Recently, noncollinear and noncoplanar spin textures have been extensively investigated as a source of rich emergent phenomena. In particular, two- or three-dimensionally modulated multiple-q spin textures often display a nontrivial topology, which enables current-induced spin manipulation or magnetic control of electron transport properties through the emergent electromagnetic field associated with quantum Berry phase or relativistic spin-orbit interactions (1–4). For example, the triple-q state represented by the superposition of helical spin textures can be often considered as a crystallized form of magnetic skyrmions, i.e., noncoplanar swirling spin texture with topologically protected particle nature (5–14). Skyrmions are now attracting attention as information carriers and are providing a natural magnonic crystal potentially suitable for magnetic data processing (15–18), and the further search of exotic multiple-q states of novel origin is highly anticipated.
The formation of triple-q skyrmion states has been experimentally observed mainly in materials with noncentrosymmetric crystal structure, where the Dzyaloshinskii-Moriya (DM) interaction is the key for the emergence of helimagnetism (16, 17). For these systems, the single-q helimagnetic order appears at zero magnetic field, and the application of a magnetic field stabilizes the hexagonal skyrmion lattice state with triple-q nature. Here, the detail of the spin texture depends on the symmetry of the crystal structure and associated DM interaction (5), and various forms of skyrmion spin textures, such as the Bloch-type one in the chiral system (7–10), the Néel-type one in the polar system (11–13), and the antivortex-type one in the D2d symmetry system (14), have recently been discovered.
On the other hand, according to the latest theories, it is also expected that nontrivial multiple-q spin orders can be stabilized even without breaking of inversion symmetry. For example, magnetic frustration between the nearest neighboring and next nearest neighboring exchange interactions on the triangular lattice is proposed to stabilize the triple-q skyrmion order under the magnetic field applied along the out-of-plane direction (19–21). Another promising approach is the employment of itinerant magnetism in the high-symmetry (hexagonal or tetragonal) lattice system (22–27), where the four-spin interaction can stabilize various multiple-q orders even in zero magnetic field.
To realize the potential multiple-q helimagnetism expected from these novel mechanisms, the search for appropriate material systems fulfilling the corresponding conditions is essential. Our target material Y3Co8Sn4 is a member of the R3M8Sn4 (R being Y or a rare earth element and M being a 3d transition metal element) family characterized by a polar hexagonal crystal structure (Fig. 1A) and an itinerant nature of the magnetism (28–30). This material family is unique because all of the abovementioned mechanisms, i.e., (i) DM interaction in noncentrosymmetric systems, (ii) frustrated exchange interactions in triangular lattice systems, and (iii) four-spin interaction in itinerant hexagonal systems, are allowed to become active in principle, depending on the relative magnitude of each interaction. For Y3Co8Sn4, the emergence of incommensurate magnetism has previously been proposed by a powder neutron diffraction study (30), while detailed information on the magnetic structure and associated mechanisms are still lacking.
(A) Crystal structure of Y3Co8Sn4. (B) Magnetic field dependence of the magnetization measured under H || (red) and H ||  (black) at 2 K. (C) Temperature dependence of magnetic susceptibility (M/H at μ0H = 0.02 T) for H || . (D and E) The integrated scattering intensity and the magnitude of the wave number for the two types of magnetic reflection, qout and qin, as a function of temperature obtained from SANS profiles. (F) Schematic illustration of the experimental geometry for SANS measurement. kin and kout are the incident and scattered neutron wave vectors, respectively. (G to J) SANS patterns measured for the (001) plane at zero magnetic field at various temperatures. The color scale indicates the scattering intensity. The integrated intensities in (D) are taken from the boxed area shown in (J). a.u., arbitrary units; f.u., formula unit.
In this work, we investigated the detailed magnetic structure for the itinerant hexagonal magnet Y3Co8Sn4, through polarized and unpolarized small-angle neutron scattering (SANS) experiments on a single-crystal specimen under various applied magnetic fields and temperatures. Our results suggest the formation of triple-q magnetic order describing in-plane vortex-like spin textures, which can be most consistently explained in terms of the four-spin interaction mechanism activated in the itinerant hexagonal systems.
Figure 1A indicates the crystal structure of Y3Co8Sn4, which belongs to the polar hexagonal space group P63mc with the polar axis along the  direction (25). Y3Co8Sn4 undergoes a ferromagnetic transition around 55 K (Fig. 1C, inset). The comparison of M-H (magnetization-magnetic field) profiles for H ||  and in Fig. 1B demonstrates that the system has an easy-plane anisotropy perpendicular to the  axis. The saturation magnetization Ms is 0.35 μB/Co at 2 K. These magnetic properties are in good agreement with previous reports on polycrystalline samples (29, 30). The temperature (T) dependence of the magnetic susceptibility (Fig. 1C) shows a notable reduction below 20 K, which implies a transition into the modulated spin state.
To elucidate the long-wavelength magnetic structure of Y3Co8Sn4, we performed SANS measurements. Figure 1 (G to J) shows the temperature dependence of the SANS patterns measured on the (001) plane under zero magnetic field. At 1.5 K (Fig. 1G), we observe a sixfold symmetric pattern with magnetic Bragg reflections aligned along the 〈〉 directions (equivalent to the a* directions) and with a wave number of qout ~0.081 A−1 that corresponds to a modulation period of 8 nm. In addition, we found a weak ring-shaped signal with a wave number of qin ~ 0.040 A−1 (i.e., modulation period of 16 nm). On increasing the temperature, the sixfold qout magnetic reflections become obscure around 14 K and vanish at T1 = 18 K, while the ring-shaped qin signal becomes stronger above 14 K and survives up to T2 = 26 K. Figure 1D shows the temperature variations of the integrated intensities for qout and qin, taken for the boxed regions shown in Fig. 1J. The intensity due to sixfold qout reflections is dominant below 13 K, while it is gradually replaced by the ring-shaped qin intensity as the temperature increases. Above T1 = 18 K, the ring-shaped qin intensity becomes dominant, and then, it gradually disappears at T2 = 26 K. Such a clear anticorrelation of intensity with temperature indicates that the two magnetic reflections qout and qin represent distinct magnetic phases and that their volume fractions vary with temperature. The distinctive nature of sixfold qout and ring-shaped qin can also be confirmed by the different temperature dependence of their wave numbers (Fig. 1E); the wave number of qout monotonically decreases toward higher T, while that of qin is almost temperature independent.
Next, we investigate the magnetic field dependence of the magnetic scattering. Figure 2 (A to D and E to H) shows the SANS patterns at 1.5 K and in a magnetic field applied along the  and axes, respectively. For both qout and qin, the integrated intensities (Fig. 2, I and K) decrease monotonically with increasing H, accompanied by gradual increase of the wave numbers (Fig. 2, J and L). The SANS pattern described by qout keeps the sixfold symmetry for both in-plane and out-of-plane orientations of H, even with a large H value close to the transition into the saturated ferromagnetic state. In contrast, the ring-shaped pattern of qin is easily transformed into a twofold pattern under an in-plane H || (see fig. S1), which indicates the alignment of modulation vector qin along the H direction.
(A to H) SANS patterns taken at 1.5 K with various magnitudes of magnetic field for H ||  (A to D) and H || (E to H). The color scale indicates the scattering intensity. (I to L) The scattering intensity and the magnitude of the wave number for two magnetic reflections qout and qin as a function of magnetic field, measured for H ||  (I and J) and H || (K and L).
To investigate the magnetic structures described by qout and qin in greater detail, we performed SANS with longitudinal neutron spin polarization analysis, using a 3He spin analyzer setup, as shown in Fig. 3C. We used a longitudinal geometry, where the incident neutron spin polarization (Sn) was aligned parallel to both the incident beam (kin) and the  axis of the sample. The longitudinal (parallel to Sn) and transverse (perpendicular to Sn) spin components of the magnetic order in the sample can be evaluated independently, since the corresponding magnetic scattering contributes to a purely non–spin-flipped (NSF) or spin-flipped (SF) response, respectively (Fig. 3D) (31). Thus, in our configuration, the SF (NSF) signal is due to spin components normal (parallel) to the  axis, which is referred to as Sxy (Sz). Figure 3 (A and B) shows polarized SANS patterns for the SF and NSF channels measured at 1.5 K under zero magnetic field. The magnetic reflections are observed in the SF channel but not in the NSF channel for both qout and qin. This proves that neither qout nor qin has a Sz component in the ground state, and therefore, the magnetic moments always lie within the (001) plane. The temperature development of each spin component for qout and qin is summarized in Fig. 3 (E and F). Note that the Sz component of qin gradually becomes finite above 14 K, which may reflect the onset of spin fluctuations close to the critical temperature T2.
(A and B) Polarized SANS patterns of (A) SF and (B) NSF channels measured at 1.5 K under zero magnetic field, which detects in-plane and out-of-plane spin components (Sxy and Sz), respectively. The color represents the scattering intensity. (C and D) Schematic illustration of (C) experimental setup and (D) the magnetic scattering selection rules. Only a local magnetization normal to q can give rise to magnetic neutron scattering. The additional selection rules provided by the longitudinal polarized beam geometry are that SF scattering arises due to Sxy spin components perpendicular to both q and kin (red arrow) and that NSF scattering arises due to Sz components || kin (blue arrow). Here, Sn represents the direction of the neutron polarization, which can be chosen experimentally to be either aligned or anti-aligned with kin. (E and F) Temperature variations of the scattering intensity of SF and NSF channels (corresponding to Sxy and Sz, respectively) for (E) qout and (F) qin. The data were integrated over the same detector regions, as shown in Fig. 1J.
On the basis of the present results, we have summarized the H-T phase diagrams for H ||  and in Fig. 4 (A and B). Considering the observed alignment of modulation vectors along the in-plane H, the ring-shaped qin pattern represents multiple domains of a single-q state with spin components modulating within the (001) plane, where the q vectors can orient randomly along any arbitrary in-plane direction (fig. S1). On the other hand, the observed SANS pattern of qout maintains a sixfold symmetry even under a large in-plane H, strongly suggestive of the formation of a triple-q spin texture.
(A and B) H-T magnetic phase diagrams derived from the field and the temperature dependence of the SANS intensity under (A) H ||  and (B) H || . (C) Triple-q spin configuration obtained by simulated annealing based on the model described by eq. S5, with the parameters α = 0.4 and K = 1.5 (). Arrows represent the xy components of the magnetic moment. (D) Simulated SANS patterns corresponding to (C), with the color indicating the square root of the spin structure factor in arbitrary units. Hexagons in (C) and (D) represent the magnetic unit cell and first Brillouin zone, respectively.
The emergence of a unique multiple-q magnetic order has been predicted theoretically from various distinctive mechanisms, such as (i) DM interaction in noncentrosymmetric systems (16, 17), (ii) frustrated exchange interactions in triangular lattice systems (19–21), and (iii) four-spin interaction in itinerant hexagonal systems (26, 27). Since Y3Co8Sn4 is characterized by a polar hexagonal crystal structure with itinerant magnetism, all of the above mechanisms may become active in principle. If the DM mechanism is responsible for the present case, then the polar symmetry of the crystal structure should favor cycloidal spin textures with magnetic moments confined within the ac plane, and application of H ||  could lead to the formation of a triple-q Néel-type skyrmion lattice state (5, 11–13, 32). Both of these magnetic orders should contain nonzero magnitude of the Sz component. However, the observed confinement of the spins within the (001) plane is inconsistent with this scenario; therefore, the DM interaction is not the main source of incommensurate magnetism in Y3Co8Sn4. Another potential source is frustrated exchange interactions, but this contribution is also determined to be minor because the Curie-Weiss temperature obtained from the temperature dependence of the inverse magnetic susceptibility is positive (i.e., ferromagnetic) and agrees well with Tc (27). Furthermore, this scheme rather assumes the frustration of short-ranged exchange interactions among localized moments, and its validity is less clear for the present case of Y3Co8Sn4 with itinerant nature of magnetism. In principle, the two mechanisms above stabilize long-wavelength multiple-q spin orders only under an out-of-plane magnetic field. Therefore, the observed triple-q spin order at H = 0 in Y3Co8Sn4 should be ascribed to a different origin.
On the basis of such an analysis, we focus on the mechanism associated with the four-spin interaction (25–27). In itinerant magnets, the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction usually favors the formation of a single-q helical spin order. On the other hand, from recent theoretical studies, it is suggested that multiple-q spin orders can be preferred because of additional four-spin interactions originating from electron hopping between four sites, when the system is characterized by a high-symmetry (i.e., hexagonal or tetragonal) crystal lattice (25–27). Since the magnetic anisotropy was not considered explicitly in previous theoretical works, we performed simulated annealing for the simple hexagonal lattice system with additional easy-plane anisotropy based on the effective spin Hamiltonian proposed in (27) (see the Supplementary Materials for details). This result of the simulations confirms that the four-spin interaction can stabilize the vortex-like triple-q spin texture with moments confined within the hexagonal lattice basal plane, as shown in Fig. 4 (C and D), which is consistent with the present SANS results. Note that Y3Co8Sn4 is characterized by a large crystallographic unit cell with four inequivalent Co sites, and the real spin texture in this compound may be more complicated due to the contribution of other factors neglected in the present model. For example, the observed appearance of a single-q order with a different modulation period at higher temperatures implies the delicate balance of competing instabilities associated with the intricate character of Fermi surfaces. Nevertheless, the main conclusion of this theoretical framework, i.e., the emergence of triple-q order with in-plane noncollinear spin textures in the ground state at H = 0, will be valid in general, irrespective of the material details.
The presently used four-spin interaction mechanism in itinerant magnets is unique, because it allows the emergence of multiple-q states even in zero magnetic field and without the necessity of inversion symmetry breaking. Recently, it has been reported that this four-spin interaction plays a key role in the emergence of a double-q square skyrmion lattice in the Fe monolayer on Ir(111), where the magnitude of the four-spin interaction turned out to be of the same order as the two-spin exchange interaction (33). Our present results for Y3Co8Sn4 suggest that a similar mechanism may be also active in single-phase bulk itinerant magnets. Note that Y3Co8Sn4 with easy-plane anisotropy is characterized by a coplanar spin texture, which is topologically trivial in terms of skyrmion number. On the other hand, the emergence of a noncoplanar multiple-q order with nonzero skyrmion number has theoretically been predicted for the system with weak or easy-axis magnetic anisotropy (26, 27), and the systematic control of magnetic anisotropy is possible for the R3M8Sn4 family due to its wide chemical tenability (28, 34, 35). Our experimental results suggest a new paradigm to realize exotic (and potentially topological) multiple-q orders and call for further exploration of other itinerant hexagonal magnets including the family of R3M8Sn4.
Single crystals of Y3Co8Sn4 were synthesized by arc-melting stoichiometric amounts of pure Y, Co, and Sn pieces, followed by slow cooling in a silica tube under vacuum. Powder x-ray diffraction analysis confirmed the single-phase nature of the crystal (fig. S4). The crystal orientation was determined by both x-ray Laue and neutron diffraction. The sample had a volume of 6 mm by 4 mm by 1 mm, with the widest surface parallel to the (001) plane.
Magnetization was measured using a SQUID magnetometer (Magnetic Property Measurement System, Quantum Design). The SANS measurements were carried out using the SANS-I and SANS-II instruments at the Swiss Spallation Neutron Source (SINQ), Paul Scherrer Institut, Switzerland, and the D33 instrument at the Institut Laue-Langevin (ILL), Grenoble, France. The wavelengths of the neutron beam were set to 5 Å (SANS-I and SANS-II) and 4.6 Å (D33). The incident beam was always directed along the  axis. The SANS diffraction patterns were obtained by summing together two-dimensional multidetector measurements taken over a range of sample + cryomagnet rotation (rocking) angles. Background data at each rocking angle were obtained for T = 60 K well above Tc and subtracted from data obtained at low T to leave only the magnetic signal. All H (magnetic field) scans were performed in the H-increasing process after zero-field cooling (ZFC), and T (temperature) scans were performed in the warming process after ZFC.
In the SANS experiments, with longitudinal polarization analysis (POLARIS) using D33 at the ILL (36), the incident neutron beam was spin polarized using an Fe-Si transmission polarizer, with the spin polarization reversible by means of an radio frequency spin flipper. The neutron spin state after scattering from the sample was analyzed using a nuclear spin-polarized 3He cell. The longitudinal neutron spin polarization axis was preserved by means of magnetic guide fields of the order of several milliteslas on the intermediate flight path between polarizer and analyzer. At the sample position, the guide field of 5 mT was applied by the cryomagnet, with the field being sufficiently low as the zero-field magnetic state of the sample was kept intact. The efficiency of the overall setup was characterized by the flipping ratio of 14. By measuring all possible spin-state combinations, corrections for the polarizing efficiency of the overall setup were taken into account in the data analysis. The polarized (and unpolarized) data reduction was performed using the GRASP software.
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/11/eaau3402/DC1
Fig. S6. Temperature dependence of the magnetic contribution for the integrated (100) peak intensity at zero field.
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Neutron-scattering experiments reveal a hexagonally crystallized form of noncollinear magnetic order in an itinerant magnet.
Neutron-scattering experiments reveal a hexagonally crystallized form of noncollinear magnetic order in an itinerant magnet.
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