Abstract
Skyrmion lattices (SkL) in centrosymmetric materials typically have a magnetic period on the nanometerscale, so that the coupling between magnetic superstructures and the underlying crystal lattice cannot be neglected. We reveal the commensurate locking of a SkL to the atomic lattice in Gd_{3}Ru_{4}Al_{12} via highresolution resonant elastic xray scattering (REXS). Weak easyplane magnetic anisotropy, demonstrated here by a combination of ferromagnetic resonance and REXS, penalizes placing a skyrmion core on a site of the atomic lattice. Under these conditions, a commensurate SkL, locked to the crystal lattice, is stable at finite temperatures – but gives way to a competing incommensurate ground state upon cooling. We discuss the role of Umklappterms in the Hamiltonian for the formation of this latticelocked state, its magnetic space group, and the role of slight discommensurations, or (line) defects in the magnetic texture. We also contrast our findings with the case of SkLs in noncentrosymmetric material platforms.
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Introduction
Magnetic skyrmion lattices (SkLs) are periodic arrays of vortexlike spin structures. In SkLs, magnetic moments are twisted into a knot, covering all directions of a sphere as we traverse a single magnetic unit cell (UC) (Fig. 1a)^{1,2,3}. These vortices were first described as topological solitons using the concepts of field theory, and such continuum models are most suitable when the magnetic UC is at least two orders of magnitude larger than the underlying crystallographic UC^{3,4,5}. With a focus on frustrated, i.e. competing, interactions, recent theoretical work^{6,7,8,9,10} has proposed SkL formation in a highsymmetry context without spinorbit driven DzyaloshinskiiMoriya interactions, paving the way for the experimental observation of SkL phases with magnetic period on the nanometerscale in centrosymmetric insulators and metals^{3,11,12,13,14}. These quasidiscrete SkLs have raised hopes of enhanced functional responses, especially those related to the interplay of emergent electromagnetic fields with conduction electron (Bloch) waves, or with incident light waves^{3,15,16,17,18,19,20}.
Evidence for coupling between the atomic lattice and skyrmion textures with lattice spacing 2 − 3 nanometers has emerged in tetragonal magnets: Centrosymmetric alloys host square and rhombic skyrmion lattices^{3,21}, and noncentrosymmetric EuNiGe_{3} exhibits a fascinating helicity reversal upon entering the SkL phase, where the magnetic texture breaks the sense of rotation prescribed by its polar structure^{13,14}. A key open challenge is the demonstration of a commensurate locking (Clocking) transition of the SkL’s spin superstructure to the underlying lattice potential in such a centrosymmetric bulk material. This phenomenon is conceptually related to instabilities of the skyrmion vortex core anticipated for spin1 systems^{22} and its observation would provide a bridge between – usually – largescale SkL spin textures in materials with broken inversion symmetry, and canted antiferromagnetism on the scale of a single unit cell. Indeed, theory predicts such Clocking based on RudermanKittelKasuyaYosida (RKKY) interactions and magnetic anisotropy, when the length scale of magnetic textures approaches the size of a crystallographic UC^{23}. Among inversion breaking material platforms, the hcpFe/Ir(111) interface has been reported to exhibit Clocking using imaging techniques, although fccFe/Ir(111) forms a latticeincommensurate structure^{16,24,25,26}. However, such locking between the periodicity of a magnetic skyrmion lattice and the underlying crystal structure has never been observed in a bulk material.
Using precise resonant xray measurements, we report a commensurate skyrmion lattice (CSkL) surrounded by incommensurate (IC) phases in bulk samples of the centrosymmetric intermetallic Gd_{3}Ru_{4}Al_{12}, locked to the distorted kagome network of magnetic gadolinium ions. We discuss this state based on (weak) singleion anisotropy K_{1}, as supported by electronspin resonance experiments.
For large classical spins, calculations on both triangular and breathing kagome lattices show that, if the singleion anisotropy is of easyplane type (easyaxis type), a commensurate skyrmion vortex may gain energy by locating its core at an interstitial site (on a lattice site)^{23,27}; an incommensurate skyrmion lattice does not benefit from this type of energy gain (Supplementary Fig. 5). We illustrate this point in the lower part of Fig. 1a, depicting a realistic magnetic structure model for the CSkL of Gd_{3}Ru_{4}Al_{12}, described by normalized vectors n(x, y), which is mapped onto a sphere using a stereographic projection, see Methods. Here, magnetic moments are conspicuously absent at the poles (Fig. 1a, upper). The sparsity of magnetic moments at the poles becomes more apparent when unfolding the sphere using a cartographic projection (Fig. 1b and Supplementary Note 1).
Results
Observation of a commensurate skyrmion lattice (CSkL)
We use elastic xray scattering in resonance with the L_{2,3} absorption edge of gadolinium, setting the sample in reflection geometry to precisely detect the magnetic period of each magnetic phase (Fig. 1c and Methods). Reporting data from a synchrotron light source, Fig. 1d depicts the core observation of this work: At moderate temperatures T = 7 − 8 K, a magnetic field B applied along the caxis drives the incommensurate proper screw ground state (ICPS) into a commensurate skyrmion lattice phase (CSkL), and again to incommensurate fanlike order (ICFan). The Clocking of the SkL was not observed in a previous study^{11}. The present and previous results are compared and summarized in Supplementary Fig. 4. Let a^{*} and c^{*} be the reciprocal space lattice constants for our target material Gd_{3}Ru_{4}Al_{12} in the hexagonal P6_{3}/mmc space group, where magnetic gadolinium ions form a kagome (star of David) lattice with a breathing distortion, corresponding to alternating bond distances. In zero magnetic field, the magnetic modulation wavevector is q = (q, 0, 0) with q ≈ 0.272 a^{*} or wavelength λ ≈ 3.7 L_{uc}, where L_{uc} is the dimension of the crystallographic UC projected parallel to q (Supplementary Fig. 1). The wavevector’s length q decreases with B in ICPS, approaching a discontinuous jump at the firstorder transition^{11} towards q = 0.25 a^{*} in CSkL. The magnetic period λ in CSkL is very close to 4 L_{uc}, with a slight offset indicated by two dashed horizontal lines in Fig. 1d. The role of this slight offset, or discommensuration λ_{disc}, is discussed below. To further support the Clocking at q = 0.25 a^{*}, we demonstrate in Supplementary Fig. 7 that the peak profiles are within less than one standard deviation from the commensurate value, and in Supplementary Fig. 10 that q has weak temperature dependence in CSkL as compared to ICPS.
In the regime labeled as CSkL in Fig. 1, previous realspace imaging experiments have observed vortex structures^{11}, and precise Hall effect measurements demonstrate that the noncoplanar magnetic state that is easily destroyed by a slight inplane magnetic field^{28}. However, neither this prior work nor the present REXS technique are able to determine whether the skyrmion core is located on a crystallographic lattice site, or on an interstitial site. This question of the relative phase shift between magnetic texture and crystal lattice can be addressed by measurements of singleion anisotropy, combined with theoretical modeling: We turn to the ferromagnetic resonance (FMR) technique in the following section.
Singleion anisotropy in Gd_{3}Ru_{4}Al_{12}
We prepared a cylindrical diskshaped sample for ferromagnetic resonance (FMR) experiments, spanned by the crystallographic a and c directions (Fig. 2a, right inset). This highly symmetric geometry allows for simple data analysis when rotating the magnetic field in the plane of the disk, see Methods. In the experiment, we drive the crystal into the fieldaligned ferromagnetic (FAFM) state with a large magnetic field, irradiate it with microwaves of frequency ν = 210 or 314 GHz, and observe a change of its reflectivity when the microwaves excite a resonance between momentup and down states (Fig. 2a, left inset).
In Fig. 2a, the anisotropic part of the free energy \({F}_{{{{\rm{anis}}}}}={K}_{1}{\cos }^{2}(\theta )\) is deduced in FAFM from a fit to the angular dependence of the FMR resonance field \({B}_{{{{\rm{res}}}}}\); we disregard anisotropy constants beyond the first order, see Methods. Using the saturation magnetization M_{S} = 7 μ_{B} / Gd^{3+}, our fit yields easyplane anisotropy with K_{1} = − 0.13 meV / Gd^{3+}. Therefore, the anisotropy field B_{ani} can be calculated as \({\mu }_{0}\left\vert 2{K}_{1}\right\vert /{M}_{{{{\rm{S}}}}}=0.74\,\)Tesla. As compared to magnetization measurements in Supplementary Fig. 11, the present FMR study yields enhanced precision; fielddependent FMR experiments separate gfactor anisotropy and exchange anisotropy from the singleion term, see Methods. Figure 2b, c illustrates the resulting isoenergy surfaces F_{anis}(θ, φ) in zero (finite) magnetic field along the caxis, respectively, where the spherical coordinates refer to the direction of the sample’s bulk magnetization M.
Anisotropy and anharmonic distortion of proper screw and skyrmion phases
Easyplane anisotropy (K_{1} < 0) can also be verified semiquantitatively by resonant elastic xray scattering in the ordered phases with periodic longrange order. Figure 1c shows the geometry of our experiment with polarization analysis: the purple scattering plane is spanned by the wavevectors k_{i} and k_{f} of the incoming and outgoing xray beams, with beam polarization ε_{i} and ε_{f}, respectively. We choose the incoming beam polarization ε_{i} to lie within the scattering plane for all our experiments (πpolarization). While the data in Fig. 1d represents a sum of scattered photons with all possible ε_{f}, we now add an analyser plate before the xray detector to separate two components of the scattered beam: \({I}_{\pi {\pi }^{{\prime} }}\) and \({I}_{\pi {\sigma }^{{\prime} }}\) with ε_{f} within or perpendicular to the scattering plane, respectively. From the scattered intensities at various magnetic reflections, we extract the ratio, see Methods,
with m_{ab} the component of the modulated magnetic moment in the scattering plane. The expected behavior of ICPS (spin plane ac) and IC cycloid (spin plane a^{*}c) is indicated in Fig. 3b by black and green dashed lines, respectively; a line with positive slope indicates ICPS character. Supplementary Fig. 3 shows representative raw data, as used to create this panel.
Beyond identifying the PS character of the spin modulation, this analysis allows us to estimate the effect of singleion anisotropy in ICPS. Specifically, we observe an elliptic distortion, i.e., a deviation from the proper screw model. Figure 3b, c displays maximum values of \(R{\sin }^{2}(2\theta )\) around 1.5 (ICPS) and 7 (CSkL), so that Eq. (1) delivers m_{ab}(q)/m_{c}(q) around 1.2 (ICPS) and 2.6 (CSkL). Figure 3b, inset, illustrates the proposed anharmonic distortion in ICPS, where magnetic moments prefer to tilt towards the basal plane to gain anisotropy energy. In Fig. 3d, a simulation of REXS anisotropy as a function of K_{1} for a spin model in both ICPS and CSkL (Supplementary Note 1) shows that the experimentally observed ellipticity is consistent with the results of FMR in Fig. 2. Given the robust observation of K_{1} < 0 via various experimental techniques, we conclude that a CSkL in magnetic space group \(P{6}_{3}{2}^{{\prime} }{2}^{{\prime} }\) and with skyrmion core between lattice sites is stable in Gd_{3}Ru_{4}Al_{12} (Supplementary Note 2).
Competition of commensurate and incommensurate phases
Figure 4a shows a magnetic field scan of xray scattering intensity at the lowest accessible temperature, T = 1.5 K; see Supplementary Fig. 4 for detailed phase diagrams and comparison to prior work. In contrast to Fig. 1d, the CSkL phase is now absent, being replaced by an incommensurate transverse conical (ICTC) state^{11}.
We compare the experimental and theoretical results for Gd_{3}Ru_{4}Al_{12} to earlier studies of magnetism in elemental rare earth metals. Various scenarios for the interplay of commensurate (C) and IC phases have been advanced^{29,30}: One typical observation in systems with strong singleion anisotropy is the appearance of spinslip structures in the IC order, and eventual squaring up at low temperatures. In this scenario, thermal fluctuations at higher T locally reduce the ordered magnetic moment, allowing for the formation of IC magnetism close to the Néel point, e.g. in elemental Holmium^{29}. Secondly, when singleion anisotropy is much weaker than exchange interactions, C orders are favored at higher temperatures where shortrange correlations dominate. However, they can give way to IC ground states as furtherneighbor exchange gains importance upon cooling (e.g. ref. ^{31}). Thirdly, in inversion breaking systems with strong DzyaloshinskiiMoriya interactions, ICC transitions appear at low temperature when solitonic distortions are introduced into an IC order via a magnetic field^{32,33,34,35}, especially if a large charge gap is opened due to nesting^{35}. The present study generalizes the second scenario to the case of complex, twisted magnetic textures, such as the CSkL, with an important role for (weak) magnetic anisotropy.
We consider the observation of Clocking induced by magnetic fields, in our experiment, based on the singleion contribution to the spin Hamiltonian (Supplementary Note 3)
where r = R + d is the position of a magnetic ion, decomposed into a unit cell coordinate R and an intracell coordinate; q and G are the momentum in the first Brillouin zone and a reciprocal lattice vector, respectively. Typically, a small number of Fourier modes q = q_{ν} and G = 0 are selected in models of incommensurate helimagnetic ordering^{12,21,28}. Meanwhile, Clocking is favored by the Umklapp terms G ≠ 0^{36,37}, representing a coupling between the primary Fourier mode of a helimagnet and its higher harmonics. Specifically for the q = 0.25 a^{*} CSkL in Gd_{3}Ru_{4}Al_{12}, \({{{\mathcal{S}}}}(3{{{\bf{q}}}})\cdot {{{\mathcal{S}}}}({{{\bf{q}}}})\) and \({{{\mathcal{S}}}}{(2{{{\bf{q}}}})}^{2}\) are the leading contributors, adding to G = (1, 0, 0) and coupling the helimagnetic order to the lattice. Application of a magnetic field to ICPS enhances the elliptic distortion as demonstrated in Fig. 3, amplifies anharmonicity, shifts q away from the value preferred by the exchange interaction and towards commensurability, and ultimately induces Clocking between the spin texture and the underlying lattice. In Supplementary Note 3, we thus derive an expression for the energetic contribution that depends on the position of the skyrmion core. Nevertheless, a full numerical treatment of Eq. (2) with large numbers of Fourier modes, to capture changes in the optimal q – as well as changes in the cycloidal / screw character of the modulation – as a function of magnetic field and temperature, remains a challenge at present (Supplementary Note 1).
To explain the effect of singleion anisotropy more intuitively, we initialize CSkL and CPS on the lattice and allow the magnetic moments to relax in the combined potential of exchange interaction, magnetic anisotropy, and external field (white/green arrows in Fig. 4b, c). The red arrows show a significant distortion of the textures, especially around the south pole. This density of strongly distorted moments is much larger for the CPS state with quasi onedimensional spin texture, which has an extended south pole region. In particular, the south pole direction of the magnetic moment sphere in Fig. 1a corresponds to a point (to a line) in case of a SkL (of a PS). We also calculated the zprojection of magnetic moments for a spin model of CPS, CSkL, and other orders, using our experimental K_{1} from FMR, thus confirming numerically the favorable pinning of the CSkL due to stronger higher harmonics (Supplementary Fig. 5).
Discussion
Figure 1d reveals a discommensuration of the magnetic lattice, i.e., a slight offset from q/a^{*} = 0.25 that indicates occasional magnetic defects. The discommensuration effect is well understood in onedimensional chain systems^{38}: For example, the introduction of small amounts of chemical disorder causes proliferation of discommensurations in spinPeierls chains, as manifested in a drop of λ_{disc} that ultimately destroys the commensurate (C) order^{39,40,41,42}. Discommensurations in two dimensions, mostly line defects, also appear for surfaceadsorbed atoms and in the multidirectional chargedensity wave state of transition metal dichalcogenides such as 2HTaSe_{2}, where CIC transitions have been extensively studied^{36,37}. Among materials with twodimensional spin textures, we compare to the CSkL observed in an interfacial system^{24,25,26}: For inversionbreaking hcpFe/Ir(111) (for centrosymmetric Gd_{3}Ru_{4}Al_{12}), the spin structure is found to be commensurate within less than 10% (within 0.7 %) of the magnetic period, and there is no (there is) evidence of CIC transitions by cooling or application of a magnetic field. Based on measurements of magnetic anisotropy and modeling, we argue here that Gd_{3}Ru_{4}Al_{12} has skyrmion cores on interstitial lattice sites, as does hcpFe/Ir(111); further that the discommensuration represents the appearance of line defects (characteristic spacing λ_{disc} = 430 nm) in the former – while phaseslip domain walls (spacing ~ 10 nm) and domains of the net magnetization (spacing ~ 30 nm) have been observed in the latter. We note that domain walls of the skyrmion helicity may appear in the bulk CSkL of centrosymmetric materials^{6,43}; these are forbidden in inversion breaking platforms such as hcpFe/Ir(111).
Finally, Fig. 4d summarizes recent progress on material search and magnetic structure studies of noncoplanar magnets with latticeaveraged net spin chirality, especially SkL host compounds^{3,13,14,44,45}. As the dimension λ of the magnetic UC increases, magnetic moments cover all directions of the S_{2} sphere, and the maximum gap in angular coverage (as defined in percent of 4π) approaches zero. For example, noncoplanar antiferromagnets on the left side of Fig. 4d leave large fractions of S_{2} out of reach of magnetic moments. Being located at the center of the plot, the CSkL in the centrosymmetric, breathing kagome magnet Gd_{3}Ru_{4}Al_{12} represents an essential link between longperiod, incommensurate magnetic textures, stabilized e.g. by DzyaloshinskiiMoriya interactions in MnSi^{3,46}, and socalled topological antiferromagnets with canted magnetic moments, such as pyrochlore systems^{44}. Although the present CSkL is locked to the crystal lattice by weak inplane magnetic anisotropy, confirmed here by ferromagnetic resonance experiments, the magnetic unit cell is large enough so that moments densely cover S_{2}, leaving only marginal gaps on the order of 5% of 4π.
Methods
Cartographic projection
For visualizing the magnetic texture in Fig. 1a, we mapped a magnetic skyrmion onto a sphere in three dimensions, and subsequently mapped from the sphere onto a planar surface using the cartographic Nicolosi globular projection. Let (θ, φ) denote a point where a line between (x, y) and the north pole of the sphere penetrates the sphere’s surface. These spherical coordinates are identified with (x, y) in Fig. 1a, upper side, while a point on the sphere is projected back onto the plane using the formalism in ref. ^{47}.
Starting from polar angles θ, φ on the sphere, the x and y coordinates of the planar projection are defined as
with functions
The sphere’s radius R and the reference point for the longitude φ_{0} can be set to 1 and 0, respectively.
Sample preparation and characterization
We prepared polycrstals of Gd_{3}Ru_{4}Al_{12} by arc melting of the constituent elements in argon atmosphere, carefully turning pellets at least three times. Subsequently, single crystals were grown from these polycrystals by the floating zone technique under argon flow. Before the melting step, the halogenlamp based furnace was evacuated to a base pressure of 8 ⋅ 10^{−4} Pa, pumping for about three hours. The growth speed in the float zoning step was 2 − 4 mm/hr. We crushed single crystalline pieces into a fine powder for xray diffraction and refined the data by the Rietveld method using the RIETAN software package^{48}. In this analysis, we did not find impurity phases with volume fraction larger than 4%. Further, singlecrystalline pieces were oriented using a Laue camera, cut with a diamond saw, and handpolished to a high sheen (~ 1 μm grit) for single crystal xray diffraction measurements. Using a microscope with Nicolet prism, singlecrystalline pieces with small amounts of RuAl_{3} impurity phase, which forms teardrop shaped inclusions of < 2% volume fraction that are hard to detect by laboratory xrays, were excluded based on patterns in the surfacereflected light. Finally, magnetization measurements (MH curve at T = 2 K, MT curve at μ_{0}H = 0.1 T) were used to confirm a systematic evolution of longrange magnetic order in these single crystals.
Ferromagnetic resonance measurements
In order to avoid unwanted shape anisotropy effects, we polished the Gd_{3}Ru_{4}Al_{12} samples cut within the ac plane into a cylindrical disc form, with ellipticity below 4 %. The ratio of diameter to thickness was 12 for the final disc, with a thickness of 120 μm.
Highfield ferromagnetic resonance (FMR) measurements were performed at École Polytechnique Fédérale de Lausanne (EPFL, Switzerland) using a home built, highsensitivity, quasioptical spectrometer operating in the range of 50 − 420 GHz. This instrument covers a broad magnetic field regime up to 16 T, using a superconducting magnet. Its variable temperature insert operates across the temperature range 1.5−300 K, using the dynamic flow of helium gas, or liquid helium, through a heat exchanger right below the sample space. For more details, see ref. ^{49}.
The overall temperature accuracy of the system is 0.1 K. The polished disclike sample is mounted on a goniometer with the ac plane coinciding with plane of rotation for the static magnetic field, the angular position of which is controllable and detectable via a potentiometer. Rotation proceeded in 5 − 10 degree steps, at sample temperature T = 2 K. The signaltonoise ratio of the spectra is improved by recording the fieldderivative of the absorbed power dP/dH using a lockin technique with magnetic field modulation. The angulardependent resonance data can be evaluated using the SmitBeliersSuhl formula as described in ref. ^{50}. Using the saturation magnetization M_{S} = 7 μ_{B} / Gd^{3+}, our fit yields the easyplane anisotropy constant K_{1} = − 1.944 ⋅ 10^{6} erg/cm^{3} ( − 0.13 meV / Gd^{3+}) and the temperature independent, isotropic gfactor 2.005.
Resonant xray scattering experiments (GdL _{2,3})
Resonant elastic xray scattering (REXS) experiments are carried out in reflection geometry at RIKEN beamline BL19LXU of SPring8 and beamline BL3A of Photon Factory, KEK, with the sample mounted inside a cryomagnet. The preparation of Sample A, used to obtain the data in Fig. 3b, is discussed in ref. ^{11}. Sample B, which was used to obtain all other data in this manuscript, has a surface perpendicular to the [110] axis, and was polished to reduce loss of intensity by diffuse scattering of xrays. The energy of incident xrays is matched to the L_{2} or L_{3} absorption edge of Gadolinium, where magnetic scattering involves virtual excitations from the 2p to the 5d atomic shells (Supplementary Fig. 2). The 5d shell is coupled to the dominant magnetic species, the halffilled 4f orbitals, by intraatomic exchange correlations.
For data in Figs. 1 and 4, which are collected at the GdL_{3} edge (E_{xray} = 7.243 keV) at BL19LXU of SPring8, we do not carry out polarization analysis of the diffracted beam. For data in Fig. 3 and Supplementary Fig. 3, which were collected at the GdL_{2} edge (E_{xray} = 7.932 keV) at BL3A of Photon Factory, \(\pi {\sigma }^{{\prime} }\) and \(\pi {\pi }^{{\prime} }\) components of the diffracted beam are separated using an analyser plate made from pyrolytic graphite (PG006).
The details of the polarization analysis in the latter case are as follows. Let k_{i} (k_{f}) be the wavevector of the incoming (outgoing) xray beam with polarization vector ε_{i} (ε_{f}), where the scattering plane is spanned by the two wavevectors; c.f. purple plane in Fig. 1c. Let z be the direction perpendicular to the scattering plane. The incident beam is πpolarized, so that ε_{i} ⊥ e_{z}, k_{i}. In the resonant elastic scattering process, the scattering crosssection \({f}_{{{{\rm{res}}}}}\) contains a term ∝ (ε_{i} × ε_{f}) ⋅ m(q), where q = k_{f} − k_{i}, and m(q) are the momentum transfer and the periodically modulated magnetic moment, respectively (Fig. 1c). In the present case, where ε_{i} ∝ k_{i} × e_{z} and the scattering plane is aligned with the crystal’s hexagonal ab plane, we separate scattered xrays with ε_{f}∥e_{z}∥c^{*} (i.e., \(\pi {\sigma }^{{\prime} }\)) and ε_{f} ∝ k_{f} × e_{z} (i.e., \(\pi {\pi }^{{\prime} }\)). Hence \({f}_{{{{\rm{res}}}}}^{\pi {\sigma }^{{\prime} }}\propto {{{{\bf{k}}}}}_{i}\cdot {{{\bf{m}}}}({{{\bf{q}}}})\) and \({f}_{{{{\rm{res}}}}}^{\pi {\pi }^{{\prime} }}\propto ({{{{\bf{k}}}}}_{i}\times {{{{\bf{k}}}}}_{f})\cdot {{{\bf{m}}}}({{{\bf{q}}}})\propto {m}_{z}({{{\bf{q}}}})\sin (2\theta )\), where 2θ is the scattering angle between k_{i} and k_{f}. The observed scattered intensities are \(I\propto {\left\vert {f}_{{{{\rm{res}}}}}\right\vert }^{2}\).
We obtain integrated intensities \({I}_{\pi {\sigma }^{{\prime} }}\) and \({I}_{\pi {\pi }^{{\prime} }}\) from Gaussian fits to linecuts in momentum space. As described, the intensity ratio \(R={I}_{\pi {\sigma }^{{\prime} }}/{I}_{\pi {\pi }^{{\prime} }}\) at a given magnetic reflection is sensitive to the cycloidal (proper screw) character of the magnetic order by virtue of being large (small) when k_{i} ⋅ (q − G′) is large (small). For proper screw order, magnetic moments arrange themselves perpendicular to the propagation direction (q − G′), with a finite projection onto both e_{z} (i.e., m_{z}) and the inplane vector e_{⊥} = (q − G′) × e_{z} (i.e., m_{ip}). Here we introduce G′, the closest reciprocal lattice vector to q, and e_{z}, a unit vector along the cdirection. It follows \(R{\sin }^{2}(2\theta )\propto {\left[{{{{\bf{k}}}}}_{i}\cdot {{{{\bf{e}}}}}_{\perp }({{{\bf{q}}}})\right]}^{2}\) for a proper screwtype order, as demonstrated for Gd_{3}Ru_{4}Al_{12} in Fig. 3b, c. Moreover, R provides information on the degree of elliptical distortion, capturing e.g. the ratio \({({m}_{y}/{m}_{z})}^{2}\) for a proper screw propagating along e_{x}, written as \({{{{\bf{e}}}}}_{y}\cdot {m}_{y}\sin (2\pi x/\lambda )+{{{{\bf{e}}}}}_{z}\cdot {m}_{z}\cos (2\pi x/\lambda )\), where e_{x}, e_{y}, e_{z} are Cartesian unit vectors aligned so that e_{x}∥q. In real materials with large (classical) magnetic moments, where the length of the moment is fixed to be spatially uniform in space, this type of deformed screw can be realized by anharmonic distortion of the proper screw texture, as shown in Fig. 3b (inset).
Data availability
The data supporting the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank Taro Nakajima for support regarding resonant elastic xray scattering (REXS) experiments at beamline BL3A of Photon Factory. Akiko Kikkawa provided guidance regarding single crystal synthesis. We acknowledge Ferenc Simon for fruitful discussions and technical support at EPFL, and Dieter Ehlers for providing his software to model the ferromagnetic resonance data. We are also grateful to Vladimir Tsurkan for polishing the samples for FMR. REXS measurements at the Institute of Material Structure Science of the High Energy Accelerator Research Organization (KEK) and at SPring8 BL19LXU were carried out under the approval of the Photon Factory program advisory committee (Proposal No. 2020G665) and under grant number 20210007, respectively. M.He. and H.A. K.v.N. acknowledge funding within the joint RFBRDFG research project contract No. 195145001 and KR2254/31. S.E. and Ma.Hi. benefited from JSPS KAKENHI Grant No. 22H04463 and 23H05431, while also acknowledging grants by the Murata Science Foundation, the Yamada Science Foundation, the Hattori Hokokai Foundation, the Iketani Science and Technology Foundation, the Mazda Foundation, the Casio Science Promotion Foundation, the Inamori Foundation, the Kenjiro Takayanagi Foundation, and the Marubun foundation through their Exchange Grant. This work is partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project numbers 518238332 and TRR 360–492547816 and by the Japan Science and Technology Agency via Core Research for Evolutional Science and Technology (CREST) Grant Nos. JPMJCR1874, JPMJCR20T1 (Japan), and by FOREST No. JPMJFR2238.
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Ma.Hi., I.K., Y.To. and T.h.A. conceived the project. Ma.Hi. synthesized and characterized the single crystals. M.He., B.G.S., H.A.K.v.N. and L.F. carried out electron spinresonance experiments at EPFL. Ma.Hi., L.S., H.O. and Y.Ta. performed REXS at SPring8. Ma.Hi., L.S., S.G., K.K., H.S., H.N. and Y.Y. carried out REXS measurements at BL3A of Photon Factory (KEK). Ma.Hi. and S.E. were in charge of spin model calculations. Ma.Hi. wrote the manuscript in close collaboration with M.M.H. and S.E., and with contributions and comments from all coauthors.
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Hirschberger, M., Szigeti, B.G., Hemmida, M. et al. Latticecommensurate skyrmion texture in a centrosymmetric breathing kagome magnet. npj Quantum Mater. 9, 45 (2024). https://doi.org/10.1038/s41535024006542
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DOI: https://doi.org/10.1038/s41535024006542
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