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Journal of Magnetism and Magnetic Materials 320 (2008) 1393–1397

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Micromagnetic simulation of the magnetic dispersion angle and effectivedamping factor for the single-phase soft magnetic films

Lin Yuan, Jianjun Jiang�, Yongjiang Di, Shaowei Bie, Huahui He

Department of Electronic Science and Technology, Huazhong University of Science and Technology, Wuhan, HuBei 430074, China

Received 7 August 2007; received in revised form 22 October 2007

Available online 4 December 2007

Abstract

The micromagnetic structure of the single-phase soft magnetic films was simulated using the model of two-dimensional hexagonal

lattices by micromagnetic method. The typical micromagnetic ripple structure of magnetic films was obtained. Thus, the magnetic

dispersion angle was calculated from the static magnetic structure of the film. Furthermore, the relationship between the magnetic

dispersion angles and the corresponding magnetic parameters of the film was discussed. The technique also demonstrated the microwave

permeability of the films and the magnetic spectra well fitted by the permeability equation, which was deduced from the

Landau–Lifshitz–Gilbert (LLG) function when the film was considered as a single domain. The fitting data of effective damping factor as

a function of the magnetic dispersion angle were investigated.

r 2007 Elsevier B.V. All rights reserved.

PACS: 75.40.Mg; 75.75.+a

Keywords: Micromagnetic; Ripple theory; Magnetic dispersion angle; Effective damping factor

1. Introduction

The soft magnetic films exist in the polycrystallinestructure [1] or the granular structure [2] in which eachgrain has random anisotropy. The static magnetic propertyof these soft magnetic films is not followed by theStoner–Wohlfarth theory [3] which is applied to the magnetwith uniaxial anisotropy. Thus, the micromagnetic rippletheory is developed by Hoffmann [4] to explain the specialmagnetic property of some soft magnetic films. From theripple theory, the micromagnetic ripple structure in whichthe magnetization shows a type of longitudinal wave likefluctuation is the typical property of such soft magneticfilms. The property of the micromagnetic ripple structure isdescribed by two parameters of the ripple wavelength andthe magnetic dispersion angle. Berkov and Gorn [5]compute the magnetization process of such films using atwo-dimensional hexagonal lattices model, and get the

- see front matter r 2007 Elsevier B.V. All rights reserved.

/j.jmmm.2007.11.025

onding author. Tel./fax: +86 27 87544472.

ddress: jiangjj@mail.hust.edu.cn (J.J. Jiang).

typical micromagnetic ripple structure. However, themagnetic corresponding length as a function of theparameter of these structures is discussed to some extent.In this work, the static magnetic structure and magnetic

dispersion angle of the single-phase soft magnetic filmswere simulated using a model of two-dimensional hexagonallattices by finite element method. Furthermore, therelationship between the magnetic dispersion angle andthe magnetic parameters including grain size, film thick-ness, local anisotropy constant and exchange constant ofmagnetic films was discussed in detail. The microwavepermeability of the films was also computed, and theeffective damping factor of the films as a function of themagnetic dispersion angle was investigated.

2. The model

The discussed model in this work was a strip film withdimension of 605 nm� 3005 nm� 10 nm, which wasformed by 7306 hexagonal grains with a side length of10 nm (grain size R of 8.66 nm) and a height of 10 nm (film

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Fig. 1. The spin configuration of xy slice of the single-phase soft magnetic film. It shows the typical magnetic ripple structure. The arrows denote the

magnetization vector. The color level denotes x component of the normalized magnetization m̄x. The magnetic parameters are: Ms ¼ 2T,

A ¼ 1.0� 10�11 J/m, K1 ¼ 20 000 J/m3, R ¼ 8.66 nm and d ¼ 10 nm.

L. Yuan et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 1393–13971394

thickness of 10 nm) (Fig. 1). Then the film was divided into126 767 tetrahedral finite elements. In this model, eachhexagonal grain was considered as one type of magneticmaterial which had an identical saturation magnetizationof 2T, exchange constant of 1.0� 10�11 J/m, dampingconstant of 0.02 and magnetocrystalline anisotropy con-stant K1 of 20 000 J/m3 but a random easy axis. Accordingto the magnetic parameter, the magnetic film could be thetypical FeCo-based soft magnetic film. At the end of thefilm in y-axis, if K1 was small, the magnetization will benon-collinear for the shape-demagnetizing energy, whichwill bring extra ferromagnetic resonance (FMR) peak orFMR linewidth. So the magnetocrystalline anisotropyconstant and the easy axis there were set to 2� 107 J/m3

and parallel to y-axis, respectively, to pin the magnetiza-tion in y-axis. In this work the codes magpar [6] was usedfor the micromagnetic calculation.

3. Results and discussion

3.1. Static magnetic property

The static magnetic structure of the film with themagnetic parameter listed in the above section is shownin Fig. 1. It shows the typical magnetic ripple structure, thehexagonal grain formed in the large coupled ellipsoid as aresult of the exchange interaction and random anisotropyinteraction. The length of the major axis of the ellipsoidwas nearly the width of the film, and the coupled length ofthe film in y-axis was nearly 70 nm.

From the linear ripple theory [4], the magnetic dispersion

angle

ffiffiffiffiffiffij2

qof the film is

ffiffiffiffiffiffij2

q¼

1

2�ffiffiffi2p

pffiffiffi24p SðMs

ffiffiffidpÞ�1=2ðAKuhÞ�3=8, (1)

where h ¼ HMs/2Ku+1, S is the structure constant of thefilm, Ms is the saturation magnetization, d is the filmthickness, A is the exchange constant, Ku is the uniaxialanisotropy constant and H is the external field. In thiswork, the magnetic dispersion angle is the root-mean-square of the angle between the magnetization and they-axis.

The magnetic dispersion angle

ffiffiffiffiffiffij2

qof the film from the

static magnetic structure shown in Fig. 1 was 2.71. Thevalue of the magnetic dispersion angle from the linearmagnetic ripple theory (Eq. (1)) was 2.51. They had a littledeviation. The reason of the deviation could be attributedto the limited grain number of the model.Next, the relationship between the magnetic dispersion

angle and the corresponding magnetic parameters of thefilms was discussed.

3.1.1. Variation of the grain size

The relationship between the magnetic dispersion angleand the grain size of the film with a thickness of 16 nm isshown in Fig. 2. The magnetic dispersion angle increasedwith the increase of the grain size. This result wasconsistent with the magnetic ripple theory. However, whenthe grain size had a bigger value, the simulation magnetic

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Fig. 3. Relationship between magnetic dispersion angle and film thick-

ness. The magnetic parameters were: Ms ¼ 2T, A ¼ 1.0� 10�11 J/m,

K1 ¼ 20 000 J/m3, R ¼ 17.32 nm and d ¼ 10–20 nm.

Fig. 4. Relationship between magnetic dispersion angle and the exchange

constant. The magnetic parameters are: Ms ¼ 2T, A ¼ 0.5� 10�11

–1.0� 10�10 J/m, K1 ¼ 20 000 J/m3, R ¼ 8.66 nm and d ¼ 10 nm.

Fig. 5. Relationship between magnetic dispersion angle and the local

anisotropy constant. The magnetic parameters are: Ms ¼ 2T, A ¼ 1.0�

10�11 J/m, K1 ¼ 10 000–50 000 J/m3, R ¼ 8.66 nm and d ¼ 10 nm.

Fig. 2. Relationship between magnetic dispersion angle and grain

size. The magnetic parameters are: Ms ¼ 2T, A ¼ 1.0� 10�11 J/m,

K1 ¼ 20 000 J/m3, R ¼ 8.66–25.98 nm and d ¼ 10 nm.

L. Yuan et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 1393–1397 1395

dispersion angle was bigger than the theory value. Thereason was that the exchange-coupled region of the filmwas reduced and the averaging phenomenon of localanisotropy was weaker when the grain size increased andclosed to the exchange length of the grain lex ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi

A=K1p

¼ 22:36 nm. This case was out of the linear rippletheory. So the magnetic dispersion angle sharply increased.

3.1.2. Variation of the film thickness

The film thickness dependence of the magnetic disper-sion angle of the film with a grain size of 17.32 nm is shownin Fig. 3. The magnetic dispersion angle decreased with theincrease of the film thickness. Moreover, an exponentialrelationship between them was observed, which was inagreement with the expectation result from the rippletheory. However, when the film thickness had a smallervalue, the simulation value had a bigger value than thetheoretical one. The reason was that the major axis lengthof coupled ellipsoid in film decreased as a result of thedecrease of the transverse stray field, when the filmthickness was decreased. The decrease of grain number ina coupled ellipsoid sharply increased the magnetic disper-sion angle, and the law had a deviation from the linear law.

3.1.3. Variation of the exchange constant

The magnetic dispersion angle as a function of theexchange constant of the film with a grain size of 17.32 nmand a film thickness of 16 nm is shown in Fig. 4. The

magnetic dispersion angle

ffiffiffiffiffiffij2

qwas nearly proportional to

A�3/8. When the exchange constant increased, the exchangelength and the grain number in a coupled region increased.The average anisotropy constant was smaller. The magne-tization at the end of the film in x-axis was tended to y-axisas a result of the volume demagnetizing field. So themagnetic dispersion angle sharply decreased.

3.1.4. Variation of the local anisotropy constant

The simulated and theory curves of the magneticdispersion angle vs. the local anisotropy constant of the

film with a grain size of 17.32 nm and a film thickness of16 nm are shown in Fig. 5. The magnetic dispersion anglelinear increased with the increase of the local anisotropyconstant and that was in good agreement with the linearripple theory.

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Fig. 8. Relationship between the effective damping factor and the

magnetic dispersion angle.

L. Yuan et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 1393–13971396

3.2. Dynamic magnetic property

When an external time-dependent excitization field wasadded to the equilibrium magnetization configuration inx-axis, the motion of the spin in the film was the Larmorprocession with damping. From the magnetization re-sponse, the magnetic spectra could be obtained using thefast Fourier transform (FFT). For the film with grain sizeof 8.66 nm and film thickness of 16 nm, the response of x

component of the normalized magnetization of the film m̄x

was obtained and is shown in Fig. 6. The magnetic spectraof the film from magnetization response using FFT and thefitting curve are shown in Fig. 7. The fitting curve wasobtained from Eq. (2), which is deduced from LLGfunction, as the magnetic structure of a film is singledomain:

mxx ¼omðo0 þ omðNy �NzÞÞ

o2fmr � o2

þ 1,

o2fmr ¼ ðo0 þ omðNy �NzÞÞðo0 þ omðNx �NzÞÞ, ð2Þ

where om ¼ m0gMs, m0 is permeability of vacuum, g is thegyromagnetic ratio of 2.21� 105m/As, o0 ¼ gHk+iao, Hk

is anisotropy field, o is the angular frequency of theexternal field, a is the Gilbert damping parameter and Nx,

Fig. 6. The external field and magnetization resonance on time

domain. The magnetic parameters are: Ms ¼ 2T, A ¼ 1.0� 10�11 J/m,

K1 ¼ 20 000 J/m3, R ¼ 8.66 nm and d ¼ 16 nm.

Fig. 7. Typical magnetic spectrum of the film and the fitting data from

Eq. (2) with an effective damping factor of 0.039.

Ny, Nz are the three diagonal components of thedemagnetizing factor tensor, respectively. Eq. (2) couldwell describe the dynamic magnetic property of the signal-phase film, though the magnetic structure was a ripplestructure installed of a single domain. But the fitting dataof damping factor had a bigger value than the settingdamping factor of 0.02. The reason was the magneticspectra broadening caused by the magnetic inhomogeneousin the single-phase film [7]. The relationship betweeneffective damping factor and magnetic dispersion anglefor the sample considered in the above section with thesame demagnetization factor is shown in Fig. 8. Theeffective damping factor was approximately linearlyincreased with the increase of the magnetic dispersionangle. This was in agreement with Chechenin’s theoryresult [8].

4. Conclusion

The static magnetic structure of the single-phase filmswas calculated using micromagnetic method, and thetypical micromagnetic ripple structure was obtained. Thus,the magnetic dispersion angle was analyzed from themicromagnetic structure. The simulation results showedthe right exponentials relation between the magneticdispersion angle and the corresponding magnetic para-meters, and were consistent with the ripple theory. Thedifference between the simulation value and the theory wassummed up by the following: (1) The local anisotropyconstant of the grains in the model was set to a big valuefor the finite model. This led to the grain number in thecoupled region being smaller than the actual case. So whensome magnetic parameter of the film was changed, thecoupled region had a large change in comparison to theactual case. Then the static property of the model was outof the linear region of the ripple theory when somemagnetic parameters were assumed. (2) The volume

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demagnetization energy of the film also brought aboutsome errors. The microwave permeability of the filmswas also obtained, and was well described by permeabi-lity equation deduced from the LLG function when thefilm was considered a single domain. The fitting dataof effective damping factor were approximately propor-tional to the magnetic dispersion angle. It consisted ofChechenin’s theory.

Acknowledgments

The financial supports from National Natural ScienceFoundation of China (Grant no. 50371029, 50771047), NewCentury Excellent Talents in University (Grant no. NCET-

04-0702) and Elitist in Natural Science Foundation of HubeiProvince (Grant no. 2005ABB002) are acknowledged.

References

[1] T. Suzuki, C.H. Wilts, J. Appl. Phys. 39 (1968) 1151.

[2] Z. Liu, D. Shindo, S. Ohnuma, H. Fujimori, J. Magn. Magn. Mater.

262 (2003) 308.

[3] E.C. Stoner, E.P. Wohlfarth, Philos. Trans. R. Soc. A 250 (1955) 975.

[4] H. Hoffmann, IEEE Trans. Magn. 4 (1968) 32.

[5] D.V. Berkov, N.L. Gorn, Phys. Rev. B 57 (1998) 14332.

[6] W. Scholz, J. Fidler, T. Schrefl, D. Suess, R. Dittrich, H. Forster,

V. Tsiantos, Comp. Mater. Sci. 28 (2003) 366.

[7] R.D. McMichael, D.J. Twisselmann, Phys. Rev. Lett. 90 (2003)

227601.

[8] N.G. Chechenin, Phys. Sol. State 46 (2004) 479.