Spin Anomalous Hall Effect (SAHE)

• SAHE is a charge-to-spin conversion mechanism in ferromagnets where a longitudinal charge current generates a transverse spin current with spin polarization aligned to the magnetization.
• It leverages spin–orbit coupling, exchange-split bands, and Berry curvature, with its magnitude critically dependent on the relative orientation of current and magnetization.
• Both first-principles calculations and experiments, such as linewidth modulation measurements, quantify SAHE via damping-like torques and efficiency ratios.

Spin anomalous Hall effect (SAHE) is a charge-to-spin conversion mechanism in ferromagnets and ferrimagnets in which a longitudinal charge current generates a transverse spin current whose spin polarization is collinear with the magnetization. In the notation used across recent spin-transport literature, SAHE is the spin analogue or spin-polarized counterpart of the anomalous Hall effect (AHE): it is rooted in spin–orbit coupling, exchange-split bands, and Berry curvature, but differs from the conventional spin Hall effect (SHE) because its spin polarization is not fixed solely by geometry or crystal symmetry. A standard form is

$$J_s^{\mathrm{SAHE}} = (\hbar/2e)\,\theta_{\mathrm{SAHE}}\,(J_c \times m),$$

with spin polarization $\sigma \parallel m$ (Damas et al., 25 Nov 2025).

1. Definition, geometry, and distinction from other Hall spin currents

SAHE is defined by a transverse spin current whose propagation direction is set by $m \times J_c$ and whose spin polarization is parallel to the source magnetization. In ferromagnetic device language, this is commonly written as

$$J_s^{\mathrm{SAHE}} = (2e/\hbar)\,\alpha_{\mathrm{SAH}}\,[p \times J_c],$$

with spin polarization along $p$, the ferromagnet’s unit magnetization vector (Seki et al., 2019). By contrast, the conventional SHE produces a transverse pure spin current with spin polarization perpendicular to both the charge current and the spin-current flow; in a thin film with normal $z\hat{\;}$, one standard form is

$$J_s^{\mathrm{SHE}} = (\hbar/2e)\,\theta_{\mathrm{SH}}\,(J_c \times z\hat{\;}).$$

The corresponding damping-like and field-like torques have the same formal structure for SHE and SAHE,

$$\tau_{DL} \propto m \times (\sigma \times m), \qquad \tau_{FL} \propto m \times \sigma,$$

but SAHE differs because $\sigma$ is set by the magnetization rather than by crystal axes alone (Damas et al., 25 Nov 2025).

In recent first-principles work on ferromagnetic alloys, SAHE is treated as one member of a three-part taxonomy comprising conventional SHE (CSHE), SAHE, and magnetic SHE (MSHE). In that classification, SAHE is time-reversal even, requires noncollinearity between $m$ and $\sigma \parallel m$0, and has the hallmark geometry of a spin current along $\sigma \parallel m$1 with spin polarization parallel to $\sigma \parallel m$2 (Zheng et al., 2023). This separates it from CSHE, which is magnetization-independent, and from MSHE, which is time-reversal odd and relaxes the usual orthogonality constraint among charge-current direction, spin-current direction, and spin-polarization axis (Zheng et al., 2023).

A basic symmetry consequence is that SAHE vanishes when the magnetization is collinear with the charge current. In the L1$\sigma \parallel m$3-FePt trilayer experiments, the orthogonal configuration $\sigma \parallel m$4 produced a robust linewidth modulation attributable to SAHE, whereas the parallel configuration $\sigma \parallel m$5 showed no measurable modulation attributable to magnetization-independent SHE or interfacial torques (Seki et al., 2019). This geometry dependence is central to experimental identification.

2. Microscopic theory and relation to Berry curvature

The theoretical treatment of SAHE is usually formulated within linear-response Kubo theory. For AHE, the conductivity is written as

$\sigma \parallel m$6

with Berry curvature accumulated over occupied bands. The spin Hall conductivity tensor for spin polarization along $\sigma \parallel m$7 is

$\sigma \parallel m$8

and, in the intrinsic clean-limit Kubo formalism,

$\sigma \parallel m$9

where the spin-current operator is $m \times J_c$0 (Zhou et al., 2021).

A particularly important decomposition separates the spin Hall response into an “intra” part tied directly to diagonal spin expectation values and an “inter” part associated with off-diagonal spin coherence. In the Fe$m \times J_c$1GeTe$m \times J_c$2 analysis, the intra spin Berry curvature yields an SAHE conductivity

$m \times J_c$3

and near Berry-curvature hot spots one may approximate

$m \times J_c$4

leading at the conductivity level to

$m \times J_c$5

This makes explicit the AHE–SAHE link through Berry curvature weighted by band spin expectation (Zhou et al., 2021).

First-principles work on L1$m \times J_c$6-type $m \times J_c$7Pt alloys uses a related Berry-curvature framework but emphasizes the ratio

$m \times J_c$8

where $m \times J_c$9 is the spin anomalous Hall conductivity and $$J_s^{\mathrm{SAHE}} = (2e/\hbar)\,\alpha_{\mathrm{SAH}}\,[p \times J_c],$$0 the anomalous Hall conductivity. In that literature, $$J_s^{\mathrm{SAHE}} = (2e/\hbar)\,\alpha_{\mathrm{SAH}}\,[p \times J_c],$$1 is interpreted as the spin polarization of the anomalous Hall current (Miura et al., 2021). The same work separates SAHE from the magnetization-independent SHE by computing

$$J_s^{\mathrm{SAHE}} = (2e/\hbar)\,\alpha_{\mathrm{SAH}}\,[p \times J_c],$$2

so that the magnetization-dependent contribution can be isolated (Miura et al., 2021).

This theoretical structure supports a common physical picture. AHE and SAHE originate from the same SOC-enabled Berry-curvature hot spots, but SAHE additionally reflects exchange-induced spin polarization and band-resolved spin filtering. The inter contribution can generate spin-current components not directly correlated with AHE, which is one reason why low-symmetry ferromagnets can produce spin polarizations beyond the simplest $$J_s^{\mathrm{SAHE}} = (2e/\hbar)\,\alpha_{\mathrm{SAH}}\,[p \times J_c],$$3 expectation (Zhou et al., 2021).

3. Experimental identification and symmetry separation

The most direct experimental evidence for SAHE has come from current-induced ferromagnetic-resonance linewidth modulation in trilayers where a ferromagnetic source layer injects spin current through a nonmagnetic spacer into a detector ferromagnet. In L1$$J_s^{\mathrm{SAHE}} = (2e/\hbar)\,\alpha_{\mathrm{SAH}}\,[p \times J_c],$$4-FePt/Cu/Ni$$J_s^{\mathrm{SAHE}} = (2e/\hbar)\,\alpha_{\mathrm{SAH}}\,[p \times J_c],$$5Fe$$J_s^{\mathrm{SAHE}} = (2e/\hbar)\,\alpha_{\mathrm{SAH}}\,[p \times J_c],$$6, the linewidth-modulation method yielded

$$J_s^{\mathrm{SAHE}} = (2e/\hbar)\,\alpha_{\mathrm{SAH}}\,[p \times J_c],$$7

with strong modulation in the orthogonal configuration and no clear modulation in the parallel configuration (Seki et al., 2019). The fitting relation used there was

$$J_s^{\mathrm{SAHE}} = (2e/\hbar)\,\alpha_{\mathrm{SAH}}\,[p \times J_c],$$8

and the same study demonstrated SAHE-induced magnetization switching at $$J_s^{\mathrm{SAHE}} = (2e/\hbar)\,\alpha_{\mathrm{SAH}}\,[p \times J_c],$$9 Oe with $p$0 A/cm$p$1 (Seki et al., 2019).

In ferrimagnetic GdFeCo/Cu/NiFe, the problem is more intricate because SHE and SAHE coexist. The experimental strategy developed there combines spin-torque ferromagnetic resonance (ST-FMR) lineshape analysis with a dc-bias method. The key observation is that the ST-FMR lineshape isolates SHE in the specific geometries used: near the magnetization compensation temperature, the GdFeCo magnetization is pinned out of plane, so the SAHE spin current along $p$2 vanishes because $p$3; far from compensation, the SAHE symmetry does not exert a torque on NiFe for the specific lineshape geometry and angles used (Damas et al., 25 Nov 2025). The dc-bias method, by contrast, modifies the NiFe dynamic susceptibility and captures both SHE and SAHE contributions through linewidth and resonance-field changes.

For the dc-bias method, the damping-like extraction uses

$p$4

and the field-like plus Oersted contribution uses

$p$5

These relations enabled quantitative separation of signs and relative magnitudes of the two symmetries (Damas et al., 25 Nov 2025).

An earlier GdFeCo characterization established the same general logic in a room-temperature FeCo-rich trilayer, emphasizing that the lineshape symmetry probes only SHE-like torques whereas dc-bias linewidth and resonance-field shifts are sensitive to both SHE and SAHE. That study reported

$p$6

and

$p$7

while symmetry analysis showed $p$8, implying a dominant negative SAHE damping-like contribution (Damas et al., 2022).

4. Representative materials and quantitative results

The current SAHE literature spans ordered ferromagnets, ferrimagnetic alloys, and van der Waals ferromagnets. The results are material-specific because they depend on crystal symmetry, magnetic order, and the orbital character of Berry-curvature hot spots.

System	Representative result	Source
L1$p$9-FePt/Cu/Ni$z\hat{\;}$0Fe$z\hat{\;}$1	$z\hat{\;}$2; SAHE-induced switching	(Seki et al., 2019)
GdFeCo/Cu/NiFe near $z\hat{\;}$3	$z\hat{\;}$4 to $z\hat{\;}$5; $z\hat{\;}$6 to $z\hat{\;}$7	(Damas et al., 25 Nov 2025)
GdFeCo/Cu/NiFe far from $z\hat{\;}$8	$z\hat{\;}$9 at 15 K to $$J_s^{\mathrm{SHE}} = (\hbar/2e)\,\theta_{\mathrm{SH}}\,(J_c \times z\hat{\;}).$$0 at 300 K	(Damas et al., 25 Nov 2025)
L1$$J_s^{\mathrm{SHE}} = (\hbar/2e)\,\theta_{\mathrm{SH}}\,(J_c \times z\hat{\;}).$$1-FePt, CoPt, NiPt	$$J_s^{\mathrm{SHE}} = (\hbar/2e)\,\theta_{\mathrm{SH}}\,(J_c \times z\hat{\;}).$$2, $$J_s^{\mathrm{SHE}} = (\hbar/2e)\,\theta_{\mathrm{SH}}\,(J_c \times z\hat{\;}).$$3, and $$J_s^{\mathrm{SHE}} = (\hbar/2e)\,\theta_{\mathrm{SH}}\,(J_c \times z\hat{\;}).$$4	(Miura et al., 2021)
Fe$$J_s^{\mathrm{SHE}} = (\hbar/2e)\,\theta_{\mathrm{SH}}\,(J_c \times z\hat{\;}).$$5Pt$$J_s^{\mathrm{SHE}} = (\hbar/2e)\,\theta_{\mathrm{SH}}\,(J_c \times z\hat{\;}).$$6, Co$$J_s^{\mathrm{SHE}} = (\hbar/2e)\,\theta_{\mathrm{SH}}\,(J_c \times z\hat{\;}).$$7Pt$$J_s^{\mathrm{SHE}} = (\hbar/2e)\,\theta_{\mathrm{SH}}\,(J_c \times z\hat{\;}).$$8, Ni$$J_s^{\mathrm{SHE}} = (\hbar/2e)\,\theta_{\mathrm{SH}}\,(J_c \times z\hat{\;}).$$9Pt$$\tau_{DL} \propto m \times (\sigma \times m), \qquad \tau_{FL} \propto m \times \sigma,$$0	$$\tau_{DL} \propto m \times (\sigma \times m), \qquad \tau_{FL} \propto m \times \sigma,$$1, $$\tau_{DL} \propto m \times (\sigma \times m), \qquad \tau_{FL} \propto m \times \sigma,$$2, and MSHE-dominant $$\tau_{DL} \propto m \times (\sigma \times m), \qquad \tau_{FL} \propto m \times \sigma,$$3 in NiPt; SAHE smaller in Pt-based alloys	(Zheng et al., 2023)
Fe$$\tau_{DL} \propto m \times (\sigma \times m), \qquad \tau_{FL} \propto m \times \sigma,$$4GeTe$$\tau_{DL} \propto m \times (\sigma \times m), \qquad \tau_{FL} \propto m \times \sigma,$$5	monolayer $$\tau_{DL} \propto m \times (\sigma \times m), \qquad \tau_{FL} \propto m \times \sigma,$$6 maximum $$\tau_{DL} \propto m \times (\sigma \times m), \qquad \tau_{FL} \propto m \times \sigma,$$7; bilayer maximum $$\tau_{DL} \propto m \times (\sigma \times m), \qquad \tau_{FL} \propto m \times \sigma,$$8; simultaneous spin-polarization components	(Zhou et al., 2021)

The L1$$\tau_{DL} \propto m \times (\sigma \times m), \qquad \tau_{FL} \propto m \times \sigma,$$9-FePt result is notable because it combines a large measured SAHE efficiency with unambiguous symmetry checks. The absence of a signal for $\sigma$0 was used to exclude a magnetization-independent SHE contribution from FePt and to rule out interface-scattering torques that would mimic SHE-like polarization (Seki et al., 2019).

The GdFeCo results are notable for explicit separation of SHE and SAHE across a compensation point. Near $\sigma$1 K, only SHE contributes to the transmitted spin current in the perpendicular geometry; far from $\sigma$2, the combined signal becomes damping-like negative, demonstrating that the SAHE-driven damping-like torque dominates and has the opposite sign to the SHE-driven term (Damas et al., 25 Nov 2025).

The first-principles $\sigma$3Pt calculations show that FePt, CoPt, and NiPt do not simply scale together. FePt has $\sigma$4 (Ω·cm)$\sigma$5 and $\sigma$6 $\sigma$7, whereas NiPt has $\sigma$8 (Ω·cm)$\sigma$9 and $m$0 $m$1, producing $m$2 (Miura et al., 2021). This sign inversion and magnitude enhancement are central to current materials-design discussions.

5. Compensation, symmetry control, and sublattice-specific transport

Ferrimagnets add a layer of complexity because the net magnetization can reverse or vanish at compensation even when the relevant electronic subsystems do not. In GdFeCo, the magnetization compensation temperature is $m$3 K, and the central experimental result is that neither the SHE nor the SAHE torque contribution reverses sign upon crossing $m$4 (Damas et al., 25 Nov 2025). The SHE-only signal remains positive across the full 15–300 K window, and the SAHE-driven damping-like term keeps the opposite sign to the SHE-driven damping-like term on both sides of compensation (Damas et al., 25 Nov 2025).

The interpretation offered for this non-inversion is sublattice-specific. The SHE is suggested to emerge predominantly from Gd 5d conduction electrons, whose contribution is magnetization-insensitive to leading order, whereas SAHE is suggested to originate predominantly from FeCo 3d exchange-split conduction electrons (Damas et al., 25 Nov 2025). In that picture, the AHE charge signal can reverse across compensation because the charge-carrier imbalance between sublattices reverses, yet the SAHE spin polarization relevant for torque generation does not invert because the FeCo 3d spin polarization retains its sign. The observed opposite sign of SAHE- and SHE-driven damping-like torques is then attributed to opposite $m$5 signs for the dominant sublattice channels (Damas et al., 25 Nov 2025). This suggests that compensation in ferrimagnets must be distinguished from a reversal of the microscopic spin-current source.

Low-symmetry ferromagnets provide a different form of control. In Fe$m$6GeTe$m$7, monolayers and bilayers were predicted to exhibit nonlinear magnetization dependence of the spin current, with simultaneous in-plane and out-of-plane spin polarizations, and bilayer Fe$m$8GeTe$m$9 can even exhibit arbitrary spin polarization because of reduced symmetry (Zhou et al., 2021). In that system, the AHE–SAHE relation is symmetry-filtered: whenever the magnetization is orthogonal to a retained mirror, AHE and hence SAHE are forbidden; away from those directions, both responses are allowed (Zhou et al., 2021). This makes SAHE not just a consequence of magnetization, but also a sensitive probe of magnetic point-group symmetry.

6. Microscopic mechanisms, unresolved issues, and device relevance

The microscopic origin of large SAHE varies across materials. In L1$\sigma \parallel m$00-type $\sigma \parallel m$01Pt alloys, the crucial first-principles observation is that the spin-down–down component of the AHC can become strongly negative in CoPt and especially NiPt. For NiPt, the spin-resolved contributions were reported as $\sigma \parallel m$02 and $\sigma \parallel m$03 (Ω·cm)$\sigma \parallel m$04, so the charge Hall channels partially cancel while the spin-current channel $\sigma \parallel m$05 is strongly amplified (Miura et al., 2021). The paper attributes this to anti-bonding Pt states around the Fermi level in the minority-spin states (Miura et al., 2021). In Pt-based disordered ferromagnetic alloys, the coexistence of CSHE, SAHE, and MSHE means that measured spin–orbit torques can mix multiple tensor channels unless magnetization-direction tests are performed (Zheng et al., 2023).

A recurrent issue is the gap between intrinsic calculations and experiment. For FePt, the intrinsic calculation gave $\sigma \parallel m$06 and, using the measured longitudinal resistivity $\sigma \parallel m$07cm, an estimated SAHE efficiency

$\sigma \parallel m$08

which is about an order of magnitude smaller than the experimental estimate of roughly $\sigma \parallel m$09 (Miura et al., 2021). The theoretical paper identifies extrinsic contributions such as skew scattering and side-jump, absent in the intrinsic calculation, as the likely source of the discrepancy (Miura et al., 2021). This is not a settled controversy so much as a clear methodological boundary between intrinsic Berry-curvature transport and experimentally realized transport.

Device relevance follows directly from torque symmetry. Because SAHE fixes the spin polarization to the source magnetization, rotating the source magnetization reorients the injected spin polarization (Seki et al., 2019). In Fe$\sigma \parallel m$10GeTe$\sigma \parallel m$11, this has been proposed as a route to out-of-plane spin polarization and field-free switching of perpendicular magnets, which is difficult to realize with conventional SHE sources whose spin polarization is symmetry-locked (Zhou et al., 2021). In ferrimagnets, the persistence of SAHE and SHE signs across compensation implies that previously reported self-torque sign changes in single-layer ferrimagnets likely reflect changes in spin-current absorption or spin dephasing rather than reversals of the generated spin currents (Damas et al., 25 Nov 2025). This suggests a design space in which composition, anisotropy, spacers, and temperature are used to tune the balance between SHE and SAHE without necessarily changing their intrinsic signs.

At the same time, several open problems remain explicitly identified in the literature. The microscopic origin of the sizable field-like torque in GdFeCo/Cu/NiFe without direct GdFeCo/NiFe contact requires further study (Damas et al., 2022). The precise sublattice-resolved mechanism of SAHE in ferrimagnets also remains to be clarified experimentally and theoretically (Damas et al., 2022). More broadly, the coexistence of CSHE, SAHE, and MSHE in ferromagnets means that quantitative extraction of a single “spin Hall angle” can be misleading unless the full tensor symmetry and magnetization geometry are controlled (Zheng et al., 2023).

In this sense, SAHE is best understood not as an isolated anomaly but as one branch of magnetization-dependent Hall spin transport. Its defining feature is simple—spin polarization parallel to magnetization—but its manifestations are governed by Berry curvature, sublattice physics, magnetic symmetry, and the coexistence of multiple spin–charge conversion channels in real materials.