Introduction

Exploration of nontrivial spin textures in magnetic metals and semiconductors is a subject of great interest in modern condensed matter science. The striking feature is that conduction electrons interacting with the spin texture often cause unprecedented electromagnetic phenomena through the quantal phase. One of the typical examples is the anomalous Hall effect (AHE)1. Conventionally, the AHE has been considered to be induced by the spontaneous magnetization in ferromagnets but is currently observed even in a variety of antiferromagnets and helimagnets2,3,4,5,6,7. For example, helimagnets such as magnetic Skyrmions show the AHE or topological Hall effect in proportion to the scalar-spin chirality (SSC) Si · (Sj × Sk), which represents the solid angle formed by three spins in a noncoplanar configuration8,9,10,11,12. From a microscopic viewpoint, this effect is usually understood from the scheme of Berry curvature for electrons; the Berry curvature (quantal phase) proportional to the SSC is induced by the long-range ordered noncoplanar spin texture, which acts as the fiction magnetic field to conduction electrons1. In particular, it has been demonstrated that the Hall response can be remarkably enhanced by tuning the real space and/or momentum space Skyrmion density13,14. The giant anomalous/topological Hall response has also received interest in terms of spintronic function such as the efficient electrical read-out of spin texture or thermoelectric energy harvesting15,16.

On the other hand, the AHE can be also triggered by fluctuating or spatially inhomogeneous spin texture that is not long-range ordered. For example, electron scattering from a single magnetic defect causes the AHE of skew scattering or side-jump mechanism17,18. In addition, it is also known that the transient noncoplanar spin texture induced by the thermal spin fluctuation triggers the AHE through the SSC. This phenomenon has been often understood from the scheme of Berry curvature mechanism1,12, but recent research proposes that the multiple skew scattering from spin clusters with noncoplanar spin texture can cause the giant AHE18,19, the anomalous Hall angle (tan θH) of which is scaled to the SSC. In particular, for the latter cases, the anomalous Hall angle comparable to or larger than the intrinsic AHE of Skrmion magnets can emerge even in the paramagnetic temperature regime far above the magnetic transition temperature20. This is in contrast with the conventional intrinsic/extrinsic AHE, which is usually enhanced in the long-range ordered state. Nevertheless, the AHE originating from spin fluctuations i.e. paramagnetic AHE is usually very small in most magnetic metals (tan θH ≤ 0.01)12,21,22,23,24,25,26,27,28 except in a few noncollinear magnets20,21. Here, we report that the collinear antiferromagnetic metal SrCo6O11 with devil’s staircase-type transition shows the large paramagnetic AHE in a moderate magnetic field (∼3 T) even far above the magnetic transition temperature, resulting in the anomalous Hall angle (tan θH) more than 0.02.

Results

Magnetic property

SrCo6O11 crystallizes in the R-type hexaferrite structure with space group P63/mmc29. In this material, the Co-sublattice can be viewed as the stacking of Kagome lattice of Co1, dimerized pillar of Co2, and triangular lattice of Co3 along the c-axis as shown in Fig. 1a. The Co1 and Co2 form the CoO6-octahedra, while the Co3 does the CoO5-bipyramids as illustrated in Fig. 1b, c, respectively. The 3d states of Co3 form a local spin moment with easy-axis anisotropy along the c-axis, while those of Co1 and Co2 take the partially filled low spin states, yielding the strongly correlated conduction electrons30,31,32,33 [see also Supplementary Fig. 1]. The conduction electrons mediate the magnetic interaction of Co3-spin through the Ruderman-Kittel-Kasuya-Yoshida (RKKY)-type interaction, resulting in the competing multiple magnetic interactions between the nearest-neighbor spins and further-neighbor spins along the c-axis34. Consequently, the collinear antiferromagnetic phases, in which spins are modulated along the c-axis but are ferromagnetically aligned within the ab-plane, occur at low temperatures. As an example, Fig. 1a shows the magnetic structure with the modulation wave number q = 1/3, where spins are aligned in the c-direction in an up-up-down order.

Fig. 1: Crystal structure and magnetic phase diagram for SrCo6O11.
figure 1

a The illustration of Co-sublattice of SrCo6O11 with up-up-down-type magnetic structure (q = 1/3)50. Arrows denote the spins on Co3-sites. The primitive vectors a, b, and c are defined in the hexagonal crystal symmetry with space group P63/mmc. b, c The illustration of CoO6-octahedra of Co1 and Co2 and CoO5-bipyramid of Co3. d The magnetic phase diagram in the temperature-magnetic field plane. Arrows denote spins of Co3. Here, q represents the wave number of magnetic modulation. Open circles and squares denote the magnetic transition at Bc1 and Bc2, respectively. Open triangles denote the transition temperature (Tp) of the incommensurate (IC) phase or q = 1/3-phase determined from the temperature dependence of M (See Supplementary Fig. 2d, e). We defined Tp at 0.1 T as Tc (=22 K). The inset shows the magnetization curve at 8 K. The horizontal dotted line denotes the magnetization plateau corresponding to q = 1/3- and q = 0-phase. e Temperature dependence of resistivity at B = 0 T.

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Figure 1d shows the magnetic phase diagram when the magnetic field (B) is applied along the c-axis. At zero magnetic field, the magnetic ordering with incommensurate (IC) modulation q ~ 1/5 appears at 22 K ( = Tc). With lowering temperature, q quasi-continuously decreases and is locked to the commensurate value of 1/6 about 12 K34. Simultaneously, various magnetic phases with different modulation periods emerge, resulting in the phase-separated state. When the magnetic field is applied along the c-axis, these states change to the q = 1/3-phase at Bc1, followed by the forced ferromagnetic phase at Bc2. These field-induced phase transitions can be seen as the magnetization jumps and plateaus as shown in the inset to Fig. 1d33,34 (see also Fig. 2a–d). Such behavior is a hallmark of the magnetic devil staircase, which originates from the competition among a large number of nearly degenerate magnetic phases with different modulation period35,36.

Fig. 2: Magnetization and electrical transport property for SrCo6O11.
figure 2

a–e The magnetization (M), f–j resistivity (\({\rho }_{{xx}}\)), k–o Hall resistivity (\({\rho }_{{yx}}\)), and p–t Hall resistivity are not proportional to M (\({\rho }_{{yx}}^{{notM}}\)) for SrCo6O11. The blue dash-dot-line and green dashed line denote the ordinary Hall resistivity (\({\rho }_{{yx}}^{N})\) and anomalous Hall resistivity proportional to M (\({\rho }_{{yx}}^{M}\)), respectively. The red dashed curve denotes the \({\rho }_{{yx}}^{N}+{\rho }_{{yx}}^{M}\) (red dashed curve in k denotes the \({\rho }_{{yx}}^{N}+{\rho }_{{yx}}^{M}\) at 22 K). \({\rho }_{{yx}}^{N}\) and \({\rho }_{{yx}}^{M}\) are omitted for clarity in l–o.

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Magneto-transport property

Figure 1e shows the temperature dependence of resistivity (\({\rho }_{{xx}}\)) at B = 0 T. The \({\rho }_{{xx}}\) shows minimal temperature dependence above Tc, suggesting the incoherent charge transport as often seen in the strongly correlated metals. At Tc, the resistivity shows a kink and steeply decreases at lower temperatures. Figure 2a–t summarizes the magnetization, magnetoresistivity, and Hall resistivity under the magnetic field. The magnetization shows jumps at Bc1 and Bc2 below Tc, which are blurred above Tc [see Fig. 2a–e]. The magnetoresistivity also shows anomalies at Bc1 and Bc2 in the temperature range 12–18 K. Furthermore, with increasing temperature from 2 K, the negative magnetoresistivity becomes gradually pronounced. In particular, the negative magnetoresistivity is observed up to around 50 K, implying that the short-ranged spin correlation remains even far above Tc.

On the other hand, the Hall resistivity (\({\rho }_{{yx}}\)) shows more peculiar temperature and field dependence [See Fig. 2k–o]. At 2 K, \({\rho }_{{yx}}\) almost linearly increases as a function of B except for a jump at Bc2. Since the magnetization is saturated above Bc2 [see Fig. 2a], the B-linear component above Bc2 is likely attributed to the ordinary Hall effect. The extrapolation of the B-linear part to zero magnetic field seems to take a finite value, suggesting the finite contribution from the anomalous Hall effect proportional to M. With increasing temperature, the slope of \({\rho }_{{yx}}\) above Bc2 gradually decreases, while that below Bc2 shows sign change above 12 K. In addition, a peak structure grows around Bc2 and becomes remarkable at higher temperatures. Above 22 K, the peak shifts to a higher-field region and gradually diminishes while broadening its width. Apparently, such behavior cannot be explained by the conventional anomalous Hall effect proportional to M. On the other hand, the ordinary Hall effect can show a peak, given that the carrier mobility is sufficiently high. This is, however, not likely, since the resistivity above Tc is in the incoherent transport regime close to the Ioffe-Regel limit (\(\sim 1\times {10}^{-3}\)Ω cm) (see Supplementary Note 1). This suggests the presence of an additional contribution other than the ordinary Hall effect and AHE proportional to M. Following the conventional wisdom7,13,20, we analyzed the Hall resistivity by assuming the following three components,

$${\rho }_{{yx}}={R}_{N}B+{S}_{A}{\rho }_{{xx}}^{\alpha }M+{\rho }_{{yx}}^{{notM}}$$
(1)

The first, second, and third terms represent the ordinary Hall component (\({\rho }_{{yx}}^{N}={R}_{N}B\)), the anomalous Hall component in proportion to M (\({\rho }_{{yx}}^{M}={S}_{A}{\rho }_{{xx}}^{\alpha }M\)), and the residual component (\({\rho }_{{yx}}^{{notM}}\)), which is neither proportional to B nor to M, respectively. Since the fitting with this model did not uniquely converge in the region below Bc2 due to the short field range of each magnetic phase [see also Supplementary Note 2], we focus on the results in the forced ferromagnetic/paramagnetic state above Bc2. Here, we assumed the exponent \(\alpha =0.4\) for \({\rho }_{{yx}}^{M}\) which corresponds to the intrinsic AHE in the incoherent transport regime1,37. Although \(\alpha\) depends on the transport regime, the analyzed results here did not significantly depend on the choice of \(\alpha\) [see Supplementary Note 3]. \({\rho }_{{yx}}^{N}\), \({\rho }_{{yx}}^{M}\) and their summation (\(={\rho }_{{yx}}^{N}+{\rho }_{{yx}}^{M}\)) at 2 K are exemplified in Fig. 2k. \({\rho }_{{yx}}\) is well reproduced by \({\rho }_{{yx}}^{N}+{\rho }_{{yx}}^{M}\), resulting in the negligible \({\rho }_{{yx}}^{{notM}}\)[see Fig. 2p]. With increasing temperatures, the difference between \({\rho }_{{yx}}\) and \({\rho }_{{yx}}^{N}+{\rho }_{{yx}}^{M}\) becomes significant especially near Bc2, resulting in a peak of \({\rho }_{{yx}}^{{notM}}\) as shown in Fig. 2q–t. Above 30 K, the peak shifts to a higher-field region and gradually diminishes. To visualize this behavior, we show the contour plot of \({\rho }_{{yx}}^{{notM}}\) on the temperature-magnetic field plane in Fig. 3a. It is evident that \({\rho }_{{yx}}^{{notM}}\) is enhanced in the wide temperature-field region near and above Tc, where the magnetization is not fully polarized, namely, the thermal spin fluctuation is remarkable.

Fig. 3: Field and temperature dependence of anomalous Hall effect for SrCo6O11.
figure 3

a The contour plot of anomalous Hall resistivity not proportional to M (\({\rho }_{{xy}}^{{notM}}\)) on the field-temperature plane. Open circles, squares, and triangles denote the magnetic transition at Bc1, Bc2, and Tp, respectively. b Temperature dependence of \({\sigma }_{{xy}}^{{notM}}\)(circles) and \({-\sigma }_{{xy}}^{M}\) at 3 T (triangles). c Anomalous Hall angle (\(\tan {{\rm{\theta }}}_{{\rm{H}}}\sim {\rho }_{{\rm{yx}}}^{{\rm{notM}}}/{\rho }_{{\rm{xx}}}\)) in various bulk oxide ferromagnets (closed circles) and noncollinear magnets (open circles)1,51,52,53,54,55,56,57,58. d Anomalous Hall conductivity not proportional to magnetization (\({\sigma }_{{xy}}^{{notM}}\)) as a function of normalized magnetization (\(M/{M}_{s}\)) with Ms being the magnetization at 2 K and 9 T.

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To quantitatively compare the temperature dependence of each term, we derived the Hall conductivity \({\sigma }_{{xy}}^{M,{notM}}[={\rho }_{{yx}}^{M,{notM}}/({\rho }_{{xx}}^{2}+{\rho }_{{yx}}^{2})]\). Figure 3b shows the temperature dependence of \({\sigma }_{{xy}}^{M}\) and \({\sigma }_{{xy}}^{{notM}}\) at 3 T. Below 10 K, \({\sigma }_{{xy}}^{{notM}}\) is much smaller than \({\sigma }_{{xy}}^{M}\). With increasing temperature, \({\sigma }_{{xy}}^{{notM}}\) increases up to nearby Tc and gradually decreases at higher temperatures, while \({\sigma }_{{xy}}^{M}\) monotonically decreases. Consequently, \({\sigma }_{{xy}}^{{notM}}\) becomes much larger than \({\sigma }_{{xy}}^{M}\) near and above Tc. It should be noted that the anomalous Hall angle \({\rho }_{{yx}}^{{notM}}/{\rho }_{{xx}}(\sim \tan {{\rm{\theta }}}_{{\rm{H}}})\) is more than 0.02 around 24 K at 3 T, which is the highest level among the bulk oxide collinear ferromagnets and antiferromagnets to the best of our knowledge [see Fig. 3c].

Anomalous Hall effect in ferromagnetic Sr0.92Ba0.08Co6O11

Interestingly, the behavior of Hall resistivity is quite different in the system without the magnetic devil’s staircases. In the R-type hexaferrite ACo6O11 (A = Ca, Sr, and Ba), the partial substitution of Sr for Ba enhances the interlayer lattice spacing [Supplementary Table 1], which results in the variation of the electronic state and thus interplane RKKY-type magnetic interactions34,38. Consequently, a single ferromagnetic transition occurs in the Ba-substituted analog of SrCo6O11 as shown in Fig. 4a, b. Figure 4d–g shows the magnetization, magnetoresistivity, and Hall resistivity for Sr0.92Ba0.08Co6O11 with Tc = 33 K [see also Supplementary Fig. 6]. The magnetization curve shows a conventional ferromagnetic behavior at 10 K without any signature of metamagnetic transition. \({\rho }_{{xx}}\) shows the negative magnetoresistivity near Tc, similar to the case of SrCo6O11. On the other hand, \({\rho }_{{yx}}\) does not show the peak structure as observed in SrCo6O11 even near and above Tc and remains to be negative. As demonstrated in Fig. 4f, \({\rho }_{{yx}}\) is well fitted by the sum of \({\rho }_{{yx}}^{N}\) and \({\rho }_{{yx}}^{M}\) in whole temperature region. In particular, \({\rho }_{{yx}}\) is almost governed by \({\rho }_{{yx}}^{M}\), resulting in the small value of \({\rho }_{{yx}}^{N}\) and \({\rho }_{{yx}}^{{notM}}\)[see also Supplementary Fig. 6]. As shown in Fig. 4c, g, it is evident that \({\rho }_{{yx}}^{{notM}}\) is much smaller than that for SrCo6O11 in all temperature/field regions. These results suggest that the remarkable AHE is triggered by the spin fluctuation inherent to the magnetic devil’s staircase.

Fig. 4: The magnetization, resistivity, and Hall resistivity for Sr0.92Ba0.08Co6O11.
figure 4

a, b Temperature dependence of magnetization (M) measured at B = 0.01 T and resistivity (\({\rho }_{{xx}}\)) at B = 0 T. c The contour plot of anomalous Hall resistivity not proportional to M (\({\rho }_{{xy}}^{{notM}}\)) on the field-temperature plane. The closed circle denotes Tc of ferromagnetic ordering. d–g The magnetization (M), resistivity (\({\rho }_{{xx}}\)), Hall resistivity (\({\rho }_{{yx}}\)), and Hall resistivity are not proportional to M (\({\rho }_{{yx}}^{{notM}}\)), respectively.

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Disordering dependence of anomalous Hall effect

The insight into the mechanism of AHE is often acquired from the scaling relation between \({\sigma }_{{xx}}\) and \({\sigma }_{{xy}}^{{notM}}\). To systematically change \({\sigma }_{{xx}}\) near Tc without substantially changing the magnetic state, we prepared two sample sets of Sr1-x(Ca1-δBaδ)xCo6O11 with x = 0.02 (set A: δ = 0.31 and set B: δ = 0.25); the co-doping of Ba and Ca enhances the non-magnetic lattice disorder while keeping the lattice constants and magnetic devil’s staircase [see Supplementary Tables 1, 2 and Supplementary Fig. 7]. As shown in Fig. 5a, the resistivity for the doped system is enhanced compared with the undoped system, while the kink at Tc remains to be observed. The inset to Fig. 5a shows the \({\rho }_{{yx}}\) at several temperatures above Tc. \({\rho }_{{yx}}\) shows a peak similar to the case of x = 0 [see also Supplementary Figs. 7 and 8]. We derived \({\sigma }_{{xy}}^{{notM}}\) as in the case of undoped systems and plotted the peak value at several temperatures near Tc (22 - 26 K) for several samples in Fig. 5b. The peak of \({\sigma }_{{xy}}^{{notM}}\) seems to be scaled to \({\sigma }_{{xx}}^{2}\).

Fig. 5: The scaling relation between σxx and σxy\notM.
figure 5

a Temperature dependence of resistivity for two set of Sr1-x(Ca1-δBaδ)xCo6O11 with x = 0.02 (A1, A2, B1 and B2). Inset shows \({\rho }_{{yx}}\) for sample B2. b The peak value of \({\sigma }_{{xy}}^{{notM}}\) near Tc (22–26 K) plotted as a function of \({\sigma }_{{xx}}\). The dashed, dash-dot, and solid lines denote \({\sigma }_{{xy}}^{{notM}}\propto {\sigma }_{{xx}}^{\beta }\) with β = 1.0, 1.6, and 2.0, respectively.

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Discussion

In general, there are several possible mechanisms for the AHE induced by thermal spin fluctuation. A possibility is the conventional extrinsic AHE from skew scattering or side-jump mechanism. \({\sigma }_{{xy}}\) of skew scattering mechanism (side-jump mechanism) is known to be proportional to \({\sigma }_{{xx}}\) (\({\sigma }_{{xx}}^{0}\))1. Moreover, the conventional extrinsic AHE is usually much smaller than the intrinsic AHE, that is, the typical anomalous Hall angle (tan θH) is less than 0.01. Therefore, this may not be likely for the present case [see also Fig. 3c and Fig. 5b]. Another possibility is the AHE due to the vector spin chirality \({S}_{i}\times {S}_{j}\)39,40,41. This mechanism, however, requires strong crystalline inhomogeneity or inversion-symmetry breaking. Since the crystal structure is derived to be centrosymmetric29, this may be also unlikely in the present case. The most plausible mechanism is the three-spin correlation process related to the SSC. In this case, the large Hall signal that is not proportional to M can be induced by a chiral spin texture or local correlation8,9,10,11,26,27,42,43,44. This often manifests itself as the anomalous Hall signal which becomes remarkable in the partially spin-polarized state near the transition temperature in ferromagnets. In Fig. 3d, we plot \({\sigma }_{{xy}}^{{notM}}\) as a function of magnetization normalized by the saturated magnetization Ms i.e. magnetization at 2 K and 9 T. \({\sigma }_{{xy}}^{{notM}}\) near and above Tc is commonly maximal at \(M/{M}_{s}\sim 0.5\), as in the case of other magnets showing the thermally induced SSC21,26.

To explore whether the thermal spin fluctuation can generate the finite SSC in SrCo6O11, we theoretically calculated the SSC using Monte Carlo simulation. To this end, we consider a Kondo lattice model with conduction electron layers [Co1] sandwiched by the layers of classical moments on AB-stacked bilayer triangular lattice [Co(3)] [see Fig. 6a, b]; the itinerant electrons interact with the localized moments by the inter-site exchange interaction. The effective spin model for the localized moments on the AB-stacked triangular lattice is assumed to be

$$\begin{array}{l}H=-{J}_{0}\sum _{{\left\langle i,j\right\rangle }_{{xy}}}{{\boldsymbol{S}}}_{i}\cdot {{\boldsymbol{S}}}_{j}-\sum _{{\left\langle i,j\right\rangle }_{{xy}}}{D}_{{ij}}\hat{z}\cdot \left({{\boldsymbol{S}}}_{i}\times {{\boldsymbol{S}}}_{j}\right)-{J}_{1}\sum _{{\left\langle i,j\right\rangle }_{z}}{{\boldsymbol{S}}}_{i}\cdot {{\boldsymbol{S}}}_{j}\\\qquad\;-{J}_{2}\sum _{{\left\langle i,j\right\rangle }_{z}}{{\boldsymbol{S}}}_{i}\cdot {{\boldsymbol{S}}}_{j}-{K}_{z}\sum _{i}{\left({S}_{i}^{z}\right)}^{2}-{h}_{z}\sum _{i}{S}_{i}^{z},\end{array}$$
(2)

which is inspired by the ANNNI model35,36. Here, \({{\boldsymbol{S}}}_{i}\) is the localized Heisenberg spin on the \(i\) th site, \({J}_{0},{J}_{1}\) and \({J}_{2}\) are in-plane nearest neighbor-, out-of-plane nearest-neighbor-, and out-of-plane second-nearest-neighbor-exchange interactions, respectively. Considering the local symmetry for the intralayer Co(3)- Co(3) bond, the in-plane Dzyaloshinskii-Moriya (DM) interaction \({D}_{{ij}}\) is assumed. \({K}_{z}\) and \({h}_{z}\) are the easy-axis anisotropy and external magnetic field along z-axis (c-axis), respectively. For the calculation, we assumed \({J}_{1}={J}_{0}=1/3\), \(-1\le {J}_{2}\le 0\), \({K}_{z}=10\) and \({D}_{{ij}}=1/6\). The statistical property of this model is studied using the Monte Carlo method with the heat bath local update method.

Fig. 6: Monte Carlo simulation of scalar-spin chirality.
figure 6

a Side view and b top view of AB-stacked triangular lattice of Co3 and Kagome layer of Co1. The yellow triangle with a dotted edge denotes an example of a triangle considering the SSC. The in-plane solid line of the Co3 triangular lattice denotes the magnetic bond with Dzyaloshinskii-Moriya (DM) interaction. c The magnetic phase diagram on the temperature vs. -J2/3J1 plane. d, e Magnetization and thermal average of scalar spin chirality for the nearest-neighbor triangles. The results in c–e are for \(N=16\times 16\times 64\) system size with \({J}_{0}=1/3\), \({J}_{1}=1/3\), \(D=1/6\), \({K}_{z}=10\).

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The magnetic phase diagram at \({h}_{z}=0\) is shown in Fig. 6c. For \({J}_{2}/3{J}_{1}=0\), a ferromagnetic phase with spontaneous magnetization along z-axis emerges at \(T=2.3{J}_{0}\). On the contrary, for \({J}_{2}/3{J}_{1}=-1\), a number of collinear antiferromagnetic orderings with easy-axis along z-direction successively emerge as the temperature decreases; the phase with modulation vector q = 7/32 parallel to the z-axis appears at \(T=2.3{J}_{0}\), that with q = 15/64 at around \(T=1.8{J}_{0}\) and that with q = 1/4 at around \(T=1.3{J}_{0}\), similar to that of the Ising spin model45. By applying the magnetic field, these antiferromagnetic phases turn into the field-forced ferromagnetic phase. As shown in Fig. 6d, the field-induced phase transition manifests itself as a jump of magnetization around \({h}_{z}={J}_{0}\) at low temperatures. At high temperatures above the antiferromagnetic transition temperature, the jump is replaced by a crossover. The calculated result is consistent with the experimental result about the appearance of temperature/field-induced transitions among magnetic phases with different q.

To study the temperature/field dependence of SSC, we focus on the thermal average of SSC for three spins on sites i, j, k

$$\chi =\left\langle {{\boldsymbol{S}}}_{i}\cdot \left({{\boldsymbol{S}}}_{j}\times {{\boldsymbol{S}}}_{k}\right)\right\rangle =\frac{1}{Z}{Tr}\left[\sum _{\left\langle i,j,k\right\rangle }{{\boldsymbol{S}}}_{i}\cdot \left({{\boldsymbol{S}}}_{j}\times {{\boldsymbol{S}}}_{k}\right)\exp (-\beta {\rm{H}})\right]$$
(3)

where the sum over \({\rm{\langle }}i,j,k{\rm{\rangle }}\) is the sum over three nearest-neighbor spins on the two adjacent Co3 layers sandwiching the conduction layer [Fig. 6a, b]. Here, β = 1/kBT, and Z is the partition function. Note that, for a translationally symmetric triangular lattice, the chirality of an upward triangle is exactly the opposite of the downward triangles. Hence, the SSC of three nearest-neighbor spins within a triangular layer does not contribute to the AHE. Therefore, we consider the SSC of three spins on two adjacent layers sandwiching a conduction layer. An example of the three spins Si (i = 1,2,3) is shown in Fig. 6a, b, in which S3 is on a different layer from S1 and S2. The thermal average of SSC is shown in Fig. 6e, in which SSC is small in the antiferromagnetic phase in the low field and low-temperature region. On the contrary, it is enhanced in the forced ferromagnetic/paramagnetic phase in the high-temperature region. In particular, the peak field of a thermal average of SSC increases with increasing temperature. This suggests that the conduction electrons of Co(1)-site acquire the finite SSC from the thermal fluctuation of Co(3)-spin in the present material, which is particularly enhanced above the magnetic transition temperature.

Finally, we discuss the mechanism of AHE from the thermally induced SSC. In the scheme of Berry curvature in momentum space, \({\sigma }_{{xy}}\) is predicted to be proportional to \({\sigma }_{{xx}}^{0}\), which is not consistent with the present result [see Fig. 5b]. The result also seems to be slightly deviate from \({\sigma }_{{xx}}^{1.6}\)-law37. On the contrary, the multiple skew scattering mechanism, topological Hall effect, and orbital Berry phase mechanism propose the \({\sigma }_{{xx}}^{2}\)-law10,19,46,47. Considering that the large \({\sigma }_{{xy}}^{{notM}}\) remains even at high temperatures where the spin correlation length would be significantly reduced, the multiple skew scattering may be most likely.

It should be noted that the large Hall effect remaining even far above the magnetic transition temperature is observed in the magnetic semiconductor EuAs and Skyrmion magnets Gd3Ru4Al1220,48. The common feature of these materials is that the long-range ordered phase below transition temperature is characterized by the noncoplanar spin structure. This is in contrast with the present material with the collinear antiferromagnetic ordering. In collinear antiferromagnets with devil’s staircase transition, it has been known that the solitonic or cluster-like spin defects such as domain walls thermally proliferate and diffuse near and above Tc35,36,49. Although such spin excitation has not been accurately captured by the current theoretical calculation, the spin-flip excitation is presumed to cause the large-angle transient noncoplanar spin texture, resulting in the SSC-related skew scattering of electrons.

In this paper, we show that the collinear antiferromagnetic metal SrCo6O11 exhibits the large anomalous Hall effect (AHE) due to the spin fluctuation of the devil’s staircase-type transition by means of transport measurement and theoretical calculation. In particular, the maximum of anomalous Hall angle exceeds 0.02, which is the highest level among the known bulk oxide collinear ferromagnets/antiferromagnets. The AHE not scaled to the magnetization becomes remarkably near and above the transition temperature (Tc) but vanishes in the field-induced fully spin-polarized state. Furthermore, such thermally induced AHE is not clearly observed in the ferromagnetic Sr0.92Ba0.08Co6O11 without the devil’s staircase transition. We also found that the anomalous hall conductivity not scaled to magnetization (\({\sigma }_{{xy}}^{{notM}}\)) is quadratically scaled to electrical conductivity, i.e., \({{\sigma }_{{xy}}^{{notM}}\propto \sigma }_{{xx}}^{2}\). These results imply that the thermally induced proliferation of solitonic/cluster-like spin defects inherent to the magnetic devil’s staircase enhances the scalar-spin-chirality skew scattering of electrons, yielding the large AHE. This work demonstrates that the large anomalous Hall effect comparable or larger than the well-known Berry curvature mechanism can be induced above the transition temperature in a collinear antiferromagnetic metal, paving a new route for high-temperature paramagnetic spintronics function such as efficient thermoelectric energy harvesting.

Methods

Sample preparation and characterization

Single crystals of R-type hexaferrite SrCo6O11 were grown under pressure by means of the cubic-anvil-type facility. The starting materials are Sr3Co2O7-δ, Sr(OH)2 · 8H2O, and KClO4 mixed in a ratio of 8:1:3. The mixture was sealed in Pt capsule and was heated up to 840 ◦C under 2 GPa. It was kept there for 10 min and then quenched to room temperature. The typical size of a crystal is about 0.3 × 0.3 × 0.1 mm with the shape of a hexagonal plate normal to the c-axis. Supplementary Fig. 1 shows the photograph of the sample. The unit cell of samples is determined by the single crystalline X-ray diffraction at room temperature (see Supplementary Table 1). Single crystalline samples of Sr0.92Ba0.08Co6O11 and Sr1–x(Ca1-δBaδ)xCo6O11 x = 0.02 (set A: δ = 0.31 and set B: δ = 0.25) are grown by similar conditions. For the latter, two series of samples (A and B) were prepared (see Supplementary Table 1 and 2). Sr-, Ca- and Ba-concentration are determined by the energy dispersive X-ray spectroscopy (EDX) analysis and scanning electron microscopy (SEM).

Measurement of resistivity, Hall resistivity, and magnetization

Measurements of resistivity and Hall resistivity were performed by standard four-terminal geometry with an indium electrode. The measurements were done by using the Physical Property Measurement System (Quantum Design) from 2 K to 300 K under the magnetic field up to 9 T. The magnetization measurements were performed by using the Dynacool System equipped with the VSM option from 2 K to 300 K under the magnetic field up to 9 T. The several samples with different shapes show nearly identical magnetization profiles, suggesting that the demagnetization factor is not significant. The magnetization measurement was performed for samples for which the transport property was measured.

Monte Carlo simulation

The magnetic phase diagram and field dependence were calculated using a Monte Carlo simulation with the standard heat-bath update method. The physical quantities were calculated using 120,000 MC steps after 20,000 steps of relaxation. The MC results were split into 6 bins for estimating the statistical error. All calculations were performed using an on-premise PC cluster at the Tokyo Institute of Technology.