research papers
Unraveling the magnetic softness in Fe–Ni–Bbased nanocrystalline material by magnetic smallangle neutron scattering
^{a}Department of Physics and Materials Science, Université du Luxembourg, 162A avenue de la Faïencerie, L1511 Luxembourg, Grand Duchy of Luxembourg, ^{b}Laboratory for Neutron Scattering, ETH Zurich and Paul Scherrer Institut, Villigen PSI 5232, Switzerland, and ^{c}Department of Materials Science and Engineering, Monash University, Clayton, Victoria 3800, Australia
^{*}Correspondence email: mathias.bersweiler@uni.lu, andreas.michels@uni.lu
Magnetic smallangle neutron scattering is employed to investigate the magnetic interactions in (Fe_{0.7}Ni_{0.3})_{86}B_{14} alloy, a HiBNANOPERMtype soft magnetic nanocrystalline material, which exhibits an ultrafine microstructure with an average grain size below 10 nm. The neutron data reveal a significant spinmisalignment scattering which is mainly related to the jump of the longitudinal magnetization at internal particle–matrix interfaces. The field dependence of the neutron data can be well described by micromagnetic smallangle neutron scattering theory. In particular, the theory explains the `cloverleaftype' angular anisotropy observed in the purely magnetic neutron scattering The presented neutron data analysis also provides access to the magnetic interaction parameters, such as the exchangestiffness constant, which plays a crucial role towards the optimization of the magnetic softness of Febased nanocrystalline materials.
Keywords: smallangle neutron scattering; micromagnetic theory; soft magnetic materials; nanocrystalline alloys; materials science; magnetic scattering; magnetic structures; inorganic materials; nanostructures.
1. Introduction
Since the pioneering work of Yoshizawa et al. (1988), the development of novel Febased nanocrystalline soft magnetic materials raised considerable interest owing to their great potential for technological applications (Petzold, 2002; Makino et al., 1997). The most well known examples are FINEMET (Yoshizawa et al., 1988), VITROPERM(Vacuumschmelze GmbH, 1993) and NANOPERMtype (Suzuki et al., 1991) soft magnetic alloys, which find widespread application as magnetic cores in highfrequency power transformers or in interface transformers in the ISDNtelecommunication network. For a brief review of the advances in Febased nanocrystalline soft magnetic alloys, we refer the reader to the article by Suzuki et al. (2019).
More recently, an ultrafinegrained microstructure combined with excellent soft magnetic properties was obtained in HiBNANOPERMtype alloys (Li et al., 2020). The magnetic softness in such materials can be attributed to the exchangeaveraging effect of the local magnetocrystalline anisotropy K_{1}. This phenomenon has been successfully modeled within the framework of the random anisotropy model (RAM) (Herzer, 1989, 1990, 2007; Suzuki et al., 1998), and becomes effective when the average grain size D is smaller than the ferromagnetic exchange length , where A_{ex} is the exchangestiffness constant and φ_{0} is a proportionality factor of the order of unity which reflects the symmetry of K_{1}. In this regime, the RAM predicts that the coercivity H_{C} scales as , where n = 3 or n = 6 depending on the nature of the magnetic anisotropy [see, for example, the work by Suzuki et al. (1998, 2019) for details]. Therefore, an improvement of the magnetic softness comes about by either reducing D and/or increasing L_{0}.
In the context of increasing L_{0}, the quantitative knowledge of A_{ex} could help to further develop novel Febased soft magnetic nanocrystalline materials. However, up to now, most of the research activities in this field are focused on the overall characterization, e.g. via hysteresisloop measurements (coercivity, saturation magnetization and permeability) and magnetic anisotropy determination (crystalline, shape or stress related) (McHenry et al., 1999; Herzer, 2013; Suzuki et al., 2019). One reason for this might be related to the fact that many of the conventional methods for measuring A_{ex} (e.g. magnetooptical, Brillouin spinwave resonance or inelastic neutron scattering) require thinfilm or singlecrystal samples.
In the present work, we employ magnetic fielddependent smallangle neutron scattering (SANS) to determine the magnetic interaction parameters in (Fe_{0.7}Ni_{0.3})_{86}B_{14} alloy, specifically, the exchangestiffness constant and the strength and spatial structure of the magnetic anisotropy and magnetostatic fields. The particular alloy under study is a promising HiBNANOPERMtype soft magnetic material, which exhibits an ultrafine microstructure with an average grain size below 10 nm (Li et al., 2020). Magnetic SANS is a unique and powerful technique to investigate the magnetism of materials on the mesoscopic length scale of ∼1–300 nm [e.g. nanorod arrays (Grigoryeva et al., 2007; Günther et al., 2014; Maurer et al., 2014), nanoparticles (Bender et al., 2019, 2020; Bersweiler et al., 2019; Zákutná et al., 2020; Kons et al., 2020; Köhler et al., 2021), INVAR alloy (Stewart et al., 2019) or nanocrystalline materials (Ito et al., 2007; Mettus & Michels, 2015; Titov et al., 2019; Oba et al., 2020; Bersweiler et al., 2021)]. For a summary of the fundamentals and the most recent applications of the magnetic SANS technique, we refer the reader to the literature (Mühlbauer et al., 2019; Michels, 2021).
This paper is organized as follows: Section 2 provides some details of the sample characterization and the neutron experiment. Section 3 summarizes the main expressions for the magnetic SANS and describes the dataanalysis procedure to obtain the exchange constant and the average magnetic anisotropy field and magnetostatic field. Section 4 presents and discusses the experimental results, while Section 5 summarizes the main findings of this study.
2. Experimental
The ultrarapidly annealed (Fe_{0.7}Ni_{0.3})_{86}B_{14} alloy (HiBNANOPERMtype) was prepared according to the synthesis process detailed by Li et al. (2020). The sample for the neutron experiment was prepared by employing the lowcapturing isotope ^{11}B as the starting material. The average crystallite size was estimated by wideangle Xray diffraction (XRD) using a Bruker D8 diffractometer in Bragg–Brentano geometry (Cu Kα radiation source). The magnetic measurements were performed at room temperature using a Cryogenic Ltd vibrating sample magnetometer equipped with a 14 T superconducting magnet and a Riken Denshi BHS40 DC hysteresis loop tracer. The crystallization and Curie temperatures were determined by means of (DTA) and thermomagnetogravimetric analysis (TMGA) on Perkin Elmer DTA/TGA 7 analyzers under a constant heating rate of 0.67 K s^{−1}. For the neutron experiments, six (Fe_{0.7}Ni_{0.3})_{86}B_{14} ribbons with a surface area of 12 × 20 mm and a thickness of ∼15 µm were stacked together, resulting in a total sample thickness of ∼90 µm. The neutron measurements were conducted at the instrument SANS1 at the Swiss Spallation Neutron Source at the Paul Scherrer Institute, Switzerland. We used an unpolarized incident neutron beam with a mean wavelength of λ = 6.0 Å and a wavelength broadening of Δλ/λ = 10% (full width at halfmaximum). All neutron measurements were conducted at room temperature and within a qrange of about 0.036 nm^{−1} ≤ q ≤ 1.16 nm^{−1}. A magnetic field H_{0} was applied perpendicular to the incident neutron beam (H_{0} ⊥ k_{0}). Neutron data were recorded by decreasing the field from the maximum field available of 8.0 to 0.02 T following the magnetization curve (see Fig. 2). The internal magnetic field H_{i} was estimated as , where M_{S} is the saturation magnetization and N_{d} is the demagnetizing factor, which was determined based on the analytical expression given for a rectangular prism (Aharoni, 1998). Neutron data reduction (corrections for background scattering and sample transmission) was conducted using the GRASP software package (Dewhurst, 2018).
3. Micromagnetic SANS theory
3.1. Unpolarized SANS
Based on the micromagnetic SANS theory for twophase particle–matrixtype ), the elastic total (nuclear + magnetic) unpolarized SANS dΣ/dΩ at momentumtransfer vector q can be formally written as (H_{0} ⊥ k_{0}):
developed by Honecker & Michels (2013where
corresponds to the (nuclear + magnetic) residual SANS
which is measured at complete magnetic saturation, anddenotes the purely magnetic SANS –(3), V is the scattering volume; b_{H} = 2.91 × 10^{8} Å^{−1} m^{−1} relates the atomic to the atomic magnetic scattering length; and represent the Fourier transforms of the nuclear scattering length density N(r) and of the magnetization vector field M(r), respectively; θ specifies the angle between H_{0} and q ≃ q{0, sin(θ), cos(θ)} in the smallangle approximation; and the asterisks (*) denote the complex conjugated quantities. is the Fourier transform of the saturation magnetization profile M_{S}(r), i.e. at complete magnetic saturation [compare Equation (2)]. For smallangle scattering, the component of the scattering vector along the incident neutron beam, here q_{x}, is smaller than the other two components q_{y} and q_{z}, so that only correlations in the plane perpendicular to the incoming neutron beam are probed.
In Equations (1)In our neutrondata analysis, to experimentally access dΣ_{mag}/dΩ, we subtracted the SANS dΣ/dΩ measured at the largest available field (approachtosaturation regime; compare Fig. 2) from measured at lower fields. This specific subtraction procedure eliminates the nuclear SANS contribution , which is field independent, and therefore
where Δ represents the differences of the Fourier components at the two selected fields (low field minus highest field).
3.2. Approachtosaturation regime
In the particular case of the approachtosaturation regime, where , and which implies therefore in Equation (4), dΣ/dΩ can be rewritten as:
where and correspond to the magnetic scattering contributions due to perturbing magnetic anisotropy fields and magnetostatic fields, respectively. More specifically, the anisotropyfield scattering function
depends of the Fourier coefficient of the magnetic anisotropy field, whereas the scattering function of the longitudinal magnetization
is related to the Fourier coefficient . For an inhomogeneous material of the NANOPERMtype, the latter quantity is related to the magnetization jump ΔM at internal (e.g. particle–matrix) interfaces. We would like to emphasize that the q dependence of S_{H} and S_{M} can often be described by a particle form factor (e.g. sphere) or a Lorentziansquared function. The corresponding (dimensionless) micromagnetic response functions R_{H} and R_{M} are given by
and
The dimensionless function depends on the internal magnetic field H_{i} and on the exchange length .
3.3. Estimation of the magnetic interaction parameters
Most of the time it is more convenient to analyze the (over 2π) azimuthally averaged SANS cross sections instead of the 2D ones. By performing an azimuthal average of the response functions [Equations (8) and (9)] with respect to the angle θ, i.e. , and by assuming S_{H} and S_{M} to be isotropic (θindependent), the SANS dΣ/dΩ can be written as:
where
and
For a given set of parameters A_{ex} and M_{S}, the numerical values of R_{H} and R_{M} are known at each value of q and H_{i}. Because of the linearity of Equation (10) in R_{H} and R_{M}, one can obtain the values of (as the intercept) and S_{H} and S_{M} (as the slopes) at each qvalue by performing a (weighted) nonnegative leastsquares fit of the azimuthally averaged SANS cross sections dΣ/dΩ measured at several H_{i}. Treating A_{ex} in the expression for as an adjustable parameter during the fitting procedure allows us to estimate this quantity. The bestfit value for A_{ex} is obtained from the minimization of the (weighted) meansquared deviation between experiment and fit:
where the indices μ and ν refer to the particular q and H_{i}values, denotes the uncertainties in the experimental data, N = N_{μ}N_{ν} corresponds to the number of data points, and dΣ^{exp}/dΩ and dΣ^{sim}/dΩ are the azimuthally averaged SANS determined from the neutron experiments and numerically computed using Equation (10), respectively. We would like to point out that the bestfit value for A_{ex} represents an average over the sample volume.
Finally, the numerical integration of the determined S_{H}(q) and S_{M}(q) over the wholeq space according to the work by Honecker & Michels (2013)
yields the meansquare anisotropy field 〈∣H_{p}∣^{2}〉 and the meansquare longitudinal magnetization fluctuation , respectively. Since the neutron experiments are performed within a finite qrange and since both integrands do not exhibit any sign of convergence, one can only obtain a lower bound for both quantities by numerical integration. Moreover, it is important to realize that the specific neutron data analysis described above does not represent a `continuous' fit of dΣ/dΩ in the conventional sense, but rather the pointbypoint reconstruction of the theoretical cross sections based on the experimental data.
4. Results and discussion
Fig. 1 displays the wideangle XRD results of the (Fe_{0.7}Ni_{0.3})_{86}B_{14} ribbons. The XRD pattern exhibits only the reflections from the f.c.c.Fe(Ni) phase, as expected for this particular composition (Li et al., 2020), and therefore confirms the highquality synthesis of the sample. The values of the lattice parameter a and the average crystallite size D were estimated from the XRD data using the LeBail fit method (LBF) implemented in the FullProf suite (RodríguezCarvajal, 1993). The bestfit values are summarized in Table 1. Both values are consistent with the data in the literature [compare the work by Anand et al. (2019) and Li et al. (2020) for a and D, respectively]. As previously discussed, the origin of the exceptionally fine microstructure observed in (Fe_{0.7}Ni_{0.3})_{86}B_{14} alloys may be qualitatively attributed to the ultrafast nucleation kinetics of the f.c.c.Fe(Ni) phase (Li et al., 2020).

Fig. 2(a) presents the positive magnetization branch on a semilogarithmic scale (measured at room temperature), while the hysteresis loop on a linear–linear scale, and between ±0.03 mT, is displayed in Fig. 2(b). The data have been normalized by the saturation magnetization M_{S}, which was estimated from the linear regression for [see inset in Fig. 2(a)]. The values of M_{S} and H_{C} (see Table 1) are in agreement with those reported in the literature (Li et al., 2020). Defining the approachtosaturation regime by M/M_{S} ≥ 90%, we can see that this regime is reached for μ_{0}H_{i} ≳ 65 mT. Moreover, the extremely small value for H_{C} combined with the high M_{S} confirms the huge potential of (Fe_{0.7}Ni_{0.3})_{86}B_{14} alloy as a soft magnetic material, and suggests that in the framework of the RAM (Herzer, 2007), H_{C} should fall into the regime where (Suzuki et al., 2019).
Fig. 3 shows the DTA and TMGA curves for the amorphous (Fe_{0.7}Ni_{0.3})_{86}B_{14} alloy. Two exothermic peaks are evident on the DTA curve reflecting the well known twostage reactions, where f.c.c.Fe(Ni) forms at the first peak followed by decomposition of the residual amorphous phase at the second peak. The sharp drop of the TMGA signal just before the second stage crystallization corresponds to the Curie temperature of the residual amorphous phase ( ≃ 720 K). This value, which reflects the exchange integral in our sample (see below), is consistent with those determined for amorphous Fe_{86}B_{14} samples prepared under similar conditions (Zang et al., 2020).
Fig. 4 (upper row) shows the experimental 2D total (nuclear + magnetic) SANS cross sections dΣ/dΩ of the (Fe_{0.7}Ni_{0.3})_{86}B_{14} ribbons at different selected fields. As can be seen, at μ_{0}H_{i} = 7.99 T (near saturation), the pattern is predominantly elongated perpendicular to the magnetic field direction. This particular feature in dΣ/dΩ is the signature of the socalled `type' angular anisotropy [compare Equation (2)]. Near saturation, the magnetic scattering resulting from the spin misalignment is small compared with that resulting from the longitudinal magnetization jump at the internal (e.g. particle–matrix) interfaces. By reducing the field, the patterns remain predominantly elongated perpendicular to the magnetic field, but at the smaller momentum transfers q an additional fielddependent signal is observed `roughly' along the diagonals of the detector, suggesting a more complex magnetization structure. Fig. 4 (middle row) presents the corresponding 2D purely magnetic SANS cross sections dΣ_{mag}/dΩ determined by subtracting dΣ/dΩ at μ_{0}H_{i} = 7.99 T from the data at lower fields. In this way, the maxima along the diagonals of the detector become more clearly visible, thereby revealing the socalled `cloverleaftype' angular anisotropy pattern. This particular feature was also previously observed in NANOPERMtype soft magnetic merials (Honecker et al., 2013), and is related to the dominant magnetostatic term S_{M} × R_{M} in the expression for dΣ_{mag}/dΩ [compare Equations (8) and (9)]. More specifically, the jump in the magnitude of the saturation magnetization at the particle–matrix interfaces, which can be of the order of 1 T in these type of alloys (Honecker et al., 2013), results in dipolar stray fields which produce spin disorder in the surroundings. Fig. 4 (lower row) displays dΣ_{mag}/dΩ computed using the micromagnetic SANS theory [Equations (5)–(9)] and the experimental parameters summarized in Table 1. As is seen, the cloverleaftype angular anisotropy experimentally observed in Fig. 4 (middle row) can be well reproduced using micromagnetic theory.
Fig. 5(a) displays the (over 2π) azimuthally averaged dΣ/dΩ, while the corresponding dΣ_{mag}/dΩ are shown in Fig. 5(b). By decreasing μ_{0}H_{i} from 7.99 T to 10 mT, the intensity of dΣ/dΩ increases by almost two orders of magnitude at the smallest momentum transfers q. By comparison to Equations (1)–(4), it appears obvious that the magnetic field dependence of dΣ/dΩ can only result from the mesoscale spin disorder (i.e. from the failure of the spins to be fully aligned along H_{0}). As is seen in Fig. 5(b), the magnitude of dΣ_{mag}/dΩ is of the same order as dΣ/dΩ, supporting the notion of dominant spinmisalignment scattering in (Fe_{0.7}Ni_{0.3})_{86}B_{14} alloy.
Fig. 6 shows the magnetic SANS results determined from the fielddependent approach described in Section 3.3. In the present case, to warrant the validity of the micromagnetic SANS theory, only dΣ/dΩ measured for μ_{0}H_{i} ≳ 65 mT (i.e. within the approachtosaturation regime, compare Fig. 2) were considered. We have also restricted our neutron data analysis to , since the magnetic SANS is expected to be fieldindependent for q ≥ q_{max} (Michels, 2021). In Fig. 6(a), we plot the (over 2π) azimuthally averaged dΣ/dΩ along with the corresponding fits based on the micromagnetic SANS theory [Equation (10), black solid lines]. It is seen that the field dependence of dΣ/dΩ over the restricted qrange can be well reproduced by the theory. Fig. 6(b) displays the (weighted) meansquared deviation between experiment and fit, χ^{2}, determined according to Equation (13), as a function of the exchangestiffness constant A_{ex}. In this way, we find A_{ex} = (10 ± 1) pJ m^{−1} (see Table 1). The comparison with previous studies is discussed in the next paragraph for more clarity. Fig. 6(c) displays the bestfit results for dΣ_{res}/dΩ, S_{H} and S_{M}. Not surprisingly, the magnitude of dΣ_{res}/dΩ (limit of dΣ/dΩ at infinite field) is smaller than the dΣ/dΩ at the largest fields [compare Fig. 6(a)], supporting the validity of the micromagnetic SANS theory. Furthermore, the magnitude of S_{H} is about two orders of magnitude smaller than S_{M}, suggesting that the magnetization jump ΔM at internal particle–matrix interfaces represents the main source of spin disorder in this material. The estimated values for the meansquare anisotropy field and the meansquare magnetostatic field in terms of Equation (14) are 0.3 and 24 mT, respectively. These values qualitatively support the notion of dominant spinmisalignment scattering due to magnetostatic fluctuations. The qdependence of S_{M} can be described using a Lorentziansquared function [blue solid line in Fig. 6(c)] from which an estimate for the magnetostatic correlation length ξ_{M} = 2.4 ± 0.2 nm is obtained. This value compares favorably with the value of l_{M} = (2A_{ex}/μ_{0}M_{S}^{2})^{1/2} = 3.7 nm [using A_{ex} = 10 pJ m^{−1} and μ_{0}M_{S} = 1.34 T (taken from Table 1)], which reflects the competition between the exchange and magnetostatic energies.
We would like to emphasize that our experimental value for A_{ex} = 10 pJ m^{−1} is about 2–3 times larger than those reported in NANOPERMtype soft magnetic materials (Honecker et al., 2013). Since the Curie temperature of the residual amorphous phase in our nanocrystalline (Fe_{0.7}Ni_{0.3})_{86}B_{14} sample is well above 700 K (see Fig. 3 and Table 1), while that of the Fe_{89}Zr_{7}B_{3}Cu_{1} sample used in the previous study (Honecker et al., 2013) was as low as 350 K, the local exchange stiffness in the grain boundary amorphous phase in HiBNANOPERMtype alloys is expected to be higher than that in NANOPERMtype alloys. This finding could explain the origin of the larger A_{ex} value reported in the present study. Therefore, one can expect an improvement of the magnetic softness in HiBNANOPERM thanks to the ensuing increase of the ferromagnetic exchange length L_{0}. It is well established that nonmagnetic and/or ferromagnetic additives and the annealing conditions strongly affect the microstructural and magnetic properties of Febased nanocrystalline materials (McHenry et al., 1999; Herzer, 2007, 2013; Suzuki et al., 2019) and therefore have a strong impact on their magnetic softness. Using A_{ex} = 10 pJ m^{−1} (this study), K_{1} ≃ 9.0 kJ m^{−3},^{1} and φ_{0} ≃ 1.5 (Herzer, 2007), we obtain L_{0} ≃ 50 nm. This value for L_{0} is in very good agreement with the typical length scale of ∼30–50 nm previously reported in soft magnetic Febased alloys. Moreover, the comparison of the average grain size D = 7 nm with the L_{0} value, here D ≪ L_{0}, also confirms that in the framework of the random anisotropy model (Herzer, 1989, 1990, 2007; Suzuki et al., 1998), the exchangeaveraged magnetic anisotropy 〈K〉 falls into the regime where 〈K〉 ∝ D^{3}. This finding is also consistent with the (experimental) D^{3}dependence of H_{C} reported in Fe–Bbased HiBNANOPERM alloys (Suzuki et al., 2019; Li et al., 2020).
5. Conclusions
We employed magnetic SANS to determine the magnetic interaction parameters in (Fe_{0.7}Ni_{0.3})_{86}B_{14} alloy, which is a HiBNANOPERMtype soft magnetic material. The analysis of the magnetic SANS data suggests the presence of strong spin misalignment on a mesoscopic length scale. In fact, the micromagnetic SANS theory provides an excellent description of the field dependence of the total (nuclear + magnetic) and purely magnetic SANS cross sections. The cloverleaftype angular anisotropy patterns observed in the magnetic SANS signal can be well reproduced by the theory. The magnitudes of the scattering functions S_{H} and S_{M} allow us to conclude that the magnetization jumps at internal particle–matrix interfaces and the ensuing dipolar stray fields are the main source of the spindisorder in this material. Our study highlights the strength of the magnetic SANS technique to characterize magnetic materials on the mesoscopic length scale. The structural and magnetic results (summarized in Table 1) provide valuable information on the (Fe_{0.7}Ni_{0.3})_{86}B_{14} ribbons, and further confirm the strong potential of Fe–Ni–Bbased HiBNANOPERMtype alloys as soft magnetic nanocrystalline materials. In the context of the random anisotropy model, we demonstrated that the magnetic softness in this system can be attributed to the combined action of the small particle size (D = 7 nm) and an increased exchange constant (A_{ex} = 10 pJ m^{−1}) resulting in an enhanced exchange correlation length L_{0}.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Footnotes
^{1}Estimated by assuming a linear regression of K_{1} in Fe_{1−x}Ni_{x} alloys for an Fe composition x between 0 and 0.4 at% [see inset in Fig. 6(b), data taken from the literature (Tarasov, 1939; Hall, 1960)].
Acknowledgements
The authors acknowledge the Swiss spallation neutron source at the Paul Scherrer Institute, Switzerland, for the provision of neutron beam time.
Funding information
Michael Adams and Andreas Michels thank the National Research Fund of Luxembourg for financial support (AFR grant No. 15639149 and CORE grant SANS4NCC).
References
Aharoni, A. (1998). J. Appl. Phys. 83, 3432–3434. Web of Science CrossRef CAS Google Scholar
Anand, K. S., Goswami, D., Jana, P. P. & Das, J. (2019). AIP Adv. 9, 055126. Google Scholar
Bender, P., Honecker, D. & Fernández Barquín, L. (2019). Appl. Phys. Lett. 115, 132406. Web of Science CrossRef Google Scholar
Bender, P., Marcano, L., Orue, I., Venero, D. A., Honecker, D., Fernández Barquín, L., Muela, A. & Luisa FdezGubieda, M. (2020). Nanoscale Adv. 2, 1115–1121. Web of Science CrossRef CAS Google Scholar
Bersweiler, M., Bender, P., Vivas, L. G., Albino, M., Petrecca, M., Mühlbauer, S., Erokhin, S., Berkov, D., Sangregorio, C. & Michels, A. (2019). Phys. Rev. B, 100, 144434. Web of Science CrossRef Google Scholar
Bersweiler, M., Pratami Sinaga, E., Peral, I., Adachi, N., Bender, P., Steinke, N. J., Gilbert, E. P., Todaka, Y., Michels, A. & Oba, Y. (2021). Phys. Rev. Mater. 5, 044409. Web of Science CrossRef Google Scholar
Dewhurst, C. D. (2018). GRASP, https://www.ill.eu/en/users/supportlabsinfrastructure/softwarescientifictools/grasp/. Google Scholar
Grigoryeva, N. A., Grigoriev, S. V., Eckerlebe, H., Eliseev, A. A., Lukashin, A. V. & Napolskii, K. S. (2007). J. Appl. Cryst. 40, s532–s536. Web of Science CrossRef CAS IUCr Journals Google Scholar
Günther, A., Bick, J.P., Szary, P., Honecker, D., Dewhurst, C. D., Keiderling, U., Feoktystov, A. V., Tschöpe, A., Birringer, R. & Michels, A. (2014). J. Appl. Cryst. 47, 992–998. Web of Science CrossRef IUCr Journals Google Scholar
Hall, R. C. (1960). J. Appl. Phys. 31, 1037–1038. CrossRef CAS Web of Science Google Scholar
Herzer, G. (1989). IEEE Trans. Magn. 25, 3327–3329. CrossRef CAS Web of Science Google Scholar
Herzer, G. (1990). IEEE Trans. Magn. 26, 1397–1402. CrossRef CAS Web of Science Google Scholar
Herzer, G. (2007). Handbook of Magnetism and Advanced Magnetic Materials, Vol. 4, edited by H. Kronmüller & S. Parkin, pp. 1882–1908. Hoboken, NJ: John Wiley Google Scholar
Herzer, G. (2013). Acta Mater. 61, 718–734. Web of Science CrossRef CAS Google Scholar
Honecker, D., Dewhurst, C. D., Suzuki, K., Erokhin, S. & Michels, A. (2013). Phys. Rev. B, 88, 094428. Web of Science CrossRef Google Scholar
Honecker, D. & Michels, A. (2013). Phys. Rev. B, 87, 224426. Web of Science CrossRef Google Scholar
Ito, N., Michels, A., Kohlbrecher, J., Garitaonandia, J. S., Suzuki, K. & Cashion, J. D. (2007). J. Magn. Magn. Mater. 316, 458–461. Web of Science CrossRef CAS Google Scholar
Köhler, T., Feoktystov, A., Petracic, O., Kentzinger, E., BhatnagarSchöffmann, T., Feygenson, M., Nandakumaran, N., Landers, J., Wende, H., Cervellino, A., Rücker, U., Kovács, A., DuninBorkowski, R. E. & Brückel, T. (2021). Nanoscale, 13, 6965–6976. Web of Science PubMed Google Scholar
Kons, C., Phan, M. H., Srikanth, H., Arena, D. A., Nemati, Z., Borchers, J. A. & Krycka, K. L. (2020). Phys. Rev. Mater. 4, 034408. Web of Science CrossRef Google Scholar
Li, Z., Parsons, R., Zang, B., Kishimoto, H., Shoji, T., Kato, A., Karel, J. & Suzuki, K. (2020). Scr. Mater. 181, 82–85. Web of Science CrossRef CAS Google Scholar
Makino, A., Hatanai, T., Naitoh, Y., Bitoh, T., Inoue, A. & Masumoto, T. (1997). IEEE Trans. Magn. 33, 3793–3798. CrossRef CAS Web of Science Google Scholar
Maurer, T., Gautrot, S., Ott, F., Chaboussant, G., Zighem, F., Cagnon, L. & Fruchart, O. (2014). Phys. Rev. B, 89, 184423. Web of Science CrossRef Google Scholar
McHenry, M. E., Willard, M. A. & Laughlin, D. E. (1999). Prog. Mater. Sci. 44, 291–433. Web of Science CrossRef CAS Google Scholar
Mettus, D. & Michels, A. (2015). J. Appl. Cryst. 48, 1437–1450. Web of Science CrossRef CAS IUCr Journals Google Scholar
Michels, A. (2021). Magnetic SmallAngle Neutron Scattering: a Probe for Mesoscale Magnetism Analysis. Oxford University Press. Google Scholar
Mühlbauer, S., Honecker, D., Périgo, A., Bergner, F., Disch, S., Heinemann, A., Erokhin, S., Berkov, D., Leighton, C., Eskildsen, M. R. & Michels, A. (2019). Rev. Mod. Phys. 91, 015004. Google Scholar
Oba, Y., Adachi, N., Todaka, Y., Gilbert, E. P. & Mamiya, H. (2020). Phys. Rev. Res. 2, 033473. Web of Science CrossRef Google Scholar
Petzold, J. (2002). J. Magn. Magn. Mater. 242–245, 84–89. Web of Science CrossRef CAS Google Scholar
RodríguezCarvajal, J. (1993). Physica B, 192, 55–69. CrossRef Web of Science Google Scholar
Stewart, J. R., Giblin, S. R., Honecker, D., Fouquet, P., Prabhakaran, D. & Taylor, J. W. (2019). J. Phys. Condens. Matter, 31, 025802. Web of Science CrossRef PubMed Google Scholar
Suzuki, K., Herzer, G. & Cadogan, J. M. (1998). J. Magn. Magn. Mater. 177–181, 949–950. Web of Science CrossRef CAS Google Scholar
Suzuki, K., Makino, A., Inoue, A. & Masumoto, T. (1991). J. Appl. Phys. 70, 6232–6237. CrossRef CAS Web of Science Google Scholar
Suzuki, K., Parsons, R., Zang, B., Onodera, K., Kishimoto, H., Shoji, T. & Kato, A. (2019). AIP Adv. 9, 035311. Google Scholar
Tarasov, L. P. (1939). Phys. Rev. 56, 1245–1246. CrossRef CAS Google Scholar
Titov, I., Barbieri, M., Bender, P., Peral, I., Kohlbrecher, J., Saito, K., Pipich, V., Yano, M. & Michels, A. (2019). Phys. Rev. Mater. 3, 084410. Web of Science CrossRef Google Scholar
Vacuumschmelze GmbH (1993). Toroidal Cores of VITROPERM, PW014. Google Scholar
Yoshizawa, Y., Oguma, S. & Yamauchi, K. (1988). J. Appl. Phys. 64, 6044–6046. CrossRef CAS Web of Science Google Scholar
Zákutná, D., Nižňanský, D., Barnsley, L. C., Babcock, E., Salhi, Z., Feoktystov, A., Honecker, D. & Disch, S. (2020). Phys. Rev. X, 10, 031019. Google Scholar
Zang, B., Parsons, R., Onodera, K., Kishimoto, H., Shoji, T., Kato, A., Garitaonandia, J. S., Liu, A. C. Y. & Suzuki, K. (2020). Phys. Rev. Mater. 4, 033404. Web of Science CrossRef Google Scholar
This is an openaccess article distributed under the terms of the Creative Commons Attribution (CCBY) Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.