Introduction

Chirality and chirality-induced novel phenomena are commonly observed and extensively studied in chiral spintronics1,2. In particular, an ultra-high tunneling magnetoresistance (TMR) of 300% and a spin polarization of 60% are obtained in chiral polymer materials due to chirality-dependent tunneling current3. In addition, a chiral anomaly in a Weyl band structure populates Weyl fermions in Mn3Sn with a specific chirality, which gives rise to a chiral current and a characteristic negative magnetoresistance4,5. The Kagome lattice of chiral AFM Mn3X (such as Mn3Sn6, Mn3Pt7, Mn3Ge8, MnGe9, Mn3Ir10) also displays a large AHE that arises from the non-zero berry phase from Bloch bands in momentum space. The AHE in high-quality Mn3Sn and Mn3Ge single crystals originates from the intrinsic mechanism11. However, the giant AHE in chiral AFM MnGe exceeds the conventional quantum limits of intrinsic AHE due to the skew scattering AHE9,12, which has a dominant contribution. Moreover, Mn3Sn and MnGe exhibit different scaling relationships9,11, with their anomalous Hall conductance linearly depending on or independent of the conductance value, respectively.

The different scaling laws in Mn3Sn and MnGe originate from two distinct physical mechanisms. The two-spin correlation13,14 is a good method to explain the large intrinsic AHE in Mn3Sn, by the vector spin chirality15,16(VSC) \({{{{{\boldsymbol{\varepsilon }}}}}}={S}_{1}{{{{{\rm{\times }}}}}}{S}_{2}+{S}_{2}{{{{{\rm{\times }}}}}}{S}_{3}+{S}_{3}{{{{{\rm{\times }}}}}}{S}_{1}\). For example, non-collinear (collinear) Mn3Pt exhibit non-zero (zero) VSC, respectively, leading to non-zero (zero) intrinsic anomalous Hall response7. In addition, Mn3Sn, Mn3Ir, and Mn3Ge (Mn3Pt, Mn3Ga) exhibit negative (positive) VSC1,2, leading to negative (positive) intrinsic anomalous Hall conductance6,7,8,11,17,18. Besides intrinsic AHE for chiral AFM, skew scattering topological AHE can be induced by the three-spin correlated scalar spin chirality16,19 (SSC):\(({{{{{{\rm{S}}}}}}}_{1}\times {{{{{{\rm{S}}}}}}}_{2})\cdot {{{{{{\rm{S}}}}}}}_{3}\). The skew scattering topological AHE exceeds the threshold value of the quantization limit 9,20,21 (~\(\frac{{{{{{{\rm{e}}}}}}}^{2}}{\bar{{{{{{\rm{h}}}}}}}{{{{{\rm{a}}}}}}}\) for MnGe). Here, a = 3.83 Å is the lattice constant for MnGe. Therefore, the intrinsic AHE in MnGe is negligible relative to the skew scattering topological AHE, resulting in an entirely different scaling relationship from Mn3Sn.

Here, we systematically study the AHE of the chiral non-collinear AFM Mn3Pt and find a universal AHE scaling law. This anomalous Hall resistivity ρAH can be described by20,22

$${\rho }_{{AH}}={a}_{{sk}}{\rho }_{{xx}}+{b}_{{in}}{\rho }_{{xx}}^{2}$$
(1)

where ρxx is the longitudinal resistivity. The first term \({a}_{{sk}}{\rho }_{{xx}}\) is the scalar spin chirality-induced skew scattering topological AHE9,12. The second term \({b}_{{in}}{\rho }_{{xx}}^{2}\) describes the intrinsic anomalous Hall effect arising from time symmetry breaking by the compensated non-collinear magnetic order8,17,18. By carefully designing an experimental procedure for Mn3Pt alloy films, we find that the AHE parameter \({a}_{{sk}}\), \({b}_{{in}}\) and sheet resistively \({\rho }_{{xx}}\) change linearly with the inverse film thickness d. The linear 1/d dependencies could be attributed to the symmetry breaking at the surface. In addition, the scaling law can explain the anomalous Hall effect in various chiral magnets (Mn3Sn6, Mn3Pt7, Mn3Ge8, MnGe9, Mn3Ir10) and should be universal for describing the AHE of chiral magnets.

Results and discussions

Chiral magnet Mn3Pt with single crystal properties and topological spin texture

Mn3Pt is a cubic chiral antiferromagnetic intermetallic compound with lattice constant a = 3.833 Å23. It exhibits chiral AFM spin order below the Néel temperature TN ≈ 474 K23,24,25. We epitaxially grow high-quality Mn3Pt thin film by sputtering on MgO (001) single-crystal substrate. Typical X-ray diffraction (XRD) spectra of MgO substrate and Mn3Pt are measured at room temperature. The out-of-plane XRD theta to 2theta scans at different planes indicate that Mn3Pt film on the MgO substrate is a single crystal (Fig. 1a). The two extra peaks of Mn3Pt (111) and (222), which indicates the establishment of the long-range chemical ordering, are located at 2θ1 = 40.5o and 2θ1 = 87.5o. 360o phi scan around Mn3Pt (111) plane and MgO (111) was measured by XRD to ensure the single-crystal property (Fig. 1b). The MgO and the Mn3Pt show fourth-degree symmetry, indicating the epitaxial growth of the sample. As a non-collinear AFM, Mn3Pt has a weak magnetic moment at room temperature due to symmetry-allowed spin canting. In addition, the spin canting induces a net moment at 0 T as shown by the measurement of the out-of-plane magnetic hysteresis loop (Fig. 1c). The direction of spin canting is not parallel to the Kagome lattice plane, which is similar to the previous report7. The corresponding spin texture at different fields is shown in the illustration figure (Fig. 1c). Here, the SSC is characterized by the stacking direction of the atoms as viewed from [111] axis. When the external magnetic field is positive (greater than the saturation field), the antiferromagnetic spin structure will tilt upward. As a sequence, the SSC will be positive and have the opposite result under negative magnetic field. In addition, when the magnetic field is zero, the net moment and SSC will be non-zero due to magnetic hysteresis effect which will induce a topological AHE.

Fig. 1: structure of Mn3Pt film.
figure 1

a Out-of-plane theta to 2theta XRD spectra of Mn3Pt at different planes. b Phi scan patterns of Mn3Pt at (111) plane and the corresponding MgO substrate. c The magnetization data was measured at 300 K with the magnetic field paralleled to the c–axis. The spin structure of cubic Mn3Pt is shown in the inset (the red sphere denotes Pt atom, the blue sphere and arrow denote Mn atom and spin direction, respectively. Mn atoms at the Kagome planes have different scalar spin chirality under different magnetic fields. d High-resolution cross-sectional scanning transmission electron microscopy (STEM) images of Mn3Pt/MgO. The bright regions correspond to the Mn3Pt monolayers. The scale bar is 1 nm. e High–resolution STEM image of the basal plane of the Mn3Pt. Blue and yellow dots represent the Pt atom and the Mn atom, respectively. The scale bar is 1 nm. f Selected–area electron diffraction (SAED) patterns of Mn3Pt obtained by transmission electron microscopy (TEM).

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The scanning transmission electron microscopy (STEM studies) further confirms the high-quality growth of chiral magnet Mn3Pt. The interlayer spacing is 3.8 Å in Mn3Pt film (Fig. 1d), which is consistent with the XRD studies. In addition, high-resolution STEM and the corresponding selected-area electron diffraction (SAED) studies reveal the single crystal feature of the chiral magnet Mn3Pt (Fig. 1e, f), which is essential for achieving the designed properties and functional spintronics devices26,27,28,29.

Bulk-like anomalous Hall effect and temperature dependence of transport properties

To better understand the topological spin chirality induced zero field Hall response. We prepare chiral magnet Mn3Pt films on MgO substrate with different thickness values by dc magnetron sputtering and measure the resistance by the standard four-point measurement set-up (Fig. 2a). At 340 K, the Anomalous Hall effect in the Mn3Pt films changes with different thickness d (Fig. 2b). The AHE coercive field decreases with the film thicknesses d, which is consistent with the reduction of the coercive field in the Mn3Pt (t nm)/STO substrate7. It is worth noting that the magnetic coercivity is much smaller than that of AHE coercivity, because the Zero magnetic moment indicates that the spin canting is zero. However, the AHE coercivity indicates that the topological AHE and the intrinsic AHE have opposite and equal contributions. The anomalous Hall resistivity (\({{\rho }}_{{AH}}=\frac{({{\rho }}_{{xy}}^{+}-{{\rho }}_{{yx}}^{-})}{2}\)) is 0.086, 0.114, 0.156, 0.346, 0.465 μΩ*cm, when the thickness is 12, 14, 18, 22, 30 nm, respectively, showing the bulk like anomalous Hall effect.

Fig. 2: Magnetic field dependence of the AHE in Mn3Pt.
figure 2

a 3D schematic of the Hall bar structure made from the Mn3Pt (yellow)/MgO (purple) stack with a top gate electrode (light gray). Standard four-point measurement setup is displayed. b Magnetic-field-dependent anomalous Hall resistance at different thicknesses for chiral magnets Mn3Pt, the measured temperature is 340 K. c Temperature dependence of \({{\rho }}_{{xx}}\) and \({{\rho }}_{{xy}}\) for 30 nm Mn3Pt. d The anomalous Hall angle \({{{{{{\rm{\theta }}}}}}}_{{AH}}=\frac{{{\rho }}_{{AH}}}{{{\rho }}_{{xx}}}\) for 30 nm Mn3Pt.

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Temperature-dependent studies of 30 nm Mn3Pt show that the anomalous Hall resistivity \({{\rho }}_{{AH}}\) becomes smaller at higher temperature and change monotonically with the temperature (Fig. 2c). The longitudinal resistivity \({{\rho }}_{{xx}}\) gradually increases at the higher temperature, indicating the metallic behavior (Fig. 2c), and there is no phase transition during our measurement. In addition, the \({{\rho }}_{{xx}}\) (T) can be fitted by a linear function of \({T}^{2}\) due to the dominant electron scattering by the spin flip30,31 mechanism. The ρAH is about three orders of magnitude smaller than the resistivity ρxx, and we can get the temperature dependence of AHE angle by the equation \({{{{{{\rm{\theta }}}}}}}_{{AH}}=\frac{{{\rho }}_{{AH}}}{{{\rho }}_{{xx}}}\,\) (Fig. 2d). The anomalous Hall angle is −0.168% at 300 K, and the magnetic moment due to spin canting still have the contribution for SSC excitation at zero field.

Universal AFM scaling law for chiral AFM Mn3Pt

Magnetic materials are the best choice to explore the AHE scaling law. Previous studies have achieved great success by attributing the AHE in ferromagnetic materials to both intrinsic and extrinsic contributions20,22. To extract the different contributions of the AHE in chiral AFM magnets with different thicknesses d, we use the AFM scaling law \({\rho }_{{AH}}={a}_{{sk}}{\rho }_{{xx}}+{b}_{{in}}{\rho }_{{xx}}^{2}\), to describe the measured data.\(\,{\rho }_{{AH}}\) is plotted as a function of \({\rho }_{{xx}}\) (Fig. 3, illustration picture), the variations of \({\rho }_{{AH}}\) and \({\rho }_{{xx}}\) are achieved by varying temperatures for a given sample. Clearly, \({a}_{{sk}}\) arises from the linear \({\rho }_{{xx}}\) dependence and \({b}_{{in}}\) arises from the quadratic \({\rho }_{{xx}}\) dependence (illustration picture). The quadratic term represents the intrinsic Hall effect, while the linear term represents the skew scattering mechanism determined by the scalar spin chirality. To extract different contributions to the AHE, the \(\frac{{\rho }_{{AH}}}{{\rho }_{{xx}}\,}\) ratio is plotted as a function of \({\rho }_{{xx}}\) (Fig. 3). In this case, the slope and intercept represent the intrinsic AHE parameter \({b}_{{in}}\) and skew scattering AHE parameter \({a}_{{sk}}\), respectively. The positive skew scattering AHE of Mn3Pt has an opposite sign to the intrinsic Hall effect, which has been predicted by recently Boltzmann transport theory in chiral magnet12. To ascertain the universal AFM scaling law for chiral AFM, we plot the AHE data for various non-collinear AFM Mn3X (X = Sn6, Pt, Ge8, Ir10, etc.) and MnGe9. For chiral AFM MnGe, the anomalous Hall angle (\({{{{{{\rm{\theta }}}}}}}_{{AH}}=\frac{{\rho }_{{AH}}}{{\rho }_{{xx}}\,}\)) is −0.18, much larger than other chiral AFM. In addition, the intrinsic Hall resistivity term (\({b}_{{in}}{\rho }_{{xx}}^{2}\)) of MnGe is negligible relative to the skew scattering topological anomalous Hall resistivity term (\({a}_{{sk}}{\rho }_{{xx}}\)). In this situation, the scaling law can be simplified to \({\rho }_{{AH}}={a}_{{sk}}{\rho }_{{xx}}\) and \({{{{{{\rm{\sigma }}}}}}}_{{AH}}={a}_{{sk}}{{{{{{\rm{\sigma }}}}}}}_{{xx}}\) [\({{{{{{\rm{\sigma }}}}}}}_{{xx}}=\frac{{\rho }_{{xx}}}{{\rho }_{{xx}}^{2}+{\rho }_{xy}^{2}}\), \({{{{{{\rm{\sigma }}}}}}}_{{AH}}=\frac{-{\rho }_{xy}}{{\rho }_{{xx}}^{2}+{\rho }_{xy}^{2}}\)]. Therefore, the anomalous Hall conductance σAH will have a linear relationship with longitudinal conductance \({{{{{{\rm{\sigma }}}}}}}_{{xx}}\) 9. On the contrary, for chiral magnets Mn3Sn and Mn3Ge without out-plane spin canting, recent work11 shows that the skew scattering AHE resistivity term (\({a}_{{sk}}{\rho }_{{xx}}\)) is negligible relative to intrinsic anomalous Hall resistivity term (\({b}_{{in}}{\rho }_{{xx}}^{2}\)), the scaling law can be simplified to \({{{{{{\rm{\sigma }}}}}}}_{{AH}}=\frac{-{\rho }_{xy}}{{\rho }_{{xx}}^{2}}={b}_{{in}}\). Therefore, the intrinsic anomalous Hall conductance \({{{{{{\rm{\sigma }}}}}}}_{{AH}}\) will be a constant value for Mn3Sn11 and the slope term \({a}_{{sk}}\) due to SSC-induced skew scattering is zero (Fig. 3). However, for the chiral AFM Mn3Ir and Mn3Pt, both the VSC-induced intrinsic AHE and SSC-induced skew scattering AHE have strong contributions, but the AHE of all the chiral AFM can be well explained by this AFM scaling law.

Fig. 3: Universal scaling law for chiral magnets.
figure 3

The various chiral AFM was plotted by \(\frac{{\rho }_{{AH}}}{{\rho }_{{xx}}}={a}_{{sk}}+{b}_{{in}}{\rho }_{{xx}}\). The slope and intercept represent the intrinsic and skew scattering anomalous Hall scaling factors, respectively. The data includes Mn3Ge11, Mn3Sn11, Mn3Ir10, MnGe9, and Mn3Pt. For Mn3Pt (12 ~ 30 nm) /MgO substrate, the inset is the measured anomalous Hall resistance \({\rho }_{{AH}}\) as a function of resistivity ρxx. Solid lines refer to the scaling law fitting of \({\rho }_{{AH}}={a}_{{sk}}{\rho }_{{xx}}+{b}_{{in}}{\rho }_{{xx}}^{2}\).

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Theoretical understanding

In most studies of the AHE scaling law, the interfacial and bulk resistivity are often equally treated9,11. However, when the film’s thickness is comparable to an electron’s mean free path, the anomalous Hall resistivity and the sheet resistivity will change with the film thickness22. \({a}_{sk}\), \({b}_{{in}}\), measured magnetizations M and sheet resistivity \({\rho }_{{xx}}\) could be described as a function of the thickness (Fig. 4a–c) which change linearly with 1/d, where the intercept and the slope correspond to bulk and surface contribution, respectively. The magnetizations M induced by the net moment increases with the thickness, which means that the spin structure will tilt larger with the thickness.

Fig. 4: Thickness dependence at room temperature.
figure 4

a–c The sheet resistivity\({\rho }_{xx}\),\(\,{a}_{{sk}}\), the measured magnetizations M and \({b}_{{in}}\) versus 1/d for Mn3Pt. The inset in (a) shows the electronic scattering at the surface and the interface. The inset of (b) illustrates that there is larger SSC excitation in thicker films. The measurements were performed at 340 K. Solid lines refer to the linear fitting results. Five samples were measured for each thickness and each error bar indicates the standard error of the mean for the five samples of each thickness.

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As a result, the SSC will increase at larger thickness, causing the larger topological anomalous Hall effect. The intrinsic AHE parameter \({b}_{{in}}\) tripled when thickness increased from 12 to 30 nm, indicating that the interfacial electron scattering can effectively control the intrinsic scattering and the presence of stronger interfacial scattering will reduce the AHE in chiral AFM. It is worth noting that this tuning effect is particularly weak when the AFM thickness is much larger than the mean free path. In addition, the anomalous Hall angle θAH achieves larger values (Fig. 4a–c) for thicker samples with larger values of ask and bin.

The thickness-controlled AHE can also be understood by the following mechanism. The larger out-of-plane anisotropy due to larger spin canting in thicker film (Fig. 2b) allows a larger SSC to be excited, as schematically illustrated in the insets of Fig. 4b. Hence, we can achieve a larger skew scattering topological AHE. To corroborate our result, we also perform first principle studies by calculating the relationship between the intrinsic anomalous Hall conductance (AHC) and SSC induced by out-of-plane spin canting. To illustrate the effect of SSC, we fix the lattice constant to be 3.83\({{{{\text{\AA }}}}}\) to exclude the influence of strain. Then, we perform the constrained magnetization calculations. The magnetic moment is initially set to a purely antiferromagnetic state, then we slightly tilt the magnetization of Mn atoms, the total net magnetization is set to the [111] direction of the Mn3Pt lattice, as shown in (Fig. 1c). We use the component of the total magnetization in the z-direction to represent the state, and the z-component varies from 0.00\({\mu }_{B}\) to 0.30\({\mu }_{B}\). The band structure does not change much in most spaces in the Brillouin zone (BZ). However, when a net magnetization exists, the bands at \(\Gamma (0,0,0)\)-\({{{{{\rm{R}}}}}}(0.5,{{{{\mathrm{0.50.5}}}}})\) path get closer and thus lead to a higher intrinsic AHC. We plot the spin polarization along the [111] direction at different k points in (Fig. 5b). The spin direction at k-points on the Γ-R path near the Fermi level have clear tendency to [111] direction. [111]-oriented net magnetization will shift the bands and gives rise to a dramatic increase of the Berry curvature at the k-point between these two bands (Fig. 5a, b). Thus, a weak net magnetization could enhance the intrinsic AHC of Mn3Pt. The calculated intrinsic contribution to the AHC versus Fermi energy with different net magnetizations is shown in (Fig. 5c). The calculation results show that the intrinsic AHC \({\sigma }_{{in}}\) is increased from 56.73 \({\Omega }^{-1}c{m}^{-1}\) to 73.90 \({\Omega }^{-1}c{m}^{-1}\) when the z–component of net magnetization increases from 0 to 0.30 \({\mu }_{B}\) (Fig. 5e, f). This calculation elucidates our result and the origin of the chirality-induced anomalous Hall effect, which has been recently reported in chiral magnets1,2,3,4,5,6,7,8,9,10,11,12,13.

Fig. 5: Theoretical calculations of Berry curvature and AHE.
figure 5

a, b Band structure of R–Γ path for Mn3Pt, the color represents the spin polarization component on [111] direction. c, d Band structure near Fermi energy (upper panel) and Berry curvature (lower panel) in atomic units along the symmetry lines. The black line represents the purely antiferromagnetic state, the red line represents the state with net magnetization along [111] direction, whose z-component is 0.30\({\mu }_{B}\). e Calculated intrinsic contribution to the AHC versus Fermi energy with different net magnetizations. f Calculated intrinsic contribution to the AHC versus different total magnetizations at the Fermi level.

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Conclusions

The AHE in chiral antiferromagnet is mainly derived from the three-spin model correlated with SSC15,16 and the two-spin model correlated with VSC13,14. In our experiment, we simultaneously discover the vector spin chirality-induced intrinsic anomalous Hall effect and the scalar spin chirality skew scattering topological AHE in chiral AFM Mn3Pt and provide a universal AHE scaling law to explain both Mn3Pt and other chiral AFMs. This work also elucidates the origin of the chirality-induced anomalous Hall effect by ab initio calculations. Our work deepens the understanding of the anomalous Hall effect in antiferromagnetic materials and facilitates the development of chiral spintronic devices.

Methods

Material growth

Mn3Pt thin flms were sputtered from a Mn3Pt target onto (001)-oriented MgO single–crystal substrates (10 × 10 × 0.5 mm3) with a base pressure of 5 × 10−6 Pa. Te deposition was performed at 600 °C. 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, Mn3Pt films were kept at 600 °C 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).

STEM

Cross-sectional wedged samples were prepared by mechanical thinning, precision polishing and ion milling. An electron-beam probe was utilized to scan thin films to achieve high resolution of local regions.

Electrical measurements

Electrical contacts onto the Mn3Pt 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 500 µA.

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 ab initio calculations were performed using the QUANTUM ESPRESSO package based on the projector augmented-wave method and a plane-wave basis set32,33. The exchange and correlation terms were described using a generalized gradient approximation in the scheme of Perdew-Burke-Ernzerhof parametrization, as implemented in the PSLIBRARY34. The fully relativistic pseudopotential was used, and the spin-orbit coupling was included in our calculation. A k-point mesh of 20 × 20 × 20 was used in the self-consistent calculations. Then the plane-wave functions were transferred to the maximally localized Wannier functions using the WANNIER90 package35. The intrinsic contribution of AHC was calculated by integrating the Berry curvature over the occupied bands through the whole Brillouin zone (BZ)36.

$${\sigma }_{{xy}}=-\frac{{e}^{2}}{{{\hslash }}}\,{\int }_{{BZ}}\frac{d{{{{{{\boldsymbol{k}}}}}}}^{3}}{{\left(2\pi \right)}^{3}}\,{\sum }_{n}{f}_{n}{\Omega }_{n}^{z}\left({{{{{\boldsymbol{k}}}}}}\right)$$
(2)

and the Berry curvature could be given in a Kubo formula term

$${\Omega }_{n}^{z}\left({{{{{\boldsymbol{k}}}}}}\right)={{{\hslash }}}^{2}{\sum }_{m\ne n}\frac{-2{Im}\left\langle {\psi }_{n{{{{{\boldsymbol{k}}}}}}}\left|{v}_{x}\right|{\psi }_{m{{{{{\boldsymbol{k}}}}}}}\right\rangle \left\langle {\psi }_{m{{{{{\boldsymbol{k}}}}}}}\left|{v}_{y}\right|{\psi }_{n{{{{{\boldsymbol{k}}}}}}}\right\rangle }{{\left({E}_{n{{{{{\boldsymbol{k}}}}}}}-{E}_{m{{{{{\boldsymbol{k}}}}}}}\right)}^{2}}$$
(3)

where \(v\) are velocity operators. An ultra-dense k-grid of 200 × 200 × 200 was employed to perform the BZ integration.