Orbital Torque in Magnetism

• Orbital torque is the current-induced torque on a magnetic layer derived from orbital angular momentum and orbital currents instead of spin currents.
• Experimental studies reveal both interfacial orbital Rashba–Edelstein and bulk orbital Hall mechanisms, underscoring the role of material interfaces and conversion efficiencies.
• Device applications leveraging orbital torque demonstrate reduced switching currents and enhanced efficiency compared to conventional spin–orbit torque systems.

Orbital torque is a current-induced torque on a magnetic layer that originates from orbital angular momentum and orbital currents rather than from spin currents. In the modern orbitronics literature, it denotes either charge-to-orbital conversion in a nonmagnetic source followed by orbital-to-spin conversion in an adjacent spin–orbit-coupled magnet, or, in rare-earth systems with finite orbital moments, a more direct coupling of injected orbital current to orbital magnetization (Go et al., 2019, Tazaki et al., 2020, Chen et al., 4 Feb 2026). Initially formulated as the orbital analogue of spin–orbit torque in nonmagnet/ferromagnet bilayers (Go et al., 2019), it is now studied in interfacial orbital Rashba–Edelstein systems, bulk orbital Hall materials, ferrimagnets, rare-earth ferromagnets, magnetic tunnel junctions, flexible heterostructures, and two-dimensional magnets (Xu et al., 2024, Ding et al., 2024, Yao et al., 11 Apr 2025, Zeer et al., 1 Feb 2025).

1. Emergence of the concept and distinction from spin–orbit torque

The original theoretical formulation considered an NM/FM bilayer in which the nonmagnet hosts an orbital Hall effect even when spin–orbit coupling is absent in the NM, while the ferromagnet supplies the spin–orbit coupling needed to convert injected orbital angular momentum into spin and then into magnetization torque through exchange (Go et al., 2019). In that construction, orbital torque has the same formal damping-like and field-like symmetries as conventional spin–orbit torque, but the source of angular momentum is orbital rather than spin.

The first explicit experimental claim of current-induced torque without spin currents was made in naturally oxidized Cu/ferromagnetic bilayers, where exceptionally large torque signals were observed despite the weak spin–orbit coupling of Cu (Tazaki et al., 2020). That work established two features that have remained central to the subject: first, the torque source can be a light metal if inversion symmetry is broken so that orbital Rashba physics or orbital Hall transport is activated; second, the damping-like torque sign can depend on the ferromagnet rather than being fixed by the source layer.

This dependence on the magnetic layer is the sharpest contrast with spin-Hall-driven spin–orbit torque. In heavy-metal SOT, the dominant damping-like sign primarily follows the spin Hall response of the source and is only weakly sensitive to the magnetic layer. In orbital torque, by contrast, the source is a charge-to-orbital converter, and the magnetic layer or interface determines how efficiently and with what sign orbital angular momentum is converted into spin or directly coupled to orbital magnetization (Lee et al., 2021, Xu et al., 2024). A persistent misconception is therefore that orbital torque eliminates spin–orbit coupling altogether. The literature instead shows that strong spin–orbit coupling is not required in the source layer, but some spin–orbit-mediated conversion step is generally required in the magnetic layer or at the interface, except in the rare-earth limit where direct orbital-moment torque becomes relevant (Ding et al., 2024, Chen et al., 4 Feb 2026).

2. Microscopic mechanisms

Two microscopic routes dominate the field. The first is interfacial: broken inversion symmetry creates an orbital Rashba texture and an orbital Rashba–Edelstein response. For naturally oxidized Cu, a representative interfacial Hamiltonian is written as

$$H_{\mathrm{OR}}=\alpha_L(L_x k_y-L_y k_x)=\alpha_L\,\hat{\boldsymbol z}\cdot(\mathbf L\times \mathbf k),$$

which produces a nonequilibrium orbital polarization $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$ and, more generally, an orbital current under applied in-plane current (Tazaki et al., 2020). This same interfacial logic underlies later analyses of CuO$_x$ and Co/Al interfaces, where orbital Rashba textures rather than bulk spin Hall currents dominate the response (Gong et al., 2023, Pezo et al., 20 Mar 2025).

The second route is bulk orbital Hall transport. In light metals and related conductors, an in-plane electric field generates a transverse orbital current according to

$J_y^{O}=\sigma_{xy}^{O}E_x,$

with $\sigma_{xy}^{O}$ the orbital Hall conductivity (Xu et al., 2024). This mechanism is emphasized in Ti, V, Cr, Zr, Ru, and SrRuO$_3$, where weak or negligible spin Hall response can coexist with strong orbital Hall transport (Lee et al., 2021, Yang et al., 2024, Vijayan et al., 22 Aug 2025, Gong et al., 26 Aug 2025).

In the standard metallic-bilayer picture, injected orbital angular momentum does not directly torque the magnetization. It is first converted into spin angular momentum by the spin–orbit coupling of the ferromagnet or by an interfacial converter, and the converted spin then couples to the local moments by exchange (Lee et al., 2021). This conversion efficiency is denoted $n_{LS}$ or $\eta_{L\to S}$ in several papers, and its dependence on the magnetic layer is a defining materials parameter (Xu et al., 2024). Rare-earth ferrimagnets and ferromagnets amplify this step because their local spin–orbit conversion can be much stronger than in $3d$ transition metals; in Gd–Co, the local conversion at Gd sites is about five times stronger and of opposite sign relative to Co (Ding et al., 2024).

A distinct limit appears in rare-earth ferromagnets with finite orbital moments. In Cr/Tb, the measured positive and giant damping-like torque cannot be reconciled with the known negative and small spin Hall effect of Cr, and the proposed interpretation is that orbital currents generated by the orbital Hall effect in Cr are injected into Tb with negligible loss and couple directly to Tb’s orbital magnetization (Chen et al., 4 Feb 2026). This suggests that the term “orbital torque” encompasses both orbital-to-spin-converted torques and, in suitable rare-earth systems, torques on orbital moments themselves.

Theoretical work has also emphasized that orbital responses in ferromagnets need not be short-ranged. In Cr/CoFe, the induced orbital angular momentum and the associated spin–orbit torque persist well beyond the spin dephasing length because nearly degenerate orbital characters create “orbital hotspots” in momentum space, suppressing the destructive interference that usually limits spin accumulation (Go et al., 2021). This long-range orbital magnetoelectric response is a central distinction between orbital and spin transport in ferromagnets.

3. Torque structure, transport phenomenology, and reciprocity

Despite the distinct microscopic origin, the torque decomposition usually mirrors conventional SOT notation. A common effective-field form is

$$\boldsymbol{\tau}=\mu_0 M_s t_F\,\mathbf m\times \mathbf H_{\mathrm{eff}},$$

with $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$0 the unit magnetization, $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$1 the saturation magnetization, and $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$2 the magnetic thickness (Tazaki et al., 2020). In an orbital-polarization picture the damping-like and field-like parts are written as

$\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$3

where $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$4 is the nonequilibrium orbital polarization (Tazaki et al., 2020). In an orbital-current picture, the same structures appear after orbital-to-spin conversion, with $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$5 replacing $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$6 in the usual SOT notation (Xu et al., 2024).

Several efficiency conventions coexist. In harmonic Hall analyses of light-metal/ferrimagnet bilayers one often defines

$\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$7

where $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$8 is the damping-like effective field and $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$9 is the current density in the light metal (Xu et al., 2024). In ST-FMR work on naturally oxidized Cu, the corresponding dimensionless torque efficiencies are denoted $_x$0 and $_x$1, with an effective conductivity $_x$2 used as a figure of merit even though the underlying mechanism is orbital rather than spin Hall (Tazaki et al., 2020). In rare-earth ferrimagnets the same object is written as an effective spin–orbital Hall angle $_x$3 (Ding et al., 2024).

A more general phenomenology treats spin and orbital chemical potentials and currents on equal footing. In that framework, orbital torque arises from a compromise between orbital current injection from the orbital source into the ferromagnet and spin current backflow from the ferromagnet back to the source, while “orbit-spin-mixing” and “spin-orbit-mixing” conductances govern orbital torque and its Onsager reciprocal, orbital pumping (Ning et al., 2024). This formulation makes explicit that OT is not only a source-layer property. The total response depends on orbital transmission through the interface, bulk or interfacial orbit-to-spin conversion, and the relative absorption and relaxation lengths of orbital and spin currents in both layers.

Reciprocity has become increasingly important. The same formalism predicts orbital pumping under magnetization dynamics (Ning et al., 2024), and in two-dimensional Janus EuSP, EuSSe, and EuSCl monolayers the calculated inverse response yields strong in-plane orbital currents with non-trivial direction of orbital polarization (Zeer et al., 1 Feb 2025). In that $_x$4 limit the steady-state torque is purely orbital in origin because the spin-flux term vanishes by construction, and the exchange torkance is essentially identical to the spin–orbital torque (Zeer et al., 1 Feb 2025).

4. Experimental identification and metrology

Because converted orbital currents generate the same damping-like and field-like torque symmetries as ordinary spin currents, OT is generally identified through material trends, thickness dependences, temperature dependences, and control experiments rather than by symmetry alone (Tazaki et al., 2020, Xu et al., 2024). ST-FMR and harmonic Hall methods dominate the field, supplemented by weak antilocalization, THz emission, and reciprocal pumping measurements.

The naturally oxidized Cu work established the template. ST-FMR on NO-Cu(10 nm)/Ni$_x$5Fe$_x$6 and NO-Cu(10 nm)/Fe strips gave $_x$7 and $_x$8 at room temperature in NO-Cu/NiFe, while $_x$9 increased strongly on cooling and exceeded $J_y^{O}=\sigma_{xy}^{O}E_x,$0 at $J_y^{O}=\sigma_{xy}^{O}E_x,$1 K, yielding $J_y^{O}=\sigma_{xy}^{O}E_x,$2 (Tazaki et al., 2020). Weak antilocalization showed that the spin–orbit scattering probability remained small, $J_y^{O}=\sigma_{xy}^{O}E_x,$3 in NO-Cu versus $J_y^{O}=\sigma_{xy}^{O}E_x,$4 in Pt, excluding a conventional heavy-metal-like spin Hall origin. Most decisively, the damping-like sign reversed between NO-Cu/NiFe and NO-Cu/Fe, contradicting the expectation that the sign is fixed by the source layer (Tazaki et al., 2020).

Bulk-OHE-dominant systems are diagnosed differently. In Ti/Fe$J_y^{O}=\sigma_{xy}^{O}E_x,$5Gd$J_y^{O}=\sigma_{xy}^{O}E_x,$6, the orbital torque efficiency increases and saturates with Ti thickness around $J_y^{O}=\sigma_{xy}^{O}E_x,$7–$J_y^{O}=\sigma_{xy}^{O}E_x,$8 nm, which indicates bulk generation rather than a purely interfacial orbital Rashba–Edelstein mechanism (Xu et al., 2024). The same study showed a four-fold increase in torque efficiency when moving from Fe$J_y^{O}=\sigma_{xy}^{O}E_x,$9Gd$\sigma_{xy}^{O}$0 to Fe$\sigma_{xy}^{O}$1Gd$\sigma_{xy}^{O}$2 while $\sigma_{xy}^{O}$3, $\sigma_{xy}^{O}$4, and $\sigma_{xy}^{O}$5 remained nearly constant, directly tying the torque to the ferrimagnet’s spin–orbit conversion rather than to the source alone (Xu et al., 2024).

Angular diagnostics have uncovered unconventional components. In FM/CuO$\sigma_{xy}^{O}$6, ST-FMR required additional $\sigma_{xy}^{O}$7 terms in both the symmetric and antisymmetric amplitudes,

$\sigma_{xy}^{O}$8

which was interpreted as evidence for a $\sigma_{xy}^{O}$9-polarized orbital current producing an out-of-plane antidamping-like orbital torque (Gong et al., 2023). A broader ST-FMR survey of SiO$_3$0/FM/NM bilayers likewise identified out-of-plane torque components through azimuthal-angular asymmetries and found systematically larger damping-like efficiencies in Ni/NM than in Py/NM for several weak-SHE materials, consistent with stronger orbital-to-spin conversion in Ni (Costa et al., 25 Mar 2026).

Multimodal cross-checks have been especially important in V. Second-harmonic Hall, ST-FMR, and THz emission showed that the damping-like torque per unit electric field remains negative for all V thicknesses, that Ni/V and Ni/Pt have opposite signs in both ST-FMR and THz polarity, and that the thickness dependence fits a drift–diffusion form with $_3$1 and $_3$2 (Vijayan et al., 22 Aug 2025). This combination of sign, thickness, and reciprocal ultrafast response is now a standard model for disentangling OHE from SHE.

5. Materials platforms and representative systems

The current literature spans interfacial orbital Rashba sources, bulk orbital Hall sources, rare-earth converters, and direct orbital-moment receptors.

Platform	Representative observation	Citation
Naturally oxidized Cu/NiFe and Cu/Fe	$_3$3 at room temperature in NO-Cu/NiFe; $_3$4; $_3$5 in NO-Cu/Fe at room temperature	(Tazaki et al., 2020)
Cr/Gd and Cr/Pt/CoFeB	$_3$6 in Gd(10)/Cr(5) at $_3$7 K; Pt insertion flips PMA switching polarity	(Lee et al., 2021)
Ti/Fe$_3$8Gd$_3$9	Nearly complete switching with $n_{LS}$0 at $n_{LS}$1 mT; at $n_{LS}$2 nm, $n_{LS}$3 and $n_{LS}$4	(Xu et al., 2024)
Gd$n_{LS}$5Co$n_{LS}$6/CuO$n_{LS}$7	$n_{LS}$8 in Gd$n_{LS}$9Co$\eta_{L\to S}$0/CuO$\eta_{L\to S}$1 rises from $\eta_{L\to S}$2 at $\eta_{L\to S}$3 K to $\eta_{L\to S}$4 at $\eta_{L\to S}$5 K	(Ding et al., 2024)
Zr/[Co/Pt]$\eta_{L\to S}$6	$\eta_{L\to S}$7; full switching at $\eta_{L\to S}$8	(Yang et al., 2024)
V-based orbital torque	$\eta_{L\to S}$9 and $3d$0	(Vijayan et al., 22 Aug 2025)
Cr/Tb	At $3d$1 K, $3d$2 Oe and $3d$3	(Chen et al., 4 Feb 2026)

Interfacial orbital Rashba systems extend the same logic to atomically sharp light-metal interfaces. At Co/Al(111), the interfacial Co layer develops a chiral in-plane orbital texture of $3d$4–$3d$5 on the Fermi surface, the integrated field-like torkance is $3d$6, and inserting a single Pt monolayer suppresses the orbital Edelstein response while reviving a sizeable damping-like torkance characteristic of Co/Pt (Pezo et al., 20 Mar 2025). This system illustrates a recurrent theme: heavy-metal insertion can enhance conventional spin-mediated torques while suppressing an interfacial orbital torque that depends on a delicate orbital Rashba texture.

Two-dimensional rare-earth Janus monolayers push the concept into a different regime. In EuSP, EuSSe, and EuSCl, the exchange torkance $3d$7 reaches almost $3d$8 and tracks a colossal current-induced orbital response on Eu $3d$9 states, while the induced Eu $$\boldsymbol{\tau}=\mu_0 M_s t_F\,\mathbf m\times \mathbf H_{\mathrm{eff}},$$0 is typically about two orders of magnitude smaller than $$\boldsymbol{\tau}=\mu_0 M_s t_F\,\mathbf m\times \mathbf H_{\mathrm{eff}},$$1 (Zeer et al., 1 Feb 2025). These materials therefore realize a limit in which the torque is not merely orbital in source but orbital in the steady-state continuity equation itself.

6. Device relevance, unresolved problems, and future directions

Several device-oriented studies show that OT is no longer only a diagnostic of orbital transport. In SOT-MRAM stacks using Ru as an orbital Hall layer and Pt as an orbital-to-spin converter, the damping-like torque efficiency is enhanced by approximately $$\boldsymbol{\tau}=\mu_0 M_s t_F\,\mathbf m\times \mathbf H_{\mathrm{eff}},$$2 relative to pure Pt, the average switching current density is reduced by approximately $$\boldsymbol{\tau}=\mu_0 M_s t_F\,\mathbf m\times \mathbf H_{\mathrm{eff}},$$3 across more than $$\boldsymbol{\tau}=\mu_0 M_s t_F\,\mathbf m\times \mathbf H_{\mathrm{eff}},$$4 devices, and switching power is reduced by more than $$\boldsymbol{\tau}=\mu_0 M_s t_F\,\mathbf m\times \mathbf H_{\mathrm{eff}},$$5 (Gupta et al., 2024). In Ru/W-driven magnetic tunnel junctions, the combination of strong orbital Hall conductivity in Ru and orbit-to-spin conversion efficiency exceeding $$\boldsymbol{\tau}=\mu_0 M_s t_F\,\mathbf m\times \mathbf H_{\mathrm{eff}},$$6 in $$\boldsymbol{\tau}=\mu_0 M_s t_F\,\mathbf m\times \mathbf H_{\mathrm{eff}},$$7-W yields $$\boldsymbol{\tau}=\mu_0 M_s t_F\,\mathbf m\times \mathbf H_{\mathrm{eff}},$$8-ps switching, a five to eight-fold reduction in driving voltages over conventional SOT-MRAM, and a minimum reported $$\boldsymbol{\tau}=\mu_0 M_s t_F\,\mathbf m\times \mathbf H_{\mathrm{eff}},$$9 fJ/bit at $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$00 ps (Yao et al., 11 Apr 2025).

Flexible orbitronics has followed rapidly. In mica/SrRuO$\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$01/CoPt, harmonic Hall measurements give $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$02, and thermally assisted switching on the low-thermal-conductivity mica substrate reduces the threshold current density to $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$03, a $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$04 reduction relative to conventional flexible spin-torque devices and a $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$05 reduction relative to the rigid SrTiO$\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$06 counterpart; the switching characteristics are maintained after $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$07 bending cycles (Gong et al., 26 Aug 2025). These results reinforce a practical advantage already noted in NO-Cu, Ru, and Zr systems: OT can combine large torque efficiency with relatively low resistive loss in light or moderately conductive materials (Tazaki et al., 2020, Yang et al., 2024, Gupta et al., 2024).

A major frontier is field-free perpendicular switching. Linear OT studies have demonstrated deterministic perpendicular switching in Ti/Fe$\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$08Gd$\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$09, V/Fe$\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$10Gd$\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$11, and Cr/Fe$\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$12Gd$\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$13 under assist fields (Xu et al., 2024), but recent theory argues that the deeper route is nonlinear orbital Hall transport. The nonlinear orbital Hall effect permits collinearly polarized orbital currents in $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$14 of the $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$15 noncentrosymmetric crystal classes, and in $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$16 of those classes the linear collinear orbital Hall response is symmetry-forbidden, making the nonlinear channel the leading source of out-of-plane orbital torque (Wang et al., 13 Nov 2025). First-principles calculations for RhSi, YPtBi, and PbTaSe$\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$17 predict large nonlinear conductivities, effective out-of-plane orbital Hall conductivities on the order of $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$18–$\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$19 for RhSi and YPtBi at $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$20, and effective out-of-plane orbital Hall angles that substantially exceed those of previously reported linear mechanisms once realistic orbital-to-spin conversion efficiencies are included (Wang et al., 13 Nov 2025).

Open problems are now well defined. Direct measurement of orbital currents and orbital accumulation remains a central challenge, explicitly noted in naturally oxidized Cu and rare-earth ferrimagnet studies (Tazaki et al., 2020, Ding et al., 2024). Quantitative determination of interfacial orbital Rashba parameters, orbital Edelstein susceptibilities, and interfacial orbit-spin-mixing conductances is incomplete (Tazaki et al., 2020, Ning et al., 2024). Bulk versus interfacial source separation remains difficult in systems where both OHE and OREE may coexist (Xu et al., 2024, Costa et al., 25 Mar 2026). The microscopic routes of orbital-to-spin conversion in ferrimagnets, especially the role of $\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$21–$\boldsymbol{\zeta}\equiv \langle \mathbf L\rangle$22 hybridization and momentum-space versus real-space pathways, remain unresolved (Xu et al., 2024). Finally, orbital pumping and nontrivial in-plane orbital-current generation, already predicted in metallic bilayers and Janus rare-earth dichalcogenides, suggest that future orbitronic devices need not be limited to current-induced switching alone (Ning et al., 2024, Zeer et al., 1 Feb 2025).

Taken together, the literature defines orbital torque not as a single mechanism but as a family of angular-momentum transfer processes in which orbital degrees of freedom are primary. What unifies these processes is the separation of generation and conversion: orbital current or orbital accumulation is created in a source layer or interface, and the magnetic layer or interface decides how that orbital angular momentum becomes a torque. This separation is the distinctive materials principle of orbitronics, and it is the basis for the field’s rapid expansion from light-metal bilayers to ferrimagnets, rare-earth ferromagnets, topological nonlinear sources, and technologically relevant memory heterostructures (Go et al., 2019, Xu et al., 2024, Yao et al., 11 Apr 2025, Wang et al., 13 Nov 2025).