稀有金属(英文版) 2020,39(01),22-27

Thickness-dependent coercivity mechanism and hysteresis loops in hard/soft magnets

Xiao-Jiao Weng Guo-Ping Zhao Hong Tang Lai-Chuan Shen Yao Xiao

College of Physics and Electronic Engineering,Sichuan Normal University

Collaborative Innovation Center for Shanxi Advanced Permanent Materials and Technology

State Key Laboratory of Metastable Materials Science and Technology,Yanshan University

作者简介：*Guo-Ping Zhao ve-mail:Zhaogp@uestc.edu.cn;

收稿日期：1 October 2018

基金：financially supported by the National Natural Science Foundation of China (Nos.51771127, 51571126 and 51772004);

Thickness-dependent coercivity mechanism and hysteresis loops in hard/soft magnets

Xiao-Jiao Weng Guo-Ping Zhao Hong Tang Lai-Chuan Shen Yao Xiao

College of Physics and Electronic Engineering,Sichuan Normal University

Collaborative Innovation Center for Shanxi Advanced Permanent Materials and Technology

State Key Laboratory of Metastable Materials Science and Technology,Yanshan University

Abstract：

Two models are established to reveal the underlying coercivity mechanism for SmCo/Fe films,where one model considers a transition layer between hard and soft layers,while the other model does not consider this layer.Based on the two models,hysteresis loops,nucleation fields and coercivity are obtained by one-dimensional(1 D) and three-dimensional(3 D) micromagnetic methods.In particular,the calculated nucleation fields(H_N)and coercivity(H_C) match very well with the experimental data.It is found that the increase in the soft phase thickness(Ls) leads to a transition of the coercivity mechanism from nucleation to pinning.Such a pinning is inherently related to nucleation and has both attributes of traditional nucleation and pinning,called as a hybrid coercivity mechanism here.It is general for all hard/soft composites and can be extended to single-phased permanent magnets where defects are inevitable.

Keyword：

Coercivity mechanism; Nucleation; Pinning; Hysteresis loops;

Received： 1 October 2018

1 Introduction

The mechanism of coercivity has been debated for manyyears [ 1, 2, 3, 4] ,but it is still not fully understood.The coercivity of the magnets in the experiments is far less than that given by the theory,especially for permanent magnets with high coercivity and crystalline anisotropy.This difference is usually attributed to crystal defects and inter-grain interactions,which play significant roles in adjusting the mechanism.Nucleation and pinning are the two maincoercivity mechanisms for permanent magnets,which are normally taken as isolated processes.However,experi-mental results demonstrate that the real mechanism is quite complicated,and in many cases,there is a hybrid one,which combines the characteristics of traditional nucleation and pinning.

Hard/soft composite magnets,also called exchangespring materials [ 5, 6, 7, 8] ,are ideal media to study the coercivity mechanism as the soft layer can be regarded as an enlarged soft defect,which manifests the role of the soft layer and reveals the underlying mechanism more clearly.Because it can effectively combine the high coercivity of the hard magnet with the high remanence of the softmagnet,hard/soft magnetic composite materials havebecome a research hot spot [ 9, 10, 11, 12, 13, 14, 15] since it was put forward in 1991 by Kneller and Hawing [ 5] .Theoretical studies suggest that the maximum energy product ((BHmax)) ofparallel-oriented composite magnets can reach 1 MJ·m^-3 [ 16] ,which is equivalent to twice that of the optimal single-phase permanent magnet.However,the energy productof the experiments is far smaller than that of the theories,resulting in an energy product paradox [ 15] .In 2011,the energy product of 0.32 MJ·m^-3 was experimentally produced by Sawatzki et al. [ 17] in epitaxial SmCos5/Fe/SmCo5 exchange spring trilayers,and then the energy product record of SmCo/Fe in 2015 was obtained by Neuet al. [ 18] ,which is 0.4 MJ·m^-3.At present,the best experimental result is 0.48 MJ·m^-3 which is obtained by Cui et al. [ 19] in Nd₂e₁₄B/FeCo anisotropic nanocom-posite films.Therefore,it is necessary to further study the coercivity mechanism of composite magnets.

In this paper,two different SmCo/Fe models have beenestablished to study the coercivity mechanism of hard/soft composite magnets.For ModelⅠa hard/soft/hard trilayer system was adopted,and we consider that defects in thehard layer can significantly reduce the effective crystalline anisotropy.For ModelⅡ,a transition layer between hard and soft layers was added in order to account for thepossible atomic diffusion effect [ 20, 21] ,which can affect the coercivity greatly.

2 Calculation models and methods

ModelⅠas shown in Fig.1,is a SmCo/Fe/SmCo trilayer,where a soft layer was sandwiched between the two hardlayers whose thickness is 24.0 nm.The directions of theeasy axes of all layers are parallel to the applied field (xaxis) within the film plane,where the effect of temperature has not been considered.For ModelⅡ,the thickness of thehard layer is reduced to 22.5 nm and a transition layer with a thickness of 3.0 nm is added between the hard and softlayers to make a complement.Similar reduction was per-formed for the soft layer thickness.As a result,both soft and hard layers were reduced by 1.5 nm in thickness.The magnetic properties within the transition layer are roughly the average of those in the hard and soft layers,as shown in Table 1.

The three-dimensional (3D) calculation,using the object-oriented micromagnetic framework (OOMMF) [ 22] software,adopts finite differential methods based on the LandauLifshitz-Gilbert dynamic equation [ 23] :

where M and M_s are the magnetizations and spontaneous magnetization (A·m^-1),respectively.t stands for time (s).H_eff is the effective field (A·m-1).γandαdenote the Landau-Lifshitz gyromagnetic ratio and dimensionless damping constant,respectively.The effective field is defined as follows:

whereμ₀ is the permeability of vacuum.And the average energy density (E) is a function of M specified by Brown's equations [ 24] :

where A and K represent the exchange constant and anisotropy constant,respectively;Hd is the demagnetization field,and n is the direction of easy axis.Equation(3) includes four energies,which are theexchange energy,anisotropy energy,applied field(Zeeman) energy and magnetostatic (demagnetization)energy,respectively.If we ignore magnetostaticinteraction,the above 3D energy can be reduced to aone-dimensional (1D) expression.In the condition,theangle changes only in the z-axis direction,thus,the energy density (F) of each region in the film plane is [ 15, 25] :

whereθis the spin angle,i.e.,angle between the magnetization and the applied field.L represents the thickness of the magnetic layer and M_s denotes the saturation magnetization.The superscripts h and s denote the hard and soft phases,respectively.A variationalmethod [ 15, 25] is used to minimize the energyexpressed in Eq.(4),which yields the equations for the angular distribution (θas a function of z) as follows:

Fig.1 Basic schemes for two trilayers in this work,where L and H represent thickness of magnet and applied magnetic field,respectively

  下载原图

Table 1 Calculation parameters for hard/soft films

where h=H/H_K represents a reduced applied field withanisotropic field H_k=2K/Ms,and is theBloch wall width.θ^s andθ^h are the directions of the magnetization in the middle of the soft layer and the surface of the hard layer,respectively.

In our calculations,for Model I,two anisotropy constants are used,which are k_h=1.30×106 and 4.0×10₆J·m^-3.The rest of the parameters are the same as those of ModelⅡ,as listed in Table 1.

3 Results and discussion

3.1 Hysteresis loops based on 1D and 3D methods for Model I

Considering that the defect of hard layer in the experiment reduces its anisotropy constant,we took a relatively small value of the anisotropy constant (k^h=1.30×10⁶ J·m^-3)to perform 1D and 3D calculations on Model I.Figure 2a,b represents 1D and 3D calculation results,respectively.It can be seen that for two calculated methods,the nucleation fields and coercivity go down with the increase in the soft layer thickness.When the thickness of the soft layer is smaller than 6 nm,the coercivity of the hard/soft trilayer is equal to the nucleation field,and the hysteresis loops show a good square degree,which is called a rigid composite magnet [ 26] .When L^s is greater than 9 nm,the pinning and the nucleation are separated.The coercivity at this time is mainly determined by the pinning,and the hysteresis loops show the characteristics of the exchange spring [ 26] .It shows that the compound magnet has a reversible reversal process.

Fig.2 Calculated hysteresis loops for Model I with various soft layer thicknesses based on a 1D (given by Eqs.(5) and (6)) and b 3D(micromagnetic software OOMMF) calculations (J representing polarization intensity)

The values of nucleation fields and coercivity obtained by 1D and 3D calculations are very close for all four soft thicknesses.When the thickness of the soft magnetic layer is 15 nm,the coercivity calculated by the 3D method is0.51 T,which is slightly smaller than that (0.52 T) calculated by 1D.And their difference is smaller than 1.9%.The nucleation fields calculated by 1D and 3D are also very close,which are 0.47 and 0.43 T,respectively.Therefore,the results obtained by the two methods are reliable.

3.2 Angle distribution for Model I

In order to more clearly show the reversal of magnetic moments in hard and soft composite magnets,we havecompared the angular distribution of the magneticmoments in the z-axis direction with L^s=15 and 21nm,respectively.As shown in Fig.3,the spin angle (θ)decreases monotonously with z increasing in differentapplied fields,indicating that the magnetic moments in the soft layer respond quickly to the applied fields.When L^s=15 nm,the composite magnet nucleates atH=-0.48 T,and the angle gradually increases as theapplied field decreases.When the applied field decreases to-0.52 T,the largest deflection angle of the composite magnet is 115.0°.If the applied field is further reduced,the magnetic moments are irreversibly reversed.When L^s=21 nm,the largest deflection angle of the compositemagnets is 132.5°when the external field is-0.41 T.It shows that the greater the soft layer thickness is,the faster the reversal of the magnetic moments is,thus the nucleation and pinning of the composite magnets will besmaller.

Fig.3 Calculated angular distributions (θ) of magenetization with L^s=15 and 21 nm for various applied fields H

3.3 Comparison of hysteresis loops calculated by 3D for two models

Experimentally,atoms at the interface between the hardand soft layers may diffuse and form a transition layer at the interface [ 15, 27] .In ModelⅡ,we added a 3 nmtransition layer in the middle of the hard and soft layers,performed a 3D simulation and compared to the results of ModelⅠwith no transition layer.Their hysteresis loops areshown in Fig.4.The anisotropy constants of the hardmagnetic layers of the two models are set to 4.0×106J·m^-3 for comparison.The results show that the hysteresis loops calculated by the two models have the same changes with the thickness of the soft layer.As Ls increases,the squareness of the hysteresis loops deteriorates,and the nucleation fields and coercivity also decrease.However,the magnetic properties are not sensitive to the hard layer thickness when L^h is much larger than its domain wall width (normally a few nanometers for a rare-earth permanent magnet) [ 15, 25] .In addition,the trilayer with a transition layer clearly has a smaller coercivity,especially for the relatively small L^s.For this case,i.e.,with a small L^s,the coercivity difference of two models is greater.A comparison of the hysteresis loops in Fig.4 shows that the transition layer does have a definite effect on the coercivity and the nucleation field.However,whether the transitionlayer is added or not,the calculated nucleation fields and coercivity have the same trend changing with the thickness of the soft layer.

Fig.4 Hysteresis loops calculated in 3D for two models at different thicknesses of soft layer

3.4 Comparisons of coercivity and nucleation fields with experimental results

Figure 5 shows the nucleation fields and coercivity based on ModelⅡand compares with experimental resultsobtained by Sawatzki et al. [ 17] .It can be seen that our calculated nucleation fields and coercivity agree well with the experimental results.When Ls is small,the pinning fields are equal to the nucleation fields.The coercivity in this case is determined by nucleation and pinning together.With the increase of Ls,the nucleation and the pinning fields monotonically decrease.When Ls is larger than6 nm,nucleation and pinning are separated,and coercivity mechanism changes from nucleation to pinning.Therefore,the soft magnetic thickness of 6 nm is a critical thickness.The experimental data given by Sawatzki et al. [ 17] prove that the coercivity mechanism is the same.The calculated values for coercivity are also consistent with the experimental results for other single-phased [ 29, 30, 31, 32] and composite permanent magnets [ 1, 2, 3, 6, 19, 27] ,indicating that the coercivity mechanism here is general.

Fig.5 Changes of nucleation fields (H_N) and coercivity (H_C) with soft layer thicknesses,where solid line is the result calculated by 3D of ModelⅡand dotted line represents experimental data of Sawatzki et al. [17] ,RM represents rigid composite region in which coercivity equals to nucleation field,while ES denotes exchange spring region in which coercivity deviates from nucleation field

4 Conclusion

The coercivity mechanism is greatly affected by the soft magnetic thickness,which is well reflected in the hysteresis loops.As the thickness of the soft magnetic layer increases,the coercivity mechanism changes from nucleation andpinning to a pure pinning.It confirms that the coercivity mechanism of hard/soft composites is a hybrid mechanism.In particular,the nucleation fields and coercivity calculated by ModelⅡ,where a transition layer is inserted between the hard and soft layers,agree quantitatively well with the experimental results.In addition,the soft defects of singlephased permanent magnets can be regarded as reduced soft magnets,so the theory of the hybrid coercivity mechanism is also applicable to single-phased permanent magnets.

Acknowledgements This work was financially supported by the National Natural Science Foundation of China (Nos. 51771127, 51571126 and 51772004).

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