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Front. Mater. Sci. https://doi.org/10.1007/s11706-019-0471-2
RESEARCH ARTICLE
Micromagnetic investigation by a simplified approach on the demagnetization field of permanent magnets with nonmagnetic phase inside Wei LI1, Lizhong ZHAO1,2, and Zhongwu LIU (✉)1 1 School of Materials Science and Engineering, South China University of Technology, Guangzhou 510640, China 2 Innovative Center for Advanced Materials, Hangzhou Dianzi University, Hangzhou 310012, China
© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019
ABSTRACT: A simplified analysis method based on micromagnetic simulation is proposed to investigate effects of nonmagnetic particles on the demagnetizing field of a permanent magnet. By applying the additivity law of the demagnetizing field, the complicated demagnetizing field of the real magnet could be analyzed by only focusing on the stray field of the reserved magnet. For a magnet with nonmagnetic particles inside, the particle size has no significant effect on the maximum value of the demagnetization field, but the area of the affected region by the particle is proportional to the particle size. A large particle produces a large affected area overlapped with those influenced by other particles, which leads to the large demagnetization field. With increasing the length of the particle along the magnetization direction, the demagnetization field on the pole surface increases. The pole surface with a convex shape will increase the demagnetization field. The demagnetizing field near the nonmagnetic particle will be further increased by the large macroscopic demagnetizing field near the pole surface. This work suggests some practical approaches to optimize the microstructure of permanent magnets. KEYWORDS: demagnetizing field; additivity; micromagnetic simulation; nonmagnetic phase; nucleation process
Contents 1 2 3 4
Introduction Simulation method The additivity of the demagnetizing field The demagnetizing field influenced by the nonmagnetic phase 4.1 Size effect 4.2 Effect of the ratio of length to diameter for the nonmagnetic phase
Received April 30, 2019; accepted June 28, 2019 E-mail: [email protected]
4.3
Effects of concavity or convexity of nonmagnetic particles 4.4 Effects of position of the nonmagnetic phase 4.5 Discussion 5 Conclusions Disclosure of potential conflicts of interests Acknowledgements References
1
Introduction
Microstructure optimization is an effective method to improve magnetic properties of NdFeB magnets [1]. As
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one of the typical structures, the grain boundary phase is the crucial factor that influences the demagnetization process and the coercivity of NdFeB magnets [2–5]. Hou et al. [6] reported that the nonmagnetic Nd-rich phase plays a critical role in the preparation of hot deformed NdFeB magnets. Zhou et al. [7] reported that rare earth element (RE)-rich phases influence magnetic properties by significantly changing the distribution of the demagnetizing field. However, the RE-rich phases in a real NdFeB magnet are in various forms [8], such as particles and continuous layers, and the induced demagnetizing fields are nonuniform and depend on the shape and the size of the RE-rich phase. Hence, the demagnetizing fields caused by these phases are very complicated, and unfortunately, it has not been fully investigated yet. Since the demagnetizing field is related to the microstructure and the shape, the calculation of the demagnetization is very difficult [9]. The distribution of the demagnetizing field in the magnet is generally nonuniform. Only in a uniformly magnetized ellipsoidal magnet, the demagnetizing field is uniform and the demagnetizing factor is one certain value. In other cases, the demagnetizing field is non-uniform and the demagnetizing factor is just an average or approximate value [10]. Moreover, nonmagnetic phases in real magnets will make the distribution of the demagnetizing field more complicated. The distribution of the magnetization and the demagnetizing field can be computed by micromagnetic simulation, which is an important theoretical tool [11–12]. However, the computation is limited by the computing resource [13]. The demagnetizing field in any position inside a magnet is produced by the whole magnetization distribution. For a magnetic system with N cells, an amount of work of O(N2) is required to calculate the demagnetizing field [14]. The three-dimensional (3D) micromagnetic models usually contain thousands or even millions of cells. Therefore, the calculation of the demagnetizing field is time-consuming and resource-consuming in the computational micromagnetism [15]. Hence, the analysis method for the demagnetizing field should be simplified to reduce the computational complexity. In this work, a simplified approach is employed to evaluate the demagnetizing field by decomposing it into several components and analyzing them individually. The demagnetizing field of permanent magnets influenced by the nonmagnetic phase is investigated in detail. The effects of the demagnetizing field on the nucleation process and the demagnetization process have been studied. The results
obtained are helpful for understanding the effects of nonmagnetic phase in the permanent magnet, such as the effect of RE-rich phases in sintered NdFeB magnets.
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Simulation method
In a magnetic system with the magnetization M occupying the volume V, the demagnetizing field at r, Hdemag(r), can be expressed as [16]: Η demag ðrÞ ¼ –
þ
#
#
Þ dτ # !V ðr – rkrÞr$Mðr 3 # –r k
#∂V ðr – r
# ÞMðr# Þ$nðr# Þ ds# kr – r# k3
(1)
where dτ′ and ds′ are the volume and the area elements, respectively. The integrals are taken over the space variable r. “∇$” is the divergence operator with respect to r′, k$k is the Euclidean norm of a 3D vector, ∂V is the boundary of V, and n(r′) is the unit outward normal at r′. In this work, the simulation of the demagnetizing field is carried out by using the micromagnetic simulation software, OOMMF (Object Oriented MicroMagnetic Framework) [17]. The discretization is realized by the finite difference method (FDM) [18]. The simulation region is a cuboid region with size of 600 nm 600 nm 1200 nm (Fig. 1). All studied models in this work are distributed inside in it. The demagnetizing fields inside and outside the magnets are both outputted and analyzed in the cuboid region. Even more, the demagnetizing fields in the space region outside the reversed model are the crucial object in Section 4 of this paper, which is used to study the influence of nonmagnetic particles on the demagnetizing field. The specific models will be described in the late discussions. The mesh size is 2 nm 2 nm 2 nm. The magnetization and easy axis are arranged along the + Z/ – Z direction. The permanent magnet used in the simulation is NdFeB. The material parameters are taken from Ref. [19] by Schrefl et al., including the anisotropy constant K1 = 4.5 MJ/m3 and the saturation magnetization Ms = 1.61 T. The magnetic parameters of the nonmagnetic phase and the space in the simulation model are set to zero. The demagnetizing fields are 3D space vectors, which are difficult to display clearly. Hence, the Z component of the demagnetizing field is chosen to represent the magnitude of the demagnetizing field in this paper. The simulation results are simplified further to facilitate the
Wei LI et al. Micromagnetic investigation by a simplified approach on the demagnetization field of ...
analysis. The Z components of the demagnetizing fields in the data section (x = 1 nm) and the data line (x = 1 nm, y = 1 nm) are plotted as the two-dimensional (2D) distribution graph and the one-dimensional (1D) distribution curve, respectively (Fig. 1).
Fig. 1 The cuboid simulation region with the size of 600 nm 600 nm 1200 nm. The origin of the coordinate is in the center of the cuboid. The simulation results in the section with blue border (x = 1 nm, named “data section”) will be used to draw the 2D distribution diagram. The simulation results in the red line (x = 1 nm, y = 1 nm, named “data line”) will be used to draw a distribution curve.
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The additivity of the demagnetizing field
Equation (1) shows that the demagnetizing field is actually the magnetic dipolar interaction between the magnetic moments. It is very important that how the demagnetizing field at r changes with M(r′), the magnetization at r′. Assuming that M(r′) are divided into M1(r′) and M2(r′), and then, Eq. (2) is obtained: H #demag ðMðr# ÞÞ ¼ H #demag ðM 1 ðr# ÞÞ þ H #demag ðM 2 ðr# ÞÞ
(2)
It shows that the demagnetizing field is additive in view of the function of M(r′). With the magnetization M(r′) divided into M1(r′) and M2(r′), the demagnetizing field generated by M(r′) is equal to the sum of the demagnetizing fields generated by M1(r′) and M2(r′). This rule is named the additivity of the demagnetizing field in this paper. It means that the demagnetizing field could be separated and analyzed individually. To verify the additivity of the demagnetizing field, several models are set up for the computation. The demagnetizing fields of the cuboid magnet with the size
3
of 600 nm 600 nm 1200 nm are simulated, and the results are shown in Panels 1–3 of Fig. 2. First, the cuboid magnet is separated into two partial magnets: One is the cuboid magnet with nonmagnetic shell and with the size of 600 nm 600 nm 1200 nm (red region in Panel 1 of Fig. 2) (the model shown above Panels 1–3 of Fig. 2); The other is the cubic-shell shape magnet (red region in Panel 2 of Fig. 2), whose shape and size are the same as the white region in Panel 1 of Fig. 2. The simulated demagnetizing fields and the stray fields are also shown in Panels 1–2 of Fig. 2. It needs to be claimed that the magnetostatic fields inside and outside a magnet are generally named the demagnetizing field and the stray field, respectively. Both fields have the same origin. For simplicity, the magnetostatic fields produced by magnets are named by the demagnetizing fields in this paper. The results indicate that the addition of the demagnetizing fields of two separated magnets (Panels 1–2 of Fig. 2) is equal to the demagnetizing field of the one combined cuboid magnet (Panel 3 of Fig. 2), which confirms the additivity of the demagnetizing field. In addition, the cuboid magnet is separated into three partial magnets shown in Panels 4–6 of Fig. 2. It indicates that the addition of the demagnetizing fields of those three parts (Panels 4–6 of Fig. 2) is also equal to the demagnetizing field of the combined cuboid magnet (Panel 3 of Fig. 2). The above results thus prove the validity of additivity of the demagnetizing field. Therefore, the demagnetizing field of the magnet can be divided into different separated parts and be analyzed individually.
4 The demagnetizing field influenced by the nonmagnetic phase The real sintered and hot deformed NdFeB magnets are composed of the RE2Fe14B phase and nonmagnetic phases. The area of the nonmagnetic phase is much smaller than the system size, hence it will not drastically change the magnetization distribution in the full range of the host magnet. Based on above analysis, due to the additivity of the demagnetizing field, the demagnetizing field for a simplified real magnet with nonmagnetic phase inside, shown in Panel 1 of Fig. 3, is equal to the addition of that of the “ideal” magnet without nonmagnetic phase (Panel 2 of Fig. 3) and that of the “reversed” magnet with magnetization Ms along the – Z direction (Panel 3 of Fig. 3). The latter has the same shape and in the same position as the nonmagnetic phase in Panel 2 of Fig. 2. As a result, the complex demagnetizing field of the real magnet could be
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Fig. 2 The demagnetizing field of the simulation models to verify the additivity. Insets show the middle sections of the models. The magnetization in red regions is Ms and the magnetization in white regions is 0. The nonmagnetic particle model and the corresponding magnet model are shown above. The vertical axes represent the Z coordinate in magnets, and the horizontal axes represent the Z component of the demagnetizing field. The curves show the demagnetizing field in the dash-dotted lines (data line in Fig. 1).
Fig. 3 The demagnetizing fields influenced by nonmagnetic phases. The middle sections of the models are shown in the insets. The magnetization in red regions is Ms, the magnetization in the white region is 0, and the magnetization in the blue region is –Ms. The vertical axes represent the Z coordinate in the magnets, and the horizontal axes represent the Z component of the demagnetizing field. The curves show the demagnetizing field in the dash-dotted lines (data line in Fig. 1).
easily understood by analyzing the demagnetizing fields of the “ideal” magnet and the “reversed” magnet respectively. In addition, in a actual magnet, the distribution of the magnetization is complex and not uniform. The nonmagnetic phases can be filled up by the magnetization with the
same magnitude and direction as that in the neighboring magnetic phases. Then the reversed magnet whose magnetization has the same magnitude and opposite direction can be used to study the influence of the nonmagnetic phase on the demagnetizing field.
Wei LI et al. Micromagnetic investigation by a simplified approach on the demagnetization field of ...
In this work, to simplify the calculations, the “ideal” magnets are set to cuboid magnets (600 nm 600 nm 1200 nm) with the uniform magnetization along + Z direction. The “reversed” magnets with the uniform magnetization along the – Z direction are set to be different in different simulation models. The difference in the demagnetizing field in different models could be analyzed by only focusing on the demagnetization field of the “reversed” magnet. This approach provides a feasible method to simulate the effects of nonmagnetic phases on the demagnetization process. Here, effects of the particle size, the ratio of length to diameter, the convexity– concavity and the position of nonmagnetic particles on the demagnetization field are studied. 4.1
Size effect
Cubic “reversed” magnets with various values of the sidelength s are used to investigate effects of the size of a
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nonmagnetic particle on the demagnetizing field, as shown in Fig. 4(a). Figure 4(b) shows the demagnetizing field along the data line, and the result indicates that the maximum values of the demagnetizing fields inside and outside the “reversed” magnet are hardly influenced by the side-length s. This is because the demagnetizing field is only determined by the shape of the “reversed” magnet. The demagnetizing field outside the reversed magnet covers a wider region with a larger size of the reversed magnet “s”, as shown in the right and the left parts of curves in Fig. 4(b). In addition, Fig. 4(c) shows the demagnetizing field variations with Z-coordinate “Z” divided by the side-length “s”, and all of the curves are coincident. It confirms that the affected region of the demagnetizing field outside the reversed magnet is proportional to the size of the reverse magnet. Hence, it indicates that in a real magnet, the affected region of nonmagnetic particles on the demagnetizing field increases with the increasing size of nonmagnetic particles.
Fig. 4 (a) The cubic reversed magnets ( –Ms) with different values of the side-length s. (b) The demagnetizing field curves in the data line. (c) The demagnetizing field curves with Z-coordinate “Z” divided by “s” correspondingly. (d) The sum of the demagnetizing field curves of the cubic reversed magnets with size s = 120 and 180 nm. The distance between them is 98 nm. The gray and light red regions represent the Z region of the demagnetizing fields with values larger than 30 kA/m (named influenced region), and the light blue region represents the overlapped region.
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Figure 4(d) shows the demagnetizing fields of two reversed magnets (s = 120 and 180 nm) with a distance of 98 nm between them. The gray and light red regions show the affected region of these two magnets. They are overlapped in the region between two magnets. The sum of the demagnetizing fields in the overlapped region (light blue region) is much higher than that in other region due to the additivity of the demagnetizing field. In addition, with the increasing size of the nonmagnetic particle, the size of the affected region increases (Figs. 4(b) and 4(c)) and the affected region is more likely to overlap with those of other nonmagnetic particles. As a result, the overlapped region exhibits a large demagnetizing field. Hence, to reduce the demagnetizing field in the magnet, the overlap region should be reduced or avoided. The effective approaches in practice include reducing the size of nonmagnetic particles and increasing the distance between nonmagnetic particles.
4.2 Effect of the ratio of length to diameter for the nonmagnetic phase
The cuboid reversed magnets with a constant side-length of 120 nm and different values of the height h in the space model are investigated (Fig. 5(a)). Figure 5(b) shows the demagnetizing field in the data line (shown in Fig. 1). The strength of the stray field outside the reversed magnet increases with the increasing h. The result indicates that the demagnetizing field near the nonmagnetic phase in a real magnet increases with the increasing ratio of length to diameter of the nonmagnetic phase. Hence, to reduce the demagnetizing field, nonmagnetic particles with long axes along the easy axis should be avoided in real magnets. For reversed magnets with the heights of h = 40 and 360 nm, the 2D distribution of the demagnetizing field in the data section are shown in Figs. 5(c) and 5(d),
Fig. 5 (a) The cuboid reversed magnets ( –Ms) with the same side-length of 120 nm and different values of the height h. (b) The demagnetizing field curves in the data line. (c)(d) 2D distribution graphs of the demagnetizing fields for h = 40 and 360 nm, respectively. The demagnetizing field is along the –Z direction in the blue regions (+ Z in the red regions). The dotted lines represent the demagnetizing field of 30 or –30 kA/m.
Wei LI et al. Micromagnetic investigation by a simplified approach on the demagnetization field of ...
respectively. The stray field of the reversed magnet is along the – Z direction in the blue regions. The gradation of color represents the magnitude of the demagnetizing field. The demagnetizing field in the dotted lines is 30 and – 30 kA/m. The dotted lines can be regarded as the outline of the demagnetizing field outside the magnet. The results show that the regions with the highest stray fields are the top and bottom surfaces of the nonmagnetic particle along the – Z direction, where the nucleation of reversed domains are easily formed during demagnetization. However, the demagnetizing field is along the + Z direction at the sides of the nonmagnetic particle. The magnetic grains here would not be the nucleation sites. Hence, the regions near the top and the bottom of nonmagnetic particles are the critical areas which deserve more attention. In addition, the additive demagnetizing field of two nonmagnetic particles is related to the relative position of the particles (Fig. 4(d)). The distance between nonmagnetic particles along the direction of the magnetization should be large enough to avoid the superposition of the demagnetizing field produced by different particles. 4.3 Effects of concavity or convexity of nonmagnetic particles
It was reported that the wettability of the nonmagnetic phases with the main phase is crucial factor to influence the demagnetization process in NdFeB magnets [20]. In fact, the different wettabilities would lead to different shapes of the nonmagnetic phase, such as the concavity and the convexity [21]. To study the influences of the concavity and the convexity on the demagnetizing field, six prismoids are removed (concave) or added (convex) at the surfaces of cubic models (200 nm 200 nm 200 nm) (Fig. 6(a)). The top and the bottom side lengths of prismoids are 40 and 196 nm, respectively. As presented in Fig. 6(b), the height of prismoids hp is set from – 80 to 80 nm. hp < 0 represents removing the prismoid (hp) from the cubic model (concave), as shown in Fig. 6(a). hp > 0 represents adding the prismoid (hp) on the cubic model (convex), as shown in Fig. 6(a). Figure 6(b) shows the 2D demagnetizing field distributions in the data section of models with different concavity and convexity (different hp). The stray field around the convex of the reversed magnet is higher than that of the concave magnet, and the affected region of the convex nonmagnetic particle is also larger. Hence, the reversed domain nucleation is likely to occur near the convex of nonmagnetic particles. As a result, for a real magnet, the
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convex nonmagnetic phase should be removed or changed to concave nonmagnetic particles in order to reduce the stray field and increase the coercivity. For NdFeB magnets, the approaches for changing the shape of nonmagnetic phases involve heat treatment, grain boundary diffusion, and improving the wettability of the nonmagnetic phase and the magnetic phase. The hp = – 80 nm model could represent the thin layer nonmagnetic grain boundary phase surrounding magnetic grains, which has been proven to be the ideal microstructure for high coercivity [22]. Figure 6(b) also indicates that the stray field near convex and concave corners in the pole surfaces is negative, while it is positive around the side surfaces. To further explain this phenomenon, the models with 2 prismoids (concave or convex) only in the pole surfaces or side surfaces were simulated. Figure 6(c) shows the 2D demagnetizing field distributions in the data section of the models. For the corners in the side surfaces, the stray field of the reversed magnet shows only minor change with that of the cube model. However, the demagnetizing field distribution would be greatly influenced by corners near the pole surfaces. With corners existing in the pole surfaces, the demagnetizing field inside the magnet increases significantly and that outside the magnet decreases significantly. Therefore, nonmagnetic particles with the concave corners in the pole surfaces will decrease the stray field, and on the contrary, convex corners will increase the stray field significantly. Therefore, the nonmagnetic phases with convex corners on the pole surface should be avoided in a real magnet. 4.4
Effects of position of the nonmagnetic phase
Figure 7 shows the demagnetizing field curves of magnets with nonmagnetic particles in different positions along the data line (as shown in Fig. 1). The additivity of the demagnetizing field is distinctly demonstrated. The demagnetizing field near the nonmagnetic particle increases with the decreasing distance between the nonmagnetic particle and the pole surface. This result suggests that nonmagnetic particles near the pole surfaces should be reduced or removed in order to enhance the coercivity of a real magnet. 4.5
Discussion
To improve magnetic properties of a permanent magnet, the microstructure and the composition in critical regions
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Fig. 6 (a) The models used to simulate concavity and convexity of reversed magnets. (b) The 2D demagnetizing field distribution in the data section of magnets (Ms) with different concavity and convexity (different hp). (c) The 2D demagnetizing field distribution in the data section of corresponding inset models with concavity and convexity in different dimensions (hp = 40 or –40 nm). The demagnetizing field is along the –Z direction in the blue regions (+ Z in the red regions). The dotted lines represent the demagnetizing field of 30 or –30 kA/m.
should be carefully controlled. The regions beside nonmagnetic phases are one kind of critical regions deserving much attention since the demagnetizing fields in these regions are significantly influenced by nonmagnetic phases. The influences of the nonmagnetic phases are generally very complicated and are related to the size, the shape, the position and the inter-distance of these phases. In the present work, it is verified that the demagnetizing field of the reversed magnet can be employed to study the influences of nonmagnetic particles based on the additivity of the demagnetizing field. The demagnetizing field in a permanent magnet is
crucial to the demagnetization process and the coercivity. Reducing the demagnetizing field can help to retard the demagnetization process and increase the coercivity [23]. Based on above results, some meaningful suggestions could be given to improve magnetic properties of NdFeBbased permanent magnets. First, the size of nonmagnetic particles (phases) should be reduced. Not only the density of nonmagnetic particles should be reduced, but also the partial accumulation of these particles should be avoided. As shown in Section 4.1, large particles and the dense distribution of particles will make the demagnetizing field overlap, and these regions with large demagnetizing fields
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The approach based on the additivity of the demagnetizing field in this work has the benefit of reducing the computational complexity of micromagnetic simulation for evaluating the demagnetizing field. Using this method, our results can show the influences of nonmagnetic particles on the demagnetizing field more clearly, which helps to understand the nucleation process and the coercivity influenced by the microstructure. The conclusions in this work provide some guidelines for optimizing the microstructure of permanent magnets. These guidelines are not only for NdFeB magnets but also for other types of magnets with nonmagnetic particles inside. Fig. 7 The demagnetizing fields along the dash-dotted lines (data line in Fig. 1) of cuboid magnets (red region, Ms) with nonmagnetic particles (white cuboid region) in different positions. The inset is the middle sections (data section in Fig. 1) of the models. The red region represents the magnets with the size of 600 nm 600 nm 1200 nm and the magnetization Ms along the + Z direction. The white cuboid region represents nonmagnetic particles with the size of 40 nm 40 nm 80 nm in different positions inside the magnet.
will serve as nucleation sites for the domain reversal. Second, improving the wettability of the nonmagnetic and the magnetic phases is preferable. With a good wettability, nonmagnetic particles will have a concave shape and the produced demagnetizing field will be low, as shown in Section 4.3. It is worth noting that above two suggestions can be actualized by grain boundary diffusion process, which has been well studied during past 15 years. This process can optimize the distribution of the grain boundary nonmagnetic phase including changing the nonmagnetic particles to thin layers and improving the wettability [24]. Various researches [25–27] have confirmed that the coercivity of NdFeB magnets can be enhanced by the grain boundary diffusion. Third, nonmagnetic rod-shaped particles with long axis along the magnetization direction should be avoided in magnets since the demagnetizing field near the pole surfaces is high (Section 4.2). Heat treatment [28] or grain boundary diffusion [29] with pressure can be feasible to change the shape of nonmagnetic particles (including the ratio of length to diameter). Finally, the microstructure in the regions near the pole surfaces is the most important for real magnets. The demagnetizing fields in these regions are larger than those in the center (Section 4.4). Hence, the influence of nonmagnetic particles on the demagnetization process will be amplified [30]. Fortunately, the microstructure optimization near the surfaces is more significant after heat treatment or grain boundary diffusion [24].
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Conclusions
Based on the additivity law of the demagnetizing field, the demagnetizing field in a magnet distributed with nonmagnetic particles can be separated into two parts, one generated by an ideal magnet without nonmagnetic particle, and another one is generated by a reversed magnet with the same shape as the nonmagnetic particle. Therefore, the complicated demagnetizing field of a real magnet could be analyzed by only focusing the stray field of the reserved magnet. Using this simplified approach, the influences of the size, the ratio of length to diameter, the shape and distribution of the nonmagnetic phase on the demagnetization field of NdFeB-based permanent magnets have been discussed. The results show that the maximum value of the demagnetization field is not significantly affected by the particle size, but the area of the affected region is proportional to the size of particles. The large particle will produce a large area overlapped with those influenced by other nonmagnetic particles, which would reduce the coercivity. The pole surfaces of nonmagnetic particles would generate larger demagnetizing field than the side surfaces. With increasing the ratio of length to diameter along the magnetization direction, the demagnetization field on the pole surface increases. Furthermore, if the shape of the pole surface has a convex corner, it will increase the demagnetization field. The demagnetizing field near the nonmagnetic particle will be further increased by the large macroscopic demagnetizing field near the pole surfaces. Nonmagnetic particles near the pole surfaces should be reduced or removed in order to enhance coercivity. This work not only provides a meaningful and simplified approach to evaluate the demagnetizing field influenced by nonmagnetic particles, but also gives some useful suggestions for the microstructure optimization of permanent magnets.
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Disclosure of potential conflicts of interests The authors declare that they have no conflicts of interest. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 51774146 and 51801047) and the Guangzhou Municipal Science and Technology Program (Grant No. 201605120111410).
Parkin S, eds. Handbook of Magnetism and Advanced Magnetic Materials. John Wiley & Sons, Ltd., 2007, doi: 10.1002/ 9780470022184.hmm201 [14] Tan X, Baras J S, Krishnaprasad P S. Fast evaluation of demagnetizing field in three-dimensional micromagnetics using multipole approximation. In: Proceedings of SPIE — The International Society for Optical Engineering, 2001, 3984: 195–
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RESEARCH ARTICLE
Micromagnetic investigation by a simplified approach on the demagnetization field of permanent magnets with nonmagnetic phase inside Wei LI1, Lizhong ZHAO1,2, and Zhongwu LIU (✉)1 1 School of Materials Science and Engineering, South China University of Technology, Guangzhou 510640, China 2 Innovative Center for Advanced Materials, Hangzhou Dianzi University, Hangzhou 310012, China
© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019
ABSTRACT: A simplified analysis method based on micromagnetic simulation is proposed to investigate effects of nonmagnetic particles on the demagnetizing field of a permanent magnet. By applying the additivity law of the demagnetizing field, the complicated demagnetizing field of the real magnet could be analyzed by only focusing on the stray field of the reserved magnet. For a magnet with nonmagnetic particles inside, the particle size has no significant effect on the maximum value of the demagnetization field, but the area of the affected region by the particle is proportional to the particle size. A large particle produces a large affected area overlapped with those influenced by other particles, which leads to the large demagnetization field. With increasing the length of the particle along the magnetization direction, the demagnetization field on the pole surface increases. The pole surface with a convex shape will increase the demagnetization field. The demagnetizing field near the nonmagnetic particle will be further increased by the large macroscopic demagnetizing field near the pole surface. This work suggests some practical approaches to optimize the microstructure of permanent magnets. KEYWORDS: demagnetizing field; additivity; micromagnetic simulation; nonmagnetic phase; nucleation process
Contents 1 2 3 4
Introduction Simulation method The additivity of the demagnetizing field The demagnetizing field influenced by the nonmagnetic phase 4.1 Size effect 4.2 Effect of the ratio of length to diameter for the nonmagnetic phase
Received April 30, 2019; accepted June 28, 2019 E-mail: [email protected]
4.3
Effects of concavity or convexity of nonmagnetic particles 4.4 Effects of position of the nonmagnetic phase 4.5 Discussion 5 Conclusions Disclosure of potential conflicts of interests Acknowledgements References
1
Introduction
Microstructure optimization is an effective method to improve magnetic properties of NdFeB magnets [1]. As
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one of the typical structures, the grain boundary phase is the crucial factor that influences the demagnetization process and the coercivity of NdFeB magnets [2–5]. Hou et al. [6] reported that the nonmagnetic Nd-rich phase plays a critical role in the preparation of hot deformed NdFeB magnets. Zhou et al. [7] reported that rare earth element (RE)-rich phases influence magnetic properties by significantly changing the distribution of the demagnetizing field. However, the RE-rich phases in a real NdFeB magnet are in various forms [8], such as particles and continuous layers, and the induced demagnetizing fields are nonuniform and depend on the shape and the size of the RE-rich phase. Hence, the demagnetizing fields caused by these phases are very complicated, and unfortunately, it has not been fully investigated yet. Since the demagnetizing field is related to the microstructure and the shape, the calculation of the demagnetization is very difficult [9]. The distribution of the demagnetizing field in the magnet is generally nonuniform. Only in a uniformly magnetized ellipsoidal magnet, the demagnetizing field is uniform and the demagnetizing factor is one certain value. In other cases, the demagnetizing field is non-uniform and the demagnetizing factor is just an average or approximate value [10]. Moreover, nonmagnetic phases in real magnets will make the distribution of the demagnetizing field more complicated. The distribution of the magnetization and the demagnetizing field can be computed by micromagnetic simulation, which is an important theoretical tool [11–12]. However, the computation is limited by the computing resource [13]. The demagnetizing field in any position inside a magnet is produced by the whole magnetization distribution. For a magnetic system with N cells, an amount of work of O(N2) is required to calculate the demagnetizing field [14]. The three-dimensional (3D) micromagnetic models usually contain thousands or even millions of cells. Therefore, the calculation of the demagnetizing field is time-consuming and resource-consuming in the computational micromagnetism [15]. Hence, the analysis method for the demagnetizing field should be simplified to reduce the computational complexity. In this work, a simplified approach is employed to evaluate the demagnetizing field by decomposing it into several components and analyzing them individually. The demagnetizing field of permanent magnets influenced by the nonmagnetic phase is investigated in detail. The effects of the demagnetizing field on the nucleation process and the demagnetization process have been studied. The results
obtained are helpful for understanding the effects of nonmagnetic phase in the permanent magnet, such as the effect of RE-rich phases in sintered NdFeB magnets.
2
Simulation method
In a magnetic system with the magnetization M occupying the volume V, the demagnetizing field at r, Hdemag(r), can be expressed as [16]: Η demag ðrÞ ¼ –
þ
#
#
Þ dτ # !V ðr – rkrÞr$Mðr 3 # –r k
#∂V ðr – r
# ÞMðr# Þ$nðr# Þ ds# kr – r# k3
(1)
where dτ′ and ds′ are the volume and the area elements, respectively. The integrals are taken over the space variable r. “∇$” is the divergence operator with respect to r′, k$k is the Euclidean norm of a 3D vector, ∂V is the boundary of V, and n(r′) is the unit outward normal at r′. In this work, the simulation of the demagnetizing field is carried out by using the micromagnetic simulation software, OOMMF (Object Oriented MicroMagnetic Framework) [17]. The discretization is realized by the finite difference method (FDM) [18]. The simulation region is a cuboid region with size of 600 nm 600 nm 1200 nm (Fig. 1). All studied models in this work are distributed inside in it. The demagnetizing fields inside and outside the magnets are both outputted and analyzed in the cuboid region. Even more, the demagnetizing fields in the space region outside the reversed model are the crucial object in Section 4 of this paper, which is used to study the influence of nonmagnetic particles on the demagnetizing field. The specific models will be described in the late discussions. The mesh size is 2 nm 2 nm 2 nm. The magnetization and easy axis are arranged along the + Z/ – Z direction. The permanent magnet used in the simulation is NdFeB. The material parameters are taken from Ref. [19] by Schrefl et al., including the anisotropy constant K1 = 4.5 MJ/m3 and the saturation magnetization Ms = 1.61 T. The magnetic parameters of the nonmagnetic phase and the space in the simulation model are set to zero. The demagnetizing fields are 3D space vectors, which are difficult to display clearly. Hence, the Z component of the demagnetizing field is chosen to represent the magnitude of the demagnetizing field in this paper. The simulation results are simplified further to facilitate the
Wei LI et al. Micromagnetic investigation by a simplified approach on the demagnetization field of ...
analysis. The Z components of the demagnetizing fields in the data section (x = 1 nm) and the data line (x = 1 nm, y = 1 nm) are plotted as the two-dimensional (2D) distribution graph and the one-dimensional (1D) distribution curve, respectively (Fig. 1).
Fig. 1 The cuboid simulation region with the size of 600 nm 600 nm 1200 nm. The origin of the coordinate is in the center of the cuboid. The simulation results in the section with blue border (x = 1 nm, named “data section”) will be used to draw the 2D distribution diagram. The simulation results in the red line (x = 1 nm, y = 1 nm, named “data line”) will be used to draw a distribution curve.
3
The additivity of the demagnetizing field
Equation (1) shows that the demagnetizing field is actually the magnetic dipolar interaction between the magnetic moments. It is very important that how the demagnetizing field at r changes with M(r′), the magnetization at r′. Assuming that M(r′) are divided into M1(r′) and M2(r′), and then, Eq. (2) is obtained: H #demag ðMðr# ÞÞ ¼ H #demag ðM 1 ðr# ÞÞ þ H #demag ðM 2 ðr# ÞÞ
(2)
It shows that the demagnetizing field is additive in view of the function of M(r′). With the magnetization M(r′) divided into M1(r′) and M2(r′), the demagnetizing field generated by M(r′) is equal to the sum of the demagnetizing fields generated by M1(r′) and M2(r′). This rule is named the additivity of the demagnetizing field in this paper. It means that the demagnetizing field could be separated and analyzed individually. To verify the additivity of the demagnetizing field, several models are set up for the computation. The demagnetizing fields of the cuboid magnet with the size
3
of 600 nm 600 nm 1200 nm are simulated, and the results are shown in Panels 1–3 of Fig. 2. First, the cuboid magnet is separated into two partial magnets: One is the cuboid magnet with nonmagnetic shell and with the size of 600 nm 600 nm 1200 nm (red region in Panel 1 of Fig. 2) (the model shown above Panels 1–3 of Fig. 2); The other is the cubic-shell shape magnet (red region in Panel 2 of Fig. 2), whose shape and size are the same as the white region in Panel 1 of Fig. 2. The simulated demagnetizing fields and the stray fields are also shown in Panels 1–2 of Fig. 2. It needs to be claimed that the magnetostatic fields inside and outside a magnet are generally named the demagnetizing field and the stray field, respectively. Both fields have the same origin. For simplicity, the magnetostatic fields produced by magnets are named by the demagnetizing fields in this paper. The results indicate that the addition of the demagnetizing fields of two separated magnets (Panels 1–2 of Fig. 2) is equal to the demagnetizing field of the one combined cuboid magnet (Panel 3 of Fig. 2), which confirms the additivity of the demagnetizing field. In addition, the cuboid magnet is separated into three partial magnets shown in Panels 4–6 of Fig. 2. It indicates that the addition of the demagnetizing fields of those three parts (Panels 4–6 of Fig. 2) is also equal to the demagnetizing field of the combined cuboid magnet (Panel 3 of Fig. 2). The above results thus prove the validity of additivity of the demagnetizing field. Therefore, the demagnetizing field of the magnet can be divided into different separated parts and be analyzed individually.
4 The demagnetizing field influenced by the nonmagnetic phase The real sintered and hot deformed NdFeB magnets are composed of the RE2Fe14B phase and nonmagnetic phases. The area of the nonmagnetic phase is much smaller than the system size, hence it will not drastically change the magnetization distribution in the full range of the host magnet. Based on above analysis, due to the additivity of the demagnetizing field, the demagnetizing field for a simplified real magnet with nonmagnetic phase inside, shown in Panel 1 of Fig. 3, is equal to the addition of that of the “ideal” magnet without nonmagnetic phase (Panel 2 of Fig. 3) and that of the “reversed” magnet with magnetization Ms along the – Z direction (Panel 3 of Fig. 3). The latter has the same shape and in the same position as the nonmagnetic phase in Panel 2 of Fig. 2. As a result, the complex demagnetizing field of the real magnet could be
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Fig. 2 The demagnetizing field of the simulation models to verify the additivity. Insets show the middle sections of the models. The magnetization in red regions is Ms and the magnetization in white regions is 0. The nonmagnetic particle model and the corresponding magnet model are shown above. The vertical axes represent the Z coordinate in magnets, and the horizontal axes represent the Z component of the demagnetizing field. The curves show the demagnetizing field in the dash-dotted lines (data line in Fig. 1).
Fig. 3 The demagnetizing fields influenced by nonmagnetic phases. The middle sections of the models are shown in the insets. The magnetization in red regions is Ms, the magnetization in the white region is 0, and the magnetization in the blue region is –Ms. The vertical axes represent the Z coordinate in the magnets, and the horizontal axes represent the Z component of the demagnetizing field. The curves show the demagnetizing field in the dash-dotted lines (data line in Fig. 1).
easily understood by analyzing the demagnetizing fields of the “ideal” magnet and the “reversed” magnet respectively. In addition, in a actual magnet, the distribution of the magnetization is complex and not uniform. The nonmagnetic phases can be filled up by the magnetization with the
same magnitude and direction as that in the neighboring magnetic phases. Then the reversed magnet whose magnetization has the same magnitude and opposite direction can be used to study the influence of the nonmagnetic phase on the demagnetizing field.
Wei LI et al. Micromagnetic investigation by a simplified approach on the demagnetization field of ...
In this work, to simplify the calculations, the “ideal” magnets are set to cuboid magnets (600 nm 600 nm 1200 nm) with the uniform magnetization along + Z direction. The “reversed” magnets with the uniform magnetization along the – Z direction are set to be different in different simulation models. The difference in the demagnetizing field in different models could be analyzed by only focusing on the demagnetization field of the “reversed” magnet. This approach provides a feasible method to simulate the effects of nonmagnetic phases on the demagnetization process. Here, effects of the particle size, the ratio of length to diameter, the convexity– concavity and the position of nonmagnetic particles on the demagnetization field are studied. 4.1
Size effect
Cubic “reversed” magnets with various values of the sidelength s are used to investigate effects of the size of a
5
nonmagnetic particle on the demagnetizing field, as shown in Fig. 4(a). Figure 4(b) shows the demagnetizing field along the data line, and the result indicates that the maximum values of the demagnetizing fields inside and outside the “reversed” magnet are hardly influenced by the side-length s. This is because the demagnetizing field is only determined by the shape of the “reversed” magnet. The demagnetizing field outside the reversed magnet covers a wider region with a larger size of the reversed magnet “s”, as shown in the right and the left parts of curves in Fig. 4(b). In addition, Fig. 4(c) shows the demagnetizing field variations with Z-coordinate “Z” divided by the side-length “s”, and all of the curves are coincident. It confirms that the affected region of the demagnetizing field outside the reversed magnet is proportional to the size of the reverse magnet. Hence, it indicates that in a real magnet, the affected region of nonmagnetic particles on the demagnetizing field increases with the increasing size of nonmagnetic particles.
Fig. 4 (a) The cubic reversed magnets ( –Ms) with different values of the side-length s. (b) The demagnetizing field curves in the data line. (c) The demagnetizing field curves with Z-coordinate “Z” divided by “s” correspondingly. (d) The sum of the demagnetizing field curves of the cubic reversed magnets with size s = 120 and 180 nm. The distance between them is 98 nm. The gray and light red regions represent the Z region of the demagnetizing fields with values larger than 30 kA/m (named influenced region), and the light blue region represents the overlapped region.
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Figure 4(d) shows the demagnetizing fields of two reversed magnets (s = 120 and 180 nm) with a distance of 98 nm between them. The gray and light red regions show the affected region of these two magnets. They are overlapped in the region between two magnets. The sum of the demagnetizing fields in the overlapped region (light blue region) is much higher than that in other region due to the additivity of the demagnetizing field. In addition, with the increasing size of the nonmagnetic particle, the size of the affected region increases (Figs. 4(b) and 4(c)) and the affected region is more likely to overlap with those of other nonmagnetic particles. As a result, the overlapped region exhibits a large demagnetizing field. Hence, to reduce the demagnetizing field in the magnet, the overlap region should be reduced or avoided. The effective approaches in practice include reducing the size of nonmagnetic particles and increasing the distance between nonmagnetic particles.
4.2 Effect of the ratio of length to diameter for the nonmagnetic phase
The cuboid reversed magnets with a constant side-length of 120 nm and different values of the height h in the space model are investigated (Fig. 5(a)). Figure 5(b) shows the demagnetizing field in the data line (shown in Fig. 1). The strength of the stray field outside the reversed magnet increases with the increasing h. The result indicates that the demagnetizing field near the nonmagnetic phase in a real magnet increases with the increasing ratio of length to diameter of the nonmagnetic phase. Hence, to reduce the demagnetizing field, nonmagnetic particles with long axes along the easy axis should be avoided in real magnets. For reversed magnets with the heights of h = 40 and 360 nm, the 2D distribution of the demagnetizing field in the data section are shown in Figs. 5(c) and 5(d),
Fig. 5 (a) The cuboid reversed magnets ( –Ms) with the same side-length of 120 nm and different values of the height h. (b) The demagnetizing field curves in the data line. (c)(d) 2D distribution graphs of the demagnetizing fields for h = 40 and 360 nm, respectively. The demagnetizing field is along the –Z direction in the blue regions (+ Z in the red regions). The dotted lines represent the demagnetizing field of 30 or –30 kA/m.
Wei LI et al. Micromagnetic investigation by a simplified approach on the demagnetization field of ...
respectively. The stray field of the reversed magnet is along the – Z direction in the blue regions. The gradation of color represents the magnitude of the demagnetizing field. The demagnetizing field in the dotted lines is 30 and – 30 kA/m. The dotted lines can be regarded as the outline of the demagnetizing field outside the magnet. The results show that the regions with the highest stray fields are the top and bottom surfaces of the nonmagnetic particle along the – Z direction, where the nucleation of reversed domains are easily formed during demagnetization. However, the demagnetizing field is along the + Z direction at the sides of the nonmagnetic particle. The magnetic grains here would not be the nucleation sites. Hence, the regions near the top and the bottom of nonmagnetic particles are the critical areas which deserve more attention. In addition, the additive demagnetizing field of two nonmagnetic particles is related to the relative position of the particles (Fig. 4(d)). The distance between nonmagnetic particles along the direction of the magnetization should be large enough to avoid the superposition of the demagnetizing field produced by different particles. 4.3 Effects of concavity or convexity of nonmagnetic particles
It was reported that the wettability of the nonmagnetic phases with the main phase is crucial factor to influence the demagnetization process in NdFeB magnets [20]. In fact, the different wettabilities would lead to different shapes of the nonmagnetic phase, such as the concavity and the convexity [21]. To study the influences of the concavity and the convexity on the demagnetizing field, six prismoids are removed (concave) or added (convex) at the surfaces of cubic models (200 nm 200 nm 200 nm) (Fig. 6(a)). The top and the bottom side lengths of prismoids are 40 and 196 nm, respectively. As presented in Fig. 6(b), the height of prismoids hp is set from – 80 to 80 nm. hp < 0 represents removing the prismoid (hp) from the cubic model (concave), as shown in Fig. 6(a). hp > 0 represents adding the prismoid (hp) on the cubic model (convex), as shown in Fig. 6(a). Figure 6(b) shows the 2D demagnetizing field distributions in the data section of models with different concavity and convexity (different hp). The stray field around the convex of the reversed magnet is higher than that of the concave magnet, and the affected region of the convex nonmagnetic particle is also larger. Hence, the reversed domain nucleation is likely to occur near the convex of nonmagnetic particles. As a result, for a real magnet, the
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convex nonmagnetic phase should be removed or changed to concave nonmagnetic particles in order to reduce the stray field and increase the coercivity. For NdFeB magnets, the approaches for changing the shape of nonmagnetic phases involve heat treatment, grain boundary diffusion, and improving the wettability of the nonmagnetic phase and the magnetic phase. The hp = – 80 nm model could represent the thin layer nonmagnetic grain boundary phase surrounding magnetic grains, which has been proven to be the ideal microstructure for high coercivity [22]. Figure 6(b) also indicates that the stray field near convex and concave corners in the pole surfaces is negative, while it is positive around the side surfaces. To further explain this phenomenon, the models with 2 prismoids (concave or convex) only in the pole surfaces or side surfaces were simulated. Figure 6(c) shows the 2D demagnetizing field distributions in the data section of the models. For the corners in the side surfaces, the stray field of the reversed magnet shows only minor change with that of the cube model. However, the demagnetizing field distribution would be greatly influenced by corners near the pole surfaces. With corners existing in the pole surfaces, the demagnetizing field inside the magnet increases significantly and that outside the magnet decreases significantly. Therefore, nonmagnetic particles with the concave corners in the pole surfaces will decrease the stray field, and on the contrary, convex corners will increase the stray field significantly. Therefore, the nonmagnetic phases with convex corners on the pole surface should be avoided in a real magnet. 4.4
Effects of position of the nonmagnetic phase
Figure 7 shows the demagnetizing field curves of magnets with nonmagnetic particles in different positions along the data line (as shown in Fig. 1). The additivity of the demagnetizing field is distinctly demonstrated. The demagnetizing field near the nonmagnetic particle increases with the decreasing distance between the nonmagnetic particle and the pole surface. This result suggests that nonmagnetic particles near the pole surfaces should be reduced or removed in order to enhance the coercivity of a real magnet. 4.5
Discussion
To improve magnetic properties of a permanent magnet, the microstructure and the composition in critical regions
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Fig. 6 (a) The models used to simulate concavity and convexity of reversed magnets. (b) The 2D demagnetizing field distribution in the data section of magnets (Ms) with different concavity and convexity (different hp). (c) The 2D demagnetizing field distribution in the data section of corresponding inset models with concavity and convexity in different dimensions (hp = 40 or –40 nm). The demagnetizing field is along the –Z direction in the blue regions (+ Z in the red regions). The dotted lines represent the demagnetizing field of 30 or –30 kA/m.
should be carefully controlled. The regions beside nonmagnetic phases are one kind of critical regions deserving much attention since the demagnetizing fields in these regions are significantly influenced by nonmagnetic phases. The influences of the nonmagnetic phases are generally very complicated and are related to the size, the shape, the position and the inter-distance of these phases. In the present work, it is verified that the demagnetizing field of the reversed magnet can be employed to study the influences of nonmagnetic particles based on the additivity of the demagnetizing field. The demagnetizing field in a permanent magnet is
crucial to the demagnetization process and the coercivity. Reducing the demagnetizing field can help to retard the demagnetization process and increase the coercivity [23]. Based on above results, some meaningful suggestions could be given to improve magnetic properties of NdFeBbased permanent magnets. First, the size of nonmagnetic particles (phases) should be reduced. Not only the density of nonmagnetic particles should be reduced, but also the partial accumulation of these particles should be avoided. As shown in Section 4.1, large particles and the dense distribution of particles will make the demagnetizing field overlap, and these regions with large demagnetizing fields
Wei LI et al. Micromagnetic investigation by a simplified approach on the demagnetization field of ...
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The approach based on the additivity of the demagnetizing field in this work has the benefit of reducing the computational complexity of micromagnetic simulation for evaluating the demagnetizing field. Using this method, our results can show the influences of nonmagnetic particles on the demagnetizing field more clearly, which helps to understand the nucleation process and the coercivity influenced by the microstructure. The conclusions in this work provide some guidelines for optimizing the microstructure of permanent magnets. These guidelines are not only for NdFeB magnets but also for other types of magnets with nonmagnetic particles inside. Fig. 7 The demagnetizing fields along the dash-dotted lines (data line in Fig. 1) of cuboid magnets (red region, Ms) with nonmagnetic particles (white cuboid region) in different positions. The inset is the middle sections (data section in Fig. 1) of the models. The red region represents the magnets with the size of 600 nm 600 nm 1200 nm and the magnetization Ms along the + Z direction. The white cuboid region represents nonmagnetic particles with the size of 40 nm 40 nm 80 nm in different positions inside the magnet.
will serve as nucleation sites for the domain reversal. Second, improving the wettability of the nonmagnetic and the magnetic phases is preferable. With a good wettability, nonmagnetic particles will have a concave shape and the produced demagnetizing field will be low, as shown in Section 4.3. It is worth noting that above two suggestions can be actualized by grain boundary diffusion process, which has been well studied during past 15 years. This process can optimize the distribution of the grain boundary nonmagnetic phase including changing the nonmagnetic particles to thin layers and improving the wettability [24]. Various researches [25–27] have confirmed that the coercivity of NdFeB magnets can be enhanced by the grain boundary diffusion. Third, nonmagnetic rod-shaped particles with long axis along the magnetization direction should be avoided in magnets since the demagnetizing field near the pole surfaces is high (Section 4.2). Heat treatment [28] or grain boundary diffusion [29] with pressure can be feasible to change the shape of nonmagnetic particles (including the ratio of length to diameter). Finally, the microstructure in the regions near the pole surfaces is the most important for real magnets. The demagnetizing fields in these regions are larger than those in the center (Section 4.4). Hence, the influence of nonmagnetic particles on the demagnetization process will be amplified [30]. Fortunately, the microstructure optimization near the surfaces is more significant after heat treatment or grain boundary diffusion [24].
5
Conclusions
Based on the additivity law of the demagnetizing field, the demagnetizing field in a magnet distributed with nonmagnetic particles can be separated into two parts, one generated by an ideal magnet without nonmagnetic particle, and another one is generated by a reversed magnet with the same shape as the nonmagnetic particle. Therefore, the complicated demagnetizing field of a real magnet could be analyzed by only focusing the stray field of the reserved magnet. Using this simplified approach, the influences of the size, the ratio of length to diameter, the shape and distribution of the nonmagnetic phase on the demagnetization field of NdFeB-based permanent magnets have been discussed. The results show that the maximum value of the demagnetization field is not significantly affected by the particle size, but the area of the affected region is proportional to the size of particles. The large particle will produce a large area overlapped with those influenced by other nonmagnetic particles, which would reduce the coercivity. The pole surfaces of nonmagnetic particles would generate larger demagnetizing field than the side surfaces. With increasing the ratio of length to diameter along the magnetization direction, the demagnetization field on the pole surface increases. Furthermore, if the shape of the pole surface has a convex corner, it will increase the demagnetization field. The demagnetizing field near the nonmagnetic particle will be further increased by the large macroscopic demagnetizing field near the pole surfaces. Nonmagnetic particles near the pole surfaces should be reduced or removed in order to enhance coercivity. This work not only provides a meaningful and simplified approach to evaluate the demagnetizing field influenced by nonmagnetic particles, but also gives some useful suggestions for the microstructure optimization of permanent magnets.
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Disclosure of potential conflicts of interests The authors declare that they have no conflicts of interest. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 51774146 and 51801047) and the Guangzhou Municipal Science and Technology Program (Grant No. 201605120111410).
Parkin S, eds. Handbook of Magnetism and Advanced Magnetic Materials. John Wiley & Sons, Ltd., 2007, doi: 10.1002/ 9780470022184.hmm201 [14] Tan X, Baras J S, Krishnaprasad P S. Fast evaluation of demagnetizing field in three-dimensional micromagnetics using multipole approximation. In: Proceedings of SPIE — The International Society for Optical Engineering, 2001, 3984: 195–
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