# Investigation on Domain Pinning Mechanism in Nanometer Hard Magnetic Materials

Authors: Yan Sun; Qingbao Ni
Journal: Advances in Condensed Matter Physics
 (2019)

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## Abstract

The domain pinning mechanism was investigated in nanometer hard magnetic materials. The pinning fields of domain wall at different inhomogeneities were studied respectively. And exchange-coupling coefficientαex was investigated too. The results showed that αex is proportional to the ratio of inhomogeneity thickness ro to wall width δm for narrow inhomogeneity, while it decreases with enhancement of ro/δm for extended inhomogeneity. At a certain value of ro, exchange-coupling coefficient and pinning fields will reach the maximum. Exchange-coupling interaction and pinning fields are greatly influenced by inhomogeneity. Control range of inhomogeneity may obtain higher coercivity.


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## Body

## 1. Introduction

In recent years, nanocomposite magnets have attracted much attention due to high theoretical energy product, which was recorded as high as 1 MJ/m3 [1]. However, an equally important uncertainty is coercivity mechanism in nanometer magnetic materials. Therefore, attention has been focused on coercivity mechanism [2–5]. Currently, there are mainly two coercivity mechanisms, nucleation and pinning. Kronmüller et al. [5] proposed that the coercivity of nanocomposite magnets was controlled by nucleation process of reverse domains and introduced a microstructural parameter αex to describe the effect of exchange-coupling interaction on coercivity, leading to μ0Hc=αkαexμ0HNmin-NeffJs (coercivity experiential formula). Zhang et al. [6] suggested both nucleation and domain–wall pinning model can be expressed as coercivity experiential formula. Experimentally, elemental additions were found to form precipitates at grain boundary [7, 8]. The properties of grain boundary are different from main phase. Therefore, there is inhomogeneous area in nanometer magnetic materials, which plays an important role in pinning domain wall. In this paper, pinning fields of domain wallHc were calculated by considering inhomogeneous area. And corresponding exchange-coupling coefficient αex was investigated too.


## 2. Calculation Model

### 2.1. Inhomogeneous Area

Experimentally, elemental additions were found to form precipitates at grain boundary [7, 8]. Properties of grain boundary are different from the main phase. Additionally, due to the exchange-coupling interaction between grains, properties of grain edge have changed too. It is assumed that grain boundary and area influenced by exchange-coupling interaction form inhomogeneous area. Referring toK1-profile given by Kronmüller [9], we assume anisotropy at inhomogeneous area is as follows: (1)K1r=K1-nK1ch2r/ro,wherero is inhomogeneity thickness.K1 is first anisotropy constant. n is a number no larger than 1, which denotes reductionK1 in inhomogeneous area. The variation of anisotropy at inhomogeneous area is shown in Figure 1.Figure 1
Variations of anisotropyK1(r) in inhomogeneous area.


### 2.2. Coercivity Theory

Inhomogeneous area is regarded as strong planar pinning centers. When inhomogeneity thicknessro is smaller than wall width δm, coercive field is given [10]:(2)Hc=1332K1Msπdδm∑1nAAi,i+1-K1iK1-NeffMs,whereAi,i+1 corresponds to the local exchange constant between adjacent layers i and i+1. Similarly, K1i corresponds to the local anisotropy constant of the ith layer. δm is the wall width. The term NeffMs takes care of demagnetization fields resulting from grain surfaces and volume charges. It is assumed that the n perturbed planes have equivalent properties (A′ and K1′ are equal). And we assume A′ of perturbed planes is equal to A. Coercivity is written as(3)Hc=1332K1Msπroδm1-K1′K1-NeffMs,where ro=md denoting inhomogeneous area thickness.Whenro is larger than δm, the coercive field is determined by (4)Hccij=12Msdγdrmax-NeffMs,where (dγ/dr)max denotes maximum slope of the wall energy given by γ=4A1K1(r). K1( r) is anisotropy at inhomogeneous area.


### 2.3. Coefficient for Exchange-Coupling Interaction

Compared with coercivity experiential formula, expressions of exchange-coupling coefficientαex are given. For thin inhomogeneity (ro<δm), exchange-coupling coefficient αex is written as (5)αex=133πroδB1-K1′K1.In this case, for K1′=K1(ro/2) and n=0.8, αex=(73/100)(πro/δB) is obtained.For thick inhomogeneity ( ro>δm), αex is written as(6)αex=nδmπrodK1rdrmax.And when n equals 0.8, αex=12δm/25πr0.


## 2.1. Inhomogeneous Area

Experimentally, elemental additions were found to form precipitates at grain boundary [7, 8]. Properties of grain boundary are different from the main phase. Additionally, due to the exchange-coupling interaction between grains, properties of grain edge have changed too. It is assumed that grain boundary and area influenced by exchange-coupling interaction form inhomogeneous area. Referring toK1-profile given by Kronmüller [9], we assume anisotropy at inhomogeneous area is as follows: (1)K1r=K1-nK1ch2r/ro,wherero is inhomogeneity thickness.K1 is first anisotropy constant. n is a number no larger than 1, which denotes reductionK1 in inhomogeneous area. The variation of anisotropy at inhomogeneous area is shown in Figure 1.Figure 1
Variations of anisotropyK1(r) in inhomogeneous area.


## 2.2. Coercivity Theory

Inhomogeneous area is regarded as strong planar pinning centers. When inhomogeneity thicknessro is smaller than wall width δm, coercive field is given [10]:(2)Hc=1332K1Msπdδm∑1nAAi,i+1-K1iK1-NeffMs,whereAi,i+1 corresponds to the local exchange constant between adjacent layers i and i+1. Similarly, K1i corresponds to the local anisotropy constant of the ith layer. δm is the wall width. The term NeffMs takes care of demagnetization fields resulting from grain surfaces and volume charges. It is assumed that the n perturbed planes have equivalent properties (A′ and K1′ are equal). And we assume A′ of perturbed planes is equal to A. Coercivity is written as(3)Hc=1332K1Msπroδm1-K1′K1-NeffMs,where ro=md denoting inhomogeneous area thickness.Whenro is larger than δm, the coercive field is determined by (4)Hccij=12Msdγdrmax-NeffMs,where (dγ/dr)max denotes maximum slope of the wall energy given by γ=4A1K1(r). K1( r) is anisotropy at inhomogeneous area.


## 2.3. Coefficient for Exchange-Coupling Interaction

Compared with coercivity experiential formula, expressions of exchange-coupling coefficientαex are given. For thin inhomogeneity (ro<δm), exchange-coupling coefficient αex is written as (5)αex=133πroδB1-K1′K1.In this case, for K1′=K1(ro/2) and n=0.8, αex=(73/100)(πro/δB) is obtained.For thick inhomogeneity ( ro>δm), αex is written as(6)αex=nδmπrodK1rdrmax.And when n equals 0.8, αex=12δm/25πr0.


## 3. Results and Discussion

We calculated coercivity and correspondingαex by using intrinsic magnetic parameters of Nd2Fe14B (K1=4.3MJ/m3[5], J=1.6T, Ms=1280KA/m[11] ).Figure1 shows variations of anisotropyK1(r) in inhomogeneous area. It can be seen thatK1(r) increases with enhancement of ratio of r to ro. The value of anisotropy of area close to grain core is larger, because it is influenced to a fewer degree by exchange-coupling interaction. It is also shown in Figure 1 thatK1(r) decreases with increasing n. n denotes the decreaseK1 in inhomogeneous area. Exchange-coupling interaction and inhomogeneity decrease anisotropy.Figure2 shows variations of exchange-coupling coefficient αex with ratio of ro to δm for narrow inhomogeneity. It can be seen that αex is proportional to ro/δm in this case. For smaller ro, exchange-coupling interaction is strong, and the effects of exchange-coupling interaction on anisotropy and coercivity are dominant. The enhancement of αex mainly results from influence of exchange-coupling interaction. Figure 3 shows variations of exchange-coupling coefficient αex with ratio of ro to δm for extended inhomogeneity. It can be seen that αex decreases with increasing of ro/δm for certain value of n. For extended inhomogeneity, the effect of defect on anisotropy and coercivity is dominant.Figure 2
Variations of exchange-coupling coefficientαex with the ratio of ro to δm for narrow inhomogeneity.Figure 3
Variation of exchange-coupling coefficientαex with the ratio of ro to δm for extended inhomogeneity.Figure4 shows dependence of αex on ratio of ro to δm. In transition region near 0.6, the broken curve indicates the transition between the approximations for narrow and extended inhomogeneities. As shown in Figure 5, the variation of coercivity and exchange-coupling coefficient is consistent. Inhomogeneity has great influence on coercivity. Magnetization reversal process is thought of as follows. Firstly, the antimagnetization core is formed in the nonuniform region. With enhancement of the antimagnetization field, domain wall displacement will encounter maximum resistance peak caused by exchange coupling interaction and inhomogeneity, and domain wall will have maximum irreversible displacement. Pinning field reaches the maximum value. With increasing inhomogeneity, area influenced by exchange-coupling interaction is larger for narrow inhomogeneity, while area influenced by exchange-coupling interaction is smaller with increasing range of inhomogeneity for extended inhomogeneity.Figure 4
Dependence ofαex on ratio of ro to δm.Figure 5
Dependence of coercivity on ratio ofro to δm.


## 4. Conclusion

It is assumed that defect and grain edge influenced by exchange-coupling interaction form inhomogeneous area. Pinning fields and exchange-coupling coefficientαex corresponding to narrow and extended inhomogeneous area are investigated respectively. By considering different inhomogenities, pinning fields and exchange-coupling coefficent were studied. While the effect of defect on them is dominant for extended inhomogeneity. Coercivity depends greatly on microstructure.



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