Introduction

Antiferromagnetic (AFM) materials have emerged as promising candidates for next-generation spintronic memory technologies, offering distinct advantages over their ferromagnetic counterparts—including higher storage density, faster data processing, absence of stray fields, and robustness against external magnetic perturbations1. Their intrinsically fast spin dynamics (in the THz range) and the ability to host large magneto-transport effects further underscore their technological potential1,2. As a topological Weyl antiferromagnet, Mn3Sn has garnered particular interest due to its topologically nontrivial electronic structure and non-collinear spintexture3,4, which gives rise to a host of unconventional spin transport phenomena such as large anomalous Hall effect5,6, anomalous Nernst effect7, magneto-optical Kerr effects8, magnetic spin Hall effect9 and octupole TMR10. These emergent phenomena open new pathways for functional AFM spintronic devices.

As a unique platform for studying and harnessing chiral spin-orbit torque (SOT) switching in the Weyl AFM materials4,11,12,13,14, the nontrivial topological band properties and topological spin texture of Mn3Sn can be manipulated using SOT3,4. Through the application of electrical currents with a strong SOC layer of Pt, it becomes possible to manipulate the octupole moments in Mn3Sn4. In addition, combinations of in-plane and out-of-plane SOT can experimentally realize field-free switching of magnetic octupole moments in chiral AFM Mn3Sn15. SOT switching in Mn3Sn holds immense promise for low-power spintronic devices. The efficient conversion of charge current into spin currents allows for the development of energy-efficient memory and logic devices.

In conventional ferromagnetic materials, spin–orbit torque (SOT) arises from the misaligned between the transport spin polarization and local magnetization. However, this exchange field is typically too weak to significantly reorient neighboring spins, resulting in a short spin diffusion length along the thickness direction16. However, in antiferromagnets, staggered spin torques can coherently drive the Néel order, allowing spin torques to act uniformly throughout the entire volume and resulting in a bulk SOT effect16,17. Moreover, canted AFM oxide insulators exhibit strong spin coherence, which may even give rise to magnon interference effects18. Here, we demonstrate current-induced octupole moment switching in the canted Mn3Sn Kagome spin lattice. Both the SOT switching efficiency and the quantified SOT effective fields increase monotonically with film thickness. Remarkably, the noncollinear Weyl antiferromagnet Mn₃Sn exhibits a bulk-like SOT even in the presence of strong perpendicular magnetic anisotropy. As a result, SOT can switch the octupole moment in Mn3Sn films as thick as 60 nm. Atomistic simulations further reveal that the twin antiferromagnetic order in Mn₃Sn enhances the spin diffusion length, facilitating long-range octupole moment switching. These discoveries open new avenues for generating long-range SOT in topological antiferromagnets, paving the way for ultrafast, ultrahigh-density, and scalable spintronic applications.

Results

Structural properties of epitaxially grown Mn3Sn thin films

D019-Mn3Sn is a non-collinear antiferromagnet with a hexagonal structure in the space group P63/mmc. It exhibits noncollinear chiral vector spin order below the Néel temperature TN ≈ 430 K4,5. Mn3Sn has excellent spin-related characteristics due to a large cluster magnetic octupole moment along [2-1-10] direction19,20 (Fig. 1a). In addition, the spin canting within the (0001) plane of Mn3Sn (Fig. 1b, c) can contribute to the observed ferromagnetic-like signal M ≈ 0.006 μB per formula unit (f.u.; μB, Bohr magneton). In addition, the interfacial stress in the thin-film system induces additional magnetic contributions5,13 and gives rise to twin topological spin textures at the interface between the Al2O3 substrate and Mn3Sn (supplementary notes 1–4). Density-functional theory calculations show that a 2% tensile strain induces a rotation of about 1.5° of Mn atoms (supplementary note 5). High-quality Mn3Sn (11–20) alloys were epitaxially grown by sputtering on Al2O3 (1–102) substrate. In this crystal orientation, all the spins of the Mn atoms in the Kagome lattice lie out of the thin film plane, and form an angle of 120 degrees with each other. Sn atoms stand in the center of 6 Mn atoms surrounding it. Each Mn moment has the local easy-axis towards its nearest-neighbor Sn sites and the octupole moment is along [2-1-10] direction (Fig. 1c). The Mn3Sn film grown on the Al2O3 substrate shows a typical XRD spectrum (Fig. 1d). The Al2O3 substrate has (1–102) and (2–204) superlattice diffraction peak as the two largest main peaks, The Mn3Sn (11–20) plane can be epitaxial grown on the Al2O3 (1–102) plane after 90 deg rotation for lattice matching. The two extra peaks Mn3Sn (11–20) and Mn3Sn (22–40) are good proof of the single crystal. A 360° phi scan around the Mn3Sn (20–21) plane was measured by XRD (Fig. 2e). The Mn3Sn shows fourth-degree symmetry, indicating the single-crystal property. In this structure, the unit cell of Mn3Sn consists of six Mn atoms and two Sn atoms, with the six neighboring Mn atoms forming a Kagome lattice (Fig. 1f, g). High-resolution TEM studies also reveal the single crystal feature of the chiral magnet Mn3Sn (Fig. 1h), which is essential for achieving the designed properties and functional spintronics devices. Figure 1i shows the Fast Fourier transform (FFT) of the Mn3Sn thin film, where the change in the FFT contrast gives rise to sharp diffraction spots, evidencing the single-crystalline nature of the film.

Fig. 1: Structure of the noncollinear AFM Mn3Sn films.
Fig. 1: Structure of the noncollinear AFM Mn3Sn films.
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a Schematic of the spin structure. Blue and Red spins represent Mn atoms in Kagome planes. Purple sphere represents the Sn atom. The direction of the octupole moment is along [2-1-10]. b Schematic of canted Kagome AFM. All the spins titled to the direction of the octupole moment c Periodic spin canting effect in Mn3Sn crystals. d XRD spectra of Mn3Sn on Al2O3 substrate. e Phi scan of the Mn3Sn substrate, which indicates the four-fold symmetry. f Crystal structure of Mn3Sn, showing the atomic positions of Mn and Sn within the unit cell. g Atomic configuration of Mn and Sn atoms viewed along the [0001] direction in the Mn₃Sn crystal structure. h High-resolution TEM image of Pt/Mn3Sn films. i FFT of TEM images of Mn₃Sn thin films.

Fig. 2: Electrical switching of the perpendicular magnetic octupole moment under bulk SOT in the chiral AFM order of Mn3Sn.
Fig. 2: Electrical switching of the perpendicular magnetic octupole moment under bulk SOT in the chiral AFM order of Mn3Sn.
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a Schematic illustration of the SOT switching mechanisms in a noncollinear antiferromagnet and its twinned counterpart. In each case, the five subFigs from left to right highlight the evolution of spin configurations during the switching process, with the rightmost Fig denoting the final stable state. Under the spin–orbit torque, the sublattice spins (blue arrows) and the magnetic octupole moment (red arrows) exhibit opposite chiralities. The directions of the damping-like (DL) effective field acting on the sublattice spins and the octupole moment are indicated by green and purple arrows, respectively. b Schematic illustrating the switching of the cluster magnetic octupole of the chiral AFM order of Mn3Sn by the SOT. Mn moments on the Kagome easy plane are shown by blue arrows. An electrical current j (purple arrow) flowing in the Pt layer generates a spin current whose polarization vector (red arrow) is perpendicular to the Kagome plane and induces the SOT on the Mn3Sn layer. c BSOT switching behavior of sputtering-grown Pt/Mn₃Sn heterostructures on Al₂O₃ substrates with varying AFM thicknesses at room temperature. d RH versus write current I of the Mn3Sn/Pt heterostructure at room temperature. The bias magnetic field of −x0.05 T is applied along the j direction. e RH versus write current I with 0.05 T bias magnetic field.

The ultralong SOT switching of octupole moment

In ferromagnetic materials, the presence of exchange splitting causes the majority (↑) and minority (↓) spin electrons at the Fermi surface to have distinct wave vectors. When a transverse spin current is injected, this difference results in rapid precession and dephasing of spin polarization as contributions from the ↑ and ↓ spin electrons. Consequently, the net transverse spin component vanishes beyond a certain depth from the surface. This characteristic decay length is known as the spin coherence length \({{{\rm{\lambda }}}}{{{\rm{c}}}}={{{\rm{\pi }}}}/|{k}_{F}^{\uparrow }-{k}_{F}^{\downarrow }|\)21. For ferromagnets like cobalt and iron, where the exchange splitting is large, λc is extremely short—on the order of a few angstroms. In addition, in fully compensated antiferromagnets where spin sublattices cancel perfectly, this coherence length diverges, rendering direct experimental observation of the spin diffusion process particularly difficult. On the non-collinear AFM, the canted antiferromagnetic order (Fig. 1b) leads to nearly equal, but not exactly identical populations of ↑ and ↓ spin electrons. As a result, the difference in their Fermi wave vectors, \(|{k}_{F}^{\uparrow }-{k}_{F}^{\downarrow }|\), is very small but finite, giving rise to an exceptionally long spin coherence length. The local moment precesses clockwise on a lattice (Fig. 2a, b), whereas it precesses more slowly than ferromagnets because the stagger moment induced by the spin canting is smaller than the ferromagnetic moment. The period (or wavelength) of spin precession in non-collinear AFMs is longer than that in ferromagnets. Therefore, the non-colinear AFM exhibits a feature of the bulk-like torque characteristic16,17,22,23.

We fabricated perpendicularly magnetic anisotropy Mn3Sn (d nm)/Pt bilayers with different AFM thickness. The in-plane current in the Pt layer can generate SOT. To detect the switched state of the octupole moment in Mn3Sn, we measure the Hall resistance RH (supplementary note 4) due to the anomalous Hall effect (AHE). Figure 2c shows RH as a function of write currents I for different AFM thickness d. A clear RH-I loop due to a reversal of octupole moment under an in-plane magnetic field Hx of 500 Oe is observed. Since non-collinear antiferromagnets exhibit a small uncompensated net moment that largely lies parallel to the octupole moment, the bipolar switching can be observed. We see that the SOT switching is clearly observed in the easy configuration with a threshold current below 60 mA. The sign of the SOT switching is determined by the bias field along the I write direction. To examine this, we measure the Hall resistance RH as a function of I write (RH–I loop) under opposite magnetic field direction (Fig. 2d, e). It was clearly shown that if the directions of I write and Hx are the same, the voltage exhibits a positive jump at critical current |Ic|. If I and Hx have opposite directions, the jump will become negative. This observation doesn’t follow the expectation of the symmetry requirement of the SOT switching of the perpendicular ferromagnetic magnetization due to the handedness anomaly effect24,25 (Fig. 2a). We observe a clear current switching Hall resistance data with different AFM thickness in the longitudinal configuration (H//I) (Fig. 2c), which is mostly determined by the anti-damping SOT26,27. The Hall signs in the H//I case for Mn3Sn/Pt bilayers indicate that the Pt layer is the source of spin currents, as it is placed on top of the Mn3Sn layer. The thickness dependences of Hall resistance are shown in Fig. 2c. Surprisingly, the Hall resistance nonmonotonically increases with increasing thickness, which can’t be explained by interfacial torques.

We also carried out experiments on the thickness dependence of the SOT efficiency for Mn3Sn alloys to find out if the long spin coherence length is a unique property of AFM. Figure 3a shows the SOT effective field for the AFM alloy as a function of thickness d. The HSOT/J shows a ‘bulk-like torque’ characteristic, indicating that the Mn3Sn alloys have a long spin coherence length. The magnitude of the out-of-plane anisotropy is about ten times greater than the in-plane anisotropy28, so a large HC/J of the chiral-spin structure in Mn3Sn can be observed. The HC is determined by the two-fold out-of-Kagome-plane anisotropy because each magnetic moment needs to precess about the easy axis and hard axis components during the current switching. Consequently, non-collinear antiferromagnets can be efficiently manipulated by low current and yet can be robust against external magnetic fields11,29. Figure 3b also shows that a transverse spin current from a Pt layer passes through a 60 nm thick Mn3Sn alloy, and the SOT switching ratio increases with the increasing thickness d.

Fig. 3: Characterizations of Mn3Sn alloy samples.
Fig. 3: Characterizations of Mn3Sn alloy samples.
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a SOT effective fields as a function of Mn3Sn alloy thickness (d). b SOT switching ratio as a function of (d). c Schematic illustration of the measurement set-up for the Mn3Sn/Pt devices. The z, y and x arrows indicate the direction of the magnetic field H, Hall current and current I, respectively. d Anomalous Hall resistivity (ρAHE) as a function of (d). e anomalous Hall resistance (RAHE) and SOT switched anomalous Hall resistance (\({R}_{{AHE}}^{{SOT}}\)) as a function of (d).

In addition, we consider the current-induced dynamics of the octupole moment in Mn3Sn through the SOT second-harmonic method. As shown in supplementary Fig. 1a, the in-plane spins are injected from the top Pt layer into Mn3Sn via the spin Hall effect. supplementary Fig. 1a, b illustrates two measurement configurations, where the injected spins are either perpendicular to the Kagome magnetization plane or lie within the Kagome plane. Under the action of the injected spin current, each sublattice moment in Mn3Sn experiences a damping-like spin–orbit torque13,15 (DL SOT), with the effective field given by HDL (A, B, C) ∝ m (A, B, C) × σ. Consequently, the impact of the SOT on the sublattice moments strongly depends on the relative orientation between the injected spins and the crystal axes. In the configuration shown in supplementary Fig. 1a, the damping-like torque τ DL (A, B, C) ∝ − m (A, B, C) × HDL (A, B, C) acts on all three sublattice moments in the same manner and along the same direction. In this case, the SOT only needs to overcome the relatively weak in-plane magnetic anisotropy. In contrast, in the configuration of supplementary Fig. 1b, the applied damping-like torque differs not only in magnitude but also in direction among the three sublattice moments, leading to a destructive superposition of torques. This corresponds to the hard configuration, where efficient detection of the second-harmonic signal cannot be achieved. It should also be noted that the effect of field-like torques is not considered here, as they do not induce switching of the magnetic octupole13.

To analyze the SOT-driven dynamics of magnetic octupoles in Mn3Sn, we measure the anomalous Hall effect (AHE) as a function of the external magnetic field and establish a comprehensive correlation between the octupole moment and the applied field, as illustrated in supplementary Fig. 2a. Through the following equation RH=R0sin(θφ) where R0 is the anomalous Hall coefficient and φoct is the angle between octupole moment moct and current I, we can precisely determine the position of the octupole. It is worth noting that the octupole position does not perfectly coincide with the magnetic field angle (supplementary Fig. 2a–c). Therefore, a representative octupole torque Loct curve (supplementary Fig. 2d) can be obtained by calculating Loct = Hmoctsin(θ−θφ), where moct denotes the octupole moment, and H is the applied field. Interestingly, we find that the twin topological spin order in Mn3Sn tilts toward the [11–20] direction, giving rise to octupole dynamics dominated by uniaxial anisotropy and twofold symmetry. Moreover, the octupole torque reaches its maximum near the [11–20] orientation. We analyzed the second-harmonic SOT signal with different thicknesses (supplementary Fig. 3a). The damping-like effective field induces a change in the octupole moment angle θφ, leading to an angular variation of the magnetic octupole under the SOT. The resulting second-harmonic signal can be expressed as: \({{{{\rm{R}}}}}_{2{{{\rm{\omega }}}}}=\frac{{{{{\rm{R}}}}}_{0}}{2}\frac{{H}_{{DL}}}{{H}_{K}+{{{\rm{H}}}}}\sin ({{{{\rm{\theta }}}}}_{{{{\rm{\varphi }}}}})\)+ C, which can be quantified by the linear relationship (supplementary Fig. 3b, c). For the damping-like torque \({\tau }_{{DL}}\) acting on the octupole dynamic, we adopt the same convention as in ferromagnets: \({\tau }_{{DL}}\) = \(-\gamma g{\mu }_{0}{{{{\bf{m}}}}}_{{{{\rm{oct}}}}}\times {H}_{{DL}}\). Figure 3d shows the thickness dependence of the SOT efficiency with \({H}_{{DL}}=\hslash {\xi }_{DL}{J}_{{HM}}/(2e{\mu }_{0}\left(3{M}_{O}\right)d)\), where ℏ, e, \({\xi }_{DL}\), \({J}_{{HM}}\) and t are the reduced Planck constant, the electron charge, the DL torque efficiency, the charge current density in the Pt layer and the thickness of Mn3Sn, respectively. The magnetization of a sublattice moment \({\mu }_{0}{M}_{O}=0.56T\) was obtained based on previous reports13. The increase in damping-like SOT efficiency with thickness indicates that the AFM possesses a long spin coherence length, consistent with a bulk-like SOT behavior. (supplementary Fig. 3d).

To investigate the presence of perpendicular magnetic anisotropy associated with the magnetic octupole, we analyze regular SOT switching in Mn3Sn/Pt at 500 Oe (Figs. 2c, 3c), since in Mn3Sn the AHE is expected to scale with the perpendicular component of the octupole polarization14,20. The symmetry of the chiral magnetic order can induce the octupole moment, which has been adopted as the magnetic order parameter to describe the anomalous Hall effect of Mn3Sn through4 RH (φoct) = R0moct,z, In addition, the octupole moment of a non-collinear antiferromagnet plays a similar role to the magnetization vector of a regular ferromagnet because the moct almost follows the applied field26. Figure 3d, e illustrates the modifications in the anomalous Hall effect induced by SOT when the applied current exceeds the critical current Ic. Therefore, we can observe the thickness dependence of SOT switched anomalous Hall resistance in Mn3Sn/Pt bilayer (Fig. 3e). The anomalous Hall resistivity also increases with increasing t, which could show a bulk-like AFM state30,31 (Fig. 3d).

To elucidate the unexpectedly high switching efficiency observed in 40 nm Mn3Sn, we combine a canting-renormalized spin-diffusion theory with large-scale atomistic spin dynamics. Figure 4a maps the calculated SOT attenuation length λSOT as a function of the spin canting angle θc, revealing an exponential suppression λSOT = λsf e−βθc² with β = 0.47 ± 0.02 (λsf is the spin-flip diffusion length). In Fig. 4b, we track two inequivalent Mn moments: spin A, situated 2 nm from the Pt interface, reverses at 1.7 ps, whereas spin B, 12 nm deeper, switches 0.23 ps later, consistent with diffusive propagation at Ds = 1.3 × 10−3 m2 s−1. A representative moment located beyond 3λSOT (Fig. 4c) merely executes a small-amplitude precession, substantiating the finite penetration depth. By repeating the simulation for 5 ≤ t ≤ 100 nm (Fig. 4d), we reproduce the experimental AHE-derived efficiency curve, which rises linearly up to 40 ± 2 nm and decays thereafter. The consistency trend between experiment and the simulations demonstrates that interfacial twin spin canting—and its concomitant reduction of the transverse-spin decay length—constitutes the essential mechanism governing bulk SOT switching in thick Mn3Sn films.

Fig. 4: Spin-diffusion theory with large-scale atomistic spin dynamics.
Fig. 4: Spin-diffusion theory with large-scale atomistic spin dynamics.
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a calculated SOT attenuation length as a function of the interfacial canting angle in Mn3Sn/Pt heterostructure. b Spin dynamics in two inequivalent Mn moments. c Time-dependent small-amplitude precession of Mn atoms. d Calculated SOT switched anomalous Hall resistance in Mn₃Sn films as a function of film thickness.

Discussion

Spin canting in antiferromagnets has led to novel phenomena such as magnon interference effects18, nonzero topological spin chirality32, and exchange-biased topological charge33. Recent real-space observations have also revealed topological spin canting at interfaces in antiferromagnets, even in collinear antiferromagnets34, which is crucial for BSOT, as well as being well-documented by our spin diffusion theory. By combining thickness-dependent SOT transport measurements and atomistic spin dynamics simulations, we show that spin current injected from a Pt overlayer can efficiently propagate through Mn3Sn films as thick as 60 nm, far exceeding the conventional spin diffusion limit observed in ferromagnets. The SOT efficiency exhibits a nonmonotonic dependence on film thickness, peaking around 40 nm, which we attribute to the interplay between spin coherence and interfacial twin spin order. Our theoretical model shows that canting-induced decoherence controls the decay length of spin torque propagation, thus establishing a unified framework to understand bulk torque generation in topological antiferromagnets. These findings not only challenge the conventional surface-dominated theory of spin-orbit torque but also lay the physical and conceptual foundation for the realization of scalable, low-power antiferromagnetic spintronic devices.

Methods

Material growth

Mn3Sn thin films were sputtered from an Mn3Sn target onto (1–102)-oriented Al2O3 single-crystal substrates (10 × 10 × 0.5 mm3) with a base pressure of 5 × 10−6 Pa. The deposition was performed at 873 K. The sputtering power and Ar gas pressure were 30 W and 0.5 Pa, respectively. The deposition rate was 1 Å s−1, as determined by X-ray reflectivity measurements. After deposition, Mn3Sn films were kept at 873 K in a vacuum for annealing for 1 h.

XRD

XRD measurements were performed by a Bruker D8 diffractometer with a five-axis configuration and Cu Kα (λ = 0.15419 nm).

Electrical measurements

Electrical contacts onto the Mn3Sn films were made by Al wires via wire bonding. Electrical measurements were performed in a Quantum Design physical property measurement system. The electrical current used for both longitudinal and Hall resistance measurements was 1000 µA. The material stack used in the electrical measurement is Mn3Sn (5, 10, 20, 40)/Pt (5), numbers in nm. Given the Hall device width of 20 um, the current densities are 0.5, 0.33, 0.2, 0.11 MA/cm2.

Magnetic measurements

Magnetic measurements were performed in a Quantum Design superconducting quantum interference device magnetometer with 10−11 A.m−2 sensitivity.

First-principles calculations

The first-principles calculations are carried out by using the Atomic orbital-Based Ab-initio Computation at UStc (ABACUS) package35,36. The exchange correlation functional was treated within the GGA/PBE37. The ion-electron interactions were described using the SG15 ONCV38, and the double-ζ plus polarization functions with a plane-wave cutoff energy of 100 Ry were employed as the NAO basis set39. The NAO bases for Mn and Sn are 4s2p2d1f and 2s2p2d1f, respectively. The total energy and forces were computed on a 5 × 5 × 6 k‑point grid, with the electronic density convergence threshold set to 1.0 × 10−5 Ry. Van der Waals interactions were incorporated via the DFT‑D3 dispersion correction40. Magnetic exchange parameters were evaluated employing the TB2J software package41. The experimentally determined lattice parameters of Mn₃Sn, a= = 5.67 Å and c = 4.53 Å, corresponding to space group No.194---were directly adopted42.

First-principles–parameterized atomistic spin-dynamics

The multi-scale workflow integrates a canting-renormalized spin-diffusion model with first-principles–parameterized atomistic spin-dynamics to reproduce the non-monotonic thickness dependence of the anomalous Hall read-out in Mn3Sn films. All input parameters are taken from high-accuracy transport, neutron-scattering and density-functional studies, while the numerical implementation follows established best practices for large-scale Landau–Lifshitz–Gilbert (LLG) simulations. Below, we detail film-growth benchmarks, the diffusion formalism, the magnetic Hamiltonian, the dynamical scheme, and the extraction of experimental observables.

Thin-film benchmark data Mn3Sn/Metal stacks exhibit a room-temperature anomalous Hall resistivity ρAHE ≈ 6 µΩ cm for thicknesses d ≈ 20–40 nm, with switching sustained up to t ≈ 100 nm. The intrinsic spin-diffusion length of nanocrystalline Mn3Sn, extracted by spin-absorption, is λsf = 0.70 ± 0.05 nm and the spin Hall angle θSH ≈ 0.11. These values are used as base parameters for the diffusive model.

Canting-dependent spin-diffusion formalism

The transverse spin accumulation μs(x,t) is treated within a one-dimensional drift–diffusion equation

$${\partial }_{t}{\mu }_{s}={D}_{s}{\partial }_{x}^{2}{\mu }_{s}-\frac{{\mu }_{s}}{{\tau }_{\phi }},$$
(1)

where Ds = ℏvF²τ/3kBT is the spin-diffusion constant (vF ≈ 1.8 × 105 m s−1 from ARPES) and τφ the dephasing time. Interface-induced canting modifies τφ via additional magnon–electron scattering channels; following κ-space perturbation theory, we write

$${\tau }_{\phi }^{-1}({\theta }_{c})={\tau }_{0}^{-1}(1+\beta {\sin }^{2}{\theta }_{c}),{\lambda }_{{{\rm{eff}}}}({\theta }_{c})=\sqrt{{D}_{s}{\tau }_{\phi }({\theta }_{c})},$$
(2)

β = 0.30–0.40 is fixed by matching the experimental peak at t ≈ 40 nm; the microscopic origin is a Dzyaloshinskii–Moriya (DM) term localized at the Pt/Mn₃Sn interface, consistent with first-principles predictions. The resulting spatial profile of the damping-like SOT current density is

$${J}_{s}(x)={J}_{s}^{0}\exp (-x/{\lambda }_{{{\rm{eff}}}}),$$
(3)

Atomistic Hamiltonian and parameterization

Each Mn moment Sᵢ occupies the Kagome lattice and evolves under

$${{{\mathscr{H}}}}=-{\sum }_{\langle {ij}\rangle }{J}_{{ij}}{{{{\bf{S}}}}}_{i}\cdot {{{{\bf{S}}}}}_{j}+{\sum }_{\langle {ij}\rangle }{{{{\bf{D}}}}}_{{ij}}\cdot ({{{{\bf{S}}}}}_{i}\times {{{{\bf{S}}}}}_{j})-K{\sum }_{i}{({{{{\bf{S}}}}}_{i}\cdot \hat{{{{\bf{n}}}}})}^{2},$$
(4)

with J₁ = 12 meV, J₂ = –4 meV, |D| = 2.0 meV and easy-plane anisotropy K = 0.05 meV, all extracted from DFT total-energy differences and inelastic-neutron dat43,44. We additionally performed DFT calculations to corroborate these parameters, obtaining J1 = 12.44 meV, J2 = 4.04 meV, and |D| = 2.2 meV. A higher-order biquadratic exchange term (B = 0.8 meV) is included to stabilize the inverse-triangular ground state in line with recent reports.

Spin-dynamics implementation

Dynamics obey the stochastic LLG equation

$${\dot{{{{\bf{S}}}}}}_{i}=-\gamma {{{{\bf{S}}}}}_{i}\times {{{{\bf{H}}}}}_{i}^{{{\rm{eff}}}}+\alpha {{{{\bf{S}}}}}_{i}\times {\dot{{{{\bf{S}}}}}}_{i}+{{{{\boldsymbol{\tau }}}}}_{{{\rm{DL}}},i},$$
(5)

integrated by a semi-implicit midpoint scheme with Δt = 0.10 fs and Gilbert damping α = 0.01. The damping-like SOT term is

$${{{{\boldsymbol{\tau }}}}}_{{{\rm{DL}}},i}=\frac{\hslash {\theta }_{{{{\rm{SH}}}}} \; {J}_{e}({x}_{i})}{2e{M}_{s}d}{{{{\bf{S}}}}}_{i}\times ({{{\boldsymbol{\sigma }}}}\times {{{{\bf{S}}}}}_{i}),$$
(6)

where Je(xi) follows the exponential profile above, σ̂ is fixed by the Pt current direction, and d = 0.25 nm is the inter-layer spacing. Simulations use 60 × 60 × Nc cells (Nc = 20–120, corresponding to 5–100 nm), periodic boundaries in-plane, free boundaries along z, and temperature T = 300 K (white-noise field satisfying the fluctuation–dissipation theorem).

Numerical platform and convergence

All simulations are performed with the open-source VAMPIRE package (v5.3) compiled with OpenMP.

Extraction of observables

Switching curves

For the representative spins A (2 nm) and B (12 nm), we record the out-of-plane component mz(t) and identify the 50% reversal point by cubic-spline interpolation; time-lags are averaged over ten stochastic realizations.

Canting profile

Instantaneous canting θc(z) is obtained from the local vector chirality χ = Si × Sj and fitted to a tanh envelope to extract the interfacial angle.

Thickness-dependent efficiency

The macroscopic Hall response is approximated by

$$\Delta {\rho }_{{{\rm{AHE}}}}\left(t\right)\propto {\int }_{\!\!\!\!0}^{d}\,{m}_{z}\left(z,{t}_{{{\rm{end}}}}\right){{{\rm{d}}}}z,$$
(7)

normalized to the maximum value to yield the efficiency η(d); simulated η(d) is plotted against experimental ρAHE(d) for direct comparison. Statistical error bars are the run-to-run standard deviation (N = 10).