Vital role of magnetocrystalline anisotropy in cubic chiral skyrmion hosts

Magnetic anisotropy is anticipated to govern the formation of exotic spin textures reported recently in cubic chiral magnets, like low-temperature tilted conical and skyrmion lattice (SkL) states and metastable SkLs with various lattice geometry. Motivated by these findings, we quantified the cubic anisotropy in a series of CoZnMn-type cubic chiral magnets. We found that the strength of anisotropy is highly enhanced towards low temperatures. Moreover, not only the magnitude but also the character of cubic anisotropy drastically varies upon changing the Co/Mn ratio. We correlate these changes with temperature- and composition-induced variations of the helical modulation vectors, deformations of skyrmions, structural rearrangements of the metastable SkLs and the large enhancement of Gilbert damping. Similar studies on magnetic anisotropy are required for the quantitative, unified description of the known stable and metastable modulated spin textures and for the systematic exploration of novel ones in this large class of skyrmion hosts.

Recently, cubic anisotropy was argued to play an essential role in the formation of a tilted conical and a low-temperature SkL state in Cu 2 OSeO 3 39-41 , another cubic chiral magnet, where a high-temperature SkL state located close to the Curie temperature, T C = 58 K, was reported earlier 15,18 . This low-temperature SkL forms when the tilted conical state gets destabilised above certain magnetic fields applied along one of the cubic 〈100〉 axes, being the easy axis of magnetic anisotropy 40,41 . Moreover, in the ultrathin-film form, cubic chiral magnets can also possess an easy-plane anisotropy in addition to the cubic anisotropies. With increasing strength of the easy-plane anisotropy, these systems are predicted to undergo a first-order phase transition from the SkL state into a square vortex-antivortex lattice 42 .
Apart from Bloch-type skyrmions formed by helical spin modulations in cubic chiral helimagnets, skyrmions and antiskyrmions can also form in crystals with axial symmetry. The lacunar spinels with a polar rhombohedral structure are the first compound family found to host a Néel-type SkL composed of spin cycloids 43 , while the antiskyrmion lattice state was first observed in Heusler compounds with non-centrosymmetric tetragonal structure 44 . In contrast to the small phase pocket of equilibrium SkLs in cubic chiral magnets, these Néel-type skyrmion and antiskyrmion lattices can be stable down to zero temperature 45,46 . In these axially symmetric materials, anisotropy emerges in two forms: (1) the anisotropy of the Dzyaloshinskii-Moriya interaction D and (2) the uniaxial magnetocrystalline anisotropy. The former confines the magnetic modulation vectors to the plane perpendicular to the unique high-symmetry axis, thereby largely increasing the thermal stability range of the SkL state, while the latter favours collinear states, hence, suppressing modulations [43][44][45][46][47][48][49][50] .
In this study, we quantify the anisotropy and analyse its impact on chiral magnetic states in a series of β-Mn-type (Co 0.5 Zn 0.5 ) 20−x Mn x compounds 51 . These materials belong to a family of cubic chiral magnets, hosting skyrmions even well above room temperature 52 . An omnifarious variety of exotic long-periodic magnetic patterns-in addition to the well-known helical, conical and hexagonal SkL-was observed in the Co-Zn-Mn family, including square and rhombic skyrmion lattices 37 , elongated skyrmions 38 , skyrmion chains, meron-antimeron lattices, chiral soliton lattices 53,54 , smectic liquid-crystalline arrangements of skyrmions 55 and domain-wall skyrmions 56 . Similar to Fe x Co 1−x Si 12,16 , their high-temperature equilibrium SkL state can be quenched via field cooling to a metastable state that persists down to low temperatures and can be robust even against the reversal of magnetic field 37,57 . The symmetry of the metastable SkL and the shape of the individual skyrmions show a large variation among these compounds when changing the Co/Mn ratio, which implies a substantial role of cubic anisotropy in determining the internal structure of individual skyrmions and the lattice geometries of these metastable states 38 . Figure 1 gives an overview of the stable and metastable magnetic phases observed as a function of temperature and composition. Just below the Curie temperature, a small pocket of SkL state emerges in all four target compounds at moderate magnetic fields. Via field cooling this thermodynamically stable hexagonal SkL state can be quenched to lower temperatures. Upon further cooling, in Co 7 Zn 7 Mn 6 , Co 8 Zn 8 Mn 4 and Co 9 Zn 9 Mn 2 , this metastable SkL gradually transforms into a square lattice 57 . In thin lamellae of Co 8 Zn 8 Mn 4 , a rectangular arrangement of elongated skyrmions was reported to form, where the elongated skyrmions co-align with the 〈100〉 cubic axes 38 . In contrast, the metastable SkL in Co 10 Zn 10 becomes rhombic with two q-vectors pointing along the 〈111〉 axes 54 . At the lowest temperatures, a spin-glass (SG) state emerges in Co 7 Zn 7 Mn 6 and Co 8 Zn 8 Mn 4 , where the magnetic modulations are disordered due to frustrated Mn spins 58 .
The stability mechanisms and underlying physics of these exotic spin textures, in particular, the influence of magnetic anisotropy, spin disorder and frustration are far from being understood. In the general model of cubic chiral magnets 29,59 , the wavelength of the chiral magnetic modulations scales with the ratio J/D, where J is the isotropic Heisenberg exchange. In addition to equilibrium spin textures, this relation was found to be also applicable for some of the metastable states: micromagnetic simulations have shown that the elongation of skyrmions, leading to a rectangular pattern in Co 8 Zn 8 Mn 4 , can be reproduced by the decrease of the J/D ratio in the presence of cubic anisotropy 60 . Easy-axis anisotropy was also suggested to stabilise domain-wall skyrmions 56 and the smectic skyrmion phase 55 . Therefore, measurements of the cubic anisotropy constants in the Co-Zn-Mn alloys are highly desired to correlate theories and experimental observations.
While these examples highlight the importance of magnetocrystalline anisotropy with regard to the stability, symmetry, and shape of individual skyrmions and SkLs, surprisingly, quantitative and systematic experimental studies on the role of anisotropy in cubic skyrmion hosts are virtually non-existing. This provides the main motivation for the present work. Here, using ferromagnetic resonance (FMR) spectroscopy, we investigate the evolution of magnetocrystalline anisotropy in (Co 0.5 Zn 0.5 ) 20−x Mn x compounds with composition (x = 0, 2, 4, 6) and temperature. Moreover, in comparison to the results of small-angle neutron scattering (SANS) experiments, we analyse if changes in anisotropy can account for changes observed in the orientation and the length of the q-vectors and may eventually lead to transformations between different SkL arrangements. This material class serves as an ideal playground for such investigations, since the material composition can be tuned whilst keeping the cubic chiral symmetry, but vastly changing the ordering temperature, the magnetic properties and-as demonstrated by this work-the strength and nature of cubic anisotropy. Figure 2 summarises the temperature dependence determined from the angular dependence of the FMR of the cubic anisotropy parameters for all the four compounds as described in detail in the Methods. As a common trend, the anisotropy energy decreases with increasing temperature and vanishes smoothly around the Curie temperature, except for Co 10 Zn 10 , where the temperature dependence is more complicated. It is important to note that the temperature dependence of the anisotropy energy density is governed by two factors, the temperature dependence of the bare microscopic anisotropies, originating from the spin-orbit coupling, and the temperature dependence of the ordered magnetic moment. Since the latter is saturated in these compounds  54 . The green area indicates the thermodynamically stable SkL pocket below the Curie temperature (blue circles and line). The regime of the metastable SkL phases with pale yellow shading, achieved via a field cooling process 37,38,54 , is separated into four regimes: (1) the transition from the equilibrium SkL into a metastable SkL, preserving the hexagonal symmetry, indicated by green triangles; (2) the crossover from the metastable hexagonal to a metastable square SkL phase in Co 7 Zn 7 Mn 6 , Co 8 Zn 8 Mn 4 and Co 9 Zn 9 Mn 2 , indicated by red squares, (3) the transformation of the metastable hexagonal into a metastable rhombic SkL state in Co 10 Zn 10 , marked by a violet rhombus and (4) the onset of a spin-glass phase (SG) in Co 7 Zn 7 Mn 6 and Co 8 Zn 8 Mn 4 , marked by light blue stars. Characteristic temperatures have only been determined for the four compounds and the lines connecting the symbols are to support an easy visualisation. (red). The inset shows the sixth-order anisotropy constant K 2 for Co 10 Zn 10 (green) and Co 9 Zn 9 Mn 2 (blue).

Magnetocrystalline anisotropy
typically below 0.6-0.8 T C , at lower temperatures the changes in K 1 and K 2 are governed by the variation of the bare anisotropy constants.
Besides its strong thermal variation, the anisotropy is sensitive to the Co/Mn ratio and drastically enhanced with increasing Co content. At low temperatures, Co 8 Zn 8 Mn 4 and Co 9 Zn 9 Mn 2 are respectively characterised by~10 and~30 times larger fourthorder cubic anisotropy (K 1 ) than Co 7 Zn 7 Mn 6 . Co 10 Zn 10 , however, does not follow this tendency, instead the K 1 anisotropy term becomes comparatively small in this compound. Increasing Co content also enhances the sixth-order cubic anisotropy (K 2 ). Reduced χ 2 statistics show that including K 2 does not improve the goodness of the fit in Co 7 Zn 7 Mn 6 and Co 8 Zn 8 Mn 4 , on the other hand, K 2 is comparable to K 1 in Co 9 Zn 9 Mn 2 and even becomes the dominant term in Co 10 Zn 10 at low temperatures. In contrast to the strong composition dependence of the anisotropy, the saturation magnetisation per magnetic ion varies by not more than 20% among the four compounds. In terms of the temperature dependence of the anisotropy, Co 10 Zn 10 is again an outlier: K 1 approaches zero around 250 K, i.e., well below the Curie temperature, where the K 2 term changes sign.
When the FM state emerges in a cubic crystal, the favoured direction of the magnetic moment is determined by the cubic anisotropy terms given in Eq. (5), assuming the magnetically induced structural distortion is negligible. When K 1 and K 2 are both positive and negative, the easy axes are the 〈100〉 or 〈111〉 directions, respectively. The former corresponds to the case of Co 7 Zn 7 Mn 6 and Co 8 Zn 8 Mn 4 , while the latter is realised in Co 10 Zn 10 . When they have opposite sign, like in Co 9 Zn 9 Mn 2 , the easy axes are determined by their ratio. Correspondingly, Co 7 Zn 7 Mn 6 , Co 8 Zn 8 Mn 4 and Co 9 Zn 9 Mn 2 are characterised by easy axes of the magnetisation along the 〈100〉 directions at all temperatures, though the anisotropy energy is considerably lowered upon approaching the Curie temperature. As an example, the anisotropy energy surface in the FM state of Co 8 Zn 8 Mn 4 is shown in Fig. 3d, as calculated using the magnetocrystalline anisotropy constants obtained from the fit of H res ðϕÞ in Fig. 3a. This energy surface, EðmÞ, represents the anisotropy of the FM state or equivalently the anisotropy experienced by an individual spin.
For a more complex magnetic texture, like the helical spin order, the magnetic anisotropy energy has to be evaluated for the whole magnetic unit cell, i.e., by summing over all spins in one period of the helix. Correspondingly, for a helix, where the q-vector is perpendicular to the spin rotation plane, the dependence of the magnetic energy on the orientation of the q-vector, EðqÞ, can be obtained by averaging the anisotropy energy of a single spin for all directions of m in the plane perpendicular to the q-vector. (In this simple approach, we treat the magnetic modulations as fully harmonic spin helices and disregard possible effects of anharmonicity.) When the magnetic easy axes are the 〈100〉 directions, as is the case in Co 8 Zn 8 Mn 4 , the preferred q-vectors also point along the 〈100〉 axes. In fact, the shapes of the two energy surfaces, EðmÞ and EðqÞ, are rather similar, as clear from Fig. 3d, g. Qualitatively, the same scenario applies also for Co 7 Zn 7 Mn 6 and Co 9 Zn 9 Mn 2 , though the weaker/stronger anisotropy in the former/ latter case leads to more spherical/rectangular energy surfaces compared to Co 8 Zn 8 Mn 4 .
In contrast, the magnetic easy axes point along the 〈111〉 directions in Co 10 Zn 10 at low temperatures, as clear from Fig. 3e. Although the helical modulation vectors also favour the 〈111〉 directions, the two energy surfaces, EðmÞ and EðqÞ, are quite different and q-vectors between 〈111〉 and 〈110〉 axes have nearly the same energy. Only the 〈100〉 axes are clearly distinguished as unfavoured directions, as can be traced in Fig. 3h. Moreover, with increasing temperature, both anisotropy terms weaken strongly and K 2 even changes sign at~250 K. Consequently, the 〈100〉 directions, being the hard axes of magnetisation at low-temperatures, nearly become the easy axes [ Fig. 3f]. In this temperature range, the eighth-order anisotropy constant K 3 (K 3 ≈ 23 kerg) becomes the dominant anisotropy term and defines the location of the shallow energy minima in the vicinity of the 〈100〉 axes. Interestingly, at these temperatures, the favoured q-vectors do not point along any of the high-symmetry directions, instead they are located around circles centred at 〈100〉 axes. In that regions, a Mexican hat-like form of EðqÞ can be discerned in Fig. 3i.

Magnetic modulations and anisotropy
In the following, we perform a comparative analysis of FMR and SANS data to reveal how the variation of cubic anisotropy with temperature and composition affects the magnetic modulations in this series of (Co 0.5 Zn 0.5 ) 20−x Mn x compounds. The SANS data are reproduced from Refs. 37,54 and re-analysed here to gain more insight into the angular distribution of the q-vectors. At low temperatures, the SANS intensity is confined to the 〈100〉 directions in Co 9 Zn 9 Mn 2 , Co 8 Zn 8 Mn 4 and Co 7 Zn 7 Mn 6 , i.e., the helical q-vectors favour these directions, as exemplified by the SANS image in Fig. 4a, recorded in the (001) plane for Co 8 Zn 8 Mn 4 . In contrast, in the SANS intensity of Co 10 Zn 10 , displayed over the (110) plane in Fig. 4b, the maxima are located at the 〈111〉 directions at low temperatures. (In both materials, the difference in the intensity of symmetry-equivalent q-vectors originates from unequal population of the corresponding helical domains.) The angular distribution of the q-vectors becomes broader with increasing temperature in both materials, as clear from the increase of the full width at half maximum (FWHM) for the SANS intensity maxima, plotted in Fig. 2. While in Co 10 Zn 10 , the FWHM is more than doubled (31 ∘ → 71 ∘ ) between 5 and 370 K, it is only increased by about 20% (23 ∘ → 29 ∘ ) in Co 8 Zn 8 Mn 4 between 50 and 200 K. In fact, the inverse of the FWHM follows a temperature dependence similar to that of the K 1 values. This shows that the anisotropy does not only select the favoured orientation of the q-vectors but also controls their spread around the preferred directions. In addition to the enhanced orientational disorder of the q-vectors towards higher temperatures, in Co 10 Zn 10 between 250 and 410 K, the favoured q-vectors move away from the 〈111〉 to 〈100〉 directions, as expected from the temperature-induced changes in EðqÞ and can be traced well in Fig. 4c.
Since apparently the helical q-vectors arrange in a way to minimise the anisotropy energy, in Fig. 4d-f, we directly compare the angular dependence of the SANS intensity and EðqÞ. In Co 8 Zn 8 Mn 4 , EðqÞ has minima along the 〈100〉 axes, coinciding with the SANS intensity maxima [ Fig. 3d]. In Co 10 Zn 10 , a similar confinement of the q-vectors by anisotropy is observed around the 〈111〉 axes at low temperature, though the width of the SANS intensity maxima are considerably broader due to the weaker anisotropy in this compound [ Fig. 4e]. Note that EðqÞ only shows little dependence in the range of ½111-½110-½111. At higher temperatures, this dependence is getting even weaker with an overall change in EðqÞ at 250 K approx. 2 kerg ⋅ cm −3 , compared to 15 kerg ⋅ cm −3 at 50 K. Consequently, though the global minima are located around circles centred at 〈100〉 axes, as shown in Fig. 3i, due to the weakness of anisotropy the helical q-vectors can get pinned along arbitrary directions, as clear from the 250 K data in Fig. 4f. The close correlation between minima of EðqÞ and maxima of the SANS intensity is observed at 410 K.

Magnetic modulations and gilbert damping
At the Curie temperature, the Gilbert-damping parameter α of these compounds ranges from 0.04 to 0.05, which is typical for ferromagnetic cobalt. Upon cooling, α is enhanced in all the four compounds; however, the strength of the increase depends strongly on the Co/Mn ratio, as clear from Fig. 5. It is nearly tripled in Co 7 Zn 7 Mn 6 , while it increases only by~50% in Co 10 Zn 10 .
Surprisingly, α and the length of the q-vectors show quantitatively similar thermal variations in all compounds except for Co 10 Zn 10 : They are both constant through the thermodynamically stable as well as the metastable hexagonal SkL phases, but exhibit a sudden uprise associated with the transformation from the metastable hexagonal to the square lattice phase. On the other hand, α and the FWHM of the q-vectors do not show a similar temperature dependence. Instead, the temperature dependence of the latter is reminiscent of that of K 1 , as seen in Fig. 2 and discussed earlier.
The strikingly similar temperature dependences of the q-vector length and the Gilbert damping imply a common underlying microscopic mechanism. A possible scenario is that antiferromagnetic correlations of Mn spins developing towards low temperatures weaken the ferromagnetic exchange (J) without significantly affecting the Dzyaloshinskii-Moriya interaction (D) between Co spins and, hence, increase the length of the q-vectors according to |q| ∝ D/J. This also explains why the increase of |q| starts at a lower temperature in Co 9 Zn 9 Mn 2 with higher Co content than in Co 8 Zn 8 Mn 4 and Co 7 Zn 7 Mn 6 , despite K 1 being larger in Co 9 Zn 9 Mn 2 than in Co 8 Zn 8 Mn 4 and Co 7 Zn 7 Mn 6 . In this picture, the lowtemperature enhancement of Gilbert damping is governed by static or dynamic disorder of Mn spins, since the Co spins are fully spin polarised in magnetic fields above 1 T, where α was determined from the FMR spectra. Thus, the parallel increase of |q| and α observed towards low temperatures can be naturally explained by the suppression of J due to increasing antiferromagnetic correlations between Mn spins. Geometrical frustration of the Mn spins occupying 8c Wyckoff positions at Mn concentrations above x = 4 was reported to drive the formation of a SG ground state 61 . A drop in α is found below the onset of the Fig. 3 Anisotropy surfaces. Angular dependence of the ferromagnetic resonance field, H res , in the ð110Þ plane is shown in a for Co 8 Zn 8 Mn 4 at 25 K and in b and c for Co 10 Zn 10 at 25 and 250 K, respectively. Black dots are the experimental data points, representing the resonance fields as fitted from the individual spectra that were recorded in 5 ∘ steps during a rotation of the magnetic field within the ð110Þ-plane, as sketched in Fig. 6. Red lines show the fit of the magnetocrystalline anisotropy using Eqs. (4) and (5). The error of the resonance field (grey coloured area) was estimated to be 5% of the linewidth of the resonance peak. d-f display the anisotropy energy surface of the ferromagnetic state, EðmÞ, calculated based on the K 1 , K 2 and K 3 values obtained from the fits in the corresponding panels above. g-i show the anisotropy energy of the helical state versus the orientation of the q-vector, EðqÞ, again calculated using the parameters obtained in the corresponding top panels. Arrows in panels d-i indicate the 〈100〉 axes. In the rotation plane of the magnetic field, the grey ð110Þ plane, 〈111〉 and 〈110〉 axes are also shown.
SG phase [see Fig. 5a, b], as observed in canonical SG materials 62 . Again, Co 10 Zn 10 is an outlier where both α and |q| exhibit little temperature dependence with no clear correlation between the two.

DISCUSSION
In the present work, we studied the evolution of the cubic anisotropy in a series of (Co 0.5 Zn 0.5 ) 20−x Mn x cubic chiral magnets with octahedral (O) symmetry. While these compounds are isostructural and the magnitude of the saturated magnetic moment does not vary by more than 20% upon changing the Co/Mn ratio, the magnetic interactions show a strong dependence on the composition, as already reflected by the large variation of the Curie temperature between 150 and 410 K. Here we show that this series offers an excellent laboratory to study the effects of anisotropy on chiral spin textures. This is because not only the magnitude of the anisotropy shows drastic variation with the Co/ Mn ratio and the temperature, but also the character of the anisotropy changes, as reflected by the reorientation of the easy axes from 〈100〉 to 〈111〉 directions with increasing Co content.
The fourth-order and-when relevant-also the sixth-order cubic anisotropies weaken towards higher temperatures in all these compounds. Remarkably, the eighth-order anisotropy also comes into play in Co 10 Zn 10 near room temperature, due to the weakness of the lower-order terms. As a consequence, the helical q-vectors in this compound experience a Mexican hat-like potential centred around the 〈100〉 directions. A similar deviation of the favoured q-vectors from high-symmetry directions has been reported in Co/Fe-doped MnSi, though in that case the effect originates from a competition of fourth-and sixth-order anisotropies allowed by the tetrahedral cubic (T) symmetry 63 . We found that the cubic anisotropy terms do not only select the favoured orientation of the q-vectors in the helical state but also control the width of their angular distribution around the preferred directions. Correspondingly, the suppression of magnetic anisotropy towards higher temperatures leads to an increasing orientational disorder of the helical q-vectors.
Besides defining the orientation of the q-vectors of the equilibrium chiral spin states, the anisotropy also plays a role in the transformation of metastable hexagonal SkL to square or rhombic lattices. In these compounds, the D/J ratio increases with lowering the temperature, as reflected by the elongation of the q-vectors upon cooling. Correspondingly, the originally closepacked hexagonal arrangement of skyrmions within the metastable SkLs becomes loose towards low temperatures, due to the conservation of the skyrmion number. Releasing the close-packing constraint leads to anisotropy-controlled rearrangements of the metastable skyrmions. In Co 9 Zn 9 Mn 2 , Co 8 Zn 8 Mn 4 and Co 7 Zn 7 Mn 6 a square lattice or a rectangular arrangement of elongated skyrmions forms with q-vectors along the 〈100〉 easy directions. In the rectangular arrangement, observed by real-space imaging in thin samples, the skyrmions are forced to elongate parallel to the magnetic easy axes. Interestingly, the Gilbert damping shows a temperature dependence similar to |q|, thus, can be used to locate these transitions.
The anisotropy plays an even more important role in the distortion of the metastable hexagonal SkL in Co 10 Zn 10 . In the other three compounds, the change in D/J determines the temperature range where the hexagonal to square lattice transition takes place  Fig. 3g-i, while the angular-dependent SANS intensity curves are obtained by radial averaging of the SANS intensities in panels a-c. and the anisotropy only governs the type of the distortion. This hierarchy is clear from the fact that the transition temperatures do not correlate with the magnitudes of the anisotropy for Co 9 Zn 9 Mn 2 , Co 8 Zn 8 Mn 4 and Co 7 Zn 7 Mn 6 . In contrast, the length of the q-vectors shows very weak temperature dependence in Co 10 Zn 10 compared to the other compounds. Still, the distortion of the metastable hexagonal SkL to a rhombic lattice with q-vectors along the 〈111〉 easy directions takes place at the highest temperature. This is likely because the rearrangement into the rhombic SkL requires a smaller distortion than to the square-lattice form and can be driven by the anisotropy.
Finally, we discuss why an anisotropy-driven low-temperature equilibrium SkL-like the one observed in Cu 2 OSeO 3 -does not emerge in Co 7 Zn 7 Mn 6 , Co 8 Zn 8 Mn 4 , Co 9 Zn 9 Mn 2 , though all four compounds have magnetic easy axes along the 〈100〉 directions. (Though in Co 7 Zn 7 Mn 6 a low-temperature equilibrium SkL has been observed, it forms due to magnetic frustration and is not stabilised by cubic anisotropy 58 .) According to the theory in Ref. 39 , the low-temperature SkL can emerge for magnetic fields applied along the 〈100〉 directions, if K 1 exceeds the critical value of K c = 0.07 μ 0 H c2 M s , where H c2 is the critical field separating the conical and the field-polarised states and M s is the saturation magnetisation. The estimated K c values, based on the saturation magnetisation and critical field values obtained in previous works 54,64 , are K c = 27 kerg ⋅ cm −3 for Co 7 Zn 7 Mn 6 , K c = 113 kerg ⋅ cm −3 for Co 8 Zn 8 Mn 4 and K c = 132 kerg ⋅ cm −3 for Co 9 Zn 9 Mn 2 . For the first two, K 1 is somewhat below the critical value needed to stabilise the low-temperature SkL. Though K 1 is somewhat larger than the critical value in Co 9 Zn 9 Mn 2 at low temperatures, its effect is weakened by the large K 2 of opposite sign. This explains why the low-temperature SkL is not realised in this Co-Zn-Mn series, though the absolute values of K 1 determined for Co 8 Zn 8 Mn 4 and Co 9 Zn 9 Mn 2 are orders of magnitude higher than in Cu 2 OSeO 3 , K 1 = 6 kerg ⋅ cm −3 65 .
In conclusion, we believe that our work, demonstrating how the quantitative determination of anisotropy terms can promote the understanding of complex spin states and related phenomena, will trigger similar experimental studies as well as theoretical approaches accounting for the effects of magnetic anisotropy on the properties of chiral magnets.

Ferromagnetic resonance
In order to determine the magnetocrystalline anisotropy energy, we studied the angular dependence of the FMR field in the field-polarised ferromagnetic state of (Co 0.5 Zn 0.5 ) 20−x Mn x with x = 0, 2, 4, 6 at temperatures ranging from 4 to 400 K. Single crystals of Co 10 Zn 10 were grown by a self-flux method from pure Co and Zn sealed in an evacuated quartz tube. Single crystals of Co 9 Zn 9 Mn 2 , Co 8 Zn 8 Mn 4 and Co 7 Zn 7 Mn 6 were grown by the Bridgman method from pure Co, Zn and Mn sealed in an evacuated quartz tube. The specimens were {110} cuts of single crystals polished to a cylindrical disc-form with a typical diameter of~1 mm and thickness of~200 μm. Field-swept FMR spectra were recorded at two different frequencies, X-band (9.4 GHz) and Q-band (34 GHz), in 5 ∘ steps upon rotation of the magnetic field in the ð110Þ plane. Since the fieldpolarised ferromagnetic state has already been achieved at 1 kOe in all the four compounds irrespective of the temperature, the FMR line, located always above 3 kOe, does not overlap with either the collective modes of the modulated states or spectral features related to metamagnetic transitions. Blue circles show the q-vector length in the SkL states, obtained during field cooling, and red squares in the helical phase, while green triangles indicate α. The green area represents the equilibrium SkL state just below the Curie temperature (blue line). The green shaded area represents the regime over which the metastable SkL preserves the hexagonal symmetry, while the red and violet shaded areas indicate the square and rhombic SkL states, respectively. The light blue shaded area stands for the spin-glass phase (see also Fig. 1). (6) Here, ϕ and ϕ H are the azimuth angles between the crystallographic x-axis and the projection of the magnetisation and field into the xy plane and θ and θ H are the polar angles between the crystallographic z-axis and the magnetisation and field. Examples of H res ðϕÞ curves and the corresponding fits are displayed in the inset of Figs. 6 and 3a-c for Co 8 Zn 8 Mn 4 at low temperature and for Co 10 Zn 10 both at low and near room temperature.

Small-angle-neutron scattering
In order to correlate changes of the anisotropy with transformations of the magnetic modulations, we also present SANS data for Co 8 Zn 8 Mn 4 and Co 10 Zn 10 that are reproduced from previous works 37,54 . SANS images were recorded in the (100) plane for Co 8 Zn 8 Mn 4 and in the (110) plane for Co 10 Zn 10 . The SANS measurements were performed at the SANS-I instrument in the Paul Scherrer Institute for Co 10 Zn 10 , Co 9 Zn 9 Mn 2 and Co 8 Zn 8 Mn 4 , and at the D33 instrument in the Institute Laue-Langevin for Co 7 Zn 7 Mn 6 . The neutron wavelength was selected to be 10 Ångstrøm. SANS intensities on the reciprocal q x q y plane were obtained summing the multiple data taken over rocking scans by subtracting nuclear and instrumental background signals. To determine the orientation of the favoured q-vectors, the angular dependence of the SANS intensity has been fitted with Gaussian peaks. The width of peaks corresponding to symmetry-equivalent directions has been averaged. In addition, we also compare the temperature dependence of Gilbert damping and the length of the q-vectors, |q|, for all the four compounds, the latter reproduced from Ref. 54 .

DATA AVAILABILITY
The data sets measured and analysed for the current study are available from the corresponding author upon reasonable request.