Unraveling the magnetic softness in Fe–Ni–B-based nanocrystalline material by magnetic small-angle neutron scattering

Magnetic field-dependent small-angle neutron scattering is employed to analyze the mesoscale magnetic interactions in a soft magnetic HiB-NANOPERM-type alloy and relate the parameters to the experimental coercivity.


Introduction
Since the pioneering work of Yoshizawa et al. (1988), the development of novel Fe-based 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 NANOPERM-type (Suzuki et al., 1991) soft magnetic alloys, which find widespread application as magnetic cores in high-frequency power transformers or in interface transformers in the ISDN-telecommunication network. For a brief review of the advances in Fe-based nanocrystalline soft magnetic alloys, we refer the reader to the article by Suzuki et al. (2019).
More recently, an ultra-fine-grained microstructure combined with excellent soft magnetic properties was obtained in HiB-NANOPERM-type alloys (Li et al., 2020). The magnetic softness in such materials can be attributed to the exchange-averaging 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(Herzer, , 1990(Herzer, , 2007Suzuki et al., 1998), and becomes effective when the average grain size D is smaller than the ferromagnetic exchange length L 0 ¼ ' 0 ðA ex =K 1 Þ 1=2 , where A ex is the exchange-stiffness constant and ' 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 H C / ðD=L 0 Þ n , where n = 3 or n = 6 depending on the nature of the magnetic anisotropy [see, for example, the work by Suzuki et al. (1998Suzuki et al. ( , 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 Fe-based 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 hysteresis-loop 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. magneto-optical, Brillouin light scattering, spin-wave resonance or inelastic neutron scattering) require thin-film or single-crystal samples.
In the present work, we employ magnetic field-dependent small-angle neutron scattering (SANS) to determine the magnetic interaction parameters in (Fe 0.7 Ni 0.3 ) 86 B 14 alloy, specifically, the exchange-stiffness constant and the strength and spatial structure of the magnetic anisotropy and magnetostatic fields. The particular alloy under study is a promising HiB-NANOPERM-type soft magnetic material, which exhibits an ultra-fine 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;Gü nther et al., 2014;Maurer et al., 2014), nanoparticles (Bender et al., 2019Bersweiler et al., 2019;Zá kutná et al., 2020;Kons et al., 2020;Kö hler 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 (Mü hlbauer 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 cross section 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.

Experimental
The ultra-rapidly annealed (Fe 0.7 Ni 0.3 ) 86 B 14 alloy (HiB-NANOPERM-type) was prepared according to the synthesis process detailed by Li et al. (2020). The sample for the neutron experiment was prepared by employing the low-capturing isotope 11 B as the starting material. The average crystallite size was estimated by wide-angle X-ray diffraction (XRD) using a Bruker D8 diffractometer in Bragg-Brentano geometry (Cu K 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 BHS-40 DC hysteresis loop tracer. The crystallization and Curie temperatures were determined by means of differential thermal analysis (DTA) and thermo-magneto-gravimetric analysis (TMGA) on Perkin Elmer DTA/TGA 7 analyzers under a constant heating rate of 0.67 K s À1 . For the neutron experiments, six (Fe 0.7 Ni 0.3 ) 86 B 14 ribbons with a surface area of 12 Â 20 mm and a thickness of $15 mm were stacked together, resulting in a total sample thickness of $90 mm. The neutron measurements were conducted at the instrument SANS-1 at the Swiss Spallation Neutron Source at the Paul Scherrer Institute, Switzerland. We used an unpolarized incident neutron beam with a mean wavelength of = 6.0 Å and a wavelength broadening of Á/ = 10% (full width at half-maximum). All neutron measurements were conducted at room temperature and within a qrange of about 0.036 nm À1 q 1.16 nm À1 . 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 H i ¼ H 0 À N d M S , 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).

Unpolarized SANS
Based on the micromagnetic SANS theory for two-phase particle-matrix-type ferromagnets developed by , the elastic total (nuclear + magnetic) unpolarized SANS cross section dAE/d at momentum-transfer vector q can be formally written as (H 0 ? k 0 ): where dAE res d ðqÞ ¼ corresponds to the (nuclear + magnetic) residual SANS cross section, which is measured at complete magnetic saturation, and denotes the purely magnetic SANS cross section. In Equations  (2)]. For small-angle 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 our neutron-data analysis, to experimentally access dAE mag /d, we subtracted the SANS cross section dAE/d measured at the largest available field (approach-to-saturation regime; compare Fig. 2) from dAE=d measured at lower fields. This specific subtraction procedure eliminates the nuclear SANS contribution / jÑ Nj 2 , which is field independent, and therefore where Á represents the differences of the Fourier components at the two selected fields (low field minus highest field).

Approach-to-saturation regime
In the particular case of the approach-to-saturation regime, whereM M z 'M M S , and which implies therefore ÁjM M z j 2 ! 0 in Equation (4), dAE/d can be re-written as: where S H ðqÞ Â R H ðq; H i Þ and S M ðqÞ Â R M ðq; H i Þ correspond to the magnetic scattering contributions due to perturbing magnetic anisotropy fields and magnetostatic fields, respectively. More specifically, the anisotropy-field scattering function depends of the Fourier coefficientH H p ðqÞ of the magnetic anisotropy field, whereas the scattering function of the longitudinal magnetization is related to the Fourier coefficientM M z / ÁM. For an inhomogeneous material of the NANOPERM-type, the latter quantity is related to the magnetization jump ÁM 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 Lorentzian-squared function. The corresponding (dimensionless) micromagnetic response functions R H and R M are given by and The dimensionless function pðq;

Estimation of the magnetic interaction parameters
Most of the time it is more convenient to analyze the (over 2) 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 , i.e. 1=ð2Þ R 2 0 ð. . .Þd, and by assuming S H and S M to be isotropic (-independent), the SANS cross section dAE/d 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 dAE res =d (as the intercept) and S H and S M (as the slopes) at each q-value by performing a (weighted) nonnegative least-squares fit of the azimuthally averaged SANS cross sections dAE/d measured at several H i . Treating A ex in the expression for pðq; H i Þ as an adjustable parameter during the fitting procedure allows us to estimate this quantity. The best-fit value for A ex is obtained from the minimization of the (weighted) mean-squared deviation between experiment and fit: where the indices and refer to the particular q and H ivalues, 2 ; denotes the uncertainties in the experimental data, N = N N corresponds to the number of data points, and dAE exp /d and dAE sim /d are the azimuthally averaged SANS cross section determined from the neutron experiments and numerically computed using Equation (10), respectively. We research papers IUCrJ (2022). 9, 65-72 would like to point out that the best-fit 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 whole-q space according to the work by yields the mean-square anisotropy field hjH p j 2 i and the mean-square longitudinal magnetization fluctuation hjM z j 2 i, respectively. Since the neutron experiments are performed within a finite q-range and since both integrands S H;M q 2 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 dAE/ d in the conventional sense, but rather the point-by-point reconstruction of the theoretical cross sections based on the experimental data.   Fig. 2(b). The data have been normalized by the saturation magnetization M S , which was estimated from the linear regression Mð1=H i Þ for 0 H i 2 ½10 T À 14 T [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 approach-tosaturation regime by M/M S ! 90%, we can see that this regime is reached for 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      and suggests that in the framework of the RAM (Herzer, 2007), H C should fall into the regime where H C / ðD=L 0 Þ 3 (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 two-stage 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 (T am C ' 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 . Fig. 4 (upper row) shows the experimental 2D total (nuclear + magnetic) SANS cross sections dAE/d of the (Fe 0.7 Ni 0.3 ) 86 B 14 ribbons at different selected fields. As can be seen, at 0 H i = 7.99 T (near saturation), the pattern is predominantly elongated perpendicular to the magnetic field direction. This particular feature in dAE/d is the signature of the so-called 'sin 2 ðÞ-type' angular anisotropy [compare Equation (2)   Experimental 2D total (nuclear + magnetic) SANS cross section dAE/d of (Fe 0.7 Ni 0.3 ) 86 B 14 alloy at the selected fields 7.99, 2.99, 0.59, 0.29 T (upper row), and the corresponding purely magnetic SANS cross section dAE mag /d (middle row). Experimental dAE mag /d were obtained by subtracting dAE/d at the (near-) saturation field of 7.99 T from the data at the lower fields. The applied (internal) magnetic field H i is horizontal in the plane of the detector ðH i ? k 0 Þ. Lower row: computed dAE mag /d based on the micromagnetic SANS theory [Equations (5)-(9)] at the same selected field values as above, and using the experimental parameters given in Table 1. Note that dAE/d and dAE mag /d are plotted in polar coordinates with q (nm À1 ), ( ) and intensity (cm À1 ). 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 field-dependent 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 dAE mag /d determined by subtracting dAE/d at 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 so-called 'clover-leaf-type' angular anisotropy pattern. This particular feature was also previously observed in NANOPERM-type soft magnetic merials , and is related to the dominant magnetostatic term S M Â R M in the expression for dAE mag /d [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 , results in dipolar stray fields which produce spin disorder in the surroundings. Fig. 4 (lower row) displays dAE mag /d computed using the micromagnetic SANS theory [Equations (5)-(9)] and the experimental parameters summarized in Table 1. As is seen, the clover-leaf-type angular anisotropy experimentally observed in Fig. 4 (middle row) can be well reproduced using micromagnetic theory. Fig. 5(a) displays the (over 2) azimuthally averaged dAE/ d, while the corresponding dAE mag /d are shown in Fig. 5(b). By decreasing 0 H i from 7.99 T to 10 mT, the intensity of dAE/ d 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 dAE/d 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 dAE mag / d is of the same order as dAE/d, supporting the notion of dominant spin-misalignment scattering in (Fe 0.7 Ni 0.3 ) 86 B 14 alloy. Fig. 6 shows the magnetic SANS results determined from the field-dependent approach described in Section 3.3. In the present case, to warrant the validity of the micromagnetic SANS theory, only dAE/d measured for 0 H i > $ 65 mT (i.e. within the approach-to-saturation regime, compare Fig. 2) were considered. We have also restricted our neutron data analysis to q q max ¼ ½ 0 M S H max 0 =ð2A ex Þ 1=2 ¼ 0:65nm À1 , since the magnetic SANS cross section is expected to be fieldindependent for q ! q max (Michels, 2021). In Fig. 6(a), we plot the (over 2) azimuthally averaged dAE/d along with the corresponding fits based on the micromagnetic SANS theory [Equation (10), black solid lines]. It is seen that the field dependence of dAE/d over the restricted q-range can be well reproduced by the theory. Fig. 6(b) displays the (weighted) mean-squared deviation between experiment and fit, 2 , determined according to Equation (13), as a function of the exchange-stiffness constant A ex . In this way, we find A ex = (10 AE 1) pJ m À1 (see Table 1). The comparison with previous studies is discussed in the next paragraph for more clarity. (a) Magnetic field dependence of the (over 2) azimuthally averaged total (nuclear + magnetic) SANS cross section dAE/d of (Fe 0.7 Ni 0.3 ) 86 B 14 alloy. (b) The corresponding purely magnetic SANS cross section dAE mag / d (log-log scale).

Figure 6
Results of the SANS data analysis of (Fe 0.7 Ni 0.3 ) 86 B 14 alloy. (a) Magnetic field dependence of the (over 2) azimuthally averaged total (nuclear + magnetic) SANS cross section dAE/d plotted in Fig. 5(a) along with the corresponding fits (black solid lines) based on the micromagnetic SANS theory [Equation (10)]. (b) Weighted mean-squared deviation between experiment and fit, 2 , determined using Equation (13) as a function of the exchangestiffness constant A ex . Inset: Fe-composition dependence of the magnetocrystalline anisotropy K 1 in Fe 1Àx Ni x alloys [data taken from the literature (Tarasov, 1939;Hall, 1960) 6(c) displays the best-fit results for dAE res /d, S H and S M . Not surprisingly, the magnitude of dAE res /d (limit of dAE/d at infinite field) is smaller than the dAE/d 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 ÁM at internal particle-matrix interfaces represents the main source of spin disorder in this material. The estimated values for the mean-square anisotropy field and the mean-square magnetostatic field in terms of Equation (14) 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 À1 is about 2-3 times larger than those reported in NANOPERM-type soft magnetic materials . 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  was as low as 350 K, the local exchange stiffness in the grain boundary amorphous phase in HiB-NANOPERMtype alloys is expected to be higher than that in NANO-PERM-type 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 HiB-NANOPERM 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 Fe-based nanocrystalline materials (McHenry et al., 1999;Herzer, 2007Herzer, , 2013Suzuki et al., 2019) and therefore have a strong impact on their magnetic softness. Using A ex = 10 pJ m À1 (this study), K 1 ' 9.0 kJ m À3 , 1 and ' 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 Fe-based 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(Herzer, , 1990(Herzer, , 2007Suzuki et al., 1998), the exchange-averaged magnetic anisotropy hKi falls into the regime where hKi / D 3 . This finding is also consistent with the (experimental) D 3dependence of H C reported in Fe-B-based HiB-NANOPERM alloys (Suzuki et al., 2019;Li et al., 2020).

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 HiB-NANOPERM-type 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 clover-leaf-type 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 spin-disorder 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-B-based HiB-NANOPERM-type 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 À1 ) 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.