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Naoki Abe, Yuya Hano, [Hiroaki Ishizuka](https://orcid.org/0000-0002-5719-4315), [Yusuke Kozuka](https://orcid.org/0000-0001-7674-600X), [Terumasa Tadano](https://orcid.org/0000-0002-8132-2161), [Yoshihiro Tsujimoto](https://orcid.org/0000-0003-2140-3362), [Kazunari Yamaura](https://orcid.org/0000-0003-0390-8244), [Shintaro Ishiwata](https://orcid.org/0000-0003-1696-2514), [Jun Fujioka](https://orcid.org/0000-0003-1340-0268)

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[Large anomalous Hall effect in spin fluctuating devil’s staircase](https://mdr.nims.go.jp/datasets/6be603c0-91b8-49b7-8fa3-9c6ff2cd5891)

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Large anomalous Hall effect in spin fluctuating devil’s staircasenpj | quantummaterials ArticlePublished in partnership with Nanjing Universityhttps://doi.org/10.1038/s41535-024-00653-3Large anomalous Hall effect in spinfluctuating devil’s staircaseCheck for updatesNaoki Abe1, Yuya Hano1, Hiroaki Ishizuka 2, Yusuke Kozuka 3, Terumasa Tadano 4,Yoshihiro Tsujimoto3, Kazunari Yamaura 3, Shintaro Ishiwata 5,6 & Jun Fujioka 1,7Electrons in metals can show a giant anomalous Hall effect (AHE) when interacting with characteristicspin texture. The AHE has been discussed in terms of scalar-spin-chirality (SSC) in long-range-ordered noncollinear spin textures typified by Skyrmion. The SSC becomes effective even in theparamagnetic state with thermal fluctuations, but the resultant AHE has been limited to be very small.Here, we report the observation of large AHE caused by the spin fluctuation near the devil’s staircasetransition in a collinear antiferromagnetic metal SrCo6O11. The AHE is prominent near and above thetransition temperature at moderate magnetic fields, where the anomalous Hall angle becomes thehighest level among known oxide collinear ferromagnets/antiferromagnets (>2%). Furthermore, theanomalous Hall conductivity is quadratically scaled to the conductivity. These results imply that thethermally induced solitonic spin defects inherent to the devil’s staircase transition promote SSC-induced skew scattering.Exploration of nontrivial spin textures in magnetic metals and semi-conductors is a subject of great interest in modern condensed matterscience. The striking feature is that conduction electrons interactingwith the spin texture often cause unprecedented electromagnetic phe-nomena through the quantal phase. One of the typical examples is theanomalous Hall effect (AHE)1. Conventionally, the AHE has beenconsidered to be induced by the spontaneous magnetization in ferro-magnets but is currently observed even in a variety of antiferromagnetsand helimagnets2–7. For example, helimagnets such as magnetic Sky-rmions show the AHE or topological Hall effect in proportion to thescalar-spin chirality (SSC) Si · (Sj × Sk), which represents the solid angleformed by three spins in a noncoplanar configuration8–12. From amicroscopic viewpoint, this effect is usually understood from thescheme of Berry curvature for electrons; the Berry curvature (quantalphase) proportional to the SSC is induced by the long-range orderednoncoplanar spin texture, which acts as the fiction magnetic field toconduction electrons1. In particular, it has been demonstrated that theHall 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 ofspintronic function such as the efficient electrical read-out of spintexture or thermoelectric energy harvesting15,16.On the other hand, the AHE can be also triggered by fluctuating orspatially inhomogeneous spin texture that is not long-range ordered. Forexample, electron scattering from a single magnetic defect causes the AHEof skew scattering or side-jumpmechanism17,18. In addition, it is also knownthat the transient noncoplanar spin texture induced by the thermal spinfluctuation triggers the AHE through the SSC. This phenomenon has beenoften understood from the scheme of Berry curvature mechanism1,12, butrecent researchproposes that themultiple skew scattering fromspin clusterswith noncoplanar spin texture can cause the giant AHE18,19, the anomalousHall angle (tan θH) of which is scaled to the SSC. In particular, for the lattercases, the anomalous Hall angle comparable to or larger than the intrinsicAHE of Skrmion magnets can emerge even in the paramagnetic tempera-ture regime far above the magnetic transition temperature20. This is incontrast with the conventional intrinsic/extrinsic AHE, which is usuallyenhanced in the long-range ordered state. Nevertheless, the AHE origi-nating from spin fluctuations i.e. paramagnetic AHE is usually very small inmost magnetic metals (tan θH ≤ 0.01)12,21–28 except in a few noncollinearmagnets20,21. Here, we report that the collinear antiferromagnetic metal1Graduate School of Science and Technology, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan. 2Department of Physics, Tokyo Institute of Technology,Meguro, Tokyo 152-8551, Japan. 3Research Center for Materials Nanoarchitechtonics (MANA), National Institute for Materials Science (NIMS), Namiki, Tsukuba,Ibaraki 305-0044, Japan. 4Research Center for Magnetic and Spintronic Materials (CMSM), National Institute for Materials Science (NIMS), Sengen, Tsukuba,Ibaraki 305-0047, Japan. 5Division of Materials Physics, Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan. 6SpintronicsResearch Network Division, Institute for Open and Transdisciplinary Research Initiatives, Osaka University, Suita, Osaka 565-0871, Japan. 7Department ofMaterials Science, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan. e-mail: fujioka@ims.tsukuba.ac.jpnpj Quantum Materials |            (2024) 9:41 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41535-024-00653-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41535-024-00653-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41535-024-00653-3&domain=pdfhttp://orcid.org/0000-0002-5719-4315http://orcid.org/0000-0002-5719-4315http://orcid.org/0000-0002-5719-4315http://orcid.org/0000-0002-5719-4315http://orcid.org/0000-0002-5719-4315http://orcid.org/0000-0001-7674-600Xhttp://orcid.org/0000-0001-7674-600Xhttp://orcid.org/0000-0001-7674-600Xhttp://orcid.org/0000-0001-7674-600Xhttp://orcid.org/0000-0001-7674-600Xhttp://orcid.org/0000-0002-8132-2161http://orcid.org/0000-0002-8132-2161http://orcid.org/0000-0002-8132-2161http://orcid.org/0000-0002-8132-2161http://orcid.org/0000-0002-8132-2161http://orcid.org/0000-0003-0390-8244http://orcid.org/0000-0003-0390-8244http://orcid.org/0000-0003-0390-8244http://orcid.org/0000-0003-0390-8244http://orcid.org/0000-0003-0390-8244http://orcid.org/0000-0003-1696-2514http://orcid.org/0000-0003-1696-2514http://orcid.org/0000-0003-1696-2514http://orcid.org/0000-0003-1696-2514http://orcid.org/0000-0003-1696-2514http://orcid.org/0000-0003-1340-0268http://orcid.org/0000-0003-1340-0268http://orcid.org/0000-0003-1340-0268http://orcid.org/0000-0003-1340-0268http://orcid.org/0000-0003-1340-0268mailto:fujioka@ims.tsukuba.ac.jpSrCo6O11withdevil’s staircase-type transition shows the largeparamagneticAHE in a moderate magnetic field (∼3 T) even far above the magnetictransition temperature, resulting in the anomalousHall angle (tan θH)morethan 0.02.ResultsMagnetic propertySrCo6O11 crystallizes in the R-type hexaferrite structure with space groupP63/mmc29. In this material, the Co-sublattice can be viewed as the stackingof Kagome lattice of Co1, dimerized pillar of Co2, and triangular lattice ofCo3 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 spinmoment with easy-axis anisotropy along the c-axis, while those of Co1 and Co2 take the par-tially filled low spin states, yielding the strongly correlated conductionelectrons30–33 [see also Supplementary Fig. 1]. The conduction electronsmediate the magnetic interaction of Co3-spin through the Ruderman-Kittel-Kasuya-Yoshida (RKKY)-type interaction, resulting in the competingmultiple magnetic interactions between the nearest-neighbor spins andfurther-neighbor spins along the c-axis34. Consequently, the collinear anti-ferromagnetic phases, in which spins aremodulated along the c-axis but areferromagnetically aligned within the ab-plane, occur at low temperatures.As an example, Fig. 1a shows the magnetic structure with the modulationwavenumberq = 1/3,where spins are aligned in the c-direction in anup-up-down order.Figure 1d shows the magnetic phase diagram when the magnetic field(B) is applied along the c-axis. At zeromagnetic field, themagnetic orderingwith incommensurate (IC)modulationq ~ 1/5 appears at 22 K ( = Tc).Withlowering temperature, q quasi-continuously decreases and is locked to thecommensurate value of 1/6 about 12 K34. Simultaneously, various magneticphases with different modulation periods emerge, resulting in the phase-separated state. When the magnetic field is applied along the c-axis, thesestates change to the q = 1/3-phase at Bc1, followed by the forced ferromag-netic phase at Bc2. These field-induced phase transitions can be seen as themagnetization jumps and plateaus as shown in the inset to Fig. 1d33,34 (seealso Fig. 2a–d). Such behavior is a hallmark of the magnetic devil staircase,which originates from the competition among a large number of nearlydegenerate magnetic phases with different modulation period35,36.Magneto-transport propertyFigure 1e shows the temperature dependence of resistivity (ρxx) at B = 0 T.The ρxx shows minimal temperature dependence above Tc, suggesting theincoherent charge transport as often seen in the strongly correlated metals.At Tc, the resistivity shows a kink and steeply decreases at lower tempera-tures. Figure 2a–t summarizes the magnetization, magnetoresistivity, andHall resistivity under themagnetic field. Themagnetization shows jumps atBc1 and Bc2 below Tc, which are blurred above Tc [see Fig. 2a–e]. Themagnetoresistivity also shows anomalies at Bc1 and Bc2 in the temperaturerange 12–18 K. Furthermore, with increasing temperature from 2 K, thenegative magnetoresistivity becomes gradually pronounced. In particular,the negativemagnetoresistivity is observed up to around 50 K, implying thatthe short-ranged spin correlation remains even far above Tc.On the other hand, the Hall resistivity (ρyx) shows more peculiartemperature and field dependence [See Fig. 2k–o]. At 2 K, ρyx almostlinearly increases as a function of B except for a jump at Bc2. Since themagnetization is saturated above Bc2 [see Fig. 2a], the B-linear com-ponent above Bc2 is likely attributed to the ordinary Hall effect. Theextrapolation of the B-linear part to zero magnetic field seems to take afinite value, suggesting the finite contribution from the anomalousHalleffect proportional toM. With increasing temperature, the slope of ρyxabove Bc2 gradually decreases, while that below Bc2 shows sign changeabove 12 K. In addition, a peak structure grows around Bc2 andbecomes remarkable at higher temperatures. Above 22 K, the peakshifts to a higher-field region and gradually diminishes while broad-ening its width. Apparently, such behavior cannot be explained by theconventional anomalous Hall effect proportional to M. On the otherhand, the ordinary Hall effect can show a peak, given that the carriermobility is sufficiently high. This is, however, not likely, since theresistivity above Tc is in the incoherent transport regime close to theIoffe-Regel limit (∼ 1× 10�3Ω cm) (see Supplementary Note 1). Thissuggests the presence of an additional contribution other than theordinary Hall effect and AHE proportional to M. Following theFig. 1 | Crystal structure and magnetic phasediagram for SrCo6O11. a The illustration of Co-sublattice of SrCo6O11 with up-up-down-typemagnetic structure (q = 1/3)50. Arrows denote thespins on Co3-sites. The primitive vectors a, b, and care defined in the hexagonal crystal symmetry withspace group P63/mmc. b, c The illustration of CoO6-octahedra of Co1 and Co2 and CoO5-bipyramid ofCo3. d The magnetic phase diagram in thetemperature-magnetic field plane. Arrows denotespins of Co3. Here, q represents the wave number ofmagnetic modulation. Open circles and squaresdenote the magnetic transition at Bc1 and Bc2,respectively. Open triangles denote the transitiontemperature (Tp) of the incommensurate (IC) phaseor q = 1/3-phase determined from the temperaturedependence ofM (See Supplementary Fig. 2d, e).WedefinedTp at 0.1 T asTc (=22 K). The inset shows themagnetization curve at 8 K. The horizontal dottedline denotes the magnetization plateau corre-sponding to q = 1/3- and q = 0-phase. eTemperaturedependence of resistivity at B = 0 T.https://doi.org/10.1038/s41535-024-00653-3 Articlenpj Quantum Materials |            (2024) 9:41 2conventional wisdom7,13,20, we analyzed theHall resistivity by assumingthe following three components,ρyx ¼ RNBþ SAραxxM þ ρnotMyx ð1ÞThe first, second, and third terms represent the ordinary Hall com-ponent (ρNyx ¼ RNB), the anomalous Hall component in proportion to M(ρMyx ¼ SAραxxM), and the residual component (ρnotMyx ), which is neitherproportional toBnor toM, respectively. Since thefittingwith thismodel didnot uniquely converge in the region below Bc2 due to the short field range ofeach magnetic phase [see also Supplementary Note 2], we focus on theresults in the forced ferromagnetic/paramagnetic state above Bc2. Here, weassumed the exponent α ¼ 0:4 for ρMyx which corresponds to the intrinsicAHE in the incoherent transport regime1,37. Although α depends on thetransport regime, the analyzed results here did not significantly depend onthe choice of α [see Supplementary Note 3]. ρNyx , ρMyx and their summation(¼ ρNyx þ ρMyx) at 2 K are exemplified in Fig. 2k. ρyx is well reproduced byρNyx þ ρMyx , resulting in the negligible ρnotMyx [see Fig. 2p]. With increasingtemperatures, the difference between ρyx and ρNyx þ ρMyx becomes significantespecially near Bc2, resulting in a peak of ρnotMyx as shown in Fig. 2q–t. Above30 K, the peak shifts to a higher-field region and gradually diminishes. Tovisualize this behavior, we show the contour plot of ρnotMyx on thetemperature-magnetic field plane in Fig. 3a. It is evident that ρnotMyx isenhanced in thewide temperature-field region near and aboveTc, where themagnetization is not fully polarized, namely, the thermal spin fluctuation isremarkable.To quantitatively compare the temperature dependence of each term,we derived theHall conductivity σM;notMxy ½¼ ρM;notMyx =ðρ2xx þ ρ2yxÞ�. Figure 3bshows the temperature dependence of σMxy and σnotMxy at 3 T. Below 10 K,σnotMxy is much smaller than σMxy . With increasing temperature, σnotMxyincreases up to nearby Tc and gradually decreases at higher temperatures,while σMxy monotonically decreases. Consequently, σnotMxy becomes muchlarger than σMxy near and above Tc. It should be noted that the anomalousHall angle ρnotMyx =ρxxð∼ tan θHÞ ismore than0.02 around24Kat 3 T,whichis the highest level among the bulk oxide collinear ferromagnets and anti-ferromagnets to the best of our knowledge [see Fig. 3c].Anomalous Hall effect in ferromagnetic Sr0.92Ba0.08Co6O11Interestingly, the behavior of Hall resistivity is quite different in the systemwithout the magnetic devil’s staircases. In the R-type hexaferrite ACo6O11(A =Ca, Sr, and Ba), the partial substitution of Sr for Ba enhances theinterlayer lattice spacing [Supplementary Table 1], which results in thevariation of the electronic state and thus interplane RKKY-type magneticinteractions34,38. Consequently, a single ferromagnetic transition occurs inthe Ba-substituted analog of SrCo6O11 as shown in Fig. 4a, b. Figure 4d–gshows the magnetization, magnetoresistivity, and Hall resistivity forSr0.92Ba0.08Co6O11 with Tc = 33 K [see also Supplementary Fig. 6]. Themagnetization curve shows a conventional ferromagnetic behavior at 10 Kwithout any signature of metamagnetic transition. ρxx shows the negativemagnetoresistivity near Tc, similar to the case of SrCo6O11. On the otherhand, ρyx does not show the peak structure as observed in SrCo6O11 evennear andaboveTc and remains tobenegative.Asdemonstrated inFig. 4f,ρyxis well fitted by the sum of ρNyx and ρMyx in whole temperature region. Inparticular, ρyx is almost governed by ρMyx , resulting in the small value of ρNyxand ρnotMyx [see also Supplementary Fig. 6]. As shown in Fig. 4c, g, it is evidentthat ρnotMyx is much smaller than that for SrCo6O11 in all temperature/fieldFig. 2 | Magnetization and electrical transport property for SrCo6O11. a–e Themagnetization (M), f–j resistivity (ρxx), k–o Hall resistivity (ρyx), and p–t Hallresistivity are not proportional toM (ρnotMyx ) for SrCo6O11. The blue dash-dot-lineand green dashed line denote the ordinary Hall resistivity (ρNyxÞ and anomalous Hallresistivity proportional toM (ρMyx), respectively. The red dashed curve denotes theρNyx þ ρMyx (red dashed curve in k denotes the ρNyx þ ρMyx at 22 K). ρNyx and ρMyx areomitted for clarity in l–o.https://doi.org/10.1038/s41535-024-00653-3 Articlenpj Quantum Materials |            (2024) 9:41 3regions. These results suggest that the remarkable AHE is triggered by thespin fluctuation inherent to the magnetic devil’s staircase.Disordering dependence of anomalous Hall effectThe insight into the mechanism of AHE is often acquired from the scalingrelation between σxx and σnotMxy . To systematically change σxx near Tcwithout substantially changing the magnetic state, we prepared two samplesets 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 latticedisorder 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 theundoped system, while the kink at Tc remains to be observed. The inset toFig. 5a shows the ρyx at several temperatures above Tc. ρyx shows a peaksimilar to the case of x = 0 [see also Supplementary Figs. 7 and 8]. Wederived σnotMxy as in the case of undopedsystems andplotted thepeak value atseveral temperatures near Tc (22 - 26 K) for several samples in Fig. 5b. Thepeak of σnotMxy seems to be scaled to σ2xx .DiscussionIn general, there are several possible mechanisms for the AHE induced bythermal spin fluctuation. A possibility is the conventional extrinsic AHEfrom skew scattering or side-jump mechanism. σxy of skew scatteringmechanism (side-jump mechanism) is known to be proportional to σxx(σ0xx)1. Moreover, the conventional extrinsic AHE is usually much smallerthan the intrinsic AHE, that is, the typical anomalous Hall angle (tan θH) isless than 0.01. Therefore, this may not be likely for the present case [see alsoFig. 3c and Fig. 5b]. Another possibility is the AHE due to the vector spinchirality Si × Sj39–41. This mechanism, however, requires strong crystallineinhomogeneity or inversion-symmetry breaking. Since the crystal structureis derived to be centrosymmetric29, this may be also unlikely in the presentcase. The most plausible mechanism is the three-spin correlation processrelated to the SSC. In this case, the large Hall signal that is not proportionaltoM can be induced by a chiral spin texture or local correlation8–11,26,27,42–44.This often manifests itself as the anomalous Hall signal which becomesremarkable in the partially spin-polarized state near the transition tem-perature in ferromagnets. In Fig. 3d, we plot σnotMxy as a function of mag-netization normalized by the saturatedmagnetizationMs i.e. magnetizationat 2 K and 9 T. σnotMxy near and above Tc is commonly maximal atM=Ms ∼ 0:5, as in the case of othermagnets showing the thermally inducedSSC21,26.To explore whether the thermal spin fluctuation can generate the finiteSSC in SrCo6O11, we theoretically calculated the SSC using Monte Carlosimulation. To this end, we consider a Kondo latticemodel with conductionelectron layers [Co1] sandwiched by the layers of classical moments onAB-stacked bilayer triangular lattice [Co(3)] [see Fig. 6a, b]; the itinerant elec-trons interact with the localized moments by the inter-site exchangeinteraction. The effective spin model for the localizedmoments on the AB-stacked triangular lattice is assumed to beH ¼ �J0Pi;jh ixy Si � Sj �Pi;jh ixy Dijẑ � Si × Sj� �� J1Pi;jh iz Si � Sj�J2Pi;jh iz Si � Sj � KzPi Szi� �2 � hzPi Szi ;ð2Þwhich is inspired by the ANNNI model35,36. Here, Si is the localized Hei-senberg spinon the i th site, J0; J1 and J2 are in-planenearest neighbor-, out-of-plane nearest-neighbor-, and out-of-plane second-nearest-neighbor-exchange interactions, respectively. Considering the local symmetry for theintralayer Co(3)- Co(3) bond, the in-plane Dzyaloshinskii-Moriya (DM)interaction Dij is assumed. Kz and hz are the easy-axis anisotropy andexternalmagneticfield along z-axis (c-axis), respectively. For the calculation,we assumed J1 ¼ J0 ¼ 1=3, �1 ≤ J2 ≤ 0, Kz ¼ 10 and Dij ¼ 1=6. Thestatistical property of this model is studied using the Monte Carlo methodwith the heat bath local update method.The magnetic phase diagram at hz ¼ 0 is shown in Fig. 6c. ForJ2=3J1 ¼ 0, a ferromagnetic phase with spontaneous magnetization alongz-axis emerges at T ¼ 2:3J0. On the contrary, for J2=3J1 ¼ �1, a numberof collinear antiferromagnetic orderings with easy-axis along z-directionsuccessively emerge as the temperature decreases; the phase with modula-tion vector q = 7/32 parallel to the z-axis appears at T ¼ 2:3J0, that withq = 15/64 at around T ¼ 1:8J0 and that with q = 1/4 at around T ¼ 1:3J0,similar to that of the Ising spinmodel45. By applying themagneticfield, theseantiferromagnetic phases turn into the field-forced ferromagnetic phase. Asshown in Fig. 6d, the field-induced phase transition manifests itself as ajump of magnetization around hz ¼ J0 at low temperatures. At highFig. 3 | Field and temperature dependence ofanomalous Hall effect for SrCo6O11. a The con-tour plot of anomalous Hall resistivity not propor-tional toM (ρnotMxy ) on the field-temperature plane.Open circles, squares, and triangles denote themagnetic transition at Bc1, Bc2, and Tp, respectively.b Temperature dependence of σnotMxy (circles) and�σMxy at 3 T (triangles). c Anomalous Hall angle(tan θH ∼ ρnotMyx =ρxx) in various bulk oxide ferro-magnets (closed circles) and noncollinear magnets(open circles)1,51–58. d Anomalous Hall conductivitynot proportional to magnetization (σnotMxy ) as afunction of normalized magnetization (M=Ms) withMs being the magnetization at 2 K and 9 T.https://doi.org/10.1038/s41535-024-00653-3 Articlenpj Quantum Materials |            (2024) 9:41 4temperatures above the antiferromagnetic transition temperature, the jumpis replaced by a crossover. The calculated result is consistent with theexperimental result about the appearance of temperature/field-inducedtransitions among magnetic phases with different q.To study the temperature/field dependence of SSC, we focus on thethermal average of SSC for three spins on sites i, j, kχ ¼ Si � Sj × Sk� �D E¼ 1ZTrXi;j;kh iSi � Sj × Sk� �expð�βHÞ2435 ð3Þwhere the sumover i; j; k is the sumover threenearest-neighbor spinson thetwo adjacent Co3 layers sandwiching the conduction layer [Fig. 6a, b].Here,β = 1/kBT, and Z is the partition function. Note that, for a translationallysymmetric triangular lattice, the chirality of anupward triangle is exactly theopposite 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 layerssandwiching 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. Thethermal average of SSC is shown in Fig. 6e, in which SSC is small in theantiferromagnetic phase in the low field and low-temperature region. Onthe contrary, it is enhanced in the forced ferromagnetic/paramagnetic phasein the high-temperature region. In particular, the peak field of a thermalaverage of SSC increases with increasing temperature. This suggests that theconduction electrons of Co(1)-site acquire the finite SSC from the thermalfluctuation of Co(3)-spin in the present material, which is particularlyenhanced above the magnetic transition temperature.Finally, we discuss themechanismofAHE from the thermally inducedSSC. In the scheme of Berry curvature inmomentum space, σxy is predictedto be proportional to σ0xx , which is not consistent with the present result [seeFig. 5b]. The result also seems to be slightly deviate from σ1:6xx -law37. On thecontrary, the multiple skew scattering mechanism, topological Hall effect,and orbital Berry phase mechanism propose the σ2xx-law10,19,46,47. Consider-ing that the large σnotMxy remains even at high temperatures where the spincorrelation length would be significantly reduced, the multiple skew scat-tering may be most likely.It should be noted that the large Hall effect remaining even far abovethe magnetic transition temperature is observed in the magnetic semi-conductor EuAs and Skyrmion magnets Gd3Ru4Al1220,48. The commonfeature of these materials is that the long-range ordered phase below tran-sition temperature is characterized by the noncoplanar spin structure. Thisis in contrast with the present material with the collinear antiferromagneticordering. In collinear antiferromagnets with devil’s staircase transition, ithas been known that the solitonic or cluster-like spin defects suchas domainwalls thermally proliferate and diffuse near and above Tc35,36,49. Althoughsuch spin excitation has not been accurately captured by the current theo-retical calculation, the spin-flip excitation is presumed to cause the large-angle transient noncoplanar spin texture, resulting in the SSC-related skewscattering of electrons.Fig. 4 | The magnetization, resistivity, and Hallresistivity for Sr0.92Ba0.08Co6O11. a, bTemperaturedependence of magnetization (M) measured atB = 0.01 T and resistivity (ρxx) at B = 0 T. c Thecontour plot of anomalous Hall resistivity not pro-portional toM (ρnotMxy ) on the field-temperatureplane. The closed circle denotes Tc of ferromagneticordering. d–g The magnetization (M), resistivity(ρxx), Hall resistivity (ρyx), and Hall resistivity arenot proportional toM (ρnotMyx ), respectively.https://doi.org/10.1038/s41535-024-00653-3 Articlenpj Quantum Materials |            (2024) 9:41 5In this paper, we show that the collinear antiferromagnetic metalSrCo6O11 exhibits the large anomalous Hall effect (AHE) due to the spinfluctuation of the devil’s staircase-type transition by means of transportmeasurement and theoretical calculation. In particular, the maximum ofanomalous Hall angle exceeds 0.02, which is the highest level among theknown bulk oxide collinear ferromagnets/antiferromagnets. The AHE notscaled to the magnetization becomes remarkably near and above the tran-sition temperature (Tc) but vanishes in thefield-induced fully spin-polarizedstate. Furthermore, such thermally induced AHE is not clearly observed inthe ferromagnetic Sr0.92Ba0.08Co6O11without the devil’s staircase transition.We also found that the anomalous hall conductivity not scaled to magne-tization (σnotMxy ) is quadratically scaled to electrical conductivity, i.e.,σnotMxy / σ2xx. These results imply that the thermally induced proliferation ofsolitonic/cluster-like spin defects inherent to the magnetic devil’s staircaseenhances the scalar-spin-chirality skew scattering of electrons, yielding thelarge AHE. This work demonstrates that the large anomalous Hall effectcomparable or larger than the well-known Berry curvature mechanism canbe induced above the transition temperature in a collinear anti-ferromagnetic metal, paving a new route for high-temperature para-magnetic spintronics function such as efficient thermoelectric energyharvesting.MethodsSample preparation and characterizationSingle crystals ofR-typehexaferrite SrCo6O11were grownunder pressure bymeans of the cubic-anvil-type facility. The startingmaterials are Sr3Co2O7-δ,Fig. 6 | Monte Carlo simulation of scalar-spinchirality. a Side view and b top view of AB-stackedtriangular lattice of Co3 and Kagome layer of Co1.The yellow triangle with a dotted edge denotes anexample of a triangle considering the SSC. The in-plane solid line of the Co3 triangular lattice denotesthe magnetic bond with Dzyaloshinskii-Moriya(DM) interaction. c Themagnetic phase diagram onthe temperature vs. -J2/3J1 plane. d, eMagnetizationand thermal average of scalar spin chirality for thenearest-neighbor triangles. The results in c–e are forN ¼ 16× 16× 64 system size with J0 ¼ 1=3,J1 ¼ 1=3, D ¼ 1=6, Kz ¼ 10.Fig. 5 | The scaling relation between σxx andσxy\notM. aTemperature dependence of resistivity fortwo set of Sr1-x(Ca1-δBaδ)xCo6O11 with x = 0.02 (A1,A2, B1 and B2). Inset shows ρyx for sample B2. bThepeak value of σnotMxy near Tc (22–26 K) plotted as afunction of σxx . The dashed, dash-dot, and solidlines denote σnotMxy / σβxx with β = 1.0, 1.6, and 2.0,respectively.https://doi.org/10.1038/s41535-024-00653-3 Articlenpj Quantum Materials |            (2024) 9:41 6Sr(OH)2 · 8H2O, and KClO4 mixed in a ratio of 8:1:3. The mixture wassealed in Pt capsule and was heated up to 840 ◦C under 2 GPa. It was keptthere for 10minand thenquenched toroomtemperature.The typical size ofa crystal is about 0.3 × 0.3 × 0.1mm with the shape of a hexagonal platenormal to the c-axis. Supplementary Fig. 1 shows the photograph of thesample. The unit cell of samples is determined by the single crystallineX-raydiffraction at room temperature (see Supplementary Table 1). Single crys-talline 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 thelatter, two series of samples (A and B) were prepared (see SupplementaryTable 1 and 2). Sr-, Ca- and Ba-concentration are determined by the energydispersive X-ray spectroscopy (EDX) analysis and scanning electronmicroscopy (SEM).Measurement of resistivity, Hall resistivity, and magnetizationMeasurements of resistivity and Hall resistivity were performed bystandard four-terminal geometry with an indium electrode. The mea-surements were done by using the Physical Property MeasurementSystem (Quantum Design) from 2 K to 300 K under the magnetic fieldup to 9 T. The magnetization measurements were performed by usingthe Dynacool System equipped with the VSM option from 2 K to 300 Kunder the magnetic field up to 9 T. The several samples with differentshapes show nearly identical magnetization profiles, suggesting that thedemagnetization factor is not significant. The magnetization mea-surement was performed for samples for which the transport propertywas measured.Monte Carlo simulationThe magnetic phase diagram and field dependence were calculated using aMonte Carlo simulation with the standard heat-bath update method. Thephysical quantities were calculated using 120,000 MC steps after20,000 steps of relaxation. The MC results were split into 6 bins for esti-mating the statistical error. All calculations were performed using an on-premise PC cluster at the Tokyo Institute of Technology.Data availabilityAll data needed to evaluate the conclusions in the paper are present in thepaper and/or the supplementary Information. Additional data requestsshould be addressed to the corresponding authors.Code availabilityThe codes used during the current study are available from the corre-sponding author upon reasonable request.Received: 18 December 2023; Accepted: 24 April 2024;References1. Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P.Anomalous Hall effect.Rev.Mod. Phys. 82, 1539 (2010).2. Schulz, T. et al. Emergent electrodynamics of skyrmions in a chiralmagnet. Nat. Phys. 8, 301 (2012).3. Feng, Z. et al. An anomalous Hall effect in altermagnetic rutheniumdioxide. Nat. Electron. 5, 735 (2022).4. Wang, M. et al. Emergent zero-field anomalous Hall effect in areconstructed rutile antiferromagnetic metal. Nat. Commun. 14,8240 (2023).5. 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N.A., Y.H., Y.T., and K.Y. grew the singlecrystalline SrCo6O11 and performed the characterizationwith the help of S.I.N.A., Y.H., and Y.K. conducted the transport/magnetization measurementand analyzed the data. H.I. andT.T. performed the numerical calculation andab initio calculation. J.F., S.I., and H.I. wrote the manuscript withcontributions from all authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41535-024-00653-3.Correspondence and requests for materials should be addressed toJun Fujioka.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in anymedium or format, as longas you give appropriate credit to the original author(s) and the source,provide a link to the Creative Commons licence, and indicate if changeswere made. 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To view a copy of thislicence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024https://doi.org/10.1038/s41535-024-00653-3 Articlenpj Quantum Materials |            (2024) 9:41 8https://doi.org/10.1038/s41535-024-00653-3http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/ Large anomalous Hall effect in spin fluctuating devil’s staircase Results Magnetic property Magneto-transport property Anomalous Hall effect in ferromagnetic Sr0.92Ba0.08Co6O11 Disordering dependence of anomalous Hall�effect Discussion Methods Sample preparation and characterization Measurement of resistivity, Hall resistivity, and magnetization Monte Carlo simulation Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information