---
title: Orbital Torque in Spintronics
url: https://www.emergentmind.com/topics/orbital-torque-ot
type: topic
---

# Orbital Torque in Spintronics

to=arxiv_search.search  天天彩票提现_json
{"query":"all:\"orbital torque\" OR ti:\"orbital torque\" OR abs:\"orbital torque\"","max_results":10,"sort_by":"submittedDate","sort_order":"descending"} +#+#+#+#+#+
to=arxiv_search.search ุ้นบาท_json
{"query":"ti:\"Long-Range Orbital Magnetoelectric Torque in Ferromagnets\" OR ti:\"Orbital Torque: Torque Generation by Orbital Current Injection\" OR ti:\"Oxide layer dependent orbital torque efficiency in ferromagnet/Cu/Oxide heterostructures\"","max_results":10,"sort_by":"relevance","sort_order":"descending"} +#+#+#+#+#+
Orbital torque (OT) is a current-induced torque on a magnetic order parameter whose proximate source is orbital angular momentum rather than a spin current. In the literature, OT denotes several closely related regimes: torque from orbital Hall currents injected from a nonmagnet into a ferromagnet, torque from interfacial orbital Rashba–Edelstein responses, and long-range orbital magnetoelectric responses inside ferromagnets. Across these formulations, the common structure is charge-to-orbital conversion in a source region, followed by orbital-to-spin conversion through spin–orbit coupling in the magnetic layer, and finally exchange transfer to the magnetization [1903.01085][2106.07928][2004.09165].

## 1. Concept and formal description

OT is defined in explicit contrast to spin-transfer torque and conventional spin–orbit torque. In the spin-current picture, a charge current first generates a spin accumulation or spin current, which then exerts torque on the magnetization. In the orbital picture, a charge current generates an orbital accumulation or orbital current, and the magnetic response emerges only after spin–orbit coupling converts orbital angular momentum into spin within the magnetic layer or at the interface [1903.01085][2307.09824].

Several papers formulate this relation at complementary levels. In transport language, the induced orbital response may be written as
\[
L_i^{\text{orb}} = \tilde{\beta}_{ij} E_j,
\]
with \(\tilde{\beta}_{ij}\) the orbital magnetoelectric tensor [2106.07928]. In interfacial notation, the spin–orbital torque is written as
\[
\boldsymbol{T}_\mathrm{SO} \approx \xi (\boldsymbol{L} \times \boldsymbol{\sigma}),
\]
where \(\xi\) is the spin–orbit-coupling strength, \(\boldsymbol{L}\) is orbital angular momentum, and \(\boldsymbol{\sigma}\) is the spin vector [2404.18055]. In the original bilayer theory of orbital-current injection, the torque on the ferromagnet is expressed through the nonequilibrium spin accumulation generated inside the ferromagnet after orbital-to-spin conversion,
\[
\mathbf{T} = \frac{J}{\hbar}\,\hat{\mathbf{M}} \times \langle \mathbf{S} \rangle^{\rm FM},
\]
with the standard field-like and damping-like decomposition following from \(\langle S_y\rangle^{\rm FM}\) and \(\langle S_x\rangle^{\rm FM}\), respectively [1903.01085].

A recurring point in the OT literature is that orbital and spin angular momenta transform similarly under symmetry operations, so torque symmetries often resemble those of ordinary SOT. The distinction is therefore microscopic rather than purely phenomenological: OT depends on orbital textures, orbital hybridization, crystal-field structure, and orbital-to-spin conversion in the magnetic detector [1903.01085][2406.19982].

## 2. Generation mechanisms and source materials

Two source mechanisms dominate the literature. The first is the bulk orbital Hall effect (OHE), in which an in-plane electric field generates a transverse orbital current even when the source layer has negligible or weak spin–orbit coupling. The second is the orbital Rashba–Edelstein effect, or more broadly orbital Rashba physics, in which inversion-symmetry breaking and orbital hybridization generate a nonequilibrium orbital accumulation at an interface [1903.01085][2307.09824].

The bulk-OHE route was established theoretically for NM/FM bilayers in which the nonmagnet can host a large orbital Hall conductivity even when \(\alpha_{\rm so}^{\rm NM}=0\). In that setting, the nonmagnet injects an orbital Hall current into the ferromagnet, and the ferromagnet’s own spin–orbit coupling converts the injected orbital angular momentum into a torque. The proposal was motivated partly by the possibility that light elements with gigantic orbital response could rival heavy-metal spin Hall sources [1903.01085].

The interfacial route is especially prominent in Cu/oxide systems. In Ta/Cu/[Ni/Co]\(_5\)/Cu–CuO\(_x\), the naturally oxidized Cu cap was reported to enhance the orbital Rashba effect through hybridization between O-2p and Cu-3d orbitals, producing an orbital current that adds to the Ta-driven SOT [2404.18055]. In CoFe/Cu/oxide heterostructures, the oxide species controls the OT efficiency through Cu–O interatomic hopping at the Cu/oxide interface; the maximum OT efficiency varies from \(\theta_{\max} \approx 0.12\) for Al\(_2\)O\(_3\) to \(\theta_{\max} \approx 0.26\) for SiO\(_2\), with the authors attributing the trend to active Cu–O hopping rather than dielectric constant [2307.09824]. In naturally oxidized Cu/ferromagnet bilayers, the current-induced torque was interpreted as arising from an orbital Rashba–Edelstein effect in Cu, with the notable observation that the torque direction depends on the ferromagnetic layer [2004.09165].

First-principles studies also identify purely interfacial orbital Rashba textures as strong torque sources. At Co/Al interfaces, state-of-the-art density-functional calculations yield large orbital Edelstein coefficients localized at the interfacial Co layer and a predominantly field-like intraband torque. The same work reports that inserting a single atomic plane of Pt between Co and Al suppresses the effect, indicating that the Co/Al orbital Rashba texture is not simply the precursor of a conventional Pt-like SOT [2503.16319].

A broader materials trend follows from these studies. OT source layers now include light metals such as Ti, V, Cr, Cu, and Zr; transition-metal oxides such as SrRuO\(_3\); and interfacial oxide systems such as Cu/CuO\(_x\) [2410.02238][2403.03043][2508.18746]. This suggests that strong bulk spin Hall conductivity is not a prerequisite for efficient current-induced torque, provided the source layer can sustain a large orbital response and the magnetic layer can convert it efficiently.

## 3. Orbital-to-spin conversion and magnetic-layer dependence

The receiving magnetic layer is not a passive sink. Multiple papers argue that OT efficiency depends critically on the magnetic layer’s spin–orbit correlation, orbital hybridization, and band-structure-resolved \(\langle \mathbf{L}\cdot\mathbf{S}\rangle\) rather than only on the source layer’s orbital Hall conductivity [2403.03043][2511.11482].

This dependence is especially explicit in Zr-based devices. In perpendicularly magnetized Zr/[Co/Pt]\(_3\), the reported orbital torque efficiency is approximately \(0.78\), whereas in Zr/CoFeB/Gd/CoFeB it is approximately \(0.04\). The stated explanation is the different spin–orbit correlation strength between the two magnetic systems, confirmed through theoretical calculations, with \(\xi_{\mathrm{OT}} = \eta_{L\!-\!S}\theta_{\mathrm{OH}}\) and \(\eta_{L\!-\!S}\) roughly proportional to \(\langle L\cdot S\rangle\) [2403.03043].

Rare-earth transition-metal ferrimagnets sharpen the same point. In Gd\(_y\)Co\(_{100-y}\)/CuO\(_x\), the effective spin-orbital Hall angle reaches up to \(-0.25\), compared with \(+0.03\) in Co/CuO\(_x\) and \(+0.13\) in Gd\(_y\)Co\(_{100-y}\)/Pt. The paper attributes this to local orbital-to-spin conversion at Gd sites that is about five times stronger and of the opposite sign relative to Co, and further reports a manyfold increase in net OT at low temperature as Gd and Co sublattices order more strongly [2406.19982].

Ferrimagnetic Fe\(_{1-x}\)Gd\(_x\) bilayers provide an experimentally direct demonstration that the magnetic layer’s SOC can control OT. In Ti/Fe\(_{1-x}\)Gd\(_x\), increasing the Gd content enhances the torque efficiency by a factor of four while coercive field, anisotropy field, and saturation magnetization remain nearly constant, a dependence identified as characteristic of OT because the orbital-to-spin conversion occurs in the ferrimagnet rather than the source metal [2410.02238].

Theoretical work on Ti/FM and Cu/FM bilayers further shows that FM dependence is not universal. In semi-realistic tight-binding calculations, the torque in Ti/FM is larger for Ni than for Co, but that trend does not necessarily hold in Cu/FM. The authors conclude that the dependence on ferromagnet species varies with the orbital current source and cannot be reduced to the separate bulk properties of the source and detector layers [2511.11482]. This result is consistent with earlier experiments on naturally oxidized Cu, where the damping-like torque reverses sign between Ni\(_{81}\)Fe\(_{19}\) and Fe, a behavior presented as counter to a pure spin-current picture and consistent with an orbital-current origin [2004.09165].

## 4. Transport range, thickness dependence, and unconventional symmetries

One of the defining claims of OT research is that orbital transport can remain effective where spin transport has already dephased. In Cr/CoFe bilayers, with an electric field applied only in the Cr layers, the induced \(\langle L_x\rangle\) in CoFe is large, non-oscillatory, and decays over about \(30\) layers, while \(\langle S_x\rangle\) is smaller, oscillatory, and decays within about \(15\) layers. The same work attributes the long range to symmetry-enforced near-degeneracies that create orbital-response hotspots and suppress destructive interference among different \(\mathbf{k}\)-states [2106.07928].

This long-range regime has a distinctive torque-thickness relation. In the same Cr/CoFe calculation, the layer-summed spin-flux contribution saturates near \(15\) CoFe layers, whereas the layer-summed spin–orbital contribution keeps increasing up to about \(30\) layers; for thicker ferromagnets, the orbital part becomes comparable to or larger than the spin-injection part, and the total damping-like torque can even change sign through cancellation [2106.07928]. The same paper reports that removing off-diagonal orbital hybridization in CoFe nearly eliminates the spin–orbital torque, identifying orbital hybridization as essential rather than incidental.

Thickness dependence is also central in interfacial Cu/CuO\(_x\) systems. In Ta/Cu/[Ni/Co]\(_5\)/Cu–CuO\(_x\), increasing the Cu–CuO\(_x\) thickness from \(4\) to \(7.5\) nm reduces the effective spin Hall angle from \(0.72\) to \(0.31\), interpreted as decay of orbital angular momentum in metallic Cu before it reaches the ferromagnet [2404.18055]. In CoFe/Cu/oxide stacks, the OT efficiency rises with Cu thickness in the thin regime, peaks near \(10\)–\(15\) nm depending on oxide, and then decreases as current shunting and orbital decay become more important [2307.09824].

A separate development concerns torque symmetry. In ferromagnetic-metal/oxidized-Cu bilayers, an out-of-plane antidamping-like orbital torque was reported from ST-FMR angular analysis, attributed to a \(z\)-polarized orbital current. The orbital-torque ratios peak for oxidized Cu thickness around \(3\) nm, and the damping parameter shows a correlated thickness dependence that the authors interpret as additional evidence for the antidamping-like character [2311.05868]. A later ST-FMR survey of SiO\(_2\)/FM/NM bilayers likewise reported an out-of-plane torque component attributed to interfacial mechanisms and to a spin-orbital polarized current along the \(z\)-direction [2603.23826].

The nonlinear regime extends these symmetry arguments further. “Out-of-Plane Nonlinear Orbital Hall Torque” proposes that a nonlinear orbital Hall effect can generate out-of-plane orbital torques across broad classes of noncentrosymmetric materials, with the nonlinear orbital response dramatically amplified by topological band degeneracies and calculated to dominate the spin response in representative topological metals RhSi, YPtBi, and PbTaSe\(_2\) [2511.10314]. This suggests that out-of-plane OT need not be restricted to special interface symmetries or linear-response geometries.

## 5. Experimental platforms and metrology

OT has been quantified mainly through harmonic Hall measurements, spin-torque ferromagnetic resonance (ST-FMR), anomalous Hall analysis, and thickness-dependent switching measurements. Harmonic Hall protocols extract \(H_{\mathrm{DL}}\) and \(H_{\mathrm{FL}}\) from first- and second-harmonic signals; ST-FMR separates symmetric and antisymmetric Lorentzian components; and macrospin anomalous-Hall analyses are used to infer effective spin Hall angles that in OT systems generally contain both spin and orbital contributions [2404.18055][2403.03043][2603.23826].

The table summarizes representative systems and reported performance metrics.

| System | Reported quantity | Key result |
|---|---|---|
| Ta/Cu/[Ni/Co]\(_5\)/Cu–CuO\(_x\)(4) | \(\theta_\mathrm{SH}^\mathrm{eff}\) | \(0.72(2)\); about ten times the Cu–CuO\(_x\)-free reference value \(0.07\) [2404.18055] |
| Zr/[Co/Pt]\(_3\) | \(\xi_{\mathrm{OT}}\), \(J_s\) | \(\xi_{\mathrm{OT}} \approx 0.78\); full switching at \(\approx 2.6\times10^6\,\mathrm{A/cm^2}\) [2403.03043] |
| Ti/Fe\(_{0.70}\)Gd\(_{0.30}\) | \(\theta_\mathrm{OH}^{\mathrm{eff}}\), \(J_\mathrm{th}\) | \(\theta_\mathrm{OH}^{\mathrm{eff}} \approx 0.1\) at \(t_\mathrm{Ti}=12\) nm; \(J_\mathrm{th}\approx 1.5\times10^7\,\mathrm{A/cm^2}\) [2410.02238] |
| Gd\(_{23}\)Co\(_{77}\)/CuO\(_x\) | \(\theta_{\mathrm{DL}}\) | \(\approx -0.02\) at \(300\) K and \(\approx -0.25\) at \(20\) K [2406.19982] |
| mica/SrRuO\(_3\)/CoPt | \(\xi_{\mathrm{DL}}\), \(J_\mathrm{th}\) | \(-0.31\); \(9.2\times10^9\,\mathrm{A/m^2}\) [2508.18746] |

Interpretation of these numbers is system-dependent. In Ta/Cu/[Ni/Co]\(_5\)/Cu–CuO\(_x\), the authors describe a collaborative action of Ta SOT and Cu/CuO\(_x\) OT rather than a pure orbital signal [2404.18055]. In Zr/[Co/Pt]\(_3\), the reported efficiency explicitly exceeds that of the W-based SOT reference in the same magnetic stack [2403.03043]. In Ti/FeGd and GdCo/CuO\(_x\), the detector layer itself is used as a diagnostic variable: changing Gd content or temperature modifies the magnetic-layer conversion efficiency, and the torque changes accordingly [2410.02238][2406.19982].

Metrological subtleties are recurrent. In Ta/Cu/[Ni/Co]\(_5\)/Cu–CuO\(_x\), the harmonic Hall analysis required correction for planar Hall contamination, and the macrospin AHR method yielded larger and more reliable \(\theta_\mathrm{SH}^\mathrm{eff}\) values [2404.18055]. In ST-FMR studies of orbital effects, angular scans are used not only to quantify damping-like efficiency but also to identify unconventional torque symmetries, including out-of-plane components [2311.05868][2603.23826].

## 6. Debates, system specificity, and future directions

OT is an active and contested subject rather than a closed taxonomy. The clearest recent controversy concerns Ta/ferromagnet bilayers. “Absence of orbital current torque in Ta/ferromagnet bilayers” reports that, after subtracting a significant thickness-dependent self-induced ST-FMR signal of the ferromagnet, Ta/Ni, Ta/Ni\(_{81}\)Fe\(_{19}\), Ta/Fe, Ta/Fe\(_{60}\)Co\(_{20}\)B\(_{20}\), and Ta/FePt all show essentially the same negative damping-like efficiency, about \(-0.03\), consistent with Ta’s spin Hall effect and not with a positive orbital-current contribution [2501.10260]. The paper argues that prior positive estimates in a specific Ni-thickness range resulted from analysis artifacts rather than orbital transport.

This negative result is system-specific rather than a general refutation of OT. The same literature set contains multiple positive claims in Cu/CuO\(_x\), Zr, Ti, Cr, SrRuO\(_3\), and rare-earth ferrimagnet platforms, often with control experiments designed to exclude self-torque or ordinary SOT as the dominant mechanism [2404.18055][2410.02238][2403.03043][2406.19982][2508.18746]. A plausible implication is that OT is highly sensitive to source-layer electronic structure, interface orbital transparency, magnetic-layer conversion efficiency, and metrological protocol, so transfer of conclusions from one material system to another is limited.

Two forward directions are prominent. One is device diversification. In a flexible mica/SrRuO\(_3\)/CoPt heterostructure, the reported OT efficiency is \(-0.31\), and a thermally assisted switching mechanism on a low-thermal-conductivity substrate yields an ultralow threshold current density of \(9.2\times10^9\,\mathrm{A/m^2}\), maintained after \(10^3\) bending cycles [2508.18746]. The other is band-structure engineering beyond linear response. The nonlinear orbital Hall proposal identifies topological band degeneracies as amplifiers of out-of-plane OT and presents topological metals as a new class of orbital-current sources [2511.10314].

Taken together, the OT literature defines a field at the intersection of transport theory, interfacial band engineering, and torque metrology. Its central claims are now specific enough to be tested comparatively: whether the source is bulk OHE or interfacial orbital Rashba physics, whether the magnetic detector enhances or suppresses orbital-to-spin conversion, whether the torque is long-ranged or interfacial, and whether the measured signal survives stringent artifact controls. That combination of microscopic specificity and experimental variability is the defining feature of orbital torque as an orbitronic concept.

Source: https://www.emergentmind.com/topics/orbital-torque-ot