Introduction

In non-centrosymmetric magnets, the interplay between ferromagnetic (FM) exchange, Dzyaloshinskii–Moriya interactions, and magnetic anisotropy, results in rich phase diagrams containing helical, conical and skyrmion lattice phases1,2,3. Skyrmion phases are the most intriguing, since each skyrmion displays a topologically nontrivial spin texture that is easily manipulated by external stimuli4,5,6. This makes skyrmions promising for diverse applications such as high-density spintronics and neuromorphic computing2,7,8.

Chiral cubic magnets such as MnSi9, FeGe10, Cu2OSeO311, and Co–Zn-Mn compounds12,13,14,15,16 are the archetypal skyrmion host materials. All of these systems display commonalities amongst their phase diagrams, including the onset of helimagnetic order on zero-field cooling (ZFC) below the critical temperature Tc, and a skyrmion lattice (SkL) phase, known as the A-phase, stabilized in a finite magnetic field (H) and over a narrow T-range directly below Tc9,11,12,13,14,15,16. The proximity of the A-phase to Tc supports theoretical suggestions that thermal fluctuations are a key ingredient for A-phase stability, this being akin to an order-by-disorder mechanism9,17,18,19,20,21. It is further proposed that quantum fluctuations can stabilize skyrmions as well22.

The Co–Zn–Mn series (Co0.5Zn0.5)20−xMnx, x = 0–20, all crystallize in the chiral cubic β-Mn-type structure (space group P4132/P4332) with 20 atoms per unit cell distributed across two Wyckoff sites, the 8c and 12d sites [Fig. 1a]. Previous work reveals that Co tends to occupy the 8c site, while Zn and Mn tend to occupy the 12d site12,15,23,24. The magnetic properties vary strongly with x; the Mn-free end member Co10Zn10 (x = 0) displays a high-T helical ordering of Co spins below Tc ~ 415 K16. As Mn substitution proceeds, Tc falls quickly to ~300 K in Co8Zn8Mn4, and ~160 K in Co7Zn7Mn6. Nonetheless all systems from 0 ≤ x ≤ 6 display an A-phase close to Tc that is stable either above, at, or below room T, depending on x12,15,16,24. The other end member Mn20 (x = 20), β-Mn itself, is an elemental geometrically frustrated spin liquid due to antiferromagnetic (AFM) interactions between Mn moments on the hyper-kagomé coordinated 12d site25,26,27. For 3 ≤ x ≤ 7 a transition to a spin glass takes place below a temperature Tg < Tc. This regime is indicative for persistent magnetic frustration effects inherited from β-Mn, that are found to co-exist with any pre-existing helical spin correlations12,13,15,16,23,24.

Fig. 1: Crystal structure, magnetic phase diagrams and muon spectroscopy of CoZnMn alloys.
figure 1

a β-Mn-type structure (space group: P4132), comprising 8c and 12d atomic sites, as viewed along the [111] axis. The bonds between Mn atoms at 12d sites are shown. Schematic T − H magnetic phase diagram in (b) Co7Zn7Mn6 and (c) Co8Zn8Mn4 obtained from SANS and susceptibility measurements13,15. Measured (symbols) and fitted (solid lines) μSR spectra at selected temperatures for Co7Zn7Mn6 (d) and Co8Zn8Mn4 (e). Corresponding temperature dependencies of relaxation rates λ(T) obtained from the fits are shown in (f) and (g) for Co7Zn7Mn6 and Co8Zn8Mn4, respectively. The temperature ranges for SkL and LTSk phases are indicated by purple and blue background colours, respectively. In (d, e), error bars indicate the statistical error. In (f, g), error bars are obtained from the fitted model described in the main text.

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Here, we focus on Co7Zn7Mn6 (x = 6), a mixed composition system displaying the unusual effects of intertwined chiral and frustrated magnetism. Remarkably this system hosts two equilibrium skyrmion phases15, these being the A-phase, and an additional low-temperature skyrmion (LTSk) phase that was unambiguously identified by small-angle neutron scattering (SANS) and Lorentz transmission electron microscopy (LTEM) and called a ‘disordered’ skyrmion phase in ref. 15. As summarized in (Fig. 1b), the LTSk phase is stable in finite H and for temperatures just above Tg ~ 35 K, this being far from the A-phase stable near Tc. With thermal fluctuations suppressed far from Tc, multiple skyrmion stability mechanisms can be supposed to exist in this single chiral magnet. In the following, we apply an arsenal of quantum beam probes (muons, neutrons, X-rays and electrons), to elucidate the microscopic magnetism and dynamics across the phase diagram of Co7Zn7Mn6, from atomic-to-nanometric length-scales, and μs-ps time scales. We find that the LTSk phase is stable over a thermal range characterized by an enhanced magnetic fluctuation rate on the μs time scale, the origin of which is connected to the geometric magnetic frustration of Mn moments on the 12d site. Our experiments thus uncover an alternative mechanism for stabilizing topological skyrmions in chiral magnets that draws upon magnetic frustration as the key ingredient.

Results and discussion

Experimental observation of magnetic fluctuations by μSR

Muon spin relaxation (μSR) experiments were performed to determine the H and T-dependent characteristic magnetic fluctuation rates in Co7Zn7Mn6. These experiments are sensitive to time scales ranging from 10−12 to 10−4 s depending on the size of the magnetic field at the muon site. Figure 1b shows μSR spectra recorded from bulk single-crystal Co7Zn7Mn6 across its entire phase diagram using the general purpose surface (GPS) instrument at the Paul Scherrer Institute. Further experimental details are given in the “Methods” section. For comparative purposes, further data were obtained similarly from the sister compound Co8Zn8Mn4 (x = 4) with Tg ~ 10 K, and in which no LTSk phase is known to exist (Fig. 1c).

Figure 1d, e shows selected time spectra measured on warming after an initial ZFC, and in zero magnetic field (ZF). For T > Tc, the T-dependent muon depolarization is dominated by fast paramagnetic spin fluctuations. As done previously for β-Mn28, we fit these high T data with a muon depolarization function A(t) described by a product of a T-independent Gaussian Kubo–Toyabe relaxation describing static nuclear moments, and another exponential term \(\exp (-{\lambda }_{{\rm{L}}}t)\) that accounts for fluctuating electronic moments: \(A(t)/{A}_{{\rm{0}}}=[1/3+2/3(1-{\sigma }^{2}{t}^{2})\cdot \exp (-{\sigma }^{2}{t}^{2}/2)]\cdot \exp (-{\lambda }_{{\rm{L}}}t)\). Here, A0 is the total asymmetry, t is time, \(\sigma ={\gamma }_{{\rm{\mu }}}\sqrt{\langle {{\Delta }}{B}_{{\rm{loc}}}^{2}\rangle }=0.359\pm 0.025\) μs−1 is the width of the local nuclear spin field distribution 〈ΔBloc〉, γμ is the muon gyromagnetic ratio, and λL is the muon relaxation rate. In the fast fluctuating limit, the correlation time of the fluctuating moment is proportional to λL.

Below Tc, the t-dependent muon depolarization in both systems reveals a slowing down of spin dynamics, particularly on cooling towards lower T. This is indicated by the initial fast depolarization of the so-called transverse part of the spectrum, followed by a slow depolarization of the longitudinal part. This separation into transverse and longitudinal relaxations far below Tc indicates clearly that each system is in a slow fluctuation regime where the magnetic fluctuation rate is much smaller than the characteristic Larmor frequency of the muon spin precession29, and can be calculated as ν = 3λL/2, where λL is the muon longitudinal relaxation rate. Within this slow fluctuation regime, the muon depolarization function that well-describes the data is given by \(A(t)/{A}_{0}={f}_{{\rm{T}}}\cdot \exp [-({\lambda }_{{\rm{T}}}t)]+{f}_{{\rm{L}}}\cdot \exp {(-{\lambda }_{{\rm{L}}}t)}^{{\beta }_{L}}.\) Here, fL and fT are respectively the longitudinal and transverse fractions of the asymmetry, λL and λT are the muon relaxation rates for each component, respectively, and βL is an exponent that indicates the distribution of depolarization rates30. Convergent fits with χ2 ≈ 1 were achieved for fitted fT = 1 − fL and fixed βL = 0.5. Notably, the fitted values of fL = 0.31 ± 0.07 are close to the value of 1/3 expected due to magnetic order in a polycrystalline sample. In the present case for the single crystal, fL ~ 1/3 due to the onset of the complex multi-domain helicoidal spin structures known to exist below Tc15.

From our fitting of all data, we obtained the T-dependence of the magnetic relaxation rates λL and λT, which inform about the spatial and temporal internal field distributions. In both compounds, fitting of the data near Tc provides a clear indication of the magnetic ordering transition [see Supplementary note 1], with the onset of the fast relaxation λT(T) [see Supplementary fig. 1] coinciding with the onset of long-range helical order below Tc13,15. Below Tc, the transverse component of the spectrum depolarizes so quickly that it becomes barely observable in the muon time window. In contrast, the fitted longitudinal component λL(T) is more revealing; Fig. 1f, g each show λL(T) to display two maxima in both ZF and a low longitudinal magnetic field (LF); one just below Tc and one just above Tg. Each maximum denotes a T range characterized by an enhanced relaxation rate. The high-T maximum is suppressed under increased LF as expected31, while the low T peak shows only a weak LF-dependence.

While a regime of enhanced magnetic relaxation underlying skyrmion phases near Tc has been commonly observed by μSR32,33,34,35,36, the existence of a second maximum just above Tg is hitherto unique to Co7Zn7Mn6 and Co8Zn8Mn4 amongst skyrmion hosts. Importantly for Co7Zn7Mn6, the low T peak in λL coincides with the thermal range of the LTSk phase, thus connecting the phase stability to the enhancemed magnetic fluctuation rate. In addition, the maximal \({\nu }_{\max }=3{\lambda }_{{\rm{Lmax}}}/2\approx 40\) MHz is considerably higher than that for Co8Zn8Mn4 (≈9 MHz) wherein no LTSk phase exists, implying LTSk phase stability to depend further on the magnitude of the low-T fluctuation rate.

The T-dependent peaks in λL(T) reported in Fig. 1f, g display clear resemblance with other magnetically frustrated systems such as re-entrant spin glasses37,38,39,40,41,42,43 and moreover other β-Mn-type compounds28,38. In binary β-Mn0.81-Ru0.19 for example, the analogous coexistence of static order and slow magnetic dynamics is substantiated by T-dependent peaks in muon relaxation rate observed near both Tg and also the Néel T44, and further by quasi-elastic neutron scattering45. This comparison implicates the physics of magnetic frustration in binary β-Mn compounds to be also present in isostructural Co7Zn7Mn6 and Co8Zn8Mn4. Moreover, since convincing models for the magnetism in frustrated β-Mn systems are derived for non-magnetic 8c sites27,46,47,48, it becomes plausible that the peak in λL(T) we observe near Tg in Co7Zn7Mn6 and Co8Zn8Mn4 are similarly due to fluctuations of frustrated Mn moments on the 12d site. The deduced importance of the Mn fluctuations is substantiated by comparing the μSR obtained from the two compounds. The peak in λL(T) just above Tg is smaller in Co8Zn8Mn4 which has a reduced Mn content compared with Co7Zn7Mn6, and thus clear signs of magnetic frustration relief in combination with the higher Tc and lower Tg.

Short-range magnetic correlations observed by diffuse neutron scattering

To obtain further evidence for the frustrated magnetism of Mn at the 12d site in Co7Zn7Mn6, we performed magnetic diffuse neutron scattering (MDNS) measurements of short-range spin correlations. The experiment integrated the spin fluctuations over energy scales of the order of THz, and thus over time scales longer than a few picoseconds. Here, microsecond spin fluctuations observed in the μSR are static within the timeframe of the present MDNS experiments, and thus they may contribute to the observed signal. Low T MDNS measurements were done on Co7Zn7Mn6 powder using the D7 spectrometer at the ILL, France49, and using an xyz polarization analysis to isolate the magnetic scattering component50 —see the “Methods” section for further experimental details. Figure 2 shows the magnetic diffuse scattering profile obtained at 2 K and in ZF. The profile contains magnetic Bragg peaks shown in light grey, arising mainly from the long-range order of Co moments at the 8c site23. The peak of MDNS shown in blue is a hallmark of short-range spin correlations often found in frustrated magnets51. Notably, the MDNS peaks near to Q = 1.6 Å−1 similarly as for other β-Mn systems with similar cubic lattice constants27,50. Higher T profiles shown in Supplementary Fig. 2 reveal the volume of MDNS falls monotonically with increasing T, and that it persists for high T > Tc.

Fig. 2: Neutron scattering from Co7Zn7Mn6 and mean-field model.
figure 2

Magnetic scattering measured at D7 from Co7Zn7Mn6 powder at T = 2 K. The entire profile is shown by blue and light grey symbols. The latter correspond to magnetic Bragg peaks, and are omitted for the MF Jnn model fitting described in the text (red line). Error bars indicate the statistical error. Inset: Sketch showing the Jnn interaction (purple bonds) between nearest-neighbours on the 12d site.

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To determine the origin of the MDNS we fit the blue data portion of the profile in Fig. 2 by a mean-field (MF) model that calculates the paramagnetic neutron scattering for AFM interacting Mn moments at the 12d sites. Further details of the calculation and our assumptions are given in the “Methods”. Optimal values of the exchange constants at short Mn–Mn distances were obtained by direct fitting to the data. The red line in Fig. 2 shows a fit to a minimal model that considers a single nearest neighbour (inset Fig. 2) AFM exchange constant Jnn ≈ −0.54 meV with a non-interacting T-dependent spin susceptibility taken into account. From extensive fitting, we find that a single parameter nearest neighbour AFM exchange constant is sufficient to optimally reproduce the experimental data with only Jnn ≈ − 0.54 meV. The inclusion of more distant exchanges does not result in a significantly improved fit, though their importance can be tested against future single-crystal measurements27 (see Supplementary note 2). Such a nearest neighbour AFM exchange interaction also gives a MF spin-pair correlation function 〈S(0)⋅S(r)〉 in reasonable agreement with that obtained directly from the data via reverse Monte Carlo (RMC) refinement (see Supplementary note 3).

Overall, the analysis suggests nearest neighbour AFM interactions between Mn moments on the 12d site as a plausible origin for the observed MDNS. It follows that due to the hyper-kagomé geometry of the 12d site, geometrical frustration is implicated as the crucial ingredient that suppresses the long-range order of Mn moments. Instead, from μSR we find these Mn moments fluctuate on μs time scales over an extended T-range down to Tg, this being consistent with their appearance as static short-range spin correlations in DNS which is sensitive to faster time scales.

Magnetic element-selective resonant X-ray scattering

To further substantiate our deductions on the existence of geometric frustration-induced fluctuations of Mn moments underlying LTSk phase stability in Co7Zn7Mn6, we next consider magnetic element-selective resonant X-ray scattering experiments performed at the L2,3 edges of Co and Mn using the XTreme beamline at the SLS facility, PSI, Switzerland52, and SEXTANTS beamline, Soleil Synchrotron, France53. Further experiment details are given in the Methods.

The T-dependent X-ray magnetic circular dichroism (XMCD) was measured on field warming at 50 kOe in the field-polarized phase. Figure 3a, b shows that at 2 K we observe a relatively strong absorption of the X-ray absorption spectroscopy (XAS)-normalized XMCD of ≈20% at the Co L3 edge, and ≈10% at the Mn L3 edge. Multiplet features observed in the Mn XMCD spectra (Fig. 3b) suggest that the Mn 2p − 3d transition results from localization of the Mn 3d electrons. Most features in the Mn spectrum are similar to those calculated for a d5 ionic state, except the small pre-edge peak at E = 637.5 eV, which may result from a crystal field effect54,55. Meanwhile, the Co L3 and L2 XMCD peak shapes are similar to the broad spectrum of metallic Co56. To evaluate a possible contribution of an oxidized surface layer, reference XAS and XMCD spectra were measured for the isostructural room-temperature chiral magnet Co9Zn9Mn2 (see Supplementary note 4). The strong FM response observed at both Co and Mn edges in this compound indicates the dominant contribution of the target CoZnMn material to the XMCD signal, compared with any contribution from the oxidized surface.

Fig. 3: X-ray spectroscopy and scattering from Co7Zn7Mn6.
figure 3

XAS and XMCD spectra of Co7Zn7Mn6 measured near (a) Co and (b) Mn L2,3 edges at H = 50 kOe. c Temperature dependence of XMCD signals ∫L3 − 2∫L2 of Co and Mn integrated across corresponding edges. Error bars indicate the statistical error. d Schematic geometry of the REXS experiment. e Zero-field REXS patterns measured at Co (E = 776.5 eV) (left panel) and Mn (E = 638.5 eV) (right panel) edges at various temperatures. The white scale bar corresponds to reciprocal space distance of 0.01 Å−1. The central part of the scattering patterns is shadowed by the beamstop.

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To estimate the orbital to spin moment ratio we follow the approach given in ref. 56 and find μl/μs = 0.260 ± 0.012 (Co) and μl/μs = 0.0107 ± 0.0045 (Mn). For the latter, we assume a correction factor of 1.47 for the spin moment57. The same sign of measured high-field XMCD signal suggests a FM orientation of the Co and Mn sub-lattices relative to each other at high H. Element-selective magnetization loops are shown in Supplementary Fig. 6. Previously reported density functional theory calculations for Co7Zn7Mn6 anticipated Mn spin magnetic moments of 3.2 μB for the 12d site, and Co moments of 1.3 μB for the 8c site23. However, the small magnitude of the XMCD signal from Mn, as well as the neutron diffraction measurements in ref. 23 suggest that Mn moments are not fully polarized even in a high external magnetic field. This scenario is consistent with the μSR data whereby there is a weak low H dependence of λL just above Tg. Energy-integrated XMCD signals at both Co and Mn edges are proportional to the corresponding spin moments ∫L3dE − 2∫L2dE ~ μs/μB and Fig. 3c shows that each becomes significantly reduced at higher T. Notably, a large increment of Mn spin moment is found below Tg when fluctuations are suppressed.

Next, we turn to transmission resonant elastic X-ray scattering (REXS) to identify the magnetic elements responsible for the nanometric helical and skyrmion phases in Co7Zn7Mn615,23. Figure 3d shows a sketch of the experimental geometry, and Fig. 3e shows REXS patterns obtained at either the Co or Mn edges. Each pattern shows magnetic X-ray scattering peaks due to helical order with Q = 0.0112 Å−1 in a similar T range above 100 K. The length of the propagation vector corresponds to a real-space periodicity of 56 nm, which is different from that in bulk samples, but consistent with Lorentz microscopy data reported for the thin plates15. Below 90 K a magnetic REXS signal could not be distinguished from the remaining background charge scattering and charge-magnetic scattering interference.

Notably, the data in Fig. 3e show that the scattering from the helical order arises due to both magnetic elements, and is clearly at the Mn L3 absorption edge (E = 638.5 eV). Since only a weak XMCD signal is found for Mn at higher T, both the optical theorem and Kramers-Kronig relations imply that both imaginary and real parts of the refractive index have small magnetic contributions58. Indeed, when shown in the same intensity scale in Figure 3e, the element-selective REXS patterns clearly demonstrate the dominant contribution of Co ions in forming the helical order. The 2p core hole absorption for the L-edge XMCD and REXS measurements occur over fs time scale thus not being sensitive to the μs fluctuation dynamics observed by the μSR.

Therefore, from the combination of XMCD and REXS data from Co7Zn7Mn6, we confirm the dominant role of Co in forming helical, and thus skyrmion structures in finite H. This implicates the majority of Mn moments instead contribute to the static short-range spin correlations observed by MDNS, and the slow μs magnetic fluctuations observed by μSR. The X-ray data nonetheless imply a minority of Mn do in fact modulate helically, presumably this is due to a small Mn occupancy at the 8c site and thus a FM coupling with Co. These results are consistent with previous studies of Co8Zn8Mn459, and ab-initio calculations23. We thus conclude that the LTSk phase in Co7Zn7Mn6 is composed principally of modulating Co moments at the 8c site, while the LTSk phase stability relies on a coupling between Co and fluctuating Mn moments that mainly occupy the 12d site, with this coupling deduced to be FM from the XMCD data.

Unimportance of low-temperature cubic anisotropy

Next we consider the importance of low T cubic anisotropies, which recent theory shows can stabilize a low T equilibrium skyrmion phase chiral magnet in Cu2OSeO360,61,62. Figure 4a, b shows magnetization curves measured on bulk single crystal Co7Zn7Mn6 for H∣∣[111], H∣∣[110], H∣∣[100]. Details of the measurement are given in the “Methods”. From these data, no clear sign of magnetic anisotropy is seen at either T = 146 K and T = 50 K, these being Ts where the A-phase and LTSk phases are stable, respectively. This contrasts with similar data from Cu2OSeO3, where the importance of low T cubic and exchange anisotropies shows up firstly as a clear anisotropy of the H-dependent magnetization between the principal crystal directions61, and secondly by the observation that the LTSk phase is energetically favoured over the helical and conical states only for H∣∣[100]60,62.

Fig. 4: Magnetization curves and Lorentz microscopy.
figure 4

Magnetization curves of bulk Co7Zn7Mn6 measured for principal crystallographic axes at (a) 146 K and (b) 50 K. Evolution of the magnetic textures in thin plate Co7Zn7Mn6 measured on zero-field warming at (c) 50 K, d 60 K, e 100 K and (f) 140 K. The white scale bar corresponds to 500 nm. The insets show corresponding fast Fourier transform patterns. Details of the LTEM measurement protocol are given in “Methods”.

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As further evidence for the weak influence of low T cubic anisotropy in Co7Zn7Mn6, we turn to real-space LTEM images of the magnetic textures in a ~150 nm thin-plate sample—see “Methods” for further experimental details. Figur 4c–f shows LTEM images obtained in H = 0 and on warming from T = 50 K to 140 K. At T = 50 K, just above Tg, Fig. 4(c) shows an image of a highly-disordered helical state. Such a state hosting a large number of topological defects can provide a fertile soil for the nucleation of the LTSk phase in finite H63. Similar disordered magnetic textures were observed after ZFC (see Supplementary note 6). On warming, this ZF texture evolves into multiple worm-like helical domains (Fig. 4d, e), before finally forming well-defined helical stripes propagating along the cubic axes at 140 K (Fig. 4f). This T-evolution of the magnetic contrast suggests that near Tg, the helical texture has multiple local energy minima, likely introduced by frustration-induced magnetic disorder, while nearer to Tc thermal energy is sufficient to drive the re-orientation of well-ordered helices along the cubic axes as preferred by magnetic anisotropy.

According to the theory given in ref. 60, the LTSk phase stabilizes when the anisotropy energy density K exceeds the critical value Kc = 0.07μ0Hc2Ms, where Hc2 is the critical field of the conical to a field-polarized phase transition, and Ms is the saturation magnetization, resulting in Kc ≈ 2.7 kJ m−3 for Co7Zn7Mn6. Recently, the value of K ≈ 0.7 kJ m−3 was determined experimentally for Co7Zn7Mn6 by means of FM resonance spectroscopy64, which is well below the theoretical threshold of the anisotropy needed to stabilize the LTSk phase.

Taken together, the magnetization data from a bulk sample, and the LTEM data from a thin plate sample, each imply a weak influence of cubic (and exchange) anisotropies at low T in Co7Zn7Mn6. In turn, this suggests such anisotropies as less important for the LTSk phase stability than the geometric magnetic frustration effects established by our experiments.

In summary, our quantum beam studies of Co7Zn7Mn6 provide convincing evidence that geometric frustration-induced magnetic fluctuations are a key ingredient for stabilizing the LTSk in this system. The crucial role of magnetic fluctuations is suggested from muon spin relaxation experiments, which reveal a pronounced increase in magnetic relaxation rate over the thermal range of LTSk phase stability. From the analysis of both muon and magnetic neutron diffuse scattering data, we connect the physical origin of the magnetic fluctuations to the geometric frustration of the Mn moments on the hyper-kagomé coordinated 12d crystallographic site. This picture is corroborated by element-selective synchrotron X-ray experiments that reveal a FM coupling between the said Mn moments, and Co moments that occupy the different 8c crystallographic site and modulate to form the helical and skyrmion phases. Finally, from both bulk magnetization and electron microscopy experiments, scant evidence is obtained for a significant role of cubic or exchange anisotropies on the magnetic properties in the thermal range of LTSk phase stability.

Overall, our data do not support an understanding of LTSk phase stability in terms of the leading theories that require either thermal fluctuations9,17,18,19,20,21 or low T cubic anisotropies60,61,62. Instead, our experiments advocate the existence of a frustration-assisted skyrmion phase stability mechanism in Co7Zn7Mn6 that is based on frustration-induced magnetic fluctuations, and which calls for a detailed theoretical analysis. The elaboration of a such a stability mechanism could find general relevance in the ongoing research into topological magnetic textures amongst the multifarious frustrated magnets.

Methods

Bulk sample preparation

Bulk single crystals of Co7Zn7Mn6 and Co8Zn8Mn4 were grown by the Bridgman method as described in refs. 13,15 and studied in previous SANS experiments. The image of the crystal structure shown in Fig. 1a was depicted using VESTA software65.

A bulk polycrystalline Co7Zn7Mn6 specimen for X-ray spectroscopy was obtained from an ingot. Thin sample preparation for the REXS experiment is described in the soft X-ray experiment section.

Co7Zn7Mn6 powder was prepared from the polycrystalline ingots for the MDNS experiment.

μSR experiment

In the present μSR experiments we have used the same single crystals of Co7Zn7Mn6 and Co8Zn8Mn4 produced by the Bridgman method as studied in the previous SANS studies refs. 13,15. The spectra were measured using the GPS muon instrument at SμS (Paul Scherrer Institute, Villigen, Switzerland)66. The samples were placed into Cu envelopes and attached to a sample holder. The crystals were oriented with [001] axis parallel to the incoming muon beam and LF. In a ZF experiment, spin-polarized muons (μ+) are implanted in a magnetic specimen at normal incidence. The spins of the implanted muons precess about the local magnetic field at the μ+ site, thus, being sensitive to the local field distribution. Statically disordered and fluctuating local magnetic moments depolarize the muons. In a LF experiment, the external magnetic field is applied in the direction of the initial polarization of the μ+. Muons implanted into the sample decay into positrons, which are emitted preferentially in the direction of the μ+ spin. In the actual ZF and LF experiments, the emitted positrons are detected along with directions both parallel and anti-parallel (forward and backward directions) to the initial muon spin direction. The observed evolution of the asymmetry A(t) (difference between the number of positrons in the forward and backward directions) depends on both static and dynamical components of the local spin arrangement in the material. The maximum asymmetry and efficiencies of the forward and backward positron detectors were determined from standard calibration measurements made in the paramagnetic state at T = 250 K and T = 320 K for Co7Zn7Mn6 and Co8Zn8Mn4, respectively, with a small applied transverse magnetic field H = 20 Oe. The μSR spectra were analyzed using the Musrfit software67.

Diffuse neutron scattering

MDNS experiments over a Q range between 0.5 and 3.8 Å−1 were carried out at D7 instrument at the Institut Laue-Langevin (Grenoble, France)50 using xyz polarization analysis to isolate the magnetic component of the scattering. A powdered Co7Zn7Mn6 sample was loaded into double-wall aluminium container and loaded into an Orange-type He cryostat. For the experiment, an incoming neutron wavelength of 3.1 Å was used.

Mean-field model

The MF calculation for paramagnetic scattering is similar to that of ref. 68 following the general formalism of ref. 69, and constitutes a phenomenological Heisenberg model of isotropically interacting local magnetic Mn moments at the 12d positions (R + ra) in the average crystal structure23,24,70. Here R is a primitive lattice translation vector, r is a vector separation between spins, and a ∈ {1, 12} are the site indices. Refinements were parameterized by a set of exchange constants, \({J}_{ab}({\bf{R}},{{\bf{R}}}^{\prime})\), while the eigenvectors and eigenvalues of the 12 × 12 reciprocal space interaction matrix \({J}_{ab}({\bf{q}})={\sum }_{{\bf{R}}}{J}_{ab}({\bf{R}},0)\exp (i{\bf{q}}\cdot {\bf{R}})\) were used to evaluate the MF expression for the diffuse scattering intensity68. To perform the spherical average needed for the powder spectrum, the binning method of ref. 71 was used. Optimal values of the exchange constants were obtained by direct fitting to the experimental data, using a simulated annealing algorithm72.

RMC analysis

The MDNS data were fitted via RMC approach68 and the distance-dependent radial spin-pair correlation function is shown in Fig. S3. Note that, due to the limited amount of information that can be extracted from the MDNS profile, for both the RMC analysis and MF calculation, we used a simplified model that assumes the MDNS arises from only a 12d site that is fully occupied by magnetic Mn atoms. While this does not represent the true situation, we expect that the near half-filling of the 12d site with Mn is sufficient to induce frustration on the atomic lengthscale that is sufficient to affect the nanoscale chiral magnetism.

Resonant soft X-ray scattering

The REXS in transmission geometry was measured using SEXTANTS beamline (Sychrotron SOLEIL, France) on Co7Zn7Mn6 lamella fabricated by focused ion beam (FIB) lithography using Hitachi NB5000 setup (Fig. S5). Commercial silicon nitride membranes from Silson Ltd (Southam, UK) were processed for soft X-ray experiments. The front side of each membrane was coated with ≈4 μm-thick layer of gold to absorb the incoming beam. An ~100 nm-thick lamella of Co7Zn7Mn6 alloy with a surface containing (001) plane was cut from the bulk single crystal. An aperture with a diameter of 3.5 μm and asymmetric shape was drilled in the gold coating of the membrane to provide the transmission through the sample (Fig. S5a). The lamella was fixed to the membrane by using single tungsten contact to reduce the influence of the tensile strain73 (Fig. S5b). The small size and asymmetric shape of the aperture were chosen for the real-space reconstructions of magnetization textures by means of phase-retrieval algorithm59,74,75. Unfortunately, poor signal-to-noise ratio due to the background charge and charge-magnetic interference scattering did not allow real-space images to be obtained.

The RESOXS chamber at SEXTANTS beamline is equipped with a high-vacuum chamber with a background pressure of 10−9 mbar53. The scattered intensity was collected by an in-vacuum charge-coupled device area x-ray detector of 2048 × 2048 pixels (Princeton Instruments). An aluminium beamstop was introduced to protect the central part of the detector for the REXS experiment. The magnetic field provided by the electromagnet was applied perpendicular to the lamella surface and parallel to the incident circularly polarized X-ray beam (Fig. S6a). A He-flow-type cryostat was used to control the sample temperature.

To subtract the background arising from the charge scattering we have measured the scattering patterns with magnetic field of H = 1.6 kOe applied perpendicular to the sample plane corresponding to the induced FM state of the system. The sample position was corrected after each temperature change by using the transmission signal detected by a photodiode.

X-ray magnetic circular dichroism

The XMCD experiment was carried out at XTreme beamline at Swiss light source (Villigen, Switzerland) equipped with a vector-field cryomagnet. The beamline is dedicated to polarization-dependent soft XAS at a high magnetic field and low temperature52.

Polarization-dependent X-ray absorption spectra were measured with energy resolution of 0.1 eV using the surface-sensitive total electron yield method near Co and Mn L2,3 edges with right and left circularly polarized Xrays. To maximize the XMCD signal, the sample was cooled down to a temperature of 2 K under a magnetic field of 50 kOe applied along the X-ray beam and [110] crystallographic axis of the sample. Prior to the experiment, the sample was polished to minimize the effect of the surface oxidation. All XAS spectra were normalized to their corresponding edge jumps between the pre-L3 and post-L2 regions. All XMCD spectra were normalized to their corresponding XAS.

Magnetization measurements

DC magnetization measurements for a Co7Zn7Mn6 single crystal were performed using the vibrating sample magnetometer mode of a superconducting quantum interference device magnetometer (SQUID MPMS3, Quantum Design). Magnetic field values in the M(H) curves (Fig. 4a, b) for H∣∣[111] are calibrated by the demagnetization factor of 0.82H. Note that the difference in saturation field values in Figs. 1b and 3a, b is because of the difference between demagnetization factors for the samples used in the M(H) and μSR measurements.

Lorentz microscopy

LTEM measurements were performed on a FIB-made ~150 nm-thick (001)-oriented lamella with a transmission electron microscope (JEM-2100F) at an acceleration voltage of 200 kV. Defocus distances for LTEM images were set at −288 μm (Fig. 4c) and −480 μm (Fig. 4d–f). A liquid-helium cooling double-tilt holder was used to obtain LTEM images at low temperature. Figure 4c–f were measured on zero-field warming after saturating Co7Zn7Mn6 sample by the magnetic field of H = 2.85 kOe applied perpendicular to the sample plane at 50 K. Therefore, a certain fraction of metastable skyrmions embedded in the helical phase is observed up to T ~120 K.