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Squeezing the periodicity of Néel-type magnetic modulations by enhanced Dzyaloshinskii-Moriya interaction of 4d electrons

Abstract

In polar magnets, such as GaV4S8, GaV4Se8 and VOSe2O5, modulated magnetic phases namely the cycloidal and the Néel-type skyrmion lattice states were identified over extended temperature ranges, even down to zero Kelvin. Our combined small-angle neutron scattering and magnetization study shows the robustness of the Néel-type magnetic modulations also against magnetic fields up to 2 T in the polar GaMo4S8. In addition to the large upper critical field, enhanced spin-orbit coupling stabilize cycloidal, Néel skyrmion lattice phases with sub-10 nm periodicity and a peculiar distribution of the magnetic modulation vectors. Moreover, we detected an additional single-q state not observed in any other polar magnets. Thus, our work demonstrates that non-centrosymmetric magnets with 4d and 5d electron systems may give rise to various highly compressed modulated states.

Introduction

In the presence of strong spin-orbit coupling (SOC) topologically non-trivial states of condensed matter emerge such as the surface states of topological insulators1,2, Dirac and Weyl fermions3,4 or Majorana particles5,6. In spin systems, the first order manifestation of the SOC is the antisymmetric Dzyaloshinskii-Moriya interaction (DMI), which gives rise to spin spirals and skyrmion lattice (SkL) states in non-centrosymmetric compounds7,8,9. The non-trivial topology of skyrmions10,11, as well as their interaction with conduction electrons12, electric fields13,14,15,16 and spin-waves17,18,19 motivated intense research exploring possible applications in next-generation data storage and microwave-frequency spintronic devices20,21,22,23.

The SkL phase was first observed in the chiral cubic helimagnet MnSi with moderate SOC24. In cubic helimagnets the symmetry-dictated form of DMI enables Bloch-type magnetic modulations, i.e., spin helices and Bloch-skyrmions. This Bloch-type SkL, comprising q-vectors perpendicular to the direction of the applied field, is stabilized by thermal fluctuations over the energetically favoured longitudinal conical state, only in the close vicinity of the Curie temperature24,25. Cubic magnetocrystalline anisotropies that are higher-order terms in the SOC determine the orientation of helical order at zero field and induce small deflections of the SkL planes for fields applied along low-symmetry crystallographic directions25,26. The increasing strength of the SOC was studied in Mn(Si1−xGex) by replacing Si with heavier Ge27. As a result, the periodicity of the magnetic modulation decreases and the SkL state is transformed to a hedgehog-lattice state. The enhanced cubic anisotropy can also lead to SkL states at low temperatures, without relying on stabilization by thermal fluctuations28,29.

For potential memory and spintronic applications a further reduction in the skyrmion size is desired, which can be achieved through the enhancement of the SOC. In the vast majority of the skyrmion host crystals reported to date, the magnetism is governed by the 3d electrons of transition metals, such as V, Mn, Fe, Co, Cu. Here, we demonstrate the emergence of particularly robust modulated magnetic phases in the 4d polar magnet, GaMo4S830,31,32,33,34. In the rhombohedral phase, our small-angle neutron scattering (SANS) data (Fig. 1c) show that the strong SOC reduces the periodicity of the magnetic structure to λ ≈ 9.8 nm, being one of the shortest modulation observed in bulk crystals hosting the SkL state due to DMI. Moreover, it modifies the distribution of the q-vectors, as illustrated in Fig. 1d, which we explain by an effective Landau theory containing a cubic anisotropy term in addition to the axial anisotropy, inherent to the rhombohedral state. Our temperature, magnetic field and angular dependent magnetization and electric polarization measurements reveal a complex phase diagram where we attributed phases to the cycloidal and SkL states and observed a third modulated magnetic phase. Moreover, we found that the modulated spin states extend up to fields as high as 2 T, which further support their robustness.

Fig. 1: A comparison of the reciprocal-space structure of the modulation wavevectors in GaV4S8 and GaMo4S8.
figure 1

a The tomographic SANS image measured at 12 K in GaV4S8 and b its graphical representation. The scattering pattern contains four rings of q-vectors represented by distinct colours in (b), each corresponding to one of the rhombohedral domain states. c The distribution of the magnetic wavevectors observed at 2 K in GaMo4S8 and d its schematic view. The rings are deflected from the \(\left\{111\right\}\)-type planes in the segments between the \(< 110 >\) directions in an alternating manner.

Spin cycloids and Néel SkL have been found recently in the polar phase of the lacunar spinels GaV4S8 and GaV4Se835,36,37,38, which are narrow-gap multiferroic semiconductors39,40. The polar rhombohedral structure (space group R3m) develops via a cooperative Jahn-Teller distortion driven by the unpaired electron of each V4 cluster with spin S = 1/2 occupying a triply degenerate orbital in the high-temperature cubic state (\(F\overline{4}3m\))39. The rhombohedral distortion can occur as elongation of the unit cell along any of the four \(< 111 >\)-type cubic directions, thus, four rhombohedral domain states, that we denote by P1−4, are present below the structural transition at ~42 K37,39,41. (Throughout this paper we use the pseudo-cubic notation.) In contrast to the chiral cubic helimagnets, in these lacunar spinels the cycloidal character of the modulations with q-vectors restricted to the plane perpendicular to the rhombohedral axis (Fig. 1a, b) enhances the stability range of the SkL phase35,37,42. In both compounds, SANS experiments revealed magnetic modulations with wavelengths of λ ~ 20 nm35,36,37,43, indicating a similar ratio of the exchange interactions and the DMIs.

In the compound GaMo4S8 studied here, the tetrahedral Mo4 clusters carry an unpaired hole with spin S = 1/239. Correspondingly, the cubic state is rhombohedrally distorted by the compression of the unit cell along one of the \(< 111 >\)-type directions31. Although, Rastogi et al., pointed out the importance of the correlation in GaMo4S8 and found a rich magnetic phase diagram below TC = 19 K long ago30, the spin ordering patterns of these phases have not been studied.

Results

Wavevector distribution of the magnetic cycloid at zero field

We explored the zero-field modulated magnetic states of GaMo4S8 by SANS experiments. (The scattering geometry is shown in Supplementary Fig. 4) The scattered intensity was recorded in zero-field at 2 K upon the 180 rotation of the sample in 1 steps, with an acquisition time of 120 s at each angle. The background signal was measured in the paramagnetic phase at 25 K following the same procedure. The scattering images were averaged over a 10 moving window in the rotation angle to improve the signal-to-noise ratio. Figure 2a–d shows the SANS images obtained on four high-symmetry planes, namely the (111), (110), \(\left(11\bar{2}\right)\) and (001) planes. A pixel-wise adaptive Wiener filter, assuming Gaussian noise, was applied for better visualization.

Fig. 2: SANS images of GaMo4S8 and reciprocal space distribution of the cycloidal wavevectors.
figure 2

In panels ad SANS images are collected at 2 K. The scale bars represent 0.5 nm−1. In eh, the q-vector distribution is shown from the direction of the neutron beam in (ad), respectively. The perturbative solution of the model potential in Eq. (1) fitted to the experimental data is visualized by solid curves plotted over the experimental data. The four different colours represent scattering from the four rhombohedral domain states.

The q-dependence of the scattering intensity in the (111) plane, averaged over the polar angles, was fitted by a Gaussian, yielding \(\left|{{{\bf{q}}}}\right|=0.64\)nm−1 for the length of the modulation vectors with a FWHM of 0.2 nm−1, which corresponds to a real-space periodicity of λ ≈ 9.8 nm. The uncertainty of \(\left|{{{\bf{q}}}}\right|\) mainly originates from the broad and anisotropic distribution of the scattering intensity. Whereas the Curie temperature is close to that of GaV4Se8, suggesting a similar strength of the symmetric exchange (J) in the two compounds, the modulation wavelength in GaMo4S8 is roughly half of that in GaV4S835 and GaV4Se837, implying a stronger DMI coupling (D), as λJ/D.

Over the wide-angle rotation experiment, each scattering image represents a planar cross section of the three-dimensional distribution of the q-vectors, where the azimuthal angle of the vertical slicing plane is varied via stepwise rotation of the sample. The 3D scattering pattern was reconstructed using the whole set of the cross section images. In order to enhance the signal-to-noise ratio and to eliminate the asymmetries of the scattering pattern introduced by imbalances between the populations of the different structural domain states, the 3D scattering pattern was symmetrized for all the symmetry operations of the cubic Td point group (for more details, see Supplementary Notes 2, 3 and Supplementary Figs. 5, 6). Figure 2e–h display the symmetrized image as viewed from the different high-symmetry directions.

It is instructive to compare the reciprocal-space q-distributions in GaV4S8 and GaMo4S8, as shown in Fig. 1a and c, respectively. The neutron scattering data collected in the zero-field cycloidal phase of GaV4S8 at 12 K is reproduced from ref. 36. In both compounds the cycloidal q-vectors are distributed over four intersecting rings corresponding to the four structural domain states. The ring structure, instead of six well-defined Bragg spots, is due to static orientational disorder of the q-vectors, as discussed in ref. 36. However, in contrast to GaV4S8, where the modulation vectors are evenly distributed over rings restricted to the \(\left\{111\right\}\)-type planes, in GaMo4S8, the four rings of the q-vectors wave out of the \(\left\{111\right\}\) planes, crossing them only along the \(< 110 >\)-type directions. This waving pattern of the q-vectors preserves the three-fold rotational symmetry of the rhombohedral structure as highlighted by the schematic image in Fig. 1d.

To explain the distribution of the q-vectors, the following effective Landau potential for the unit vector \(\hat{{{{\bf{q}}}}}\) is considered with its x, y, z components defined in the cubic setting,

$${{{\mathcal{V}}}}(\hat{{{{\bf{q}}}}})={(\hat{{{{\bf{n}}}}}\hat{{{{\bf{q}}}}})}^{2}+\alpha \left({\hat{q}}_{x}^{4}+{\hat{q}}_{y}^{4}+{\hat{q}}_{z}^{4}\right)+....$$
(1)

The first term describes the uniaxial anisotropy emerging in the rhombohedral phase with the \(\hat{{{{\bf{n}}}}}\) unit vector parallel to any of the four \(< 111 >\)-type polar axes, and the second term is the lowest-order term compatible with the cubic Td symmetry. The length of the q-vectors is essentially fixed by the DMI, q ~ D/J, which is consistent with the experimental SANS data. The coefficient α parameterizing the relative strengths of the two terms in Eq. (1) is thus effectively of second order in the SOC. The first term favours the confinement of the q-vectors normal to the polar axes, \(\hat{{{{\bf{n}}}}}\), as imposed by the DMI. The waving of the q-vectors out of the {111} planes is captured by the second term. On the microscopic level it represents magnetocrystalline contributions to the Ginzburg-Landau theory for the magnetization that are effectively of fourth order in the SOC.

The minimal-energy solutions to Eq. (1) are sought by parametrizing \(\hat{{{{\bf{q}}}}}\) in spherical coordinates. The polar angle, Θ is chosen with respect to the polar axis \(\hat{{{{\bf{n}}}}}\parallel < 111 >\) of each domain. The azimuthal angle, Φ is enclosed between the in-plane component of \(\hat{{{{\bf{q}}}}}\) and one of the corresponding \(< 1\bar{1}0 >\) directions. In the limit of weak SOC, α is negligible and the wavevectors are basically in-plane (Θ ≈ π/2), since the spirals are degenerate with respect to the azimuthal angle Φ. This gives rise to an equal distribution on circles, as observed for GaV4S8 in Fig. 1a. Expanding the potential in Θ around π/2 up to the second order and minimizing one obtains for the deviations from the circle

$${{\Theta }}({{\Phi }})=\frac{\pi }{2}+\frac{\sqrt{2}}{3}\frac{\alpha }{1+\alpha }\sin (3{{\Phi }}),$$
(2)

that describes the waving of q in GaMo4S8. The least-square fitting of the data to Eq. (2) yields α = −0.14 ± 0.003 and q = 0.64 nm−1 ± 10−3 (λ = 9.81 nm). As shown in Fig. 2e–h, the fitted model is in excellent agreement with the SANS data, indicating the significant influence of cubic anisotropies in GaMo4S8 as opposed to GaV4S8 in accord with the stronger atomic SOC of Mo. We note that for GaV4S8 we find α = 0 ± 0.04 within the experimental precision. The uncertainty is mainly determined by the image preconditioning rather than the accuracy of the fitting, for details see Supplementary Notes 2 and 3.

The tilting of q out of the {111} planes as well as higher-order SOC represented by additional terms \({\hat{q}}_{x}^{6}+{\hat{q}}_{y}^{6}+{\hat{q}}_{z}^{6}\) in the Landau potential break the degeneracy of q-vectors around the ring and favour either \(\langle 1\bar{1}0\rangle\) or \(\langle 11\bar{2}\rangle\)-type directions. However, within our experimental accuracy, we were not able to resolve an enhanced intensity along any of these directions, see Supplementary Notes 3 and Supplementary Fig. 3.

Magnetic phase diagram at finite magnetic fields

We explore the modulated magnetic states stabilized by the interplay of external magnetic fields and the strong SOC of GaMo4S8. Due to the polar symmetry of this compound, the phase diagram may depend on the angle between the polar axis and the field, thus, we measured longitudinal magnetic and differential magnetoelectric susceptibility at 10 K while the field was rotated in finite steps within the (1\(\bar{1}\)0) plane between successive field sweeps. The obtained data are respectively shown in Fig. 3a, b as colour maps over the field magnitude-orientation plane, where the angle ϕ was measured from the [111] axis as sketched in Fig. 3c. The critical fields identified as peaks or sharp steps in the susceptibility and magnetocurrent curves are indicated by lines in Fig. 3a, b. (Field scans measured at a few high symmetry directions are shown in Supplementary Figs. 2 and 3.)

Fig. 3: Angular dependence of the magnetic phase diagram at 10 K.
figure 3

Angular dependence of the magnetic susceptibility and magnetocurrent measured at 10 K are displayed in panel a and b, respectively. Panel c illustrates the orientations of the four polar axes (P1−4) and the plane where the magnetic field was rotated by angle ϕ. The lines represent the magnetic phase boundaries for the different domains (white: P1, black: P2, grey: P3 and P4). Dotted lines in panel (a) represent a splitting of the phase boundary, while dashed lines in panel (b) indicate regions where the anomalies could not unambiguously be identified. The magnetic phase diagram constructed based on the angular dependence of these anomalies at 10 K is shown in panel d. cyc, SkL and 1q state respectively label the cycloidal, skyrmion lattice and a modulated phase described by a single q-vector. H and H are field components parallel and perpendicular to the polar axis, respectively.

Compared to the early magnetization study performed on powder samples30, in the single crystal sample, we resolved more than two phase transitions strongly depending on the direction of the field. The interpretation of the complicated angular-dependent pattern of the anomalies can be simplified by assuming that all four polar domain states are present in the sample. Anomalies and their replica appearing at positions shifted by ~109 are attributed to the P1 (white lines) and the P2 (black lines) domains since both polar axes lie within the rotation plane and they span ~109. The remaining anomaly (grey line) should correspond to both the P3 and P4 domains, when the field is rotated in the high-symmetry (1\(\bar{1}\)0) plane. The field direction [001] is even more special as the polar axes span the same 55 with the field in all domains, thus, the lines should intersect each other. Such an intersection occurs at ~1.5 T where the small deviations likely caused by a slight misorientation of the sample. However, in the same angular range phase boundaries are split around 0.75 T indicated by dotted lines in Fig. 3a. We found that anomalies observed on the different domains collapse into a common phase diagram when plotted on the H - H plane, where H and H represent the field component parallel and perpendicular to the polar axis, respectively (Fig. 3d). This confirms that the sample is in a polar multidomain state and the magnetic state stabilized by the field depends only on the angle between the polar axis and the field. We note that the finite magnetocurrent signal detected in all phases implies that the modulated magnetic states couple to the electric polarization as observed in the sister compounds GaV4S8 and GaV4Se838,41,44. This magnetoelectric coupling is allowed by the polar symmetry of GaMo4S841, and the magnetic field-induced changes of the spin texture modify the polarization texture.

In order to elucidate the nature of the various phases, we explored their properties for certain directions of the applied field in more details. We studied the magnetic field-induced changes in spin textures at 10 K by SANS experiments performed for fields parallel to the [111] axis on the (111) scattering plane. A representative set of SANS images are displayed in Fig. 4a. The smeared wavy ring of intensity present in low magnetic fields, see Fig. 2a, continuously evolves to well-defined Bragg spots in moderate fields pointing to a field-driven enhancement of the magnetic correlation length. To make our analysis more quantitative the field dependence of the scattering intensity and the average length of the q-vectors are plotted in Fig. 4c, d, respectively, in comparison with the susceptibility shown in Fig. 4b. The grey and red curves are obtained by fitting a Gaussian peak on the azimuthally averaged scattering intensity around the \(\langle 1\bar{1}0\rangle\) and \(\langle 11\bar{2}\rangle\)-type directions highlighted by grey and red sectors in Fig. 4a.

Fig. 4: Magnetic field dependence of the spin texture at 10 K.
figure 4

SANS images acquired at 10 K in increasing magnetic fields are shown in panel a. The scale bar represents 0.5 nm−1. Both the neutron beam and the direction of the magnetic field was parallel to the [111] direction. The magnetic field dependence of the susceptibility shown in panel b is compared with the scattering intensity (panel c) and the position of the intensity peak (panel d) measured in the grey and red sections of the SANS patterns. The coloured stripes (same colour coding as in Fig. 3) at the top of panel (b) represent the sequence of phase transitions in P1 and P2−4 domains with polar axes spanning 0 and ~71 with respect to the field.

The phase transitions are marked by clear anomalies of the scattering parameters. As we have demonstrated above, only the P1 domain, responsible for the red ring in Figs. 1d and 2, scatters neutrons into the red sections whereas all domains contribute to the intensity detected in the grey regions. Remarkably, our measurements evidence that modulated magnetic structures are extremely robust against a magnetic field, i.e., they extend up to 1.3 and 1.8 T for fields parallel to and inclined at 71 from the polar axis, respectively. Such exceptional stability of modulated bulk phases has been observed so far only in centrosymmetric materials, hosting nearly atomic-scale skyrmions due to exchange frustration45,46. As the modulated phases are expected to be stable up to a critical field of order HFMD2/J, this finding corroborates that the DMI is the strongest in GaMo4S8 among the lacunar spinels known to host SkLs, likely due to enhancement of the SOC from 3d to 4d electrons. Interestingly, after a low-field decrease q increases in higher fields in the grey region, which is unusual among skyrmion host spiral magnets. Since q does not change above 1.3 T, where only the P2−4 domains contribute to the SANS intensity, the anomalous field induced shortening of the magnetic periodicity occurs in the P1 domain, i.e., for field parallel to the polar axis.

Since the DMI pattern dictated by the C3v symmetry induces cycloidal modulation9 we assign the zero field ground state of GaMo4S8 to a single-q cycloidal state (orange region in Fig. 3d) in analogy with GaV4S8 and GaV4Se8. Although it is well established both theoretically and experimentally that the cycloidal phase is the zero field state in polar magnets of Cnv symmetry, future polarized neutron diffraction experiments may be performed to directly prove this. The hexagonal SANS pattern with minimal intensity in the red sections, that is observed above ~0.75 T, imply the formation of a SkL state or can alternatively manifest the coexistence of cycloidal domains with q-vectors pointing to the different \(\langle 1\bar{1}0\rangle\)-type directions. Following the correspondence with the phase diagram of the sister compounds we rather attributed the purple phase pocket in Fig. 3d to the SkL state. The robustness of this phase against strong tilting of the magnetic field from the polar axes implies an easy-axis type magnetic anisotropy, thus, it has the same character as in GaV4S842,47, which is in agreement with the predictions of recent first-principles calculations48,49.

At finite fields between the cycloidal and the field-polarized FM states for fields perpendicular to the polar axis we detected an additional phase (coloured in blue in Fig. 3d) that is not present in the former material-specific model calculations42,48,49,50,51. We labelled it as 1q state since it is a modulated phase according to SANS, however, we cannot unambiguously determine its internal spin structure. As it possesses a finite magnetization it is likely to be a fan or a conical phase, which is not present in any other lacunar spinels. The stability range of this intermediate state narrows as the field is rotated toward the polar axis, however, it can separate the cycloidal and SkL phases even in parallel fields. The anomalous increase in q above 0.5 T also corresponds to the emergence of this single-q state. Although in the sister compounds the emergence of magnetic states at the polar domain walls (DWs) has been reported44,52, we rather assigned the 1q state to a bulk phase. At the onset of the 1q state, we found robust anomalies in all experiments including SANS. Since SANS does not possess the sensitivity to detect DW-confined states with small volume fraction, as found in GaV4Se844, we believe the 1q state corresponds to a new bulk phase.

The temperature-field phase diagram is mapped by susceptibility measurements below TC = 19 K as shown in Fig. 5a, b for fields along [111] and [001] directions, respectively. (Field sweeps measured at constant temperatures are presented in Supplementary Fig. 1.) Schematic phase diagrams deduced for a polar monodomain sample are presented in Fig. 5c and d for fields respectively spanning zero and 55 with the polar axis. The latter case, H[001] likely provides a representative phase diagram for the whole angular range 55–90 as evidenced for 10 K in Fig. 3d. All phases observed at 10 K including the SkL state extend down to the lowest temperatures. In case of strong easy-axis anisotropy, modulated phases are stabilized only by thermal fluctuations as shown for GaV4S836, thus, the presence of cycloidal and SkL phases at low temperatures implies weaker easy-axis anisotropy in GaMo4S8. The single-q state enters between the cycloidal and the SkL phases only below 13 K for fields applied along the polar axis, and its field stability range grows toward low temperatures. Moreover, below 6 K additional anomalies of the susceptibility appear suggesting the emergence of a new phase not present at 10 K in Fig. 3.

Fig. 5: Magnetic field and temperature dependence of the phase diagram for field parallel to the [111] and [001] directions.
figure 5

The magnetic suscpetibility as a function of magnetic field and temperature is shown for field parallel to the [111] and [001] directions in panel a and b, respectively. The direction of the magnetic field with respect to the four polar axes (P1−4) is indicated above the colour plots. The lines indicate anomalies of the susceptibility corresponding to phase transitions. For H[111] transitions occurring in the P1 domain are highlighted by white lines whereas phase boundaries of the P2−4 domains are drawn in black. For H[001] all four domains are indistinguishable. Panel c and d display schematic phase diagrams deduced for a polar monodomain sample for fields respectively spanning zero and 55 with the polar axis.

Discussion

The main features of the phase diagram and their assignment are consistent with recent theoretical studies of GaMo4S8 that combines DFT calculations and Monte-Carlo simulations48,49,50. These works predict only two modulated structures, the cycloidal and the Néel-type SkL states for field parallel to the polar axis. With a moderate strength of uniaxial anisotropy, these states were found to be stable down to the lowest temperatures. Although the absolute values of the interaction strength largely vary between the different calculations all of them predict a critical temperature consistent with the experiments. The ratio of the relevant DMI and the exchange interactions determining the periodicity of the cycloidal order is similar in refs. 48,50, and the calculated periodicity (about λ ~ 10 nm) is close to the experimentally observed one. According to Nikolaev and Solovyev48, the small periodicity of the cycloidal and SkL phases is not only the consequence of the enhanced DMI due to stronger SOC, but the isotropic exchange term is also reduced from GaV4S8 to GaMo4S8 due to weaker screening. However, further theoretical efforts are required to understand the emergence of the 1q state between the cycloidal and the SkL phases as well as the additional low-temperature phases. An important direction of future research is, in particular, the theoretical investigation of a micromagnetic model that is able to account both for the waving of the wavevector captured by Eq. (1) as well as the magnetic phase diagram observed experimentally.

In conclusion, we studied the magnetically ordered phases of a 4d cluster magnet GaMo4S8 by SANS, magnetization and magneto-current measurements. We found modulated magnetic states with sub-10 nm periodicity that can be attributed to the stronger DMI due to the enhanced SOC of GaMo4S8, with respect to typical 3d transition metal-based skyrmion hosts. The q-space distribution of the modulation vectors is markedly deformed, which is explained in terms of higher-order anisotropies becoming important in this 4d compound. In finite fields, a series of phase transitions are observed, which is assigned to the transformation of the cycloidal state to the SkL and a new single-q state. Moreover, these modulated spin textures are coupled to the ferroelectric polarization as evidenced by our magneto-current measurements. The exceptional stability of the modulated states against magnetic fields also indicates the importance of SOC in GaMo4S8. Our findings imply that a remarkable scaling down of the skyrmion size in bulk non-centrosymmetric materials can be achieved by exploring the plethora of 4d and 5d magnets.

Methods

Synthesis

Single crystals of GaMo4S8 (typical size 0.5–1 mm) were grown by the flux method53. A mixture of Mo, MoS2 and Ga2S3 with the molar ratio of 11:3:4 was pressed into a pellet, loaded in an alumina crucible, sealed in a molybdenum tube by arc melting under argon atmosphere, and heated quickly up to 1873 K and cooled slowly to 1273 K.

Small-angle neutron scattering

SANS experiments were performed at the Oak-Ridge National Laboratory High-Flux Isotope Reactor, using the General-Purpose Small-Angle Neutron Scattering Diffractometer54,55. A single crystal with a mass of m = 112 mg was mounted onto a rotatable sample stick with its \(\left[1\bar{1}0\right]\) cubic direction parallel to the rotation axis. A neutron wavelength of λn= 6 Å with Δλn/λn= 0.13 broadening was used with the detector set to a distance of 5 m from the sample, employing a collimator of the same length.

Magnetization and polarization measurements

The magnetization measurements were performed using a Quantum Design MPMS. The longitudinal magnetic susceptibility calculated from static magnetization measurements as well as the magnetocurrent were measured at 10 K while the field was rotated in fine steps within the (1\(\bar{1}\)0) plane between successive field sweeps. The step size of the rotation was 1 and 2.5 degrees in the case of the magnetocurrent and the magnetization measurements, respectively. Magnetocurrent (j = dP/dt) measurements were carried out using a Keysight Electrometer. The sample was contacted on parallel (111) faces, and correspondingly, the magnetic field-induced changes in the electric polarization component parallel to the [111] direction was detected. The magnetoelectric susceptibility was calculated from the magnetocurrent by division through the constant magnetic field sweep rate of 1.2 T/min. The field dependence of the magnetization was measured in decreasing temperature steps.

Data availability

Data associated with figures are provided with this paper. The data and the data evaluation code used in the tomographic analysis of the zero-field SANS data is publicly available at GitHub: https://github.com/Buadam/SANS_Reciprocal_Space_Tomography. All other data that support the plots within this paper are available from the corresponding authors upon reasonable request.

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Acknowledgements

We thank S. Picozzi, and S. Dong for useful discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) via the Transregional Research Collaboration TRR 80: From Electronic Correlations to Functionality (Augsburg-Munich-Stuttgart) and via the DFG Priority Program SPP2137, Skyrmionics, under Grant No. KE 2370/1-1. M.G. is supported by DFG through project A07 of SFB 1143 (project-ID 247310070), DFG projects No. 270344603 and 324327023. S.B. acknowledges support by National Research, Development and Innovation Office - NKFIH, FK 153003, Bolyai 00318/20/11 and by the BME-Nanotechnology and Materials Science FIKP grant of EMMI (BME FIKP-NAT). This research was supported by the Ministry of Innovation and Technology and the National Research, Development and Innovation Office within the Quantum Information National Laboratory of Hungary. D.S. acknowledges the FWF Austrian Science Fund Grants No. I 2816-N27 and TAI 334-N. This research used resources at the High Flux Isotope Reactor, a Department of Energy (DOE) Office of Science User Facility operated by the Oak Ridge National Laboratory.

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T.W., Y.T. and H.N. synthesized and characterized the crystal; Á.B., K.G., L.F.K. and L.B. measured magnetization; Á.B., D.S., and L.D.-S. performed the SANS experiments; K.G. performed angular dependent magnetization and polarization measurements; Á.B., I.K. and S.B. analyzed the SANS results; M.A. and M.G. developed the theory; Á.B., I.K., M.G. and S.B. wrote the manuscript with contributions from K.G.; I.K. and S.B. planned the project.

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Correspondence to Ádám Butykai or Sándor Bordács.

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Butykai, Á., Geirhos, K., Szaller, D. et al. Squeezing the periodicity of Néel-type magnetic modulations by enhanced Dzyaloshinskii-Moriya interaction of 4d electrons. npj Quantum Mater. 7, 26 (2022). https://doi.org/10.1038/s41535-022-00432-y

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