# Fileset

[s41467-023-43543-1.pdf](https://mdr.nims.go.jp/filesets/397046c1-17b7-4bbf-adb1-6dc49fc2916e/download)

## Creator

[Hiroki Koizumi](https://orcid.org/0000-0002-2293-9428), [Yuichi Yamasaki](https://orcid.org/0000-0002-8560-3462), [Hideto Yanagihara](https://orcid.org/0000-0001-5500-017X)

## Rights

[Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/)

## Other metadata

[Quadrupole anomalous Hall effect in magnetically induced electron nematic state](https://mdr.nims.go.jp/datasets/022b08e2-5afd-4376-96e0-ccd91fe63854)

## Fulltext

Quadrupole anomalous Hall effect in magnetically induced electron nematic stateArticle https://doi.org/10.1038/s41467-023-43543-1Quadrupole anomalous Hall effect inmagnetically induced electron nematic stateHiroki Koizumi 1,2,3 , Yuichi Yamasaki 4 & Hideto Yanagihara 1,5Berry phases in both momentum and real space cause transverse motion initinerant electrons, manifesting various off-diagonal transport effect suchanomalous and topological Hall effects. Although these Hall effects are iso-tropic within the plane perpendicular to the fictitious magnetic field, here, wereport the manifestation of the anisotropic linear anomalous Hall effect (AHE)in the spinel oxide NiCo2O4 epitaxial film. The unconventional Hall effectindicates a quadrupole dependence on the in-plane current direction beingadded to the uniform AHE. Moreover, its sign can be manipulated just bymagnetic-field cooling. The anisotropic effect is attributed to an electronnematic state originating from a deformed electronic state owing to anextended magnetic toroidal quadrupole and ferrimagnetic order.Diverse physical properties in condensed matter systems are repre-sented by tensor objects connecting two or more measurablequantities1. According to Neumann’s principle, the physical propertytensor must be invariant under the system’s symmetry operation2,3. Inmagnets, the magnetic point group, particularly, the tensor transfor-mation under the time reversal operations (T ), should be considered.As for the conductivity of magnetically ordered materials, an electricfield E induced by anapplied electric currentwith density J is written asEj =ρijðH,ΣÞ Ji. The electric resistivity tensor ρij explicitly expressesexternal magnetic field H and spin configuration Σ4–6. Concerning theT operation on the resistivity tensor, Onsager’s theorem7 gives areciprocal relation ρijðH,ΣÞ= T ρjiðH,ΣÞ=ρjið�H,� ΣÞ. Hall resistivity,an antisymmetric component of transverse resistivity (TR) withrespect to H and M, can lead to8ρxyðHz ,MzÞ= � ρyxðHz ,Mz Þ, ð1Þindicating that J k xðyÞ induces E k yð�xÞ in the presence of Hz andMz.Hence, theHall voltage is isotropic and independent of the direction ofJ; i. e. rotational symmetry is preserved within the normal plane to Hand M, as shown in Fig. 1a9.In contrast, since a symmetric TR ρij, an even-function componentof resistivity toH andM, does not always satisfy Eq. (1), it occasionallyexhibits anisotropic behaviour depending on the direction of J5. Anexample is the anisotropic electronic state such as an electronicnematic state10,11. The symmetric TRwill appearwhen J is applied alongthe off-principal crystallographic axes. Thus, the response is useful foraccurately detecting electronic state deformations12. Another intri-guing property is the spin Hall effect in an anisotropic spin splitting.For instance, the extendedmagnetic toroidal quadrupole (MTQ) orderwith C4T symmetry, i. e. a combination operation of 90∘ rotation (C4)and T , causes a quadrupole (d-wave) shape spin splitting13. The spin-by-spin anisotropic electronic state will produce an anisotropic spinresistivity tensor within the xy-plane, ρzijðΣÞ=ρzjiðΣÞ, with appliedT -even spin current Jzs , as shown in Fig. 1c14–16. However, MTQ orderforbids the charge current of AHE by its symmetry.From Onsager’s theorem, the Hall effect which is an antisym-metric TR for Hz and Mz is isotropic with respect to the currentdirection, while a symmetric TR can be anisotropic5. Indeed, a planarHall effect where TRs are anisotropic depending on the angle betweenJ andM shows a symmetric response to M. However, here, we show amanifestation of an unconventional anisotropic Hall effect even forcharge current in a conical ferrimagnet composed ofMTQ, as shown inFig. 1c. At first glance, the behaviour violates Onsager’s theorem butcan be elucidated without contradiction by assuming symmetric MTQupon Mz reversal.Received: 7 May 2023Accepted: 13 November 2023Check for updates1Department of Applied Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan. 2Research Center for Magnetic and Spintronic Materials (CMSM),National Institute forMaterials Science (NIMS), Tsukuba, Ibaraki 305-0047, Japan. 3Center for Science and Innovation inSpintronics (CSIS), TohokuUniversity,Sendai 980-8577, Japan. 4Research and Services Division of Materials Data and Integrated System (MaDIS), National Institute for Materials Science (NIMS),Tsukuba, Ibaraki 305-0047, Japan. 5Tsukuba ResearchCenter for EnergyMaterials Science (TREMS), University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan.e-mail: hiroki.koizumi.d7@tohoku.ac.jp; YAMASAKI.Yuichi@nims.go.jp; yanagihara.hideto.fm@u.tsukuba.ac.jpNature Communications |         (2023) 14:8074 11234567890():,;1234567890():,;http://orcid.org/0000-0002-2293-9428http://orcid.org/0000-0002-2293-9428http://orcid.org/0000-0002-2293-9428http://orcid.org/0000-0002-2293-9428http://orcid.org/0000-0002-2293-9428http://orcid.org/0000-0002-8560-3462http://orcid.org/0000-0002-8560-3462http://orcid.org/0000-0002-8560-3462http://orcid.org/0000-0002-8560-3462http://orcid.org/0000-0002-8560-3462http://orcid.org/0000-0001-5500-017Xhttp://orcid.org/0000-0001-5500-017Xhttp://orcid.org/0000-0001-5500-017Xhttp://orcid.org/0000-0001-5500-017Xhttp://orcid.org/0000-0001-5500-017Xhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43543-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43543-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43543-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43543-1&domain=pdfmailto:hiroki.koizumi.d7@tohoku.ac.jpmailto:YAMASAKI.Yuichi@nims.go.jpmailto:yanagihara.hideto.fm@u.tsukuba.ac.jpThe anisotropic Hall effects were demonstrated on epitaxialNiCo2O4(001) films on MgAl2O4(001) substrates. NiCo2O4 is a con-ductive inverse spinel oxide (Fd�3m) exhibiting ferrimagnetism with aNéel temperature as high as TN ≈ 400 K17. Stoichiometric thin filmsexhibit perpendicular magnetic anisotropy at room temperature18–20.Moreover, sharp out-of-plane magnetisation switchings are observedin the magnetic field dependence of AHE over the entire temperaturerange18,21. In contrast, the anti-site Ni3+ distribution, which can bemanipulated by the O2 flow rate during thin film deposition22, changesthemagnetic anisotropy fromperpendicular to an easy-conemagneticanisotropy at low-temperatures20. In the present study, we found thatsuch anti-site NiCo2O4 thin film realises a conical ferrimagnetic struc-ture composed of the out-of-plane ferromagnetic and in-plane MTQcomponents.ResultsDependenceof anomalousHall effect on thedirectionof currentapplicationWe first show experimental evidence for the anisotropic Hall effectmeasured at 5 K. Prior to the low-temperature measurements, thesample was cooled under a low positive or negative magnetic field(μ0HFC = ± 0.1 T), apparently perpendicular to the film plane fromroom temperature. Figure 2a–h show the applied magnetic field H k½001� dependence of Hall effects with changing J direction. TheHall resistivities ρOij (i , j = x : ½100� , y : ½010�, x0 : ½110�, y0 : ½�110�) wereextracted by antisymmetrisation analysis (raw data and symmetric TRρEij are shown in Supplementary Figs. 1 and 2, respectively). The Halleffects measured after a field cooling (FC) procedure with the positiveand negative HFC are indicated as ρO+ij and ρO�ij , shown in Fig. 2a–h,respectively. In the high-H region ðjμ0Hj > 1 TÞ, all Hall resistivitycurves behave similarly with a coincident saturation of 5.25μΩcm.However, they show peculiar behaviour different from the conven-tional AHE in the low-H region. These results are reminiscent of thetopological Hall effect23,24; however, they exhibit anisotropic beha-viours that are dependent on the direction J; ρO±x0y0 and�ρO±y0x0 are similarto the conventional AHE proportional to Mz [see SupplementaryFig. 8a]. Whereas there are distinct additional contributions in ρO±xyand �ρO±yx .The Hall effects can be decomposed into isotropic and aniso-tropic components in the applied J direction. Figure 2i, j show theanisotropic ρOA+xy and isotropic ρOI +xy components derived, respec-tively, which are half of the difference and average between ρO+xyand �ρO+yx . The observed largest value of ρOA+xy is 6.9μΩcm, which is~1.3 times that of the saturation value of conventional AHE, as shown inFig. 2i, j, respectively. In contrast, the isotropic ρOI+xy seems to coincidewith ρO±x0y0 . Hence, the anisotropic Hall effect has cos 2φ response forthe current angle φ by the [100] axis. When HFC is inverted, the ani-sotropic ρOA�xy is completely sign-reversed within the experimentalerror, as shown in Fig. 2k. In the case of zero-field cooling (ZFC), theanisotropic Hall effect cancels, and the curve is identical to ρOI ±xy ,indicating that the cooling process does not affect the Mz-H curve, asshown in Fig. 2j. To generalise the above results, the series of Hallresistivity can be expressed asρO± ðφÞ=ρOI ±ρOA cos 2φ ð2Þwith ρOI and ρOA being the isotropic and anisotropic components ofHall effect, respectively. The plus/minus sign indicates the sign ofcooling magnetic field [see also Supplementary Fig. 3]. At first glance,such an anisotropic result appears to involve a planar Hall effectcaused by assuming the presence of in-planemagnetisation. However,it cannot explain the antisymmetric behaviour with respect to Hz andHFC. The possibility of AHE from an extrinsic origin, such as theorientated magnetic domains25 or phase separation26, is also excludedin the present sample for the same reason. Hereafter, the anisotropicHall effect will be referred to as a quadrupole AHE (QuadAHE) todistinguish it from the conventional AHE and the planar Hall effect.Temperature dependence of the anisotropic anomalousHall effectSince the QuadAHE responds at low-H and is reversed by HFC, it isassumed to originate from a nontrivial magnetic structure. We discusspossible magnetic structure coupling with the anisotropic electronicstate in the tetragonally distorted spinel structure NiCo2O4 based ontemperature dependence [Fig. 3a,b] and symmetry of resistivities. Thespinel structure consists of a diamond lattice at the A-site and a pyr-ochlore lattice at the B-site (see Supplementary Fig. 7e). Because of theantiferromagnetic interaction between A- and B-sites, JAB, the Néel-type collinear ferrimagnetic order is realised with the perpendicularmagnetic anisotropy at room temperature18,19,21. With decreasing tem-perature, there is a transition from the Néel-type to a non-collinearmagnetic structurewith a canted spin at the A-site owing to the changein spin anisotropy from the perpendicular to the easy-cone magneticanisotropy20. Such transformation is identified in the conventional AHEshapes owing to the suppressed M around zero magnetic fields, asshown in Fig. 3a. At high temperatures, M saturates at low fields, andFig. 1 | Correspondence among magnetotransport phenomena, centrosym-metric extendedmagneticmultipolesonpyrochlore lattice, andorbital shapesfor magnetic (red and blue) and toroidal (green and magenta) charges. Iso-tropic anomalous Hall effect occurs on (a) magnetic dipole order (ferromagnet).b Extended magnetic toroidal quadrupole order allows anisotropic spin Hall effectbut no electric charge Hall effect. c Conical magnetic structure comprising themagnetic toroidal quadrupole and magnetic dipole shows anisotropic electriccharge Hall effect.Article https://doi.org/10.1038/s41467-023-43543-1Nature Communications |         (2023) 14:8074 2the remanence is almost identical to the saturation but decreases atlower temperatures. Fig. 3c shows the temperature dependence of thesquareness ratio defined as the ratio of AHE remanence to saturation. Itis 1 in the higher temperature collinear phase and declines at tem-peratures below the magnetic transition temperature TS ≈ 130 K.Concurrent with the magnetic transformation, the symmetric TRappears below TS, as shown in Fig. 3d [see also Supplementary Fig. 4].The symmetric and anisotropic TR, ρExyðHÞ=ρEyxðHÞ, as seen in Sup-plementary Fig. 2, suggests that the electronic state is anisotropicwithin the (001) plane. However, it is unaffected by the reversal of HFCand ascribed to different origins from the QuadAHE. This is presumedto be an electronic state change originating from a coplanar magneticstructure, the so-called Yafet-Kittel-type two-dimensional magneticstructurewith cantedmagneticmomenton theA-site27. The symmetricand anisotropic TR ρExy can be regarded as the planar Hall effect ori-ginating from the in-plane antiferromagnetic component of thecoplanar Yafet-Kittel-type magnetic structure. In contrast, the Qua-dAHE emerges below TQ ≈ 80 K, as shown in Fig. 3b,d, showing varia-tions of ρOAxy and temperature dependence of remanence. There is noapparent anomaly in the squareness ratio and symmetric TR at TQ,hence the QuadAHE is attributed to a furthermagnetic transformationfrom the Yafet-Kittel-typemagnetic structures on the B-site atTQ. Sucha magnetic structure change from collinear to multistep non-collinearhas not been observed in stoichiometric samples18,21,28. The resultssuggest that the easy-cone magnetic anisotropy due to the anti-siteNi3+ distribution in the present sample is essential for structuralchanges.DiscussionMagnetic structure based on the symmetry of anomalousHall effectAt high temperatures and high-H collinear ferrimagnetic states, theB-site magnetic structure belongs to the magnetic point group of4=mm0m0, indicating an isotropic electronic state within the film plane(Supplementary Table I), consistent with the conventional AHE. Incontrast, the QuadAHE, in Eq. (2), in the low-H region suggests amagnetic structure with C4T symmetry as in the in-plane anti-ferromagnetic component. Such a structure is realised in a pyrochlorelattice composed of the spinel B-site. There are 12 orthonormal irre-ducible magnetic structures in the pyrochlore lattice characterised byextended multiples of spatial inversion symmetry, namely threemagnetic dipole, four magnetic octupole and five MTQmoments (seeRefs. 29,30). The T1gmagnetic octupole with 4=mm0m0 magnetic pointgroup would induce AHE, whereas the A2g octupole, known as the all-in-all-out structure, will not induce AHE due to its m�3m0 symmetry.However, bi-axial epitaxial strain in thin films can make the magneticoctupole component nonzero in the all-in-all-out structure, thusinducing AHE31. In the present epitaxial in-plane tensile strain, there arefour possible in-plane antiferromagnetic bases, MTQ Tv ð4=mmmÞ,MTQ ~Tuð40=mmm0Þ, MTQ Txyð40=mm0mÞ, and magnetic octupole-2 -1 0 1 2-12-10-8-6-4-2024681012(O�xy�(�O�yx))(��cm)0H (T)0HFC=�0.1 T1 2(e)�(g)� OAyx2-2 -1 0 1 2-12-10-8-6-4-2024681012(d)�(b)0H (T)(O+xy�(�O+yx))(��cm)0HFC=�0.1 T1 2 OAxy2-2 -1 0 1 2-12-10-8-6-4-20246810121 2ZFC0H (T)(O+xy�(�O+yx))(��cm)(d)�(b)OIxy2ijk-2 -1 0 1 2-12-10-8-6-4-20246810120H (T)O ij(��cm)0HFC=�0.1 TJ//[010]� O�yx-2 -1 0 1 2-12-10-8-6-4-20246810120H (T)O ij(��cm)0HFC=�0.1 TJ//[010]� O+yx-2 -1 0 1 2-12-10-8-6-4-20246810120H (T)O ij(��cm)0HFC=�0.1 TJ//[110]� O�y'x'-2 -1 0 1 2-12-10-8-6-4-20246810120H (T)O ij(��cm)0HFC=�0.1 TJ//[110]O�x'y'-2 -1 0 1 2-12-10-8-6-4-2024681012� �9T� �9T0H (T)O ij(��cm)0HFC=�0.1 TJ//[110]O+x'y'-2 -1 0 1 2-12-10-8-6-4-20246810120H (T)O ij(��cm)0HFC=�0.1 TJ//[110]� O+y'x'-2 -1 0 1 2-12-10-8-6-4-20246810120H (T)O ij(��cm)0HFC=�0.1 TJ//[100]O�xy-2 -1 0 1 2-12-10-8-6-4-20246810120H (T)O ij(��cm)0HFC=�0.1 TJ//[100]O+xya b cd ef g hFig. 2 | Magnetic field dependence of Hall effect ρOij , antisymmetric componentof transverse resistivity, with changing applied current J direction at 5 K.a–d, h, g and f, e indicate Hall resistivities for J jj y0 : ½�110�, y : ½010� x0 : ½110�, andx : ½100�, respectively, measured after positive (negative) field cooling (HFC). Thecentral figure shows thedirections of the applied current and inducedelectric field;blue and red arrows represent the conventional and unconventional TRs observedin the low-field region, respectively. The measurements setup, e.g. Hall bar shapeand current direction, is illustrated in the inset schematic of each figure. i Extractedanisotropic and (j) isotropic components of ρOij for μ0HFC = + 0.1 T. k Anisotropiccomponent for μ0HFC = −0.1 T.Article https://doi.org/10.1038/s41467-023-43543-1Nature Communications |         (2023) 14:8074 3Mαz ð4=mm0m0Þ, as shown in Supplementary Fig. 5a. Here, MTQ ~Tucorresponds to a linear combination ofMTQTu andmagneticoctupoleMxyz, as shown in Fig. 3f. Among these four components, 40=mm0m or40=mmm0 possibly elucidate QuadAHE because of C4T symmetry.Considering the existence of the (110) mirror symmetry with T inQuadAHE, MTQ ~Tu (40=mmm0) reasonably accounts for the experi-mental results.T -odd magnetic orders with spatial inversion cause symmetricband dispersions with spin splitting13. The band dispersion of the fer-romagnet, i.e. the magnetic dipole order, shows a uniform spinpolarised structure, as shown in Fig. 3e. The T -odd MTQ ~Tu moment40=mmm0 demonstrates the quadrupole-type (d-wave like) spin split-ting; the up and down spin bands are elliptical and rotate at 90∘ fromeach other, as shown in Fig. 3f. In the pure MTQ antiferromagneticorder, the spin Hall effect is allowed16, but the electric charge Halleffect is prohibited owing to C4T symmetry. In contrast, in a conicalmagnetic structure consisting of magnetic dipole and MTQ, C4 sym-metry is broken. The charge and spinHall effects appearbecauseof theanisotropic Fermi surface, namely the liquid crystal-like electronnematic state, originating from the differences in the spin density, asshown in Fig. 3g.Magnetotransport model for QuadAHE. To approach the magneto-transport phenomena, we consider a minimal model Hamiltonian inthe MTQ ~Tu conical magnetic order written asH=Xkσσ 0_22mk2σ0 + 2~tukxkyσz� �+mzσz" #cykσckσ0 ð3Þwhere cykσ (ckσ) is the creation (annihilation) operator of an electronwith wave vector k and spin σi ði=0; x; y; zÞ13; mz and ~tu denote themolecular fields from the magnetic dipole Mz and MTQ ~Tu orders,respectively. Here, we only consider the kz =0 plane for simplicity.kxkyσz corresponds to the quadrupole-type spin splitting due to theC4T symmetry. Then, Boltzmann’s transport equation gives the chargeconductivity asσkσ?� �= σD�nð1 +~t2uÞ+2Δn~tu sin 2φ2Δn~tu cos 2φ !, ð4Þwith the average (�n) and difference (Δn) of electron number for up anddown spins. Assuming jσ?j≪jσkj, the anisotropic resistivities areobtained asρ=ρkρ?� �= � 2ησ2Dsin 2φcos 2φ� �, ð5Þwhere η =~tuΔn and σD is isotropic longitudinal conductivity. Thecos2φ response reproduces the angle dependence of the observedQuadAHE. SinceΔn and ~tu are T -odd, ρ⊥ is proportional to the T -evenquantity η. If ~tu is not reversed withMz reversal, namely preserving thein-plane antiferromagnetic structurewhile reversing theperpendicularferromagnetic component Mz, QuadAHE exhibits an antisymmetricresponse with respect to Mz. Figure 4a shows schematics of theelectronic bands and magnetic structure modified by the magneticfield change. At high magnetic fields, the band dispersion is isotropicowing to the collinear ferrimagnetic structure; in contrast, under weakmagnetic fields, theMTQ ~Tu conical magnetic structure contributes toFig. 3 | Temperature dependence of anomalousHall effect (AHE) and electronicstates modulated by magnetic order. Magnetic field dependence of (a) normal-ised AHE after ZFC and (b) extracted anisotropic components for J k ½100� mea-sured at selected temperatures. The inset of (a) indicates the ratio of saturation andremanence of AHE, defined as a squareness ratio. Temperature dependence ofc squareness ratio and d symmetric TR and anisotropic Hall effect, ρE and ρOA,respectively. The insets of (c) indicate the assumed magnetic structure for the A-and B-sites, drawn by blue and green spheres, respectively. Magnetic structure onpyrochlore lattice and band dispersion for (e) magnetic dipole Mz [4=mm0m0], (f)magnetic toroidal quadrupole ~Tu [40=mmm0], and (g) conical magnetMz + ~Tu [m0m0m].Article https://doi.org/10.1038/s41467-023-43543-1Nature Communications |         (2023) 14:8074 4the anisotropic band structure. Assuming symmetric MTQ ~Tu withrespect to Mz, only the band sizes of the up and down spins change,contributing to QuadAHE reversal. This model also suggests that thetransverse magnetoresistance (MR) effect, i.e. change of longitudinalresistance by perpendicularH, exhibits an antisymmetric MR effect toMz32–34 and an anisotropic response to applied electric current.Figure 4b indicates themagnetic field dependence of antisymmetrisedMR along the y0 : ½�110� direction (φ = 135°). The antisymmetricresistivity, ρO+y0y0 ð�Hz ,�Mz Þ≈� ρO+y0y0 ðHz ,MzÞ, can be recognised in thelow-H region, and its sign is reversed by the reversal of HFC,i. e. ρO�y0y0 ðHz ,MzÞ≈� ρO+y0y0 ðHz ,Mz Þ and QuadAHE.Magnetic domain control of extended MTQ moment by magnetic-field cooling. The QuadAHE is fully sign-reversed by the inversion ofHFC [Fig. 2i,k], indicating the absence of complexities, such as multipleMTQdomains. During the isothermal reversal ofMz, theMTQ ~Tu single-domain feature, realised using the magnetic-field cooling procedure,was nearly maintained. It suggests thatMTQ andmagnetic dipole formindependent domain walls without coupling. These behaviours are incontrast with conical magnet multiferroics, wherein electric and mag-neticfields are required to realise the single domain and an inversion ofspin helicity occurs with the reversal of Mz35,36. Figure 4c depicts fourpossible MTQ ~Tu conical magnetic structures, accessed using HFC andisothermal reversal of Hz with respect to the sign of ~tu, mz, and η. Thedirectional alignment of electron nematic state can be manipulated byMz reversal due to the symmetric property of MTQ.MTQselection bymagnetic field cooling canbe interpreted by theDzyaloshinskii-Moriya (DM) interactions37,38. In the pyrochlore struc-ture, a nonzero DM vector exists on bonds between the B-site ions39,40.For every bond, the magnitude and sign of DM interaction energy inthe MTQ conical magnetic order are indicated by its radii and colours,respectively. The four MTQ conical states energetically degenerate ifthe conical axis is parallel to the z-axis. However, for non-equivalentmagnitudes of DM vectors owing to the epitaxial strain, thedegeneracy is liftedwhen the conical axis is tilted toward the 〈110〉 axes(details are provided in Supplementary Note 8). The most stable MTQconical state is uniquely determined by the sign of Mz and the tiltingdirection (½110� or ½1�10�). Namely, the single MTQ domain is realised byslightly tilted HFC applied on the magnetic structure transition duringcooling, and its sign is determinedby the tilteddirection and signofHz.In the actual experiment, we apply HFC in the z direction; however, thein-plane magnetic field component may be attributed to the mis-alignment of partially unadjusted equipment. This conjecture is sup-ported by the fact that the selected MTQ sign reverses when rotatingthe sample by 90∘ in the experimental arrangement corresponding tothe rotation of the in-planeHFC component between the ½110� and [1�10]axis [Supplementary Fig. 6].The easy-cone magnetic anisotropy and DM interaction manifestthe magnetic structural change from ferromagnetic to MTQ conicalmagnetic structure on pyrochlore lattice in the anti-site NiCo2O4 thinfilm. Since the energy scales of magnetic anisotropy and DM interac-tions are weaker than the antiferromagnetic superexchangeinteractions41, QuadAHE is suppressed at lower magnetic fields com-pared to the AHE in frustrated magnetic pyrochlore materials8,42,43.However, the experimental results confirm the existence of the sameQuadAHE curve even after returning from a higher magnetic field of 9T to a lower magnetic field (<0:3T). Upon applying a magnetic field,the MTQ conical magnetic structure gradually changes with decreas-ing cone angle; however, it could not becomecompletely collinear dueto the influence of the remaining easy-cone magnetic anisotropy,preserving stable MTQ sign information.Consideration by Onsager’s reciprocal theoremFinally, we reconsider the Onsager reciprocal theorem in the Qua-dAHE. Within the same sign of HFC, the observed Hall resistivity ρijsatisfies theOnsager reciprocal relation of Eq. (1)when ði, jÞ= ðx0, y0Þ butnot when ði , jÞ= ðx,yÞ. However, considering the T -oddMTQ ~Tu whosesign is determined by that of HFC, the Onsager reciprocal theorem canFig. 4 | Schematics of extented magnetic toroidal quadrupole conical orderdependent onmagnetic field. a Assumedmagnetic field dependence ofmagneticstructure, electronic band structure, and corresponding anisotropic Hall effect.b Magnetic field dependence of antisymmetric longitudinal magnetoresistancewith J k ½�110� and H k ½001�. c Relations of the four degenerated conical magneticstructures and magnetic fields. The signs of mz and ~tu are determined by the zcomponent and the direction of the in-plane component of the magnetic field HFCapplied upon the magnetic transition from the B-site collinear to the MTQ conicalmagnetic structure. After realising the single domain MTQ conical magneticstructure, the isothermal magnetic field reversal inverts Mz and not MTQ ~Tu.Article https://doi.org/10.1038/s41467-023-43543-1Nature Communications |         (2023) 14:8074 5be satisfied by extending ρijðH,mz ,~tuÞ= ρjið�H,�mz ,� ~tuÞ. In otherwords, the Onsager reciprocal theorem is certainly satisfied for the Toperation which reverses all the spin directions. When the Hall resis-tivity is T -odd, it should be isotropic such as that in the conventionalAHE. From contrapositive reasoning, the manifestation of anisotropicQuadAHE requires a term proportional to the T -even quantity. Thiswould correspond to the interference term η, as in Eq. (5). Therefore,from the point of Onsager reciprocal theorem, the existence of sym-metric MTQ to Mz reversal is mandatory for antisymmetric QuadAHE.For the longitudinalMR, theOnsager reciprocal theorem indicates thatρijðH,ΣÞ= ρijð�H, � ΣÞ, suggesting that the antisymmetric MR is for-bidden. However, the existence of the symmetric MTQ makes anti-symmetric MR admissible.In summary, we unveiled the unconventional AHE and MR, that isantisymmetric with respect toMz and anisotropic Hall effect dependingon the applied current direction. The results are explained by the elec-tron nematic state induced by the quadrupolar spin-split band structureand energy shift due to MTQ and magnetic dipole order, respectively.The MTQ conical can be manipulated by isothermal field inversion andmagnetic-field cooling. Though the anisotropic Hall effect and anti-symmetric magnetoresistivity seem to violate the Onsager reciprocaltheorem, it can be understood considering a resistivity proportional toT -even interference terms between the MTQ and magnetic dipole, i.e.η=~tuΔn, which is symmetric with T operation but antisymmetric withrespect toMz. Thedegeneracyof the conicalMTQstructure canbe liftedby the tilted magnetic field applied on cooling due to the difference inthe DM interaction by the thin film epitaxial compressive strain.The results are expected to pave the way for new emergentproperties with potential applications to spintronic devices such as amultivalued memory device using the electron nematic state and canopen a new research field of the Hall effect originating from extendedmagnetic multipoles. Although the present study infers the realisationof electron nematic state44–46 from QuadAHE, for example, an angle-resolved photo-emission spectroscopy measurement will providemore direct evidence of distorted Fermi surface and clarify a differentorigin than the nematic state observed in Sr3Ru2O7 and othersystems47,48. The MTQ conical magnetic structure of NiCo2O4 is notdirectly determined in the current study. Since it is a thin-film sample,it is difficult to analyse the magnetic structure by neutron scatteringwhich requires the volume of the sample. In addition, since the mag-netic anisotropy depends on the anti-site Ni3+ distribution, the specificgrowth conditions under which the MTQ conical magnetic structurerealises have not yet been found. Additional research is required togather more evidence that will further support the present model.MethodsSample preparation and measurementsEpitaxial NiCo2O4 films with 50 nm thickness were grown on MgAl2O4substrate by the reactive RF magnetron sputtering technique (ES-250MB: Eiko Engineering Co. Ltd.). We used a 2-inch alloy target with anominal composition of Ni:Co = 1:2. The growth conditions of NiCo2O4films were Ar and O2 flow rates of 10 and 5.0 sccm, respectively, aprocess temperature of 300 °C, and a working pressure of 1.5 Pa.Finally, we cooled the NiCo2O4 film to room temperature under anoxygen pressure of 0.8 Pa.For investigating electric properties, the film was patterned intoHall bars by photolithography and Ar ion milling. Next, Cr and Auelectrode layers were sputtered for longitudinal resistivity (LR) and TRmeasurements. The shapes of Hall bars are shown in each figure (Fig. 2and Supplementary Fig. 3) and have sizes of 20μm×300μm and200μm× 1400μm. The different devices/electrodes are isolated bythe Ar ion milling technique. Since the substrate MgAl2O4 is an insu-lator, each hole bar is electrically separated by removing the NiCo2O4film other than the hole bar by themilling. The electric properties weremeasured with a physical property measurement system using a DCcurrent source (Keithley 6221) and nanovoltmeter (Keithley 2182). Formeasuring the anisotropy of TR, electric currents are appliedalong J k ½100�, k ½110�, k ½010�, and k ½�110�.Data availabilityThe data that support the findings of this study are available from thecorresponding author upon request.References1. Nye, J. &Nye, P. Physical Properties of Crystals: Their Representationby Tensors and Matrices (Oxford Science Publications ClarendonPress, 1985).2. Birss, R. R. Symmetry and Magnetism (Elsevier Science & Technol-ogy, 1966).3. Newnham, R. Properties of Materials: Anisotropy, Symmetry, Struc-ture (OUP Oxford, 2004).4. Grimmer, H. General relations for transport properties in magneti-cally ordered crystals. Acta. Crystallogr. Sect. A 49, 763–771 (1993).5. Grimmer, H. Thermoelectric transport properties in magneticallyordered crystals. Acta. Crystallogr. Sect. A 73, 333–345 (2017).6. Gallego, S. V., Etxebarria, J., Elcoro, L., Tasci, E. S. & Perez-Mato, J.M. Automatic calculation of symmetry-adapted tensors inmagneticand non-magnetic materials: a new tool of the Bilbao crystal-lographic server. Acta. Crystallogr. Sect. A 75, 438–447 (2019).7. Onsager, L. Reciprocal relations in irreversible processes. i. Phys.Rev. 37, 405–426 (1931).8. Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P.Anomalous hall effect. Rev. Mod. Phys. 82, 1539–1592 (2010).9. Yanagihara, H. & Salamon, M. B. Skyrmion strings and the anom-alous Hall Effect in CrO2. Phys. Rev. Lett. 89, 187201 (2002).10. Grigera, S. A. et al. Disorder-sensitive phase formation linked tometamagnetic quantum criticality. Science 306, 1154–1157 (2004).11. Yamase, H., Oganesyan, V. & Metzner, W. Mean-field theory forsymmetry-breaking fermi surface deformations on a square lattice.Phys. Rev. B 72, 035114 (2005).12. Wu, J., Bollinger, A. T., He, X. & Božović, I. Spontaneous breaking ofrotational symmetry in copper oxide superconductors. Nature 547,432–435 (2017).13. Hayami, S., Yatsushiro, M., Yanagi, Y. & Kusunose, H. Classificationof atomic-scalemultipoles under crystallographic point groups andapplication to linear response tensors. Phys. Rev. B 98,165110 (2018).14. Seemann, M., Ködderitzsch, D., Wimmer, S. & Ebert, H. Symmetry-imposed shape of linear response tensors. Phys. Rev. B 92,155138 (2015).15. Naka, M. et al. Spin current generation in organic antiferromagnets.Nat. Commun. 10, 4305 (2019).16. González-Hernández, R. et al. Efficient electrical spin splitter basedonnonrelativistic collinear antiferromagnetism. Phys. Rev. Lett. 126,127701 (2021).17. Battle, P., Cheetham, A. & Goodenough, J. A neutron diffractionstudy of the ferrimagnetic spinel NiCo2O4.Mater. Res. Bull. 14, 1013– 1024 (1979).18. Chen, X. et al. Magnetotransport anomaly in room-temperatureferrimagnetic NiCo2O4 thin films. Adv. Mater. 31, 1805260 (2019).19. Kan, D. et al. Spin and orbitalmagneticmoments in perpendicularlymagnetized Ni1−xCo2+yO4−z epitaxial thin films: Effects of site-dependent cation valence states. Phys. Rev. B 101, 224434 (2020).20. Koizumi, H. et al. Spin reorientation in tetragonally distorted spineloxide NiCo2O4 epitaxial films. Phys. Rev. B 104, 014422 (2021).21. Chen, X. et al. Anomalous Hall effect and perpendicular magneticanisotropy in ultrathin ferrimagnetic NiCo2O4 films.Appl. Phys. Lett.120, 242401 (2022).22. Shen, Y., Kan, D., Tan, Z.,Wakabayashi, Y. & Shimakawa, Y. Tuningofferrimagnetism and perpendicular magnetic anisotropy in NiCo2O4Article https://doi.org/10.1038/s41467-023-43543-1Nature Communications |         (2023) 14:8074 6epitaxial films by the cation distribution. Phys. Rev. B 101,094412 (2020).23. Nagaosa, N. & Tokura, Y. Topological properties and dynamics ofmagnetic skyrmions. Nat. Nanotechnol. 8, 899–911 (2013).24. Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronicproperties. Rev. Mod. Phys. 82, 1959–2007 (2010).25. Dho, J. & Kim, J. Magnetic domain structure of the ferrimagnetic(001) NiCo2O4 film with perpendicular magnetic anisotropy.Thin Solid Films 756, 139361 (2022).26. Xue, M. et al. Transport anomaly in perpendicular magnetic aniso-tropic NiCo2O4 thin films with column-like phase separation. ACSAppl. Electron. Mater. 2, 3964–3970 (2020).27. Yafet, Y. & Kittel, C. Antiferromagnetic arrangements in ferrites.Phys. Rev. 87, 290–294 (1952).28. Kan, D., Xie, L. & Shimakawa, Y. Scaling of the anomalousHall effectin perpendicularly magnetized epitaxial films of the ferrimagnetNiCo2O4. Phys. Rev. B 104, 134407 (2021).29. Wills, A. S. et al. Magnetic ordering in Gd2Sn2O7: the archetypalHeisenberg pyrochlore antiferromagnet. J. Phys. Condens. Matter18, L37–L42 (2006).30. Suzuki, M.-T. et al. Multipole expansion for magnetic structures: ageneration scheme for a symmetry-adapted orthonormal basis setin the crystallographic point group. Phys. Rev. B 99, 174407 (2019).31. Kim, W. J. et al. Strain engineering of the magnetic multipolemoments and anomalous Hall effect in pyrochlore iridate thin films.Sci. Adv. 6, 254–288 (2020).32. Fujita, T. C. et al. Odd-parity magnetoresistance in pyrochlore iri-date thin films with broken time-reversal symmetry. Sci. Rep. 5,9711 (2015).33. Hirose, H. T., Yamaura, J.-I. & Hiroi, Z. Robust ferromagnetism car-ried by antiferromagnetic domain walls. Sci. Rep. 7, 42440 (2017).34. Wang, Y. et al. Antisymmetric linear magnetoresistance and theplanar hall effect. Nat. Commun. 11, 216 (2020).35. Yamasaki, Y. et al. Magnetic reversal of the ferroelectric polarizationin a multiferroic spinel oxide. Phys. Rev. Lett. 96, 207204 (2006).36. Choi, Y. J. et al. Thermally or magnetically induced polarizationreversal in the multiferroic CoCr2O4. Phys. Rev. Lett. 102,067601 (2009).37. Dzyaloshinsky, I. A thermodynamic theory of “weak" ferromagnet-ism of antiferromagnetics. J. Phys. Chem. Solids 4, 241–255 (1958).38. Moriya, T. Anisotropic superexchange interaction and weak ferro-magnetism. Phys. Rev. 120, 91 (1960).39. Onose, Y. et al. Observation of themagnon hall effect.Science329,297–299 (2010).40. Xiang, H. J. et al. Single-ion anisotropy, Dzyaloshinskii-Moriyainteraction, and negative magnetoresistance of the spin-12 pyro-chlore R2V2O7. Phys. Rev. B 83, 174402 (2011).41. Moriya, T. New mechanism of anisotropic superexchange interac-tion. Phys. Rev. Lett. 4, 228–230 (1960).42. Taguchi, Y., Oohara, Y., Yoshizawa, H., Nagaosa, N. & Tokura, Y. Spinchirality, berry phase, and anomalous hall effect in a frustratedferromagnet. Science 291, 2573–2576 (2001).43. Machida, Y. et al. Unconventional anomalous hall effect enhancedby a noncoplanar spin texture in the frustrated kondo latticePr2Ir2O7. Phys. Rev. Lett. 98, 057203 (2007).44. Fernandes, R. M., Abrahams, E. & Schmalian, J. Anisotropic in-planeresistivity in the nematic phase of the iron pnictides. Phys. Rev. Lett.107, 217002 (2011).45. Wårdh, J. et al. Colossal transverse magnetoresistance due tonematic superconducting phase fluctuations in a copper oxide.PNAS Nexus 2, pgad255 (2023).46. Wu, J., Bollinger, A. T., He, X. & Božović, I. Detecting electronicnematicity by the angle-resolved transverse resistivity measure-ments. J. Supercond. Nov. Magn. 32, 1623–1628 (2019).47. Borzi, R. A. et al. Formation of a nematic fluid at high fields inSr3Ru2O7. Science 315, 214–217 (2007).48. Marshall, P. B., Ahadi, K., Kim, H. & Stemmer, S. Electron nematicfluid in a strained Sr3Ru2O7 film. Phys. Rev. B 97, 155160 (2018).AcknowledgementsThe authors thank T. Arima and H. Nakao for the productive discussion.Microfabricationof thisworkwas carriedout under theMEXTprogramofAdvanced Research Infrastructure for Materials and Nanotechnology inJapan (JPMXP1222BA0017) and the cooperation of T. Kashiwagi andNanotechnology Platform at the University of Tsukuba. This work wasperformed under the approval of the “Photon Factory Program AdvisoryCommittee" (proposals No.2017G602 and No. 2016S2-005). H.K. thanksY. Hatsugai, S. Kuroda and S. Mitani for their useful comments. Thisproject is partly supported by the Japan Society for the Promotion ofScience (JSPS) KAKENHI (23H01842: H.Y., 22H04966: H.Y.,21H01750:H.Y. and 19H04399: Y.Y.) and TIA-Kakehashi (TK22-023: H.Y.and TK23-017: H.Y.). This work is also partially supported by PRESTO(JPMJPR177A: Y.Y.) and CREST(JPMJCR1861: Y.Y.), Japan Science andTechnology Agency (JST). H.K. acknowledges partly supports of Grant-in-Aid for JSPS Fellows (20J10749:H.K. and 22J00871:H.K.).Author contributionsH.K. fabricated thin films, performed experiments, and collected data.H.K., Y.Y. H.Y. discussed the results, wrote the paper, and preparedfigures.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-43543-1.Correspondence and requests for materials should be addressed toHiroki Koizumi, Yuichi Yamasaki or Hideto Yanagihara.Peer review information Nature Communications thanks Yimin Xiong,and the other, anonymous, reviewers for their contribution to the peerreview of this work. A peer review file is available.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jur-isdictional 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 any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate ifchanges were made. The images or other third party material in thisarticle are included in the article’s Creative Commons license, unlessindicated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons license and your intendeduse is not permitted by statutory regulation or exceeds the permitteduse, you will need to obtain permission directly from the copyrightholder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2023Article https://doi.org/10.1038/s41467-023-43543-1Nature Communications |         (2023) 14:8074 7https://doi.org/10.1038/s41467-023-43543-1http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Quadrupole anomalous Hall effect in magnetically induced electron nematic�state Results Dependence of anomalous Hall effect on the direction of current application Temperature dependence of the anisotropic anomalous Hall�effect Discussion Magnetic structure based on the symmetry of anomalous Hall�effect Magnetotransport model for QuadAHE Magnetic domain control of extended MTQ moment by magnetic-field cooling Consideration by Onsager’s reciprocal theorem Methods Sample preparation and measurements Data availability References Acknowledgements Author contributions Competing interests Additional information