# Emergent topological spin structures in a centrosymmetric cubic perovskite

###### Abstract

The skyrmion crystal (SkX) characterized by a multiple- helical spin modulation has been reported as a unique topological state that competes with the single- helimagnetic order in noncentrosymmetric materials. Here we report the discovery of a rich variety of multiple- helimagnetic spin structures in the centrosymmetric cubic perovskite SrFeO. On the basis of neutron diffraction measurements, we have identified two types of robust multiple- topological spin structures that appear in the absence of external magnetic fields: an anisotropic double- spin spiral and an isotropic quadruple- spiral hosting a three-dimensional lattice of hedgehog singularities. The present system not only diversifies the family of SkX host materials, but furthermore provides an experimental missing link between centrosymmetric lattices and topological helimagnetic order. It also offers perspectives for integration of SkXs into oxide electronic devices.

^{†}

^{†}preprint: APS/123-QED

The discovery of novel magnetic spin structures has the potential to open new fields in condensed matter physics. This is exemplified by magnetic skyrmions with a vortex-like spin configuration, which has led to a multitude of possible applications of topological spin textures in spintronics Bogdanov ; Robler ; Muhlbauer ; Nagaosa_Tokura ; Kanazawa_Adv . Highly symmetric crystal lattices allow magnetically ordered states with different equivalent propagation vectors , and complex mesoscopic superstructures can emerge from superpositions of several such degenerate states. The ”skyrmion crystal” (SkX), a multi- superposition of magnetic skyrmions, has garnered particular recent attention because of its intriguing connection to topological spin and charge transport phenomena MnSi_PRL ; MnGe_PRL ; Jonietz ; Mochizuki .

So far, SkXs have mostly been reported for non-centrosymmetric lattices, with details depending on the symmetry of the underlying crystal lattice, the magnetic anisotropy, and the relative strength of the competing interactions; i.e., the ferromagnetic exchange interaction and the Dzyaloshinskii-Moriya (DM) interaction Kanazawa_Adv . Three types of two-dimensional SkX characterized by multiple coplanar vectors have been reported: (i) a Bloch-type SkX formed by superposing three proper-screw spin modulations, which has been found in chiral helimagnets such as B20 compounds (MnSi, FeGe, etc.)Muhlbauer ; Yu_Nature ; Munzer_FeCoSi ; Yu_NMat , CuOSeOSeki , and Co-Zn-Mn alloys Tokunaga , (ii) a Néel-type SkX formed by three cycloidal spin modulations found in polar helimagnets, like GaV(S,Se) and VOSeOIstevan ; Fujima ; Bordacs ; Kurumaji , and (iii) an antiskyrmion crystal formed by three spiral spin modulations found in a Mn-Pt-Sn inverse Heusler compound with symmetry Nayak . Recently, a three-dimensional topological spin structure generated by triple- vectors that are orthogonal to each other has been tentatively identified in the B20 compound MnGe with a relatively strong DM interaction MnGe_PRB . This spin structure has hedgehog and antihedgehog singularities, where the associated emergent magnetic monopole and antimonopole manifest themself as a source of anomalous magnetotransport phenomena MnGe_PRL ; MnGe_NCom . However, as the experiments were performed on polycrystalline samples, a detailed characterization of these structures has not been possible.

In noncentrosymmetric helimagnets, the DM interaction originating from spin-orbit coupling apparently plays an important role in the formation of a SkX by selecting both the helicity and vorticity for each skyrmion MnGe_PRL ; MnGe_PRB ; MnGe_NCom . On the other hand, the emergence of SkXs has been theoretically predicted also to occur in magnetically frustrated centrosymmetric helimagnets with high lattice symmetry Okubo ; Mostovoy_NCom ; Wang ; Batista ; Ozawa ; Hayami . In the absence of the DM interaction, these helimagnets have the potential to show rich topological spin textures due to fewer constraints on the spin helix. There exist a large number of centrosymmetric helimagnets, some of which show multiple- spin modulations such as the rare-earth magnets Forgan ; Jensen . However, the presence of topologically nontrivial helimagnetic phases in centrosymmetric systems remains to be explored.

The simple cubic perovskite SrFeO with crystal structure displayed in Fig. 1(a) is known to host a helimagnetic order below 130 K with metallic conductivity MacChesney ; Takeda_screw . The origin of the helimagnetic order in SrFeO and related iron oxides has been discussed in terms of the competition between the nearest-neighbor and the further-neighbor interactions Kim_PRL or the double-exchange mechanism considering the itinerant oxygen holes Mostovoy ; Mostovoy2017 . Early neutron diffraction data were described in terms of a single- proper-screw spiral with propagation vector along [111] or equivalent directions of the cubic lattice Takeda_screw . Recently, however, SrFeO was shown to display a rich variety of helimagnetic phases depending on temperature and external magnetic field as shown in Fig. 1(a) Ishiwata_SFO ; Reehuis . Among them, Phases I and II are extraordinary in the sense that they exhibit a large unconventional Hall effect Hayashi ; Ishiwata_SFO ; Chakraverty . The presence of sharp phase transitions with unusual transport signatures indicates well-ordered magnetic superstructures, rather than an incoherent superposition of domains of the single- structure with different propagation vectors. In both Phase I and Phase II, the Hall resistivity as a function of along [111] increases nonlinearly and reaches a maximum below the phase boundary to Phase IV or Phase V. While this behavior implies the emergence of noncoplanar and/or topological spin textures with scalar spin chirality MnSi_PRL ; MnGe_PRL , the magnetic structures within each phase have remained elusive. In this work, on the basis of comprehensive single crystal neutron diffraction studies, we reveal that the magnetic structures of the mysterious Phases I and II in the centrosymmetric cubic perovskite SrFeO are indeed topological in nature, being identified as anisotropic double- and isotropic quadruple- helimagnetic structures, respectively.

Figures 1(b) and 1(c) illustrate the spin structures reproduced by the superposition of the double- and quadruple- magnetic modulations, which we propose in this study as models for Phases I and II, respectively. Note that we tentatively adopt the same helicity and phase for each spin modulation. Owing to the cubic symmetry of the crystal, there are four -vectors of , , and , where 0.13; in this paper, we refer to them as , , and , respectively, as described in Fig. 1(a). As explained later in detail, Phase I can be described as a double- structure encompassing proper-screw and cycloidal modulations with slightly different -vectors, so that the overall symmetry of the superstructure is reduced (see supplementary information for details). Here we define the topological number as the integral of the solid angle made by the three adjacent spins around the singular point as described in Ref. Nagaosa_Tokura . Following this definition, the anisotropic double- spin spiral in Phase I turns out to be topologically nontrivial as shown in Fig. 1(b), where noncoplanar vortex-like spin configurations can be found. As for Phase II, 4 equivalent proper-screw-type spin modulations yield a face-centered cubic lattice of topological singularities, at which the hedgehog/anti-hedgehog spin texture acts as the source/sink of the emergent magnetic fields (Fig. 1(c)), i.e., an emergent magnetic monopole/antimonopole. The application of external magnetic fields changes the relative positions of the magnetic monopoles and antimonopoles, producing the effective magnetic flux necessary for the topological Hall effectMnGe_NCom .

First, let us identify the multiple- state in Phases I and II on the basis of results of high-resolution neutron diffraction measurements for SrFeO at the WISH (Wide angle In a Single Histogram) diffractometer. Figure 2(a) shows integrated intensities of incommensurate magnetic reflections near the reciprocal lattice point, which were measured on heating in zero field after field cooling (FC) with an external field of 7 T along the [111] axis. The data labeled show the intensity corresponding to the magnetic modulation vector . In the simple single- proper-screw spin structure, there exist four kinds of domains in each of which only one of the -vectors is selected (the helicity is not considered here). If this were the case for Phase I, one of the domains with the propagation vector parallel to would be selected through the FC process due to the difference in the Zeeman energy, which would give rise to the nonvanishing scattering intensity only for . As seen in Fig. 2(a), however, nonzero scattering intensities are found not only for but for . Thus, the possibility of the single- proper-screw spin structure can be ruled out for Phase I and likewise for Phase II. It should be noted for Phase I that the scattering intensity for is much larger than for . This is consistent with the anisotropic double- spin structure, in which one of the -vectors with proper-screw-type spin spiral tends to be aligned along the external field, thus yielding a magnetic modulation with large amplitude. We found that all the scattering intensities are comparable to each other in Phase II, suggesting that the field-oriented anisotropic magnetic structure in Phase I disappears upon the first-order phase transition to Phase II.

Having confirmed the multiple- state in Phases I and II, we measured the dependence of the magnetic scattering intensity at the selected temperatures after zero-field cooling (ZFC) as shown in Fig. 2(b). At 50 K and zero field in Phase I, the scattering intensities for are distributed in a certain range, reflecting the multidomain state of the anisotropic multiple- spin structure. As is increased beyond 5 T, the intensity only for parallel to becomes larger, and the others become smaller. The significant -induced change with a large hysteresis at low is ascribable to the domain reorientation, as also suggested by the previous report on the magnetoresistance anomaly measured after ZFC (see the shaded area in Figs. 1(a) and 2(b)) Ishiwata_SFO . For Phase II, on the other hand, the scattering intensities for are comparable to each other and -dependent anomalies and hysteresis are absent (see Fig. 2(c)), being consistent with the presumed isotropic quadruple- helimagnetic structure. Upon an increase in to 12 T that induces the II-IV phase transition, the intensity distribution tends to become the one expected for the single- state. However, since the maximum of 12 T is located near the phase boundary and the scattering intensities for remain nonzero, further experiments with larger are indispensable to characterize the spin structure of Phase IV.

To further characterize the multiple- spin structure and the domain state in Phase I, we measured small-angle neutron scattering (SANS) for and after FC and ZFC. The SANS experiments were performed for SrFeCoO having essentially the same phase diagram as SrFeO (see Fig. 4(a)) Long . Figures 3(d) and 3(g) respectively show the magnetic reflections around and measured at 3 K in Phase I after FC. The scattering profile for revealed that the magnetic modulation wave vector is no longer described by the simple , but split into three peaks indexed as (= ), and , where . On the other hand, a single peak is found for . However, this reflection is also slightly shifted from the [1] direction, and assigned as (= ). The slightly different -vector may reflect the influence of small, anisotropic terms in the spin Hamiltonian (see the discussion below). The observation of the triplet peaks around after FC with parallel to [111] is a signature of the three -dependent domains (with each domain containing four kinds of helicity) of the anisotropic double--helimagnetic structure with and . The azimuthal directions of these propagation vectors deviate from [111] and [1], respectively, so that Phase I can accommodate the angular mismatch between and . In fact, we confirmed that the intensities for the magnetic scattering along the three 111 equivalents, [1], [], and [1], are significantly different even after the FC process, reflecting the imbalance of the three domains with the different directions (see Fig. S2). As the temperature increases through the I-II phase transition at 90 K, the directions of all propagation vectors become parallel to the 111 equivalents, consistent with the proposal for the isotropic quadruple- helmagnetic structure in Phase II. (see Figs. 3(d-i)). The schematic representations of the observed scattering peaks for Phases I and II are displayed in Figs. 3(a-c). The numbers of the domain types of Phase I in the ZFC and FC states are 48 and 12, respectively, the latter of which is obtained by considering the three-fold symmetry around [111], and the helicity degree of freedom associated with the two kinds of spin spirals. At least at low temperature, magnetic fields of order 10 T therefore mostly select different equivalent domains without modifying the magnetic structure substantially.

Next, we performed polarized SANS experiments to learn microscopically how the spins are twisting along [1], which is essentially equivalent to [] and [1] with respect to the angle relative to . Here, we assume three possible arrangements of spin spirals: i) vertical-cycloid type with the spin-spiral plane parallel to and (Fig. 4(d)); ii) proper-screw type with the spin-spiral plane normal to (Fig. 4(e)); and iii) horizontal-cycloid type with the spin-spiral plane parallel to and normal to (Fig. 4(f)). When considering that only the component of the magnetic moments perpendicular to the scattering vector causes neutron scattering, the effective spin components for each spin spiral can be described as shown at the right end of Figs. 4(d-f). Nevertheless, by using spin polarized SANS, we can distinguish between the three types of spin spiral. Since the non-spin-flip (NSF) and the spin-flip (SF) geometries detect only the spin components parallel and normal to , respectively, the intensity ratios of NSF and SF scattering at and are expected to display the dependence represented by the relative lengths of the blue and red arrows in Figs. 4(d-f). The measurements were performed after FC in Phase I, so that 6 kinds of domains with nearly parallel to are selected. The inset of Fig. 4(a) shows the -scan profiles of the magnetic scattering for at 3 K and 0.3 T, which were normalized by the flipping ratios. As shown in Figs. 4(b) and 4(c), the normalized scattering intensity for the NSF geometry is much larger than that for the SF geometry in Phase I, whereas the scattering intensities for both geometries are nearly the same in Phase II. This result indicates that Phase I and Phase II encompass vertical-cycloid and proper-screw states, respectively, propagating along [1]. In Phase V at 7 T, the scattering intensity for the SF geometry is larger than that for the NSF geometry, implying that the spin arrangement propagating along [1] is in the horizontal-cycloid type. This result indicates that Phase V has multiple- spin spirals as well, calling for further experiments to identify the spin structure.

To summarize, we have identified two kinds of topological spin structures in SrFeO, that appear robustly even without external magnetic fields. The anisotropic double- spin structure in Phase I manifests itself as a unique topological order reflecting the versatility of the centrosymmetric lattice that permits various types of spin spiral. However, this spin texture cannot explain the observed topological Hall resistivity, because the expected direction of the emergent magnetic flux is perpendicular to the external magnetic field. Future research will have to assess whether the discrepancy between the expected and observed directions of the emergent magnetic flux in Phase I arises from an additional internal spin modulation that is not resolved in the current experiment. On the other hand, Phase II can be described in terms of an isotropic quadruple- spin spiral, that presumably yields emergent magnetic monopoles as reported for the noncentrosymmetric helimagnet MnGe MnGe_PRB .

We now turn to the mechanisms underlying the observed cascade of magnetic phase transitions. The fact that the diffraction patterns at the lowest temperature in ZFC and FC states are closely similar implies that Phase I is not stabilized by the Zeeman interaction, but rather by anisotropic terms in the zero-field spin Hamiltonian (such as magneto-crystalline anisotropies generated by the spin-orbit coupling) that go beyond the primary isotropic double-exchange and/or superexchange interactions. With increasing temperature, the corresponding free-energy gain could be offset by the higher entropy of the more symmetric quadruple- structure, leading to the observed phase transition between Phases I and II. Recently, Monte Carlo simulations based on a Kondo lattice model with a biquadratic interaction defined in momentum space have indicated various multiple- phases on a centrosymmetric lattice with rotational symmetryHayami . Although this model does not apply directly to our system, this theoretical work and our experimental results provide a timely showcase of the rich variety of multiple- helimagnetic phases with topological singularities that can be expected to emerge ubiquitously in frustrated itinerant magnets with high lattice symmetry, even without the DM interaction. Moreover, perovskite-type oxides are a broad class of materials that already find many applications especially in the form of heterostructures enabling the interplay between topological magnetism and other collective quantum phenomena. The discovery of topological spin order in SrFeO is therefore a milestone for integrating potential topological magnetic states into existing device architectures.

### Appendix: Materials and Methods

Single crystals of SrFeO and SrFeCoO were obtained by a high-pressure oxygen annealing for the large single crystals of the oxygen-deficient perovskite with brownmillerite-type structure as described in Refs. Ishiwata_SFO and Long_JPCM . The orientation of the single crystal with dimensions of about 3 3 2 mm was checked by x-ray Laue diffractions.

The temperature and magnetic field variations of the neutron diffraction intensities shown in Fig. 2 were measured for SrFeO with a cold neutron time-of-flight diffractometer, WISH (Wide angle In a Single Histogram) Chapon , at the ISIS Facility in UK. The single crystal SrFeO was loaded into a vertical-field cryomagnet, whose maximum field was 13.5 T. The cubic [111] axis was set to be parallel to the magnetic field. The cryomagnet has a large vertical opening angle from -5 to 10, which enabled us to measure the out-of-plane incommensurate magnetic reflections near the (1, , 0) reciprocal lattice point under magnetic field along the [111] direction. Small angle neutron scattering (SANS) measurements were carried out for SrFeCoO with a vertical-field cryomagnet (7 T in maximum) at the MIRA beamline in FRM II and a horizontal-field cryomagnet (6.8 T in maximum) at the SINQ beamline in Paul Scherrer Institute Georgii1 ; Georgii2 . We employed an experimental setup of the horizontal configuration (see Fig. S1(a)), in which the sample can be rotated by 360 around the [2] axis ( axis), and vertical configuration (see Fig. S1(b)), in which the sample can be rotated by 360° around the [111] axis ( axis). In both experimental configurations, magnetic field was applied parallel to the [111] axis and perpendicular to the incident neutron beam. Experimental data taken at MIRA were collected both in polarized and unpolarized modes at a wavelength = 4.5 Å, and those at SINQ were collected in unpolarized mode at = 4.7 Å. The polarization of neutron spins was generated by the magnetic field gradient near the vertical superconducting magnet. The flipping ratios for 0.3 T and 7 T are 4.9 and 3.0, respectively.

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## References

## Acknowledgments

The authors thank M. Mostovoy, S. Hayami, Y. Motome, N. Nagaosa, and M. Takano for useful comments, and thank N. Egetenmeyer for her kind experimental support. This work is partly supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science, Japan (Kakenhi No. 17H01195 and No. 16K17736), JST PRESTO Hyper-nano-space design toward Innovative Functionality (Grant No. JPMJPR1412), and Asahi Glass Foundation. D. S. I. acknowledges support from the German Research Foundation (DFG) through the Collaborative Research Center SFB 1143 in Dresden (project C03). J. S. W. acknowledges support from Swiss National Science Foundation (SNSF) via the Sinergia network ’NanoSkyrmionics’ (grant CRSII5-171003), and the SNSF project grant No. 153451. B.K. acknowledges support from the DFG through the Collaborative Research Center TRR80.