Net Dzyaloshinskii–Moriya interaction in defect-enriched ferromagnet

The Dzyaloshinskii–Moriya interaction (DMI), which typically occurs in lattices without space inversion symmetry, can also be induced in a highly symmetric lattice by local symmetry breaking due to any lattice defect. We recently presented an experimental study of polarized small angle neutron scattering (SANS) on the nanocrystalline soft magnet Vitroperm (Fe73Si16B7Nb3Cu1), where the interface between the FeSi nanoparticles and the amorphous magnetic matrix serves as such a defect. The SANS cross sections exhibited the polarization-dependent asymmetric term originating from the DMI. One would naturally expect the defects characterized by a positive and a negative DMI constant D to be randomly distributed and this DMI-induced asymmetry to disappear. Thus, the observation of such an asymmetry indicates that there exists an extra symmetry breaking. In the present work we experimentally explore the possible causes by measuring the DMI-induced asymmetry in the SANS cross sections of the Vitroperm sample tilted in different directions with respect to the external magnetic field. Furthermore, we analyzed the scattered neutron beam using a spin filter based on polarized protons and confirm that the asymmetric DMI signal originates from the difference between the two spin-flip scattering cross-sections.

It has been shown that spin-polarized small-angle neutron scattering (SANS) is a powerful technique to study the DMI in defect-rich polycrystalline materials, due to the unique dependence of the SANS cross section on the chiral magnetism [12]. Recently, the DMI in such defect-rich materials: nanocrystalline Tb and Ho, mechanically-deformed Co, as well as a commercial soft magnetic material Vitroperm (Fe 73 Si 16 B 7 Nb 3 Cu 1 ) has been investigated [13,14]. These experiments provide strong evidence that the DMI is of generic relevance for the magnetic microstructure of defect-rich ferromagnets. For Vitroperm, the polarized SANS cross sections were measured with an external magnetic field parallel to the macroscopic easy axis and perpendicular to the incoming neutron beam. The difference between flipper-on and flipper-off SANS signal as a function of the applied magnetic field is shown in figure 3 in [14]. A strong left-right asymmetry was observed between 14.5 mT and 25 mT and could be related to the predicted expression for the chiral function (see equation (4) and [12]). The observation is explained by the DMI caused by the lack of structural inversion symmetry at microstructural defect sites in the material; in our case, most likely the interface between the FeSi nanoparticles and the amorphous magnetic matrix. Furthermore we were able to estimate the strength of the DMI constant D by fitting the data to the theory described in [12].
However, if the defects in the material would be randomly distributed with some defects, whose chirality are characterized by a positive D (right-handed) and some others by a negative D (left-handed) [17], then one may expect that the net DMI-induced asymmetry would disappear. Therefore, the observation of a left-right asymmetric SANS signal indicates that there exists an extra symmetry breaking. This manuscript presents our experimental study searching for the source of the symmetry breaking.
We performed systematic half-polarized SANS measurements for different orientations of the sample with respect to the magnetic field and characterized the change of the DMIinduced asymmetry in the signal. As a possible explanation for the observed different signs of the asymmetry, we suggest that it could be caused by the combined effect of the demagnetizing field and the applied magnetic field. We back this conjecture with a set of simulations. In addition, we performed a fully polarization analyzed SANS experiment with a spin analyzer based on polarized protons [18]. The results confirm the theory [12] (see section 3) that the observed asymmetric signal [13,14] originates from the difference between the two spin-flip scattering cross sections.

Sample
Exactly the same vitroperm (Fe 73 Si 16 B 7 Nb 3 Cu 1 ) sample that was studied in [14] was used for the experiments in this paper. It is a commercially available nanocrystalline soft magnet, consisting of a distribution of FeSi nano-precipitates (∼10 nm [14]) in an amorphous magnetic matrix, with a permeability of µ = 1.5 × 10 5 and a saturation magnetization of µ 0 M s = 1.2 T, supplied by Vacuumschmelze GmbH, Hanau, Germany 5 . The desired nanocrystalline structure (FeSi grains) was generated by annealing the initially amorphous microstructured sample at around 500 • C-600 • C. The magnetization curve of the vitroperm sample has been reported in [14] and confirmed the low coercivity, related to the small crystalline grain size with a mean diameter of 10-20 nm. Spin-polarized SANS is therefore an ideal technique to investigate magnetic structures of this size [19]. A stack of 20 sheets of 25 × 35 mm 2 with a thickness of about 30 µm each was used for the SANS measurements. The total sample thickness has been carefully optimized in a depolarization analysis [20], in which we show that the depolarization of the sample can be crucial for the SANS signal of interest.

SANS
The polarized SANS experiment was performed at the SANS I instrument [21] at the continuous spallation neutron source SINQ at the Paul Scherrer Institute (PSI), Switzerland. The mean wavelength of the incident neutron beam was λ = 5.63 Å with a wavelength spread of ∆λ/λ = 10% (FWHM) and polarized by means of a V-shaped Fe/Si supermirror transmission polarizer to P = 98%. Using an adiabatic spin flipper, the polarization of the neutron beam could be reversed with an efficiency of ϵ = 99%. The external magnetic field µ 0 H 0 = 17 mT was applied by an electromagnet horizontally and perpendicular to the wave vector k 0 of the incident neutron beam. According to the magnetization curve given by figure 1 in [14], the sample is in the nearsaturation regime, so we can apply the theory developed in [12] (see below). Although there would only be a single magnetic domain under this near-saturation regime, the contrast of the local magnetization density difference between the nanoprecipitates and the amorphous matrix can be detected by SANS.
We define the x-axis along k 0 and the z-axis along H 0 , hence the y-axis is the vertical axis. The same stack of 20 sheets of Vitroperm, as used in [14], was used for the neutron experiments. For half-polarized SANS, the sample was mounted at room temperature on a circular plate, which is then fixed on a vertical sample holder. With this design, there were two axes, about which we could rotate our sample-the horizontal e x ∥ k 0 and the vertical e y (see figure 1). The rotation angle about the former was defined as ψ, while about the latter as φ. Then we set ψ = 0, φ = 0 corresponding to the magnetic easy axis of the sample parallel to the external magnetic field and sample sheet facing perpendicularly to the neutron beam direction. We further define θ as the angle between the scattering vector q and e z . For each measurement the sample was first saturated at a field of 0.8 T in order to follow the same magnetic hysteresis curve and then measured at 17 mT.

Spin filter for polarization analysis
In order to confirm that the DMI signal is from the spinflip SANS cross sections, a fully polarization analyzed SANS experiment was performed using a spin analyzer allowing us to separate the contribution of different SANS cross-sections. The spin analyzer based on polarized protons in a naphthalene single crystal [18,22] was used to analyze the spin states of the scattered neutron beam. The naphthalene spin filter crystal of 6.0 × 4.9 × 4.0 mm 3 size doped with 2.5 × 10 5 mol mol −1 d-pentacene was polarized in the laboratory using triplet dynamic nuclear polarization [23] and then transported to the neutron scattering facility and installed in the electromagnet at the SANS instrument [18]. After set-up of the spin filter, its flipping ratio and the corresponding filter polarization were determined, where the polarization homogeneity was measured moving the sample vertically behind a 1 mm slit aperture. As shown in figure 2, the filter was homogeneously polarized with an average proton polarization of 67% corresponding to a flipping ratio of ∼3.3. NMR measurements before and after the transport confirmed that there was no discernible loss of polarization of the spin filter.
The same stack of 20 sheets of Vitroperm sample was then mounted at room temperature on a sample holder fixed to the vacuum tube of the spin filter cryostat at a distance of ∼3.5 cm to the filter within the homogeneous field region of the electromagnet. The same magnetic field also maintained the spin filter polarization and the relaxation time was measured to be >700 h at 17 mT. A 4 × 4 mm 2 cadmium aperture, placed inside the cryostat at a distance of 14 mm from the filter crystal,  defined the analyzer cross section. For the background measurements the magnetic field was kept at 0.8 T, while for each sample measurement the sample was also first saturated at a field of 0.8 T and then measured at 17 mT. In order to measure all four spin channels, the naphthalene polarization was reversed during the experiment by an adiabatic fast passage (AFP) with an efficiency of ∼99%, as determined by both NMR and neutron transmission. The spin filter performance was very stable throughout the experiment of more than two days, as can be seen from the polarization evolution presented in figure 3.

Polarized SANS cross section
Based on pioneering early work [24,25], the elastic polarized SANS cross-sections (POLARIS) has been carefully studied and presented in [9]. We summarize them here to facilitate the discussion and interpretation of our experiments. For the specific neutron scattering geometry used, where the incident neutron beam is perpendicular to the externally applied magnetic field (k 0 ⊥ H 0 ), the cross sections can be expressed as [14] dΣ and where the first superscript (e.g. '+') that is attached to dΣ/dΩ refers to the spin state of the incident neutrons, whereas the second one (e.g. '−') specifies the spin state of the scattered neutrons. N and M are the Fourier coefficients of the nuclear scattering length density and the magnetization vector field respectively. V is the scattering volume and the constant b H = 2.91 × 10 8 A −1 m −1 relates the atomic magnetic moment µ a to the atomic magnetic scattering length b m = b H µ a [9,26]. The chiral function can be written as Typically in highly symmetric systems, the magnetization coefficients M x,y,z are real and the chiral function χ vanishes. However, for structures lacking inversion symmetry [12], M x,y,z can have an imaginary part, hence the chiral function χ(q) becomes non-zero.
Using the micromagnetic continuum theory, the chiral function can be expressed as [12] 2i where H p denotes the anisotropy-field Fourier coefficient functions. The function p(q, Unlike all the other scattering cross section terms, which are point symmetric, i.e. Σ(q) = Σ(−q), the chiral term is rather inversion anti-symmetric, i.e. 2i χ(q) = −2i χ(−q).
The chiral function can be experimentally determined by measuring the difference between the two spin-flip cross sections Experiments where only the incident neutron beam is polarized, while the scattered neutrons are not spin analyzed, provide the half-polarized SANS cross sections (SANSPOL) Here, too, the difference between these yields information on the chiral magnetism Indeed, polarized SANS is a particularly powerful technique to study the DMI induced by defects. The difference in the two spin-flip neutron cross sections is a key signature of chiral magnetism.

Half-polarized SANS study of the DMI symmetry breaking
In a previous study [14] we have observed a non-pointsymmetric scattering in the difference between the two halfpolarized SANS cross sections (∆Σ = Σ + − Σ − ). This indicates that the defects characterized by a positive D and the others by a negative D [17] are not randomly distributed.
To identify the origin of this symmetry breaking, polarized SANS measurements were performed with a 6 mm diameter circular beam defining aperture at µ 0 H 0 = 17 mT as a function of the sample rotation angles ψ (about e x ) and φ (about e y ). The detector was set at 11 m from the sample position with 11 m beam collimation. Figure 4 shows the contrast signals ∆Σ(q) = Σ + − Σ − as a function of φ while keeping ψ = 0. Figures 5 and 6 show the azimuthally averaged ∆Σ(q) (θ > 0 horizontal sector (±20 • ) minus the θ < 0 horizontal sector (±20 • )) as well as the signal sum as a function of φ. We observe that the left right asymmetric signal is reversed when φ changes sign. The sample was rotated about the y-axis, thus M y should be invariant for a rotation of ±φ, while M x (along the beam direction and perpendicular to the external field) should be inverted  due to symmetry. This can explain, according to the first term in equation (3) which is along the z-direction of the magnetic field and which we mainly observe, the inversion of the asymmetry ∆Σ for a rotation about the y-axis of ±φ. Under the symmetric condition ψ = 0, φ = 0, the asymmetric SANS signal vanishes. This indicates that under this condition the additional symmetry breaking disappears and the defects characterized by a positive D or a negative D are randomly distributed and no net effect can be observed. Similar experiments were performed with ψ = 5 • , 10 • and 90 • and displayed in figure 6. Once again, the asymmetry in the SANS signal is reversed when the sign of φ is changed, and this asymmetric signal disappears when φ = 0 • . The asymmetry is particularly strong for ψ = 90 • , where the sample magnetic easy axis is aligned along the vertical y-axis during the variation of φ. Even for this case (when rotating about the magnetic easy axis) a reversion of the asymmetric SANS signal was observed, which proves that the extra symmetry breaking leading to a net observed asymmetric SANS signal is not related to the combined action of the macroscopic texture axis and the applied magnetic field as postulated in [14].
From the q-dependence of the asymmetric signal (see figure 5) it can be seen that for q > 0.006 Å −1 no peak structure is observed and the signal decays with a q −4 power function. This indicates that the chiral magnetic structure is larger than a scale of 2π/q ∼ 100 nm, which is much larger than the size of a single nano-precipitate (∼10 nm [14]). Therefore, the chiral magnetic structure must either be a combined effect of several nano-precipitates or of a few very  large ones. To better investigate the scale of this structure, we performed a polarized SANS measurement in a smaller q-region with higher q-resolution at 20 m detector distance and the corresponding neutron collimation. We aligned the sample with ψ = 90 • , φ = 10 • to enhance the DMI asymmetry. Figure 7 shows the contrast signal ∆Σ = Σ + − Σ − at the same magnetic field µ 0 H 0 = 17 mT, whereas the q dependence is shown in figure 8. No clear scattering peak is observed and the chiral magnetic structure is expected to be larger than a scale of 200 nm (2π/q), much larger than the mean nano-precipitate size. Furthermore, the higher q-resolution also allowed us to study the angular (θ) dependence of the asymmetric scattering pattern in more detail. In particular, we were looking for the sin θ cos 2 θ term predicted by the theoretical model [12], which should lead to a splitting of the asymmetric DMI scattering peak at high magnetic fields, but has not been observed in former defect-induced DMI experiments [13,14]. Figure 9 shows the asymmetric DMI signal as a function of θ (sum of the two ±10 • horizontal sectors and considering the different signs). Thanks to the high q resolution, an indication (>3σ) of the splitting of the peak is observed. However the position of the peak does not agree with the theoretical prediction at θ = ±45 • . This suggests that the micromagnetic theory for the defect-induced DMI may need further improvement to understand the experimental results.

Discussion on the origin of a net DMI
Combining all the experimental results we propose the following qualitative explanation of how the rotation of the sample breaks the symmetry and induces a preferred chiral magnetic structure that can be detected by neutrons. When the ferromagnetic sample is rotated in the external magnetic field, the demagnetizing field H d changes, which depends on the magnetization and the geometry of the ferromagnet. In order to calculate the effect, we approximate the magnetization in the sample as a continuum and consider the demagnetizing field as a macroscopic anisotropy. The microscopic magnetic moments of local spins forming the chiral magnetic structure are then determined by the total magnetic field including this macroscopic demagnetizing field. Since for each measurement the sample was first saturated at a field of 0.8 T and then brought to 17 mT, which is still close to full saturation (see figure 1 in [14]), we assume the magnetization of the sample M s to be parallel to the external field H 0 . The demagnetizing field H d can now be calculated for the 2-d condition with different rotations angles φ by solving Maxwell's equations using Matlab ® packages. As an example, figure 10 shows the demagnetizing field in the sample at an external field φ = 10 • out of plane. It is clear that the demagnetizing field is mostly perpendicular to the plane of the sample, due to the sample dimensions. More simulation results for different angles φ are given in the supporting information. Figure 11 then sketches the geometry of the neutron experiment together with the magnetic fields to illustrate the induced net chiral magnetic structure for different sample orientations, particularly for different φ angles. The local magnetic flux density B is tilted by the demagnetizing field towards opposite directions for φ and −φ. When the local magnetic moments are canted according to the magnetic flux density (compare figure 5 in [12]), the chirality for the two orientations (φ and −φ) is actually inverted on the projection along the neutron beam direction k 0 , which determines the detected neutron cross section. When the sample is strictly aligned with the external magnetic field in plane, no net DMI can be detected by neutrons.
Interestingly, the DMI induced neutron differential cross section ∆Σ is most pronounced at φ ∼ ±10 • (see figures 4 and 6) instead of a higher angle where the demagnetizing field has a larger component perpendicular the external field (see supporting information). Experimentally we have also shown that the size of the chiral magnetic structure is much larger than the nano-precipitate size. We suspect that these could be due to the DMI competing with the exchange interaction, as illustrated in figure 12, and with the thermal energy k B T. Since the sample is ferromagnetic, the neighboring spins tend to align in parallel to reduce the exchange energy. In this scenario, the exchange interaction favors the alignment of the neighboring chiral structures to the opposite chirality, which however can result in an increase in the DMI energy. Alternatively, decreasing the DMI interaction may lead to an increase of the exchange interaction energy. The two cases are demonstrated in figure 12. Therefore, the problem is more complicated than just minimizing the DMI energy potential, but rather the total energy, and may lead to the most pronounced DMI effect at φ = ±10 • . Naturally, the distribution of the precipitates (e.g. the distance between precipitates) should also make an influence. Furthermore, the energy must be compared with the thermal energy k B T. We conjecture that only the DMI of a chiral structure supported by either multiple nano-precipitates or a few large precipitates can lead to an energy larger than k B T so that the asymmetric occupation probability for the two directions of chirality can be seen. To verify these conjectures, comprehensive micromagnetic theoretical modeling and further experimental investigations, e.g. at low temperatures, have to be carried out.

Fully polarization analyzed SANS study of the DMI in Vitroperm
According to the theory [12], the asymmetric signal observed in the contrast between the two half polarized SANS cross sections Σ + − Σ − originates from the difference between the two spin-flip scattering cross sections Σ +− − Σ −+ . This can only be confirmed by performing an experiment with a full polarization analysis using a spin analyzer, which was in our case a spin filter based on polarized protons [18]. The sample was measured with the rotation angles ψ = 0 • , φ ∼ −10 • (compare figure 4). As we are more interested in the spin-flip channel, about three times (calculated according to the flipping ratio) more statistics was taken for the spin-flip data than the non-spin-flip data. Figure 13 shows the background subtracted and spin leakage corrected intensity plots of the four spin dependent cross sections on a logarithmic scale. For data treatment we used the approach described in [27], while neglecting the spin polarizer and flipper inefficiencies. Note that the spin channels with neutrons scattered to the '−' state show a stronger homogeneous background, but it is still an order of magnitude weaker than the signal, which is due to a non-perfect background subtraction. We can very well understand the scattering patterns by comparing them to equations (1) and (2). In the present experiment the sample is close to saturation but M x and M y are non-zero. Therefore, besides the | M z | 2 sin 4 θ term in the non-spin-flip channel and the | M z | 2 sin 2 θ cos 2 θ term in the spin-flip channel, we also observe the other terms containing M x and M y . In particular a strong scattering at small q (q < 0.015 Å −1 ) is observed in the spin-flip cross sections, where also the DMI induced chiral signal is expected (see equation (3)). However, most of the terms can be easily removed by subtracting the two spin-flip cross sections, leaving only the chiral term that we are interested in. Figure 14 shows the difference between the two spin flip cross sections ∆Σ = Σ +− − Σ −+ . Note, even though the measured differential cross section ∆Σ is attenuated by the neutron depolarization caused by the sample, it is still linear to the actual differential cross section [20]. A strongly left right asymmetric signal is observed. As pointed out in the previous section, the advantage of performing a full polarization analysis is that the nuclear-magnetic interference scattering,  which exhibits a spin dependence, is only observed in the nonspin-flip channels and no other term than the chiral function (see equation (3)) shows a spin dependence in the spin-flip channels. The result confirms the theoretical prediction [12] that the defect related DMI, caused by the lack of inversion symmetry, induces an asymmetry between the two spin-flip cross sections, which is also the source of the asymmetric signal observed in the half-polarized SANS experiment (see figure 4).

Discussions and conclusions
The existence of chiral magnetism related to the DMI induced by local inverse symmetry breaking near the defects in the Vitroperm (Fe 73 Si 16 B 7 Nb 3 Cu 1 ) nanocrystalline ferromagnet was confirmed by half-polarized SANS experiments in [14]. However, if these DMI-related local defects, which can be characterized by D and −D, are randomly distributed, there should be no net effect in the difference between the two halfpolarized SANS cross sections ∆Σ that would indicate a symmetry breaking. In this paper, we present our experimental study to find the source of symmetry breaking using polarized SANS, a powerful technique to study defect induced DMI. With half-polarized SANS experiments, we show that the sample geometry with respect to the external magnetic field is crucial in producing a net DMI effect. When the sample, which is a stack of 20 sheets of 25 × 35 mm 2 with a thickness of about 30 µm each, is well aligned in plane with the magnetic field e z , the net DMI effect is zero. This is in agreement with our perception of randomly distributed defects characterized by D and −D (different chirality). When rotating the sample plane (of the sheet) about e y out of the magnetic field, a net DMI signal can be detected and the sign depends on the rotation direction. The effect persists when rotating about the macroscopic texture axis, proving that the origin of the imbalance of D and −D is not relevant for this, contrary to our initial assumption [14].
On a macroscopic level, when rotating about the vertical y-axis by ±φ, the following relations should hold for the magnetization according to symmetry: M z (φ) = M z (−φ), M y (φ) = M y (−φ) and M x (φ) = − M x (−φ). Based on equation (3), it is clear that the chiral function should be inverted for ±φ as we have found experimentally. However, the microscopic origin of the DMI, in particular the symmetry breaking leading to an imbalance between chiral structures characterized by D and −D, is still unclear. We propose that the imbalance may be caused by the change of the demagnetizing field when rotating the sample. We treat the demagnetizing field as a macroscopic anisotropy on the spins on the nanoprecipitate interface. We show that the chirality for the projection of the induced chiral magnetic structure along the neutron beam direction is inverted when rotating the sample, which is detected by neutrons.
Furthermore, with a high q-resolution half polarized SANS measurement, an indication of peak splitting of the DMI asymmetry was observed in the ∆Σ (see figure 9). A splitting by the sin θ cos 2 θ term has been predicted in [12], which has never been observed before. However we note that splitting is not exactly at the predicted q. In combination with all the experimental observations, we believe that the a better theoretical understanding of the micromagnetism is required for the defect-induced DMI and further experiments are needed.
In addition, the full polarization analysis of the SANS on the Vitroperm sample confirms that the asymmetric signal observed in the difference between the half polarized SANS cross sections Σ + − Σ + (see figures 5 and 6) is indeed caused by the difference between the two spin-flip cross sections ∆Σ = Σ +− − Σ −+ , as predicted by the theory [12]. This asymmetric signal is explained by the defect-induced DMI originating from the lack of the structural inversion symmetry, where in our system most probably the interface between the FeSi nanoparticles and the amorphous magnetic matrix serves as the defect. The q dependence of ∆Σ suggests that this effect goes beyond a single precipitate and involves either multiple nano-precipitates or a few very large precipitates that can create an asymmetric occupation probability for the two directions of chirality, which can be detected by neutrons. The experiment further shows that polarized SANS, especially with a full polarization analysis, is extremely powerful in investigating defect-induced DMI. Considering that the defect-induced DMI is most likely a general effect in defectrich ferromagnets [13,14], we believe that this topic deserves further SANS studies with full polarization analysis.

Data availability statement
The data that support the findings of this study are openly accessible at https://zenodo.org/record/7941723#. ZGOjLnbMKUl.