Spontaneous reversal of spin chirality and competing phases in the topological magnet EuAl₄

Abstract

Materials exhibiting a spontaneous reversal of spin chirality have the potential to drive the widespread adoption of chiral magnets in spintronic devices. Unlike the majority of chiral magnets that require the application of an external field to reverse the spin chirality, we observe the spin chirality to spontaneously reverse in the topological magnet EuAl₄. Using resonant elastic x-ray scattering we demonstrate that all four magnetic phases in EuAl₄ are single-k, where the first two magnetic phases are characterized by spin density wave order and the last two by helical spin order. A single spin chirality was stabilised across the 1mm² sample, and the reversal of spin chirality occurred whilst maintaining a helical magnetic structure. At the onset of the helical magnetism, the crystal symmetry lowers to a chiral monoclinic space group, explaining the asymmetry in the chiral spin order, and establishing a mechanism by which the spin chirality could reverse via magnetostructural coupling.

Introduction

The bivalent nature of chirality, i.e. a chiral system is either left or right-handed, lends itself well to the encoding of binary information for classical computation. This is reflected in a surge in interest to control the chiral magnetic states of a material through electronic and spin degrees of freedom, for instance via non-collinear or polarised spin currents^1,2,3,4. Chiral magnetism can occur in inversion symmetric crystals, for which competing exchange interactions are suggested to play a role in stabilising the chiral order^5,6,7, or in non-centrosymmetric crystals that allow for Dzylaoshinskii-Moriya exchange interactions, which gives rise to the spin chirality^8,9,10. In the latter case it is often the coupling between the macroscopic order parameter that breaks the inversion symmetry, and spin that allows for the field-controlled reversal of the chirality¹¹. Reversing the spin chirality in the absence of an applied field could have far reaching consequences for the design and development of future spintronic devices, just as materials that exhibit a spontaneous rotation of the magnetisation have had^12,13,14.

Besides the complex and intriguing zero field magnetism, demonstrated by a cascade of four different phase transitions in a temperature interval of only 5 K, EuAl₄ is also shown to stabilise seven different magnetic phases and four different skyrmion lattices under an applied magnetic field^7,15,16. EuAl₄ is expected to be centrosymmetric, and hence the existence of chiral topological objects such as skymrions, was explained not by the Dzyaloshinskii Moriya interaction, but Ruderman-Kittel-Kasuya-Yosida (RKKY) and frustrated itinerant interactions^7,17.

EuAl₄ crystallises in the tetragonal space group I4/mmm at room temperature. The Eu²⁺ ions, Wyckoff position 2a, form two-dimensional square layers in the ab plane. Neighbouring Eu layers, which are separated along c, are related by the I-centering translation¹⁸. We label the Eu ion at the vertex Eu11, and the Eu ion at the body centre as Eu12. Below T_CDW = 145 K the onset of charge density wave order occurs with propagation vector (0,0,τ) τ = 0.1781(3)^19,20,21. Single crystal neutron diffraction studies demonstrated that the four magnetically ordered phases of EuAl₄, which we label as AFM1, AFM2, AFM3 and AFM4 with onset temperatures of T₁ = 15.4 K, T₂ = 13.2 K, T₃ = 12.2 K and T₄ ∼10 K respectively, ordered with multiple magnetic propagation vectors²². However, the nature of the magnetic phases remained an open question, as did the physical mechanisms driving the changing modulations.

In this manuscript we solve the nature of each of the four magnetically phases of EuAl₄ using resonant elastic X-ray scattering. We show that all four phases are single-k, where the AFM1 and AFM2 phases are characterised by spin density wave order, and the AFM3 and AFM4 phases by helical spin order. Through the use of circular polarised X-rays we find that the system selects to order in a single chiral magnetic domain in both the AFM3 and AFM4 phases, which is stabilised over a 1 mm² sample. Unlike other chiral magnets that have a single spin chiral ground state, we observed a spontaneous reversal of the spin chirality at T₄ across the entire sample, whilst maintaining a helical magnetic structure. We demonstrate that at the onset of the helical magnetic order at T₃, the crystal structure symmetry lowers to chiral monoclinic, which explains the asymmetry in the chiral spin order, and establishes a mechanism by which the spin chirality could reverse via magnetostructural coupling. Despite the isotropic nature of the Eu²⁺ ions we find that the EuAl₄ adopts anistropic spin structures, where the in-plane component of the spin moment is orientated perpendicular to the direction of the propagation vector, which we demonstrate is favoured by magnetic dipolar interactions.

Results

Resonant elastic X-ray scattering (REXS) measurements were performed at the I16 beamline at Diamond Light Source²³ on an as grown 1 mm³ sample of EuAl₄. The beamline was aligned at the Eu L₃ edge, 6.972 keV. One of the key challenges that arises in determining the magnetic structure in the presence of multiple magnetic modulation vectors is to be able to distinguish whether these magnetic propagation vectors modulate the same magnetic phase, a multi-k structure, or whether each magnetic modulation vector corresponds to a distinct single-k domain that is separated spatially in the crystal²⁴. In principle one can distinguish between these two scenarios if the incident beam on the crystal is smaller than the size of the domains, and thus by probing different positions on the sample, one may either observe a spatial separation of the various magnetic propagation vectors (single-k) or the existence of the magnetic propagation vectors at all positions on the sample (multi-k). In general, the size of domains depends on a number of factors, such as the thermal cycling of the sample and local strain effects, so it is challenging to predict the expected size of the domains, and without a-priori knowledge of the domain size or distribution it becomes necessary to rely on other phenomenological and experimental methods to distinguish between the two scenarios²⁴. According to Landau’s theory of phase transitions²⁵, for a multi-k structure one can expect the presence of free energy coupling terms that combine arms of the star of k with a secondary lattice modulation imposed by translation and time reversal symmetry. In this regard, one can search for these secondary distortions and the absence of such satellites is a good (but not definite) indication that the system is single-k.

Magnetic structures

Below T₁, we observed the onset of two modulation vectors; ${{{\bf{k}}}}_{{{\rm{qq}}}}\,$= (0.097(5),0.093(5),0) and ${{{\bf{k}}}}_{{{\rm{q}}}\bar{{{\rm{q}}}}}$ = (0.092(5),-0.087(5),0), as shown in Fig. 1a, which lie along the $\triangle$ line of symmetry (α,α,0) in the Brillouin Zone. Below T₂, an additional two propagation vectors appeared, ${{{\bf{k}}}}_{{{\rm{q}}}0}^{2}$ = (0.175(5),0,0) and ${{{\bf{k}}}}_{0{{\rm{q}}}}^{2}$ = (0,0.178(5),0). The associated reflections were of approximately equal intensity, but considerably weaker, by at least two orders of magnitude compared with those related to the ${{{\bf{k}}}}_{{{\rm{qq}}}}$ and ${{{\bf{k}}}}_{{{\rm{q}}}\bar{{{\rm{q}}}}}$ propagation vectors. Below T₃, the intensity of reflections measured with propagation vectors ${{{\bf{k}}}}_{{{\rm{qq}}}}$ and ${{{\bf{k}}}}_{{{\rm{q}}}\bar{{{\rm{q}}}}}$ rapidly reduced to zero. Concomitant with this was a large increase in scattering measured from reflections with propagation vector ${{{\bf{k}}}}_{0{{\rm{q}}}}^{3}$, where the magnitude of the propagation vector jumps from 0.175(5) to 0.165(5), as shown in Fig. 1. Below T₄, ${{{\bf{k}}}}_{0{{\rm{q}}}}^{4}$ changes from 0.165(5) to 0.188(5), whilst remaining along the same direction, with a noticeable increase in the intensity of the scattering. The above reflections were identified to be magnetic in origin from a measurement of the polarisation and energy dependence, shown in Fig. S1 (Supplementary Note 2) of the Supplementary Material (SM). These observations are in agreement with the results of the single crystal neutron diffraction experiments²², with the exception that we observed the ${{{\bf{k}}}}_{{{\rm{q}}}0}^{2}$ and ${{{\bf{k}}}}_{0{{\rm{q}}}}^{2}$ to be present in the AFM2 phase.

Fig. 1: Temperature dependence of the magnetic satellites of EuAl₄.

Temperature dependence of (a) the integrated intensity and (b) the magnitude of the propagation vector along the h and k directions for the ${{{\boldsymbol{k}}}}_{{qq}}$, ${{{\boldsymbol{k}}}}_{q\bar{q}}$, ${{{\boldsymbol{k}}}}_{q0}^{2}$, ${{{\boldsymbol{k}}}}_{0q}^{2}$, ${{{\boldsymbol{k}}}}_{0q}^{3}$ ${and}$ ${{{\boldsymbol{k}}}}_{0q}^{4}$ magnetic satellites of the (0,0,8) reflection. c–f The magnetic satellites of the (0,0,8) reflection shown in the (h k 0) plane for each of the magnetically ordered phases, where the intensity of the magnetic satellites is represented by the size of the circular markers plotted according to a logarithmic scale. The (0,0,8) reflection is also shown for reference. The black crosses indicate the expected position of the magnetic satellite based on the experimentally measured position of the corresponding +k satellite. g–i Line cuts of the magnetic satellites measured about the (0,0,8) reflection at various temperatures. We note that this data was collected at a position in the sample where the ${{{\boldsymbol{k}}}}_{q0}^{3}$ ${and}$ ${{{\boldsymbol{k}}}}_{q0}^{4}$ magnetic satellites were not present. Data for each of the magnetic satellites is given in a different colour for clarity and ease of data interpretation; red for ${{{\boldsymbol{k}}}}_{{qq}}$, orange for ${{{\boldsymbol{k}}}}_{q\bar{q}}$, purple for ${{{\boldsymbol{k}}}}_{q0}^{2}$, blue for ${{{\boldsymbol{k}}}}_{0q}^{2}$, green for ${{{\boldsymbol{k}}}}_{0q}^{3}$ and black for ${{{\boldsymbol{k}}}}_{0q}^{4}$ magnetic satellite.

To identify the orientation of the spins we performed azimuthal dependencies using linearly polarised light²⁶. The azimuthal dependencies were repeated using incident circular polarised light to measure the collinearity of the spin structure. A contrast in the intensity of the magnetic satellites when probed with oppositely polarised circular light would occur if the spins are in a non-collinear arrangement, whilst an absence of any contrast implies the spin structure is collinear or equally populated by non-collinear inversion domains. In order to probe the spatial arrangement of the modulation vectors in each of the phases, and thus attempt to clarify its multi-k vs single-k nature, the incident X-ray beam was focused to 100 µm by 37 µm spot, and the response across the sample was measured in each of the magnetically ordered phases. A schematic of the REXS experimental setup is given in Fig. 2a.

Fig. 2: Maps showing the variation in the intensity of the magnetic satellites and azimuthal dependencies collected on EuAl₄ in the AFM1 phase, together with the resonant elastic X-ray experimental setup used to collect the data.

a A schematic diagram of the REXS experimental setup used to collect the maps shown in (b, c). Also shown is the azimuthal angle Ψ for a specular reflection, i.e the (0,0,8) reflection, which is given by the red circular arrow. The variation in intensity of the (b) ${{{\boldsymbol{k}}}}_{{qq}}$ and (c) ${{{\boldsymbol{k}}}}_{q\bar{q}}$ magnetic satellites of the (0,0,8) reflection taken as the beam was translated across the sample at 13.7 K in the AFM1 phase. d The scattered intensity from the ${{{\boldsymbol{k}}}}_{{qq}}$ magnetic satellite of the (0,0,8) reflection as a function of azimuth collected with incident σ, circular right (⟳) and circular left (⟲) polarised light in the AFM1 phase. The markers represent the normalised data, and the lines represent the fitted azimuthal dependencies using a SDW with m|| $a\bar{b}$.

AFM1 phase

In the AFM1 phase, the ${{{\bf{k}}}}_{{{\rm{qq}}}}$ and ${{{\bf{k}}}}_{{{\rm{q}}}\bar{{{\rm{q}}}}}$ magnetic satellites of the (0,0,8) reflection were present across the entire sample, as shown in Fig. 2b, c. Even though we did not observe any spatial segregation of the two satellites, we did observe that the ratio of their scattered intensity varied significantly across the sample, which implies the magnetic structure was not multi-k, a multi-k structure would give rise to a constant ratio. Furthermore, for a multi-k solution one would expect structural modulations with propagation vector ${{{\bf{k}}}}_{{{\rm{qq}}}}$ + ${{{\bf{k}}}}_{{{\rm{q}}}\bar{{{\rm{q}}}}}$²⁷$,$ as this would couple the two magnetic propagation vectors in a trilinear free energy invariant that does not break translational symmetry, which was not observed. Together this suggests that the AFM1 phase is single-k, where the ${{{\bf{k}}}}_{{{\rm{qq}}}}$ and ${{{\bf{k}}}}_{{{\rm{q}}}\bar{{{\rm{q}}}}}$ correspond to different domains that were smaller than the focused beam spot. Azimuthal scans measured on the magnetic satellite of the ${{{\bf{k}}}}_{{{\rm{qq}}}}$ are shown in Fig. 2d. The absence of any contrast in the intensity of the satellites when probed with incident circular light, suggests that the spin structure is collinear. The three possible spin structures determined from symmetry, details of which are given by Sec. S1 and Tables. S1, S2 of the SM, were used to fit the data with a single variable parameter, and the only structure to give an excellent fit to the data, Fig. 2d, was a spin density wave where the moments are aligned in the ab plane perpendicular to the direction of the propagation vector.

AFM2 phase

In the AFM2 phase, the ${{{\bf{k}}}}_{{{\rm{qq}}}}$ and ${{{\bf{k}}}}_{{{\rm{q}}}\bar{{{\rm{q}}}}}$, ${{{\bf{k}}}}_{{{\rm{q}}}0}^{2}$ and ${{{\bf{k}}}}_{0{{\rm{q}}}}^{2}$ were observed to exist at all points on the sample, Fig. 3a–c. As ${{{\bf{k}}}}_{{{\rm{q}}}0}^{2}$ and ${{{\bf{k}}}}_{0{{\rm{q}}}}^{2}$ are equal to ${{{\bf{k}}}}_{{{\rm{qq}}}}$ + ${{{\bf{k}}}}_{{{\rm{q}}}\bar{{{\rm{q}}}}}$ and ${{{\bf{k}}}}_{{{\rm{qq}}}}$ - ${{{\bf{k}}}}_{{{\rm{q}}}\bar{{{\rm{q}}}}}$ respectively, within the experimental error, these could be the peaks that would appear in the presence of a multi-k structure. However, as they are magnetic and not structural, it would necessitate the appearance of magnetism at the Γ-point so that the free energy term coupling these order parameters is fourth power, and therefore invariant upon time-reversal. Physically this would manifest as ferromagnetic order, which was not observed from magnetometry data^15,28 or from previous single crystal neutron diffraction studies²² indicating that together ${{{\bf{k}}}}_{{{\rm{qq}}}}$ and ${{{\bf{k}}}}_{{{\rm{q}}}\bar{{{\rm{q}}}}}$, do not form a multi-k structure, and instead correspond to two different domains, as was the case for the AFM1 phase. Therefore, the AFM2 phase is characterised by two distinct magnetic orders that coexist, and which are spatially separated in the sample. Such a phase separation has also been observed in another member of this family of intermetallics, EuAl₂Ga₂²⁶. Azimuthal scans measured on the magnetic satellites of the (0,0,8) show that the magnetic ordering associated with propagation vector ${{{\bf{k}}}}_{{{\rm{qq}}}}$ (and ${{{\bf{k}}}}_{{{\rm{q}}}\bar{{{\rm{q}}}}}$) was still a SDW m||${{\rm{a}}}\bar{{{\rm{b}}}}$ (m||ab), as shown in Fig. 3d,e. The ${{{\bf{k}}}}_{{{\rm{q}}}0}^{2}$ and ${{{\bf{k}}}}_{0{{\rm{q}}}}^{2}$ also correspond to different domains as the ratio of their scattered intensity varied significantly across the sample, Fig. 3c. The azimuthal data collected on the ${{{\bf{k}}}}_{{{\rm{q}}}0}^{2}$ (${{{\bf{k}}}}_{0{{\rm{q}}}}^{2}$) satellites of the (0,0,8) reflection could be fit with two possible spin structures as shown in Fig. 3f, g; a SDW m||c with a single variable parameter or a helix with m||bc (m||ac) with three variable parameters. In the case of the helix the data were fit assuming the presence of equal populations of both inversion domains, as there was no contrast in the scattered intensity when probed with circular right and left polarised light, and by assuming a highly elliptical helix where the ratio of the fourier components ${F}_{{{\rm{y}}}({{\rm{x}}})}/{F}_{{{\rm{z}}}}$ was between 0.2 and 0.3, which suggests that the SDW m||c is the more likely solution.

Fig. 3: Maps showing the variation in the intensity of the magnetic satellites and azimuthal dependencies collected on EuAl₄ in the AFM2 phase using resonant elastic X-ray scattering.

The variation in intensity of the (a) ${{{\bf{k}}}}_{{{\rm{q}}}0}^{2}$ and (b) ${{{\bf{k}}}}_{0{{\rm{q}}}}^{2}$ magnetic satellites of the (0,0,8) reflection taken as the beam was translated across the sample at 12.7 K in the AFM2 phase. c Shows the difference in intensity of the ${{{\bf{k}}}}_{{{\rm{q}}}0}^{2}$ and ${{{\bf{k}}}}_{0{{\rm{q}}}}^{2}$, normalised by the sum of their intensity taken at 12.7 K in the AFM2 phase. d–g The scattered intensity from the ${{{\bf{k}}}}_{{{\rm{qq}}}}$, ${{{\bf{k}}}}_{{{\rm{q}}}\bar{{{\rm{q}}}}}$, ${{{\bf{k}}}}_{{{\rm{q}}}0}^{2}$ and ${{{\bf{k}}}}_{0{{\rm{q}}}}^{2}$ magnetic satellite of the (0,0,8) reflection as a function of azimuth collected with incident σ, circular right (⟳) and circular left (⟲) polarised light in the AFM2 phase at 12.7 K. The markers represent the normalised data, and the lines represent the fitted magnetic structure solutions.

AFM3 phase

While the ${{{\bf{k}}}}_{{{\rm{q}}}0}^{2}$ and ${{{\bf{k}}}}_{0{{\rm{q}}}}^{2}$ propagation vectors were approximately of equal intensity in the AFM2 phase, in the AFM3 phase the intensity of ${{{\bf{k}}}}_{{{\rm{q}}}0}^{3}$ was considerably reduced, by a factor of 5 compared with the ${{{\bf{k}}}}_{0{{\rm{q}}}}^{3}$ and was not present across the entire sample, it could only be found in a 300 µm by 300 µm region. Each of the magnetic structure solutions identified by symmetry, Tables S3, S4 of the SM, was used to fit the azimuthal data collected on the ${{{\bf{k}}}}_{0{{\rm{q}}}}^{3}$ satellite of the (0,0,8) reflection, and an excellent fit was only achieved using a helical magnetic structure model with moments oriented in the ac plane, as shown in Fig. 4a. To fit the azimuthal data consisting of 66 data points, we used 2 parameters: the amplitude of the oscillation of the helix and its ellipticity. An excellent fit was achieved with an ellipticity of 1.4 (${F}_{{{\rm{x}}}}$/${F}_{{{\rm{z}}}}$).

Fig. 4: Maps showing the variation in the intensity of the magnetic satellites and azimuthal dependencies collected on EuAl₄ in the AFM3 and AFM4 phase using resonant elastic X-ray scattering.

The scattered intensity from the ${{{\boldsymbol{k}}}}_{0q}^{3}$ and ${{{\boldsymbol{k}}}}_{0q}^{4}$ satellite of the (0,0,8) reflection collected as a function of azimuth in (a) the AFM3 phase at 10.6 K and (b) the AFM4 phase at 7 K. The incident polarisation of light was either σ, circular right (⟳) or circular left (⟲) polarized. The data, represented by the circular markers, are normalised against the maximum intensity collected in the ⟳ channel at 7 K. The solid lines represent the fit to the data using a helical magnetic structure with moments oriented in the ac plane. The variation in the difference in intensity of the ${{{\boldsymbol{k}}}}_{0q}$ satellite of the (0,0,8) reflection at ψ = −90°, collected with incident ⟳ and ⟲ light as the beam was translated across the sample at (c) 10.6 K in the AFM3 phase and (d) 7 K in the AFM4 phase. The bottom panels show the same as the top panels except plotting the response of the incident ⟲ light and not the difference.

AFM4 phase

To fit the azimuthal scan of the ${{{\bf{k}}}}_{0{{\rm{q}}}}^{4}$ satellite of the (0,0,8) reflection collected in the AFM4 phase, we repeated the procedure outlined above, and found that the best fit was still a helical magnetic structure solution with moments orientated in the ac plane, but with a reversal of the chirality, such that now the best fit to the azimuthal data, Fig. 4b, was achieved with an ellipticity of −1.32 (${F}_{{{\rm{x}}}}$/${F}_{{{\rm{z}}}}$). As the X-ray beam does not probe the entire sample, we questioned whether we had measured a reversal in chirality or a shift in the magnetic chiral domain pattern that may occur with temperature. To clarify this matter we rastered the entire sample with the focused beam spot, first using circular right, and then with circular left polarised light at 7 K and 10.6 K, representative of the AFM4 and AFM3 phases respectively. We found there to be a small variation in the difference in intensity scattered from the circular light, but largely the sign of the difference was constant across the entire sample for a given temperature, as shown in Fig. 4c, d. Upon warming to the AFM3 phase, the sign of the difference flipped, demonstrating that the magnetic chirality across the whole sample had reversed upon warming! We also observed striped features, Fig. 4c, d, which we demonstrate arises from a change of the crystal structure symmetry to monoclinic below T₃, as discussed further in subsequent sections. A representation of the spin order stabilised in each of the magnetically ordered phases of EuAl₄ is shown in Fig. 5, together with the crystallographic I4/mmm unit cell.

Fig. 5: Experimentally determined spin ordering in the four magnetically ordered phases of EuAl₄.

a–e The spin order stabilised in EuAl₄ in each of the four magnetically ordered phases is given for the ${{{\boldsymbol{k}}}}_{{qq}}$ and ${{{\boldsymbol{k}}}}_{0q}$ propagation vectors. The Eu spins are represented by the black solid arrows and the Eu and Al ions are omitted for clarity. To the right of each spin ordering a representation of the direction of the propagation vector, given by the red arrow, and the spin direction, by the blue arrows, is also shown for the ab plane. f The crystallographic I4/mmm unit cell of EuAl₄ is shown for two different projections with the Eu ions shown by the solid purple circles, and the Al ions by the blue solid circles.

Magnetic anisotropy

For the rare earth ions, with the exception of Gd³⁺ that has a half-filled 4f sub-shell, anisotropic interactions typically arise from perturbative effects of the crystalline electric field on the free ion wavefunctions of the rare earth ions^29,30, where L and S are evaluated using the Russell-Saunders coupling scheme. Eu²⁺, which is isoelectronic to Gd³⁺, also has S = 7/2 and L = 0, and is thus expected to be isotropic with respect to its crystallographic environment. However the adoption of anisotropic magnetic structures clearly points to the presence of anisotropy. One could make the argument that the assumption of a fully quenched orbital angular momentum may not be true, and hence there may be a small finite 4f spin–orbit coupling interaction, however it would not explain the temperature dependent change in anisotropy. Instead, we adopt the theory used to explain the magnetocrystalline anisotropy present in elemental Gd³¹.

Treating the 4f states as a spin-polarised core that is well localised, and the 5d and 6s as the hybridised valence conduction bands, which from angle-resolved photoelectron spectroscopy measurements³² and numerous first principles calculations^31,33 has show to be a good assumption, it was shown that the calculated easy axis in elemental Gd is indirectly dependent on the magnitude of the magnetic moment developed on the 4f sites^31,34. The exchange field from the 4f ions can split the 5d-6s conduction bands, giving rise to a valence band moment, and a spin-orbit interaction from the conduction electrons that can act as a source of magnetocrystalline anisotropy³¹. The spin–orbit coupling is directly dependent on the exchange splitting of the conduction band, which in turn is dependent on the magnitude of the 4f moment. Hence, a change in the ordered 4f moment can dramatically change the spin-orbit coupling, and therefore the magnetic anisotropy. While no calculations have been performed for Eu²⁺, this source of magnetic anisotropy is expected to be present for all rare earths, but is never observed owing to the dominance of crystal electric field effects for cases where L ≠ 0³¹. In EuAl₄ the valence bands are expected to be dominated by the Eu atomic orbitals³², and hence is not expected to change the arguments above.

Dipolar calculations

Besides the spin orbit coupling from the 5d–6s bands, the 4f dipolar interactions are also expected to contribute to the magnetic anisotropy³¹. We emphasise that the dipolar interaction is unlikely to drive the ordering of the Eu ions in EuAl₄ as the dipolar energy is on the order of ∼1 K, however it can be relevant in choosing the magnetic anisotropy in the material, given that no dominant sources of anisotropy are expected to be present. We calculated the dipolar energy, given in Eq. 1³⁵, for each of the different spin structure solutions, whilst varying both the magnitude of the Eu moment and the ratio of the a/b lattice parameters, and the results are shown in Fig. 6.

$${E}_{{{\rm{dip}}}}=\frac{{\mu }_{0}}{4\pi {r}^{3}}\left[{{{\bf{\mu }}}}_{1}\cdot {{{\bf{\mu }}}}_{2}-\frac{3}{{r}^{2}}\,\left({{{\bf{\mu }}}}_{1}\cdot {{\bf{r}}}\right)\left({{{\bf{\mu }}}}_{2}\cdot {{\bf{r}}}\right)\right]$$

(1)

Fig. 6: Calculated dipolar energy for various magnetic structures for the k_qq and k_oq propagation vectors of EuAl₄.

The dipole energy of EuAl₄ calculated for the Eu1 ion taking into account all Eu ions in two unit cells along each crystallographic direction, as a function of the magnitude of the Eu1 moment and the ratio of the b/a lattice parameters, for each of the symmetry allowed spin configurations for the ${{{\boldsymbol{k}}}}_{{qq}}$ and ${{{\boldsymbol{k}}}}_{0q}$ propagation vector. The calculations were evaluated for ${{{\boldsymbol{k}}}}_{{qq}}$ = (0.09,0.09,0) and ${{{\boldsymbol{k}}}}_{0q}$ = (0,0.19,0).

In performing these calculations, we assumed that the Eu ions do not contribute to the periodic distortion of the lattice that gives rise to the CDW satellites, and instead it is suggested that the Al ions that displace^21,22. We note changing the magnitude of ${{{\bf{k}}}}_{0{{\rm{q}}}}$ did not change the qualitative results of the calculations, and hence these results are valid for the AFM2, AFM3 and AFM4 phases.

Figure 6 shows that for the ${{{\bf{k}}}}_{{{\rm{qq}}}}$ propagation vector, the spin configuration that minimises the dipolar energy is a SDW with the moments orientated perpendicular to the direction of the propagation vector in the ab plane, i.e. for m||$a\bar{b}$, as observed experimentally. For the ${{{\bf{k}}}}_{0{{\rm{q}}}}$ propagation vector, the spin configuration that minimises the dipolar energy is a SDW m||a. These calculations show that the magnetic dipole interaction selects the moment in the ab plane to be orientated perpendicular to the direction of the propagation vector, in agreement with our experimental observations. Whilst dipolar interactions are sufficient to explain the observed anisotropy of the spin structures associated with the ${{{\bf{k}}}}_{{{\rm{qq}}}}$ propagation vector in the AFM1 and AFM2 phases, it cannot explain why an additional component along c is favoured for the ${{{\bf{k}}}}_{0{{\rm{q}}}}$ propagation vector, indicating other anisotropic terms become energetically favourable as the temperature is lowered below T₃, such as those discussed in the preceding paragraph. The dipolar interaction is dependent on the magnitude of the magnetic moments and the distance between them, and is therefore applicable to other systems that order in the same symmetry and with similar sized moments, such as the square-net systems GdRu₂Si₂³⁶ and EuGa₂Al₂²⁶, where the in-plane moment has indeed been observed to order perpendicular to the direction of the propagation vector, further validating our calculations.

Structural and symmetry changes

Below T₃ we observed a change to the average crystal structure, from a splitting of the (1,0,7) reflection in the hk plane as shown in Fig. 7a. No further discontinuous changes to the Bragg reflections were observed below T₄, however large changes to the intensity of the CDW satellites were observed. To measure the CDW reflections we realigned the beamline to 6.5 keV, just below the Eu L₃ edge, so that we could only be sensitive to changes associated with the ionic displacements of the CDW order. The intensity of the CDW reflections underwent small shifts between 20 K (PM) and T₃, and larger changes below T₄. Of particular importance were reflections such as the (1,0,9+τ) that had finite intensity in all magnetically ordered phases but tended to zero in the AFM4 phase, and the (0,0,6+τ) for which the intensity significantly increased in the AFM4 phase, as shown in Fig. 7b, c. A refinement of the CDW structure was not possible from the data collected, and hence neither was a complete determination of the space group symmetry.

Fig. 7: Temperature dependence of the (1,0,7) reflection, charge density wave reflections and maps of the variation in intensity of the (1,0,7) reflection across the EuAl₄ sample.

a Reciprocal space maps showing the (1,0,7) reflection in the hk plane as the sample was warmed. b Variation in the intensity of CDW reflections (h,k,l+τ) as the sample was warmed through each of the magnetically ordered phase. The intensity of the reflection is represented by both the size and colour of the marker. c Rocking curves of the (0,0,6+τ), (0,0,8+τ), (1,0,9+τ) and (1,1,6+τ) CDW reflections collected upon warming between 8 and 20 K. The variation in intensity of the (1,0,7) reflection collected as the beam was translated across the sample at (d) 300 K in the PM phase and (e) at 8 K in the AFM4 phase.

Nevertheless, from a phenomenological point of view, an asymmetry between two spin chiral states would not occur unless the crystal structure itself was chiral or polar. Indeed, an analysis of the symmetry of the magnetic and nuclear structures suggests that the symmetry is lowered to a chiral monoclinic space group below T₃ as we outline below. The average structure of the CDW below T_CDW was reported to be described by the Fmmm²¹ or by the Immm³⁷ space groups. Both these subgroups derive from the I4/mmm parent structure from the Λ₅ irreducible representation (irrep) with different order parameter directions (ODP) (a,a,a,−a) and (a,a,0,0) respectively. To describe the total symmetry of the AFM3 and AFM4 phases is necessary to combine the symmetry requirement of both the CDW and helical phases. This will result in a (3 + 2)D magnetic superspace group able to correctly describe the system symmetry. Since there is an uncertainty regarding the order parameter direction of the CDW phase is necessary to consider both the Fmmm and Immm reported structures. The magnetic helical structure, which is present in the AFM3 and AFM4 phases, is described by the action of two irreps of the parent I4/mmm structure, namely mΣ₃ and mΣ₄, as shown in Supplementary Note 1 of the SM. Considering only the magnetic degree of freedom, the point group symmetry of the magnetic structure is 222, which reflects the chiral nature of the helical phase. After combining the possible reported structures of the CDW phase, together with the helical magnetic structure, as described in detail in Supplementary Note 3 of the SM, one obtains three possible magnetic superspace groups that can describe the symmetry of the AFM3 and AFM4 phases. Of these three magnetic superspace groups, only the two monoclinic groups would change the OPD of the CDW distortion that is consistent with the observed splitting of the nuclear reflections and changes to the magnetic peaks. For instance, the monoclinic strain allowed by these symmetry groups would give rise to two monoclinic domains in a single chiral magnetic domain, which can account for the stripe contrast observed in Fig. 4c, d and Fig. 7e, as well as explaining the nuclear reflection splitting shown in Fig. 7a. We note that this lowering of symmetry to monoclinic is a tendency observed in numerous square-net systems that have charge density wave modulations^{38,39,40,41,42}. Importantly it is worth noticing that for both monoclinic space groups, the symmetry of the CDW (without considering the magnetic degrees of freedom) is now chiral, having point group 222 or 2. In inversion symmetric crystals one expects to establish equally populated chiral magnetic domains that restores the global parity of the crystal⁸, whilst for non-centrosymmetric crystals an asymmetry in the chiral domain population is expected, and it is often the case that a single magnetic chiral domain is stabilised^8,9,43. Hence, the observation of a single chiral magnetic domain in EuAl₄ is further evidence of the breaking of inversion symmetry in this crystal below T₃. This change in the CDW symmetry at T₃, and its now chiral nature is important in explaining the chirality reversal observed at T₄ as will be discussed in the next section.

Reversal of the spin chirality

The establishment of a chiral symmetry below T₃ can break the degeneracy between the two chiral states of the magnetic helix. Without the chiral or polar structural distortions, the two chiral states of the magnetic helix represent degenerate chiral domains. A change to the polarity or chirality of the crystal structure could then induce a change to the magnetic chirality, if the magnetic and structural order parameters were coupled²⁶. It is also possible that we observe a new type of phase transition with a spontaneous change of magnetic chirality, but this scenario again requires the asymmetry between the two chiral state of the magnetic helix and therefore the chiral or polar crystal structure. An alternative mechanism to reverse the spin chirality is the competition between Dzylaoshinskii Moriya and dipolar interactions tuned by RKKY interactions, which has been shown to reverse the chirality of domain walls⁴⁴.

Discussion

Resonant elastic X-ray scattering was used to solve the nature of the four magnetically ordered phases of EuAl₄. We show that in the AFM1 phase EuAl₄ orders to form a SDW with moments aligned in the ab plane, perpendicular to the propagation vector. The AFM2 phase is characterised by the coexistence of two distinct magnetic orders, the SDW present in the AFM1 phase, and a SDW with m||c, which are spatially separated in the sample. In the AFM3 phase the ordering of the magnetic modulations change, and a magnetic helix of a single chiral domain is stabilised, where one component of the moment is in the ab plane, perpendicular to the propagation vector, and the second is along c. We find that in every magnetically ordered phase of EuAl₄ the in-plane moment is perpendicular to the orientation of the magnetic propagation vector, which we demonstrate is favoured by magnetic dipolar interactions. This source of anisotropy may be relevant to other square-net rare-earth intermetallic system in which the magnetic ions adopt a similar arrangement, such as in GdRu₂Si₂, EuGa₂Al₂ and GdSbTe. Below T₄ the spin chirality of the helix reverses, in the absence of any change to the spin ordering of the sample, and where only a single spin chirality is selected in both the AFM3 and AFM4 phases across the entire 1mm² sample. These experimental results demonstrate the spontaneous reversal of spin chirality. Switching the spin chirality in the absence of an external field would reduce the energy barrier for switching between two bistable chiral states, and thus substantially broaden the chiral spintronic devices that could be engineered. Furthermore, it may also have wide implications for the future design and development of spintronic materials. In future studies it would be of interest to demonstrate the reversal of spin chirality in EuAl₄ through transport measurements, such as with the use of polarised spin currents; a necessary first step towards furthering the technological use of this material. In demonstrating that the system lowers to chiral monoclinic at the onset of the magnetic helical ordering we are able to explain the observed asymmetry in the chiral spin states, and provide a possible mechanism by which the switching of the spin chirality could occur, i.e. via magnetostructural coupling. Although we note that it is also possible that we may have observed a new type of magnetic phase transition. Key to the design of new materials that may also exhibit such a spontaneous reversal of spin chirality would be determining the mechanism driving such a phase transition, which would also be of interest to explore further in future studies.

Methods

REXS measurements were performed in the vertical scattering geometry with an incident beam size of 180 µm in the horizontal direction and ∼30 µm in the vertical. To measure the variation in intensity across the sample, for instance the data presented in Figs. 2b, c, 3a, c and 4c, d we focused the beam using slits and the beam projection on the sample was 100 µm by 37 µm. The sample was mounted so that the (0,0,l) was specular, and the azimuthal reference was (h,0,0), such that at ϕ = 0 the a axis was parallel to the incident beam. An ARS closed cycle refrigerator was used to cool the sample from room temperature to 5 K. A Pilatus 100 K area detector was used to measure the diffracted intensity when it was not analysed into the σ and π channels. To analyse the diffracted beam a Cu (220) crystal orientated close to 44.66° with respect to the incident beam was used together with a Merlin area detector. We note that at 6.972 keV the attenuation length of the X-rays is expected to be approximately 10 µm for a grazing incident angle greater than 40°, which serves as an estimate for the X-ray penetration depth.

The incident polarisation of light was tuned between linear horizontal (σ), linear vertical (π) and two circular polarisations of opposite handedness, circular right (⟳) and circular left (⟲), using the light transmitted through a 1 mm diamond (0,0,1) plate orientated close to the (1,1,1) reflection in the asymmetric Laue mode. We note that the conversion of the light from linear vertical to circular is not perfect, and thus a fraction of the light remains σ polarised. Polarised light can be described by the three real Stokes parameters, ${P}_{1}$, ${P}_{2}$ and ${P}_{3},$ which form the vector P⁴⁵. For perfectly circular polarised light P = (0, ±1, 0). We fitted the following expression, P = (0, ${P}_{2}$, $\sqrt{1\,-\,{P}_{2}^{2}}$), which assumes the beam is partially circular polarised and partially linearly polarised, and determined the value of P₂ to be 0.88.