2130 IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 9, SEPTEMBER 2008

Micromagnetic Simulation of a Magnetic Film With Surface RoughnessLin Yuan, Jian-Jun Jiang, Zhongyou Wang, Bin Tian, and Huahui He

Department of Electronic Science and Technology, Huazhong University of Science and Technology, Wuhan 430-074, China

We have simulated micromagnetically the static magnetic structure and magnetic spectrum of a magnetic film with surface roughness,using a finite stripe film model. We found that the rough surface can lead to a ripple magnetic structure, which results in increasing thedamping factor of the film, and that the damping factor increases with the increase of the root-mean-square roughness. For a particularfilm with a given rough surface, the damping factor first decreases because of a decrease of the magnetic dispersion, then increasesbecause of the surface-roughness-induced demagnetizing effect with increasing the external field.

Index Terms—Damping factor, micromagnetic, surface roughness.

I. INTRODUCTION

THE soft magnetic thin films used in high-frequency fieldhave been extensively studied because of their potential

impact on electromagnetic wave absorbing, wireless com-munication, and magnetic recording industry. As the filmthickness is reduced, the magnetic properties are expected to bestrongly influenced by surfaces roughness. For static magneticproperties, the magnetic domain wall thickness, domain size,coercivity [1], in-plane demagnetizing factor [2], [3], and mag-netic anisotropy [4] are influenced by the surface roughness.Simulations by Zhao and Liu et al. [5], [6] using the MonteCarlo method have shown that the surface magnetic orderingtemperature decreases toward the bulk Curie temperature withincreasing roughness, and a strong anisotropy is induced by sur-face roughness. For dynamic magnetic properties, Craus et al.[7] study the influence of surface roughness on magnetizationdynamics of soft magnetic Fe–Zr–N thin films. The imaginarypart of the permeability shows a two-peak structure witha bigger root-mean-square (rms) roughness. Rantschler et al.[8] find that increasing the interface roughness increases thedamping factor from ferromagnetic resonance (FMR) experi-ment. In this work, the magnetic film with surface roughnessusing the micromagnetic method is extensively investigated.First, the static magnetic structure of the film is obtained and themagnetic spectrum is further calculated. Then, the relationshipbetween damping factor and the rms roughness is analyzed.Next, for a certain rms roughness the damping factor of thefilm as a function of the external field is presented. Finally, themicrowave permeability spectra for different correlation lengthof the rough surface are examined.

II. MICROMAGNETIC THEORY

In micromagnetics, the magnetic material is consideredfrom a macroscopic point of view and the magnetic polar-ization is assumed to be a continuous function in space. Themagnetodynamics of the magnetization is described by the

Digital Object Identifier 10.1109/TMAG.2008.2000942

Landau– Lifshitz–Gilbert (LLG) equation

(1)

(2)

where is the magnetization vector, is the time, is theGilbert damping parameter, is the gyromagnetic ratio of

m/As, is the saturation magnetization, isthe effective magnetic field, is the external magnetic field,

is the demagnetization field, is the exchange constant,is the first-order magnetocrystalline anisotropy constant,

and is the easy axis.The dynamic permeability is calculated by applying an ex-

ternal time-dependent excitation field to a sample with an equi-librium magnetization configuration. Actually, there are someexternal magnetic forms, such as in [9] and [10]. In this study,the external magnetic field has the same form as [9] for themodels are similar

(3)

where is the amplitude of the external field of 1000 A/m.Then the permeability is given by

(4)

where are the Fourier transform of the magne-tization resonance and the external field, respectively. And isthe direction of the external field. Calculations in this work areperformed using the GNU codes magpar [11].

III. MODELING FORMULATION

The discussed model in this study is a finite stripe film, witha dimension of 3000 nm 15 000 nm 100–140 nm, as shownin Fig. 1. The film is divided into 197 072 tetrahedral finite el-ements. There are five nodes in the direction of film thickness.In the middle of the film the stripe is considered as a material,which has saturation magnetization of 2 T, magnetocrystallineconstant of 4000 J/m , easy axis of -axis, exchange con-stant of J/m, and damping factor of 0.01. At theend of the film in -axis, if is small, the magnetization willbe noncollinear for the shape-demagnetizing energy, which will

0018-9464/$25.00 © 2008 IEEE

YUAN et al.: MICROMAGNETIC SIMULATION OF A MAGNETIC FILM WITH SURFACE ROUGHNESS 2131

Fig. 1. (Color online) The static magnetic structure of the film with rms rough-ness of 4.58 nm and � = 300 nm. (a) Sketch map of the model and the magneti-zation distribution of the rough surface. The color level denotes the x componentof the normalized magnetizationm . (b) The magnetization distribution in sliceof x = 0. � is defined by the distance between five nodes in y-axis.

bring extra ferromagnetic resonance peak or FMR linewidth.So the magnetocrystalline anisotropy constant and the easy axisthere are set to J/m and parallel to the -axis, respec-tively, to pin the magnetization there in -axis.

In one side of the film, we add a random fluctuation to theextension of the film in -direction with amplitude of 10 nmwhich leads to a rough surface with rms roughness of 4.58 nm.And the correlation length (defined in Fig. 1) is 300 nm. For thesake of adjusting the rms roughness, the scale of the roughnesssurface is changed while the width and the length of the film arefixed. The scale of the model in -axis and -axis are changedsimilarly in order to adjust , hence, the rms roughness and filmthickness are fixed.

IV. RESULTS AND DISCUSSION

Fig. 1 shows the micromagnetic structure of the film with rmsroughness of4.58 nm and of300 nm.FromFig. 1(a), the magne-tization in-plane exhibits a ripple structure. On the rough surface,the magnetization is parallel to the surface, as shown in [4], [12].And it is the reason for the ripple magnetic structure. In addition,thereisnodomainwall inthemagnetizationofthefilmbutnearlyasingle domain structure, and the magnetization in one tetrahedralmesh slowly changes. Therefore, the total free energy calculationof film is accurate, though the mesh size is big.

Based on the static magnetic structure, the imaginary part ofthe permeability shows a large peak as shown in Fig. 2. Usingthe curve fit method the fitting result shows the magnetic spectracan be well fit by (5). And as a result of the curve fit the dampingfactor can be obtained [13]. The following formula is deducedfrom LLG equation as a magnetic film is single-domain

(5)

Fig. 2. (Color online) The imaginary part of the permeability of the film withdifferent rms roughness.

TABLE IMAGNETIC DISPERSION ANGLE IN x-AXIS m ; z-AXIS m AND

DAMPING FACTOR VERSUS DIFFERENT SAMPLE WITH DIFFERENT

SURFACE TOPOGRAPHY AND NEARLY EQUAL RMS ROUGHNESS

where is the angular frequencyof the external field, is the anisotropy field,are the three diagonal components of the demagnetizing factortensor, respectively.

To investigate the influence of rough surface topography onthe permeability spectra, the models with different rms rough-ness of 4.51–4.58 nm and of 300 nm are generated by fourrandom processes. The results are shown in Table I. The mag-netic dispersion angle [14] and damping factor have a slightchange for different samples. In the next content, the rough sur-face topography of the film is the same as that of sample 1.

The imaginary part of permeability of the film with dif-ferent rms roughness is shown in Fig. 2. The peak frequencies of

increase with the increase of the rms roughness, which is dueto the increase of the film thickness and the decrease of the de-magnetizing factor in -axis. From the static magnetic structure,the magnetic dispersion angle of the film is obtained and shown

in Fig. 3. The magnetic dispersion angle in-plane and that

out-of-plane increase with increasing rms roughness. Itcould interpret by the increase of the average slope of the rough

surface. And has a bigger value than . The reason isthat the magnetization deviation from -axis in-plane just needsto conquer the transverse demagnetizing energy, but the devia-tion from -axis out-of-plane needs to conquer the shape demag-netizing energy in -axis. The former is easier than the latter. So

has a big value. The damping factor of the film increaseswith the increase of the rms roughness. This result is in agree-ment with the experiment in [8]. Herein, we interpret it by theincrease of the magnetic dispersion of the film as the rms rough-ness of the film increases.

2132 IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 9, SEPTEMBER 2008

Fig. 3. The magnetic dispersion angle and the damping factor of the film withdifferent rms roughness.

Fig. 4. (Color online) The magnetization distribution in slice of x = 0 withdifferent external field, (a) 0 kA/m, (b) 100 kA/m, (c) 500 kA/m. Increasing theexternal field spin the rough surface magnetization to y-axis.

When an external field is added to the film in -axis, the staticmagnetic structure and permeability of the film with rms rough-ness of 9.17 nm and of 300 nm is estimated. Fig. 4 showsthe static magnetic structure with different external field. Therough surface magnetization is also parallel to the rough surfaceat external field of 0 kA/m. And the rough surface magnetiza-tion tends to rotate in -axis with increasing the external field.

This can also be seen in Fig. 5. decreases with the increase

of external field. Fig. 5 also shows , damping factor, and

the demagnetizing energy. decreases more sharply than

. When the external field increases, the damping factor de-creases initially for the decrease of the magnetic dispersion. Atthe same time, the demagnetizing energy increases for the in-crease of the demagnetizing factor in-plane caused by the roughsurface and the rotating of the magnetization to -axis. Whenthe external field increases continually, the magnetic dispersionangle decreases and the demagnetizing energy increases. Butthe damping factor increases. It could interpret by the FMRbroadening for the random, enhanced stray field in rough sur-face when the magnetization tends to -axis. That could be areason for explaining the experiment in [15], [16]. For the filmwith rms roughness of 4.58 nm and of 300 nm, the relationshipbetween damping factor and external field is investigated and

Fig. 5. The dispersion angle, damping factor, and demagnetizing energy of thefilm with rms roughness of 9.17 nm and � of 300 nm as a function of the externalfield.

Fig. 6. The dispersion angle, damping factor, and demagnetizing energy of thefilm with rms roughness of 4.58 nm and � of 300 nm as a function of the externalfield.

Fig. 7. The magnetic dispersion angle in-plane (Mx), out-of-plane (Mz), anddamping factor versus correlation length.

shown in Fig. 6. The film has a small magnetic dispersion com-pared with the film with roughness of 9.17 nm, so the externalfield where the damping factor begins to increase is smaller.

Finally, the permeability spectra of the film with different cor-relation length and rms roughness of 4.58 nm are investigated.Fig. 7 shows the magnetic dispersion angle and the dampingfactor of film. The magnetic dispersion angle increases with thedecrease of the correlation length. When the magnetic disper-sion is large of 20.3 , the magnetic spectra show a big dampingfactor instead of a two-peak structure [7] and are in agreementwith (5). The two-peak structure magnetic spectra probably ap-pear in the more actual calculation when the random anisotropyand rough surface are considered simultaneously. But that needs

YUAN et al.: MICROMAGNETIC SIMULATION OF A MAGNETIC FILM WITH SURFACE ROUGHNESS 2133

huge computation time for the different scale between the grainsize of about 10 nm and the roughness correlation length of hun-dreds of nanometers.

V. CONCLUSION

The static magnetic structure and the magnetic spectra of thefinite size film with rough surface are investigated by micro-magnetic method. The surface magnetization is parallel to thesurface plane which brings the ripple structure. The dampingfactor is in proportion with the average slope of the rough sur-face. For a certain film, the damping factor of the film decreasesfirst, and then increases with increasing external field. That is theresult of the competition between the decrease of the magneticdispersion angle and the increase of surface-roughness-inducedrandom stray field with increasing external field. The formerleads to decrease the damping factor and the latter borders theFMR linewidth.

ACKNOWLEDGMENT

This work was supported by National Natural Science Foun-dation of China under Grant 50371029, 50771047, New CenturyExcellent Talents in University under Grant NCET-04-0702,and Elitist in Natural Science Foundation of Hubei Provinceunder Grant 2005ABB002.

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Manuscript received July 11, 2007; revised May 18, 2008. Published August20, 2008 (projected). Corresponding author: J. Jiang (e-mail: jiangjj@mail.hust.edu.cn).