Research Article | Open Access

# Nanocompositional Electron Microscopic Analysis and Role of Grain Boundary Phase of Isotropically Oriented Nd-Fe-B Magnets

**Academic Editor:**Jamal Berakdar

#### Abstract

Nanoanalytical TEM characterization in combination with finite element micromagnetic modelling clarifies the impact of the grain misalignment and grain boundary nanocomposition on the coercive field and gives guidelines how to improve coercivity in Nd-Fe-B based magnets. The nanoprobe electron energy loss spectroscopy measurements obtained an asymmetric composition profile of the Fe-content across the grain boundary phase in isotropically oriented melt-spun magnets and showed an enrichment of iron up to 60 at% in the Nd-containing grain boundaries close to Nd_{2}Fe_{14}B grain surfaces parallel to the -axis and a reduced iron content up to 35% close to grain surfaces perpendicular to the -axis. The numerical micromagnetic simulations on isotropically oriented magnets using realistic model structures from the TEM results reveal a complex magnetization reversal starting at the grain boundary phase and show that the coercive field increases compared to directly coupled grains with no grain boundary phase independently of the grain boundary thickness. This behaviour is contrary to the one in aligned anisotropic magnets, where the coercive field decreases compared to directly coupled grains with an increasing grain boundary thickness, if value is > 0.2 T, and the magnetization reversal and expansion of reversed magnetic domains primarily start as Bloch domain wall at grain boundaries at the prismatic planes parallel to the -axis and secondly as Néel domain wall at the basal planes perpendicular to the -axis. In summary our study shows an increase of coercive field in isotropically oriented Nd-Fe-B magnets for GB layer thickness > 5 nm and an average value of the GB layer < 0.8 T compared to the magnet with perfectly aligned grains.

#### 1. Introduction

The increasing demand of high-performance rare earth permanent magnets with a high coercive field and an energy density product value suitable for large scale applications in wind turbines and electrically powered automotive devices led to the development of heavy rare earth lean/rare earth-free Nd-Fe-B based magnets and to the optimization of the complex multiphase microstructure of the magnets [1]. The hard magnetic properties are primarily controlled by the size, shape, and misalignment of the hard magnetic grains and their distributions and secondarily by the occurrence of other nonmagnetic and soft magnetic phases [2–4]. In addition, the coercive field also strongly depends on the intergranular grain boundary (GB) phases separating the hard magnetic grains [5, 6]. The role of dopant elements, the thickness, and magnetic properties of the GB-phases have extensively been studied during the last 30 years [7, 8]. Local changes of the exchange coupling between grains and the decrease of the anisotropy field and demagnetizing field at/near intergranular phases considerably reduce the overall coercive field. First principles ab initio calculations claimed that even an antiparallel exchange coupling between a crystalline *α*-Fe phase and the prismatic planes of Nd_{2}Fe_{14}B would be energetically favorable, while a positive exchange-coupling constant was predicted in the Nd_{2}Fe_{14}B /*α*-Fe interface [9].

Advances in electron microscopic characterization technology have greatly improved the ability to quantify real microstructures found in Nd-Fe-B magnets. These techniques, in combination with finite element micromagnetic modelling, are improving the understanding of magnetization reversal processes and coercivity mechanisms. Micromagnetic simulations give a deep insight into the mechanisms that cause magnetization reversal at external fields well below the anisotropy field [10]. Nowadays, the new nanoanalytical electron microscopic techniques with atomic resolution allow the creation of precise microstructural models suitable for the numerical micromagnetic calculation of the demagnetization curve including the coercive field value. A recent high resolution TEM/STEM investigation of the intergranular GB-phase of a large grained, anisotropic sintered heavy rare earth-free Nd‐Fe‐B magnet with grain sizes up to several microns revealed a difference in composition for grain boundaries parallel (large Fe‐content) and perpendicular (low Fe-content) to the alignment direction [11]. This combined TEM/STEM and micromagnetic study of the anisotropic nature of grain boundaries shows a decrease of the coercive field with an increasing thickness of the grain boundary layer.

Two quite distinct methods are in commercial use for producing Nd-Fe-B magnets: the rapid-solidification technique of melt spinning and the traditional powder-metallurgy-sintering approach. The present study compares different microstructures of various melt-spun materials with isotropically oriented hard magnetic grains with a grain size ranging from 20 nm to 100 nm. The melt-spinning procedure involves the ejection of a molten starting alloy through a crucible orifice onto the surface of a substrate copper disc with a high rotating speed [12]. The microstructure and magnetic properties of melt-spun neodymium-iron-boron ribbons are sensitively dependent on the quench rate. The resulting hysteretic properties of an individual magnet material strongly depend on their nominal composition, microstructure, and processing parameters [13]. Melt-spun magnet materials have widely been used for bonded and hot deformed type magnets so far. Hot-pressed melt-spun nanocrystalline heavy rare earth-free Nd-Fe-B magnets are promising candidates for a low cost solution for applications that require thermal stability up to 175°C–200°C [14].

The aim of the present paper is to determine the influence of the grain size, orientation of grains, and nanocomposition of GBs on the coercive field and magnetization reversal behaviour by a combined TEM/STEM and micromagnetic study with special emphasis on the nanoanalytical, high resolution EELS characterization of isotropically oriented GBs. The microstructural model structure based on an anisotropic compositional behaviour of GBs parallel and perpendicular to the easy axis of the grains which is used for the numerical micromagnetic simulations has been derived from the detailed nanoanalytical TEM/STEM analysis.

#### 2. Materials

In the present study we investigated the microstructure of three rapidly quenched Nd-Fe-B ribbons in a nanoanalytical TEM/STEM study, which were provided by Magnequench Technology Center, Singapore. The isotropic RE-rich two-phase ribbon (MQU-F) with the nominal chemical composition (Pr,Nd)_{13.6}Fe_{73.6}Co_{6.6}Ga_{0.6}B_{5.6} [15] has a distinct 3 nm–6 nm thick RE-rich GB-phase separating the isotropically oriented equiaxed and platelet shaped Nd-Fe-B grains. The isotropic fine grained ribbon (MQP-B+) with the nominal chemical composition Nd_{12.4}Fe_{77.3}Co_{5.2}B_{5.2} [16] is enriched in “Fe + Co” and possesses therefore a 1 nm–3 nm thin “Fe + Co”-rich GB-phase separating the isotropically orientated equiaxed Nd-Fe-B grains. In comparison an isotropically oriented and large grained nanocomposite with additional soft magnetic *α*-Fe and Nb-granular phases and without a GB-phase between the hard magnetic grains has been investigated.

#### 3. Methods

The nanoanalytical and structural investigations of the rapidly quenched Nd-Fe-B permanent magnet materials have been carried out with an analytical field emission transmission electron microscope (TEM) (FEI Tecnai F20) at 200 kV, which is equipped with a silicon drift energy dispersive X-ray (EDX) detector, a Gatan GIF Tridiem image filter and electron energy loss spectrometer (EELS) and a high angle annular dark field (HAADF) detector. Conventional sample TEM preparation including cutting, polishing, and ion milling in a Precision Ion Polishing System (PIPS) from Gatan was conducted. The structural investigations were performed with Fast Fourier Transformation (FFT) of high resolution TEM/STEM (HRTEM) images and selected area electron diffraction (SAED).

EELS experiments were conducted to accurately determine the relative chemical composition of the intergranular phases via the -factor method. This method calculates the relative atomic percentage of an element (e.g., Nd) with respect to another element (e.g., Fe) from the ratio of their edge intensities in the EELS (or EDX) spectrum via the -factor (e.g., (Nd/Fe)), which was derived from the measurement of a standard specimen (e.g., Nd_{2}Fe_{14}B single crystal). TEM specimens with a relative thickness , where is the absolute specimen thickness and the mean free path in the specimen, were used in these experiments. Firstly, the -factors of Pr/Fe and Nd/Fe were calculated from EELS spectra of single crystalline Pr_{2}Fe_{14}B and Nd_{2}Fe_{14}B standards. Secondly, the background in the EELS spectra was fitted with a power-law function and subtracted, which resulted in the edge intensities of the elements. Thirdly, the relative atomic composition was calculated from the edge intensities via the -factors. The determination of the relative chemical composition via the -factor method is accurate for with a relative error of 5% [17]. An optimized background model was used to measure the Fe- ionization edge due to its close vicinity to the F-K edge and the Nd- ionization edge due to its close vicinity to the Pr- edge [18]. To avoid the development of an oxidized layer on the surface of the TEM specimen, precise precautions were taken. The influence of the electron beam broadening and the tilt of the GBs with respect to the incident electron beam on the chemical composition of 2 nm–6 nm thin GBs, as described in our previous publication [11], were taken into account. The higher yield in the elastic scattering events in EELS with respect to EDX [19] leads to a shorter acquisition time of each spectrum in a line scan. This is an advantage especially in the chemical analysis of thin GBs in thin (<50 nm) TEM specimens.

The finite element software package FEMME, which is a hybrid finite element/boundary element method code, was used for the numerical micromagnetic simulations [20]. On each point of the finite element mesh the Landau-Lifshitz-Gilbert equation is being solved [21]. Besides the intrinsic magnetic properties, namely, the exchange constant* A*, the saturation polarization , and the uniaxial magnetocrystalline anisotropy constant , also the direction of the easy axis (direction of ) of a volume of a phase, which can be set with the polar angle and the azimuthal angle , is an input parameter for the simulation. was set to zero in the GBs, since it is expected to have a negligibly small or zero value. The long range demagnetizing field and the direct exchange coupling between neighbouring atomic moments in the hard magnetic grains and soft magnetic grain boundary layers strongly influence the magnetization reversal. Besides the exchange and the demagnetizing field, the magnetocrystalline anisotropy and the misorientation of the individual grains also contribute to the resulting magnetization reversal and coercivity [10].

Realistic finite element granular structures based on TEM investigations of melt-spun Nd-Fe-B magnets have been generated using the Voronoi algorithm [22]. This algorithm creates a unique volume decomposition based on a set of seeding points, similar to the Wigner-Seitz cell construction. We used the voro++ code [23] to create a Voronoi structure of equiaxed grains. The output from voro++ acts as an input for a Salome [24] script that creates a finite element discretization (mesh) of the granular structure. Two finite element model structures were created, one with directly coupled grains and one with a grain boundary phase with an approximate thickness of 10% of the grain size (Figures 1 and 2). The distribution of the easy axes of an isotropically orientated magnet is equal to the random distribution of points on a half sphere with a calculated azimuthal angle and polar angle , where and have to be chosen from random variates between 0 and 1. This results in an average misorientation angle ° and a projection of the magnetization parallel to the external field of 0.5 [25, 26].

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For a clear distinction between GBs parallel and perpendicular to the external field and the -axis of the adjacent grains a simple two-grain model structure with an edge length of 40 nm was created and meshed with the software package GID version 12.0.4 [27] (Figure 3). Two Nd_{2}Fe_{14}B grains are separated by a GB-phase consisting of two equally thick GB-volumes with a total GB thickness between 2, 4, 5, 6, and 8 nm. All model structures were discretized with a 0.5 nm–2.5 nm mesh size, where the mesh tessellation was chosen in a way to ensure that the smallest GB volume has at least one central node surrounded with the nearest neighbours corresponding to GB material.

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#### 4. Results and Discussion

##### 4.1. TEM/STEM Characterization and Micromagnetic Simulation

###### 4.1.1. Isotropic RE-Rich Two-Phase Melt-Spun Ribbon (MQU-F)

The polycrystalline microstructure of a rapidly quenched MQU-F ribbon with isotropic orientated -axis of hard magnetic Nd-Fe-B grains with a size ranging from 20 nm to over 100 nm is shown in the TEM bright field (BF) and HAADF images of Figure 4. The contrast of the TEM-BF image is originated by the combination of orientation/diffraction contrast and absorption contrast, which depends on the thickness and average density of the TEM specimen leading to the bright contrast of the GB-phase. A HAADF image is generated in the STEM mode and the origin of the images contrast depends on the chosen camera length. At a cameral length (cl) below ≈ 80 mm the intensity distribution in the HAADF image mainly consists of the average atomic number of the probed volume (-contrast) and the thickness of the specimen [28]. The GB-phase shows a double contrast with a dark interface to the adjacent grains and a bright center in the HAADF image in Figure 4(b). The HAADF intensity profile along the EELS-1 line scan and dependence (-contrast) are shown in the insert in Figure 4(b). The -contrast was calculated from the atomic percentage of the elements measured with EELS (Figure 7(a)). The dark interface between the grains and the GB is enriched in “Fe + Co” and contains less “Pr + Nd,” leading to a lower average atomic number. The -axis of elongated grains was always found to be perpendicular to the longer edge of the grains.

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The hard magnetic Nd-Fe-B grains are separated by a 3 nm–6 nm thick rare earth- (RE-) rich GB-phase and near GB junctions by the cubic -(Pr,Nd)_{2}O_{3} phase, which also has previously been reported in literature [2, 7, 11, 29–32]. The weakly paramagnetic -(Pr,Nd)_{2}O_{3} phase has only a negligible influence on the magnetization reversal compared to the soft ferromagnetic GB-phases. Dopants like Al, Ga, and Cu influence the liquid phase during sintering [3]. Ga-atoms were dissolved in the hard magnetic grains and GBs partially replacing the Fe-atoms during rapid quenching, since their amount is too low to form separate phases. The amorphous oxygen containing RE-rich GB-phase, shown in the HRTEM image in Figure 5, has an approximate composition of (Pr,Nd)_{41}(Fe,Co)_{49}O_{6}F_{4}. The RE/Fe ratio is in agreement with the composition of Nd_{48}Fe_{48}Cu_{4} reported by Sasaki et al. [33]. A combined STEM and three-dimensional atom probe tomography (3D-AP) study of sintered Nd-Fe-B magnets reported a chemical composition of the Nd enriched amorphous GB-phase of Nd_{30}Fe_{45}Cu_{24.1}B_{0.9} [34]. Sepehri-Amin et al. [35] produced a ferromagnetic Nd_{30}Fe_{66}B_{3}Cu_{1} thin film, whose chemical composition was derived from a laser assisted 3D-AP investigation of GB-phases of sintered Nd-Fe-B magnets. Woodcock et al. [36] reported of an amorphous oxide containing RE-rich GB-phase in a hot deformed Nd-Fe-Co-B-Ga magnet. The (001), (112), and (101) lattice fringes of the hard magnetic Nd_{2}Fe_{14}B grains are visible in Figure 5.

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Crystalline GBs with a lattice plane spacing of 0.31 nm and 0.34 nm are shown in the HRTEM image in Figure 6. The (001), (114), and (111) lattice fringes of the hard magnetic grains are visible. Sasaki et al. [37] reported about a crystalline GB-phase with a RE content of 60 at% in Nd_{13.5}Pr_{0.2}Dy_{0.2}Tb_{0.2}Fe_{76.0}Co_{1.8}B_{6.6}Cu_{0.1}Al_{0.5}Ni_{0.4}O_{0.5} sintered magnet with a high energy product investigated with STEM methods. Another 3D-AP study [33] of a sintered Nd-Fe-B magnet reported about a crystalline GB with Nd-content of 55 at%. A crystalline 5 nm–10 nm thick Cu enriched cubic c-Nd_{2}O_{3 }GB-phase in Nd_{12.0}Dy_{2.7}Fe_{76.3}Cu_{0.4}B_{6.0 }M_{2.6} (M = Al, Co, and Nb) sintered Nd-Fe-B magnet was reported by Kim et al. [38]. A crystalline Nd enriched Nd_{16.4}Fe_{71.8}Co_{7.8}B_{3.5}Ga_{0.5 }GB-phase in Nd-Fe-B magnets subjected to a hydrogen-disproportion-desorption-recombination process was reported in 3D-AP study [39].

In a previous study we have shown [11] that in an aligned sintered magnet the GBs perpendicular (-GB) to the alignment direction of the magnet have a higher RE content (up to 60 at%) than the GBs parallel (-GB) to the alignment direction (RE content below 30 at%). GBs with intermediate misorientation to the alignment direction (-GB) show a chemical composition corresponding to an average of - and -GB. In sintered anisotropic magnets pure - and -GBs are common, but in melt-spun isotropic magnet materials the GB is a mix of - and -GB in general, due to the strong misalignment of the neighbouring grains. The EELS-1 line scan starts from a 2-14-1 grain into a -GB, resulting in a strong gradient of the chemical composition, and continues from the -GB into a grain with approximately 45° misorientation of the -axis with respect to the surface normal of the GB (Figures 4(b) and 7(a)). This correlates with a gradual change of the chemical composition. The EELS-2 line scan starts in a grain whose -axis is orientated perpendicular to the surface normal of the GB resulting in a slow change in chemical composition (Figures 6 and 7(b)). Since the -axis of the second grain is orientated parallel to the surface normal of the GB the change in chemical composition is faster. The faster change in the chemical composition from a -GB with respect to the -GB is shown in the EELS-3 line scan (Figures 6 and 7(c)).

The average “Fe + Co” concentration of the GB-phase in the investigated MQU-F ribbon is 55 at%, if only “Fe + Co” and “Pr + Nd” elements are considered. According to the magnetic phase diagram of Nd_{100−x}Fe_{x} which was recently published by Sakuma et al. [40] we assumed for the GB-phase a magnetic saturation polarization of 0.43 T and calculated an exchange stiffness constant of 1.0 pJ/T. The relation between and the exchange constant was used, as suggested by Kronmüller and Fähnle [41].

Using the Voronoi model structure of isotropically orientated Nd_{2}Fe_{14}B grains (Figure 1) with an average grain size of 50 nm and a GB-phase with a thickness of 4 nm–6 nm (Figures 5 and 6) we calculated the demagnetization curves obtained from the numerical finite element micromagnetic simulations depending on the coupling between the grains and the degree of misorientation of the grains. Figure 8 shows a high accordance of the coercive field between the measured value and the randomly misoriented grains. It should be noted that for the simulated demagnetization curve (sm-GB_60°) the remanence gets underestimated in the simulation with a perfectly isotropic distribution of the -axes (°). In addition Figure 8 shows that the simulations for directly coupled Nd_{2}Fe_{14}B grains (no-GB-phase) underestimate the coercive field by 1.5 T (°). The simulation with a smaller degree of misalignment of the hard magnetic grains (°) reveals the significant increase of and with respect to the perfectly isotropically oriented case (°). This is in agreement with the Stoner-Wohlfarth model of noninteracting single-domain particles [26], where is increasing by of the anisotropy field , which corresponds to T in Nd_{2}Fe_{14}B, if is reduced from 60° to 45°. The reduction of with rising value of is attenuated in the simulations with a ferromagnetic GB-phase. The higher value of the simulation with ° with respect to the simulation with ° is explained by the higher value of the component of the polarization parallel to the applied field direction (-direction).

###### 4.1.2. Isotropic Fine Grained Melt-Spun Ribbon (MQP-B+)

The small grained microstructure of the sample MQP-B+ is shown in the TEM-BF image of Figure 9(a). The isotropic orientation of the -axes of the Nd-Fe-B grains with a grain size ranging from 15 nm to 50 nm is displayed in the medium angle annular dark field image (MAADF) of Figure 9(b), which is generated at a higher camera length ( mm) compared to the HAADF image. The MAADF contrast generation is similar to the one of a TEM-BF image. The insert in Figure 9(a) shows EELS line scan across a 3 nm thick “Pr + Nd” enriched GB-phase.

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Under the assumption that all boron is bound in the Nd_{2}(Fe,Co)_{14}B phase the chemical composition of the intergranular GB-phases has been calculated from the nominal composition Nd_{12.4}(Fe,Co)_{82.5}B_{5.2} to be Nd_{17}(Fe,Co)_{83}. This corresponds to 12 at% of the total composition. With the approximation of 30 nm large rhombic dodecahedron shaped grains separated by a 2 nm–3 nm thick GB-phase the volume fraction of the GB-phase is 21%. The chemical composition of the GB measured by EELS is Nd_{20}(Fe,Co)_{77}O_{3}. These results are in good agreement with experiments with an Auger Microprobe spectrometer [42].

The micromagnetic simulations were carried out with the Voronoi model structure with isotropically orientated grains (Figure 1) with an average grain size of 35 nm and a soft magnetic GB-phase with a thickness of 2 nm–4 nm and average values for T and pJ/m, which is similar as described for the MQU-F sample. The simulated coercive field value is in good agreement with the measured value (Figure 10). Due to the high value of the GB the coercive field value (sm-GB) is only slightly increased with respect to of the simulation from directly coupled Nd_{2}Fe_{14}B grains (no-GB).

###### 4.1.3. Isotropic Large Grained Nanocomposite with *α*-Fe and Nb-Containing Granular Phases

The large grained microstructure of the exchange coupled nanocomposite with isotropically orientated Nd-Fe-B grains and a grain size ranging from 30 nm to 150 nm is shown in the TEM-BF image of Figure 11(a). The insert in Figure 11(a) is EELS line scan across a GB of two Nd_{2}Fe_{14}B grains with no detected intergranular GB-phase. Besides the hard magnetic 2-14-1 phase the soft ferromagnetic *α*-Fe and the weakly antiferromagnetic Fe_{2}Nb phase ( K) [43] are shown in the HRTEM image in Figure 11(b).

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A large area EDX mapping in the HAADF image in Figure 12(b)–12(e) was used to determine the areal fraction of the identified granular phases (Figure 12(a)). Besides the *α*-Fe phase another soft magnetic Nb_{6}Fe_{76}B_{18} ( T, mT) phase which was formed by rapid quenching [44] was identified. Table 1 summarizes the lattice parameter, space groups, and prototypes of the analyzed phases which were used to identify the phases in the HRTEM images. The bright areas in the Fe-K map (Figure 12(c)) correspond to the *α*-Fe phase. The Fe_{2}Nb phase is located at the high intensities of the Nb-K map (Figure 12(d)) and the Nb_{6}Fe_{76}B_{18} phase at the more dull yellow regions. The location of the 2-14-1 phase is clearly visible in the bright areas in the Nd-L map (Figure 12(e)).

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A Voronoi model structure with 29 directly coupled grains (Figure 2) with an average size of 60 nm was used to simulate the hysteretic properties. Corresponding to the analyzed volume distribution of the phases we assumed 21 (72%) Nd_{2}Fe_{14}B grains, 4 (14%) *α*-Fe grains, and 4 (14%) Nb_{6}Fe_{76}B_{18} grains. The magnetic properties of the phases are summarized in Table 2. All values were set to zero except in the hard magnetic Nd_{2}Fe_{14}B phase.

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The Néel temperature of the weakly antiferromagnetic Fe_{2}Nb phase is ≈ 270 K and therefore we assumed nonmagnetic properties for the simulation at room temperature. |

The measured demagnetization curve and the simulated curves of directly coupled grains with an average grain misorientation of 45° and 60° are shown in Figure 13. For the realistic phase distribution the calculated coercive field is slightly underestimated in the simulation compared to the measured value. One reason for this discrepancy is relatively small sample area where the areal distribution was acquired, with respect to the whole ribbon volume. A higher quality of the random distribution of the granular phases would be achieved in a model with a larger number of grains. The model with 29 directly coupled Nd_{2}Fe_{14}B grains overestimates both and significantly. The strong decrease of in the model structure with the realistic assumption of soft magnetic grains, compared to the case of only hard magnetic Nd_{2}Fe_{14}B grains, was also reported in a detailed micromagnetic study of Nd-Fe-B magnet with soft magnetic granular phases [45].

##### 4.2. Micromagnetic Simulations of the Switching Field of Randomly Orientated Grains

The orientation relation of grain boundaries of adjacent grains and their composition close to their grain surfaces with respect to the alignment direction of the magnet and external field direction influence the resulting magnetic switching field and coercive field, respectively. Using the two-grain (2-G) model structure of Figure 3 we compare in Figure 14 three different configurations which possibly occur in anisotropically and isotropically oriented magnets. The first and second case in Figure 14 show a pure -GB and pure -GB, commonly found in anisotropic aligned sintered Nd-Fe-B magnets. The external field is parallel to direction in both cases. The third case shows -GB facing the lower grain and -GB facing the upper grain and is parallel to , typically found in isotropically oriented melt-spun Nd-Fe-B magnets.

values for - and -GB were calculated from the chemical composition obtained from TEM/EELS measurements of GBs in anisotropic sintered Nd-Fe-B magnets [11]. The measured “Fe + Co” concentrations of the GBs in melt-spun magnets (Figures 7 and 9(a)) and the corresponding and values are summarized in Table 3.

The micromagnetic simulations show that the switching field depends on both, the GB thickness and value of the GB layer (Figure 15(a)). For small value of the -GB (<0.2 T) slightly increases with rising GB thickness (-GB). For high value of the -GB (1.0 T) is significantly lower with rising GB thickness (-GB). In both cases the external field is parallel to direction. This behaviour is typical for anisotropic magnets with perfectly aligned grains. In the isotropic case (-GB), with , the switching field value slightly decreases with rising GB thickness (Figure 15(a)). For a GB thickness > 5 nm the anisotropic -GB ( ) has a lower compared to the isotropic -GB ( ). This is an explanation for the trend of higher values of magnets with higher misorientation degree, which contradicts the results formulated by Stoner and Wohlfarth [26] for noninteracting grains or particles but agrees with experimental results [52] and previous simulations [11]. In comparison, the dependence of the switching field of a 2-G model structure with averaged homogeneous magnetic properties in the GB layer T and pJ/m and T and pJ/m, respectively, is shown in Figure 15(a) (dotted lines). With a low value (0.43 T) of the GB layer and is above the value of the anisotropic -GB ( ). The switching field value of the averaged GB ( ) with a of 1.10 T is below of the -GB ( ) for all GB thicknesses. At a GB thickness of about 4 nm the -GB and the homogeneous GB with a of 0.43 T have approximately the same switching field values. Therefore it is justified to use a single phased GB with homogeneous magnetic properties in isotropic multigrain finite element model structures. Figure 15(b) shows that the switching field constantly decreases from 0.44 to 0.33 for GB thickness ranging from 2 nm to 20 nm with rising value of the GB layer, as shown in the results of the simulations of the 2-G model structure with an averaged homogeneous GB composition ( ). It should be noted that the switching field values of the isotropic cases -GB , GB (0.43 T) , and GB(1.10 T) (Figure 15(a)) and 2, 8, and 20 nm GB (Figure 15(b)) are more or less independent of the GB layer thickness.

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During the magnetization reversal processes different types of domain wall (DW) types, such as Bloch and Néel DWs, are formed in perfectly aligned magnets depending on the orientation of the GB with respect to the -axis of the adjacent grains and the direction of the external field. The calculated demagnetization curves for the pure -GB with // and T (Table 3) and for the pure -GB with // and T and a GB thickness of 8 nm are shown in Figure 16. As a result of the large difference in and values the coercive field for - and -GB varies from 2.7 T to 6.5 T. The -GB shows a 12% higher coercive field, if the magnetic properties of - and -GB are the same. This difference is originated by the different total energies for the formation of a Bloch domain wall (DW) (-GB) and a Néel DW (-GB) with an additional stray field contribution.

The magnetization of the -GB rotates in the perpendicular direction with respect to the adjacent grains at a relatively small external field of 0.95 T (Figure 17①). Two Néel DWs are formed, whereby the magnetization within the center of the GB is antiparallel to one of the adjacent grains, until being at a high external field value of 6.45 T (Figure 17②). The high value of the necessary external field is originated by the large formation energy of a Néel DW due to the strong stray field occurring along the whole interfaces between the GB and the neighbouring grains.

The magnetization reversal state ③ is typical for a Bloch DW nucleated in the -GB (Figure 18③). Since the magnetization vector has a degree of freedom to rotate along the -axis with relatively low activation energy, the -GB switches at a lower external field of 3.78 T and finally forms two Bloch DWs at the interfaces with the hard magnetic grains (Figure 18④). The formation energy of the stray-field-free Bloch DWs is smaller than the one of the Néel DWs. In general the DWs are complex magnetization transitions between neighbouring magnetic domains. Their energy, thickness, and shape depend on various parameters such as the intrinsic magnetic properties and the shape of the magnetic material. The complex structure of DWs can only be calculated numerically by means of micromagnetic simulations [53].

The saturation polarization and the thickness of the GB layer have been varied using the isotropic Voronoi model structure of Figure 1 in order to verify the results of the 2-G model structure of Figure 15 with a realistic model structure with averaged homogeneous magnetic properties. At a small value of and the GB magnetically decouples the isotropically orientated hard magnetic grains leading to an increase of with respect to direct coupled Nd_{2}Fe_{14}B grains (Figure 19(a)). This behaviour is strongly pronounced in the MQU-F magnet material and also present in the MQP-B+ ribbon. As and of the GB-phase rise, decreases linearly due to stronger coupling of the hard magnetic grains and the higher probability of a nucleation of a reverse magnetic domain in the GB. Simultaneously the remanence increases because of the stronger remanence enhancement effect of the coupled Nd-Fe-B grains [54]. At a GB thickness of 5 nm and grain size of 50 nm the coercive fields for the model structures with and without a GB-phase are equal at T ( pJ/m) and equal at T ( pJ/m) for a GB thickness of 3 nm and a grain size of 30 nm (Figure 19(a)). The further increase in and leads to a reduction of with respect to directly coupled Nd_{2}Fe_{14}B grains. In these simulations the ratio between the grain size and the GB thickness was kept constant. This accredits the significant difference in of the 30 nm G_3 nm GB and 50 nm G_5 nm GB simulations. This influence of the grain size is approximately equal to the difference of the calculated values of the simulations of the model structure of directly coupled grains without a GB-phase (dotted lines in Figure 19(a)). Bance et al. [55] showed that the decrease of with increasing grain size in hard magnets is caused by the nonuniform magnetostatic field in the polyhedral grains. In summary the results from the 2-G model structure that is mostly independent of the GB thickness in isotropically oriented Nd-Fe-B magnets were also verified with the realistic Voronoi model structure calculations.

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The dependence of on the GB properties is more strongly pronounced in aligned Nd-Fe-B magnets. Figure 19(b) compares the results of simulations using the Voronoi model structure of Figure 1 with an average grain misalignment °. We observed that the decrease of with rising grain size is less pronounced in the simulations of anisotropically oriented directly coupled Nd-Fe-B grains (dotted lines in Figure 19(b)). Secondly, the GB thickness has a stronger influence on the reduction of in anisotropic magnets, which is shown in the greater difference in the values of the 30 nm G_3 nm GB and 50 nm G_5 nm GB simulations compared to the directly coupled simulations (no-GB). This is in accordance with our recently published results of the strong decrease of with rising GB thickness in anisotropic Nd-Fe-B magnets [11]. It should be emphasized that the presence of a soft magnetic GB layer always leads to a reduction of the coercive field in aligned magnet, if the saturation polarization of the GB is T ( pJ/m). The decrease of with rising of the GB layer shows a nonlinear behaviour in anisotropically oriented grains, compared to the linear decrease in the isotropic case.

#### 5. Conclusion

The TEM/EELS analysis of nanocrystalline Nd-Fe-B based magnet materials revealed an asymmetric composition profile of the Fe- and the Nd-content across the grain boundary phase in isotropically oriented melt-spun magnets. We found an enrichment of iron up to 60 at% in the Nd-containing grain boundaries close to the prismatic Nd_{2}Fe_{14}B grain surfaces and a reduced iron content up to 35% close to basal grain surfaces perpendicular to the -axis. Numerical micromagnetic simulations based on granular Voronoi model structures showed that the coercive field strongly depends on the average Fe-content and the saturation polarization and exchange stiffness constant of the GB-phase as well as on the GB thickness and grain orientation. In general, the coercive field is significantly increased, if the Fe-content of the GBs, especially parallel to the -direction of the hard magnetic 2-14-1 grains, is reduced. Our simulations predicted an increase of the coercive field of isotropically oriented magnets with a soft magnetic GB-phase independently of the grain boundary thickness between 2 nm and 20 nm for T compared to directly coupled 2-14-1 grains with no-GB-phase. Contrary to this result we have demonstrated that the coercive field of anisotropic, aligned magnets significantly decreases for soft magnetic GB-phases with T and GB thickness of 3 nm–5 nm compared to directly coupled 2-14-1 grains. Moreover a rising GB thickness > 4 nm further leads to a significant reduction in coercive field in anisotropic aligned magnets.

We have demonstrated that numerical micromagnetic simulations perfectly predict the hysteretic properties of Nd_{2}Fe_{14}B based magnets compared to measured properties, if their microstructure has been quantitatively analyzed by TEM/EELS investigations.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The authors are grateful to Dr. P. Toson for helpful discussions and Dr. D. Brown from Magnequench Technology Center Singapore, for providing the rapidly quenched Nd-Fe-B magnet samples. The TEM/STEM investigations were carried out using facilities at the University Service Center for Transmission Electron Microscopy, Vienna University of Technology, Austria. The authors acknowledge the TU Wien University Library for financial support through its Open Access Funding Programme.

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#### Copyright

Copyright © 2017 Gregor A. Zickler and Josef Fidler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.