}_{ij} j_j.$$8 valley overwhelmingly dominant because $$\langle L_i\rangle = \alpha^L_{ij} E_j,\qquad M_i = \chi^{OE}_{ij} E_j,\qquad M_i = \beta^{Mj}_{ij} j_j.$$9 (Sahu et al., 13 Nov 2025).

In ferroelectric GeTe, the polar displacement of Ge and Te along $M$0 produces bulk inversion asymmetry and Rashba-like splittings near $M$1 and $M$2. There, the Fermi-surface orbital texture exceeds the spin texture by one order of magnitude and remains largely unchanged when SOC is removed, demonstrating that the OEE is driven primarily by the intrinsic orbital character of Bloch states in an inversion-broken crystal rather than by spin physics (Leiva-Montecinos et al., 27 May 2025).

At metallic interfaces, the mechanism can be interfacial hybridization rather than bulk polarity. In Co/Al, first-principles calculations reveal a large helical orbital texture localized at the interfacial Co layer. Its origin is an orbital Rashba effect generated by hybridization between Al $M$3 and Co $M$4 states; it exists already without SOC, whereas the transverse spin texture vanishes without SOC. SOC adds a smaller higher-order orbital winding and converts the orbital accumulation into torque (Nikolaev et al., 2024).

Chiral systems provide a distinct route. In analytically solvable helical models motivated by tellurium, the orbital angular momentum is directly generated by the crystal chirality through intersite circulation along the helix. The resulting orbital Edelstein susceptibility is odd under reversal of chirality and can exceed the spin contribution by orders of magnitude (Göbel et al., 7 Feb 2025). Chiral carbon nanotubes exhibit an analogous “chirality-induced orbital selectivity”: the orbital Edelstein susceptibility is an odd function of chirality angle, proportional to radius, and for metallic tubes near the Fermi level grows quadratically with energy (Göbel et al., 10 Apr 2025).

The OEE can also originate from neither atomic orbital character nor bulk chirality, but from real-space geometry at edges. In slab calculations based entirely on $M$5 orbitals, edge-localized states on zigzag or irregular boundaries acquire itinerant orbital angular momentum through self-rotation of the wavepacket, whereas straight edges can yield vanishing edge OAM texture. This establishes an edge-state OEE from inter-atomic orbital motion and shows that bulk topology alone does not determine the effect (Lee et al., 2024).

A further nonrelativistic mechanism arises from an asymmetric scalar potential gradient. In a diffusive parabolic-band model with $M$6, the orbital magnetic moment operator $M$7 combines with the density gradient to produce

$M$8

so the OEE scales as $M$9 and does not rely on SOC (Ado et al., 2024).

3. Response theory: modern orbital magnetization, Boltzmann transport, and Kubo formulations

The orbital quantity entering OEE is the Bloch-state orbital magnetic moment or its equivalent orbital angular momentum expectation value. A standard modern-theory expression is

$j$0

which appears in normal-state OEE studies of chiral crystals, bilayer Rashba systems, gated TMDs, and Floquet-driven jacutingaite (Yoda et al., 2017). In some multiorbital materials, the orbital response is instead evaluated from Wannier-interpolated orbital angular momentum operators within the modern theory of orbital magnetization, as in GeTe (Leiva-Montecinos et al., 27 May 2025). In others, especially $j$1-orbital systems such as TMDs, the atom-centered approximation is used and is argued to be accurate near the band edges that dominate transport (Sahu et al., 13 Nov 2025).

Within semiclassical Boltzmann theory at $j$2, a representative OEE response takes the Fermi-surface form

$j$3

with $j$4 for orbital or spin channels. Only states at the Fermi surface contribute. This formulation underlies the OEE calculations in Janus TMDs and related 2D systems (Sahu et al., 13 Nov 2025). Closely related expressions are used in oxide 2DEGs and GeTe, where the nonequilibrium distribution is written as $j$5, with $j$6 under a constant-$j$7 approximation (Johansson et al., 2020).

Kubo formulations are needed when interband coherence matters. In Co/Al, the OEE susceptibility is decomposed into intraband and interband terms,

$j$8

with the intraband contribution tracking the Fermi-surface shift of a preexisting orbital texture, and the interband term capturing coherence between Bloch states (Nikolaev et al., 2024). In optically driven jacutingaite, the practical Kubo-like expression explicitly separates intraband and interband processes with different phenomenological scattering times $j$9 and $\tau$0; in that model, the interband OEE vanishes because the off-diagonal matrix element of $\tau$1 is zero, so the orbital response is purely intraband (Bau et al., 9 May 2025).

The superconducting counterpart replaces dissipative transport by superflow. In multiorbital noncentrosymmetric superconductors, a phase gradient $\tau$2 produces a nondissipative orbital magnetization,

$\tau$3

which defines an orbital Edelstein coefficient $\tau$4. Here the relevant matrix elements are interband velocity–orbital-angular-momentum terms in the Bogoliubov–de Gennes spectrum, and avoided crossings strongly enhance the effect (Chirolli et al., 2021).

4. Materials realizations and quantitative trends

The OEE has been reported theoretically or experimentally across a wide set of platforms, but the dominant mechanism and the relative size of the orbital and spin channels differ substantially.

Platform	Dominant mechanism	Reported hallmark
Janus monolayer TMDs	$\tau$5-driven orbital Rashba texture	OEE exceeds SEE by roughly an order of magnitude; $\tau$6 valley dominates
Bulk GeTe	ferroelectric inversion breaking and bulk orbital texture	orbital moment surpasses spin moment by one order of magnitude; OEE remains largely unaffected without SOC
AlO$\tau$7/SrTiO$\tau$8 2DEG	multi-orbital $\tau$9 Rashba-like bands	OEE exceeds SEE by more than one order of magnitude
Co/Al heterostructure	interfacial helical orbital texture without SOC	$M/j$0 and estimated $M/j$1
Noncentrosymmetric superconductors	orbital Rashba coupling under supercurrent	orbital magnetization about $M/j$2 to $M/j$3 the spin magnetization
Oxidized Cu orbital Rashba system	nonlocal direct/inverse orbital-charge conversion	equal-magnitude nonlocal voltages $M/j$4; orbital decay length about $M/j$5–$M/j$6

These trends are documented, respectively, in Janus TMDs (Sahu et al., 13 Nov 2025), GeTe (Leiva-Montecinos et al., 27 May 2025), AlO$M/j$7/SrTiO$M/j$8 (Johansson et al., 2020), Co/Al (Nikolaev et al., 2024), noncentrosymmetric superconductors (Chirolli et al., 2021), and nonlocal oxidized Cu devices (Gao et al., 16 Feb 2025).

In Janus TMDs, the valley dependence is particularly explicit. Representative orbital Rashba constants at $M/j$9 are $\tau$0 for MoSSe and $\tau$1 for WSTe, while the corresponding $\tau$2 values are much smaller. This is why the $\tau$3 pocket dominates the OEE. Across the studied Janus compounds, Te-containing systems enhance either orbital Rashba strength, spin Rashba strength, or both, and the paper identifies WSTe as optimal for simultaneously large OEE and SEE (Sahu et al., 13 Nov 2025).

In bulk GeTe, the effective Rashba parameters near $\tau$4 are $\tau$5, $\tau$6, $\tau$7, and $\tau$8 along $\tau$9, with similar values along $m_n(\mathbf{k})=-m_n(-\mathbf{k})$0. The computed orbital susceptibility $m_n(\mathbf{k})=-m_n(-\mathbf{k})$1 is about one order of magnitude larger than the spin counterpart over a broad energy window, and it does not change sign where the spin response does, because both bands in a Rashba-split pair contribute with the same orbital rotation sense (Leiva-Montecinos et al., 27 May 2025).

At the Co/Al interface, the large orbital response is tied to a very specific layer selectivity: the interfacial Co layer hosts the strongest orbital texture and the largest $m_n(\mathbf{k})=-m_n(-\mathbf{k})$2. The estimated field-like effective field $m_n(\mathbf{k})=-m_n(-\mathbf{k})$3 is comparable to the torque enhancement observed experimentally when thin Al is inserted into Co/Pt-type stacks, which the authors interpret as an interfacial orbital Rashba–Edelstein contribution (Nikolaev et al., 2024).

In the superconducting setting, the magnitude can be even more extreme. For representative parameters $m_n(\mathbf{k})=-m_n(-\mathbf{k})$4, $m_n(\mathbf{k})=-m_n(-\mathbf{k})$5, $m_n(\mathbf{k})=-m_n(-\mathbf{k})$6, and $m_n(\mathbf{k})=-m_n(-\mathbf{k})$7, the orbital magnetization exceeds the spin contribution by about $m_n(\mathbf{k})=-m_n(-\mathbf{k})$8 for $m_n(\mathbf{k})=-m_n(-\mathbf{k})$9 and about $m_n(\mathbf{k})=m_n(-\mathbf{k})$0 for $m_n(\mathbf{k})=m_n(-\mathbf{k})$1, with sign reversal controlled by the chemical potential relative to avoided crossings (Chirolli et al., 2021).

5. Nonlinear, optical, and ultrafast extensions

The OEE is not restricted to steady linear dc transport. Several recent works extend it into nonlinear, Floquet, and ultrafast regimes.

On Au(001), a femtosecond laser pulse drives a nonlinear orbital Edelstein response that is intrinsically time dependent and nonperturbative in field amplitude. For a linearly polarized $m_n(\mathbf{k})=m_n(-\mathbf{k})$2 pulse with a $m_n(\mathbf{k})=m_n(-\mathbf{k})$3 envelope, the induced $m_n(\mathbf{k})=m_n(-\mathbf{k})$4 and $m_n(\mathbf{k})=m_n(-\mathbf{k})$5 are symmetry-allowed only when Rashba SOC is present, show a delayed maximum at about $m_n(\mathbf{k})=m_n(-\mathbf{k})$6 after the pulse center, and display strong post-pulse beating. The associated longitudinal orbital current $m_n(\mathbf{k})=m_n(-\mathbf{k})$7 oscillates at $m_n(\mathbf{k})=m_n(-\mathbf{k})$8, and transverse orbital Hall currents exhibit markedly different temporal behavior from their spin analogues (Busch et al., 4 May 2025).

Few-layer WTe$m_n(\mathbf{k})=m_n(-\mathbf{k})$9 realizes a nonlinear Edelstein effect in transport rather than pump–probe spectroscopy. Under an ac current at frequency $-\mathbf{k}$00, an out-of-plane magnetization with a second-harmonic response at $-\mathbf{k}$01 is detected using an Fe$-\mathbf{k}$02GeTe$-\mathbf{k}$03 electrode. The signal scales quadratically with current and is explained by a field-induced correction to the orbital magnetic moment built from the Berry-connection polarizability tensor,

$-\mathbf{k}$04

which generates a nonlinear orbital magnetization $-\mathbf{k}$05. In this setting, the orbital channel is identified as the primary origin of the nonlinear response, with spin emerging secondarily through SOC (Ye et al., 2024).

Floquet engineering in monolayer jacutingaite Pt$-\mathbf{k}$06HgSe$-\mathbf{k}$07 reveals a different role for OEE: it becomes a probe of light-induced topological phase transitions. Circularly polarized light generates a Floquet mass $-\mathbf{k}$08 that competes with the intrinsic spin-orbit mass in the Dirac Hamiltonian. The spin Edelstein conductivity shows a pronounced discontinuity across the transition, whereas the orbital Edelstein susceptibility vanishes at the onset of the semimetallic regime because the Bloch orbital magnetic moment is proportional to the Dirac mass and therefore goes to zero when a selected spin–valley sector becomes gapless (Bau et al., 9 May 2025).

The superconducting OEE is likewise a nonlinear extension in a broader sense: it is nondissipative and controlled by the superfluid phase rather than by Joule transport. Real-space phase inhomogeneities can modulate the orbital Edelstein signal on the coherence-length scale and create domains with opposite orbital-moment orientation, a behavior without a normal-state counterpart (Chirolli et al., 2021).

6. Detection, reciprocity, and device relevance

Several experimental strategies target the current-induced orbital magnetization directly. Magneto-optical Kerr effect, x-ray magnetic circular dichroism, NV-center magnetometry, torque magnetometry, and time-resolved ARPES or XMCD all appear in the literature as suitable probes, with the preferred method depending on whether the target is a dc orbital magnetization, an ultrafast orbital texture, or a local edge accumulation (Leiva-Montecinos et al., 27 May 2025).

A major advance is the nonlocal electrical detection of the reciprocal orbital Edelstein effect in oxidized Cu-based orbital Rashba devices. In Al$-\mathbf{k}$09O$-\mathbf{k}$10/CuO$-\mathbf{k}$11/Cu trilayers, direct and inverse orbital-charge conversion were probed in reciprocal geometries and yielded equal-magnitude nonlocal resistance changes,

$-\mathbf{k}$12

at $-\mathbf{k}$13 and room temperature, thereby confirming Onsager reciprocity. The extracted lateral orbital decay length is about $-\mathbf{k}$14–$-\mathbf{k}$15 at room temperature, essentially independent of Cu thickness, and decreases on cooling in a way opposite to conventional spin diffusion in Cu (Gao et al., 16 Feb 2025).

Interface orbitronics provides a second application direction. In Pt/CuO$-\mathbf{k}$16/YIG devices, interfacial orbital Rashba–Edelstein conversion dramatically enhances magnon injection and detection efficiencies. At $-\mathbf{k}$17, the first-harmonic nonlocal magnon signal is enhanced by about $-\mathbf{k}$18, and the second-harmonic signal by about $-\mathbf{k}$19, relative to Pt-only electrodes. The analysis also indicates that the inverse orbital Rashba Edelstein efficiency is about $-\mathbf{k}$20 times the direct one in that geometry (Mendoza-Rodarte et al., 2024).

The device implications extend beyond magnonics. Janus TMD studies explicitly propose orbital torque in ferromagnet/Janus bilayers, magnetoresistive signatures of orbital Rashba–Edelstein transport, and valley-optical responses tied to the distinction between $-\mathbf{k}$21 and $-\mathbf{k}$22 orbital textures (Sahu et al., 13 Nov 2025). Co/Al work connects the OEE to large field-like torques in all-metallic heterostructures with light elements (Nikolaev et al., 2024). Chiral-system studies frame the OEE as the microscopic origin of chirality-induced orbital selectivity and as a precursor to chirality-induced spin selectivity once orbital angular momentum is converted to spin in contacts or adjacent materials (Göbel et al., 7 Feb 2025).

7. Conceptual issues, limitations, and open directions

Two conceptual points recur across the literature. First, SOC is neither universally necessary nor universally irrelevant. The balance depends on the mechanism. GeTe, Janus TMDs without the spin channel, chiral tellurium models, chiral CNTs, density-wave systems, scalar-potential gradients, and edge-state OEE all demonstrate SOC-independent or primarily nonrelativistic orbital conversion (Leiva-Montecinos et al., 27 May 2025). Au(001), by contrast, is an explicit case where the chosen geometry makes the OEE Rashba-enabled, so switching off RSOC eliminates the effect (Busch et al., 4 May 2025).

Second, bulk topology does not by itself guarantee an OEE. The edge-state study shows that two slabs with the same bulk Hamiltonian can differ sharply: straight edges may show no itinerant OAM texture, whereas zigzag or irregular edges exhibit a clear edge OEE. The authors therefore state that there is no bulk-boundary correspondence for the accumulation process (Lee et al., 2024). A related nuance appears in transport reciprocity: nonlocal oxidized Cu devices confirm Onsager reciprocity for charge–orbital interconversion, but Pt/CuO$-\mathbf{k}$23 magnon devices display a disparity between direct and inverse efficiencies because the relevant interfacial conversion chain includes additional relaxation and conversion processes (Gao et al., 16 Feb 2025).

Methodologically, most normal-state studies rely on the relaxation-time approximation, often with constant $-\mathbf{k}$24, zero temperature, and no explicit treatment of disorder, phonons, or electron-electron scattering. Several works also omit Berry-curvature density-of-states terms because the Edelstein response is treated as a Fermi-surface phenomenon, or they rely on the atom-centered approximation when the states are sufficiently atomic-like. These assumptions are appropriate in some platforms and questionable in others, especially where intersite circulation or strong disorder dominate (Sahu et al., 13 Nov 2025).

Open problems are therefore well defined rather than speculative. They include quantitative disorder and temperature scaling of OEE; microscopic theory for the orbital decay length in oxidized Cu and related orbital Rashba systems; disentangling surface and bulk contributions in ferroelectric Rashba semiconductors; quantitative calibration of orbital-to-spin conversion in chiral conductors and nanotubes; and systematic control of edge geometry in topological and higher-order topological materials to optimize itinerant edge OEE (Gao et al., 16 Feb 2025). A plausible implication is that future progress will depend less on identifying a single “universal” OEE mechanism than on matching symmetry, orbital texture, scattering regime, and readout channel within each material class.