Micromagnetic simulation of effect of stress-induced anisotropy in soft magnetic thinfilmsDaniel Z. Bai, Jian-Gang Zhu, Winnie Yu, and James A. Bain

Citation: Journal of Applied Physics 95, 6864 (2004); doi: 10.1063/1.1667445 View online: http://dx.doi.org/10.1063/1.1667445 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/95/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Kerr microscopy studies of the effects of bending stress on galfenola) J. Appl. Phys. 115, 17E310 (2014); 10.1063/1.4865468 Direct observation of an anisotropic in-plane residual stress induced by B addition as an origin of high magneticanisotropy field of Ru/FeCoB film J. Appl. Phys. 107, 09A323 (2010); 10.1063/1.3350899 On annealing-induced amorphization and anisotropy in a ferromagnetic Fe-based film: A magnetic and propertystudy Appl. Phys. Lett. 88, 012510 (2006); 10.1063/1.2161938 Soft magnetic properties of Co–Fe–Zr–B–Al–O films J. Appl. Phys. 91, 8450 (2002); 10.1063/1.1447521 Barkhausen noise in soft amorphous magnetic materials under applied stress J. Appl. Phys. 85, 5196 (1999); 10.1063/1.369122

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Micromagnetic simulation of effect of stress-induced anisotropyin soft magnetic thin films

Daniel Z. Bai,a) Jian-Gang Zhu, Winnie Yu, and James A. BainData Storage Systems Center, Department of Electrical and Computer Engineering, Carnegie MellonUniversity, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213

~Presented on 7 January 2004!

The effect of stress in soft magnetic thin films, in particular, on the in-plane anisotropy, has beenstudied, based on the analysis of the magnetoelastic energy associated with the stress. The easy-axisdirections and the effective anisotropy constants have been identified, as functions of the stress, themagnetostriction coefficients of the material, and the growth texture of the film. The magnetoelasticenergy has been combined into an existing micromagnetic model to simulate the magnetization ofthin films with various materials, stress states, and growth textures. Simulation results of staticmagnetic domain structures are in agreement with the theoretical predictions. ©2004 AmericanInstitute of Physics.@DOI: 10.1063/1.1667445#

I. INTRODUCTION

Stress in a soft magnetic thin film could greatly affectthe magnetic properties of the film. Theoretical1 andexperimental1,2 studies have shown the stress-induced per-pendicular anisotropy and consequently the stripe domains insoft magnetic thin films. In real devices such as a recordinghead, a particular in-plane anisotropy easy axis is often de-sired and it could be achieved by careful control of thestress.3 The stress-induced anisotropy depends not only onthe magnitude and direction of the stress, but also on themagnetostriction coefficients of the materials, as well as onthe crystallographic growth texture of the film. In this article,the stress-induced in-plane anisotropy in soft magnetic thinfilms will be first derived, based on an analytical analysis ofthe magnetoelastic energy. Micromagnetic models includingthe stress effect for recording media with hexagonal crystalstructure have been developed previously.4,5 Here, we willextend a micromagnetic model6 to include this effect for cu-bic structured soft magnetic materials with different crystal-lographic textures. Simulation results of static magnetic do-mains in thin-film soft magnetic elements under variouscombinations of stress, materials, and growth textures willthen be presented.

II. THEORY

In this section, the stress-induced in-plane anisotropywill be derived analytically. The derivation is similar to thatby Zou et al.1 Considered here is a soft magnetic polycrys-talline thin film with a cubic crystal structure and magneto-striction coefficientsl111 and l100. Although there is a netcontribution to the total perpendicular anisotropy from thecrystal anisotropy constantsK1 and K2 for textured films,1

the net in-plane crystal anisotropy, which is the average overall the grains randomly orientated in the plane, will be zero,therefore it is not considered. For a cubic structured single

crystal grain with uniform magnetizationM under a uniaxialstresss, the magnetoelastic energy density is given by

Es52 32l100s~a1

2g121a2

2g221a3

2g32!

23l111s~a1a2g1g21a2a3g2g31a3a1g3g1!,

~1!

where (a1 ,a2 ,a3) and (g1 ,g2 ,g3) are the directions ofMand s, respectively, both with respect to the three primarycubic crystal axes of the grain. However, the directions of thestress and the magnetization are usually known only withrespect to the film, specifically in the derivation here, bothwill be assumed in the plane. Therefore, two coordinate sys-tems are needed: the global systemoxyz, which is fixed tothe film and the local systemo8x8y8z8, which is local toeach of the grains in the film. The latter has three Cartesianaxes of the@100#, @010#, and @001# directions of the localcrystal grain. Shown in Fig. 1 is an illustration of the twocoordinate systems as well asM ands. In general,

S xyzD 5AS x8

y8z8D 5S l 1 m1 n1

l 2 m2 n2

l 3 m3 n3

D S x8y8z8D , ~2!

a!Electronic mail: zbai@andrew.cmu.eduFIG. 1. The two coordinate systems and the relative orientation of the stressand the magnetization of a grain in the thin film.

JOURNAL OF APPLIED PHYSICS VOLUME 95, NUMBER 11 1 JUNE 2004

68640021-8979/2004/95(11)/6864/3/$22.00 © 2004 American Institute of Physics

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whereA is the transformation matrix relating the coordinatesof a vector in the two systems. The elements ofA are definedas follows:

l 15cosc cosw2cosu sinc sinw,

l 252cosc sinw2cosu sinc cosw,

l 35sinu sinc,

m15sinc cosw1cosu cosc sinw,

m252sinc sinw1cosu cosc cosw,

m352sinu cosc,

n15sinu sinw,

n25sinu cosw,

n35cosu, ~3!

whereu, c, andw are the three Euler angles through whichthe coordinate basiso8x8y8z8 is transformed intooxyz.

To find the stress-induced anisotropy constant, one needsan expression of the average magnetoelastic energy densityas a function ofv, the angle betweenM and s. Withoutlosing generality,s can be assumed in the direction ofy, asshown in Fig. 1. Therefore, in the global coordinates,s5(0,1,0), andM5(sinv,cosv,0). By transformingM ands into the local systems, applying Eq.~1!, and averagingover all possible grain orientations within the texture con-straints, the average magnetoelastic energy density will havethe following form:

Es5Ks sin2 v1const. ~4!

For a polycrystalline film without texture,u varies ran-domly ;~0,p! and c and w;(0,2p) from one grain to an-other.

For 100 texture, the film normalz is @001# direction,u50, c andw are random;~0, 2p!. For 110 texture, the filmnormal z is @110# direction,u5p/2, c53p/4, andw is ran-dom ;~0, 2p!. For 111 texture, the film normalz is @111#direction,u5sin21(A2/3), c53p/4, andw is random;~0,2p!, whereKs is the stress-induced in-plane anisotropy con-stant, which is texture dependent.

The results of the stress-induced in-plane anisotropy fordifferent textures are summarized in Table I. It can be seen

that, except for 100 texture,l111 has more weight thanl100

in determining the strength of the anisotropy. In the casel1115l1005l, the anisotropy constant becomesKs5 3

2ls,independent of the film textures, which is consistent with theresult of an isotropic magnetostrictive material.7 Also notethat, depending on the sign of the stress~positive is tensileand negative is compressive! as well asl111 andl100 of thematerial, the value ofKs in Eq. ~4! can be either positive ornegative. The easy axis is along the direction ofs for theformer case while orthogonal tos in the plane for the lattercase. In other words, changing the stress from tensile to com-pressive or vice versa and maintaining the magnitude of thestress will turn the easy axis to the perpendicular directionwith equal magnitude ofKs . As an example, Table I alsolists the in-plane anisotropy constants for Fe65Co35 alloy andFe thin films, respectively, both undergoing a 1 GPa uniaxialtensile stress, with parametersl1115103.731026 and l100

517.531026 for Fe65Co35, and l111522131026 andl10052131026 for Fe.

FIG. 2. The magnetic domains of a Fe65Co35 thin-film element. From top tobottom:~a! stress free,~b! horizontal compressive stress,~c! vertical tensilestress, and~d! horizontal compressive stress. The stress magnitudes are all 1GPa. The film textures are 111 for~a!–~c! and random for~d!.

TABLE I. Effective in-plane anisotropy constants for different textures.

Texture Ks Fe65Co35 Fe

100sS34 l1111

3

4l100D 90.9 0

110sS 15

16l1111

9

16l100D 107 27.9

111sS l1111

1

2l100D 112 210.5

RandomsS 9

10l1111

3

5l100D 103 26.3

Note: The numbers shown for the two materials are theKs values under a 1GPa uniaxial tensile stress, in unit of 104 erg/cm3.

6865J. Appl. Phys., Vol. 95, No. 11, Part 2, 1 June 2004 Bai et al.

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III. MODELING

Based on the analysis just presented, we have combinedthe magnetoelastic energy term into an existing micromag-netic model6 to simulate the effect of stress on thin-film mag-netization. In the model, a thin-film element is discretizedinto 10 nm310 nm310 nm cubes, each assumed a crystal-lite with crystallographic orientation following the desiredtextures. Therefore, there is a unique coordinate transformmatrix associated with each of the grains throughout the en-tire film. The stress and magnetization are first transformedinto each of the individual local coordinate systems in orderto use Eq.~1!. All subsequent computations of the magneti-zation are then done in the local system. The final result willbe inversely transformed to the global system. In all cases,the stress is assumed unchanged throughout the simulation;the initial magnetization state is assumed to be two equal-sized antiparallel domains~top and bottom half! along thelong axis separated by a 180° domain wall along the centerline; and the size of the film element simulated is 1.28mm30.64mm with thickness of 80 nm.

Figure 2 shows the simulation results of static magneticdomains of a Fe65Co35 thin film element under differentstress states. When the film is stress free, the magnetic do-mains are fully determined by minimizing the demagnetiza-tion energy, thus forming the closure domain structure@Fig.2~a!#. Under a 1 GPa uniform tensile stress along the verticaldirection, the anisotropyKs is able to turn the major do-mains into the vertical direction@Fig. 2~c!#. As mentionedearlier, a 1 GPa uniform compressive stress in the horizontaldirection should have the same effect, and this is confirmedin Fig. 2~b!. As shown in Table I, the effect of texture is notsignificant onKs for Fe65Co35. As an example, Fig. 2~d!shows the domains of a random textured film, which is al-most identical to Fig. 2~b!.

The situation for Fe is, however, quite different, due tothe different magnetostriction coefficients, as seen in Table I.The negative sign ofKs indicates the opposite behavior ofthe anisotropy to that of Fe65Co35. The magnitude ofKs isalso much smaller, therefore, to see the same effect, a much-higher stress is needed. Figures 3~a! and 3~b! are the simu-lation results for Fe, both consistent with Table I.

When an equal biaxial stress exists in the film, the effec-tive in-plane anisotropy is zero, since the total magnetoelas-tic energy will be independent of the direction of magnetiza-tion. However, if the stresses along the two perpendiculardirections are not equal, the effective anisotropy easy axiswill be along that of the stronger one, with the magnitudebeing the difference of the two.7 Figure 4 shows the magne-tization pattern of a Fe65Co35 film on a Si substrate with a0.6% uniform strain. The stresses are obtained from a finiteelement simulation, which are compressive with mean valuesof 2790 MPa along the long axis and2520 MPa along theshort axis. As a result, there is an effective easy axis alongthe short-axis direction. The relatively irregular shape of thedomains is due to the nonuniformity of the stress, and thesmaller volume of the majority domains is due to the lowerstress therefore weaker anisotropy.

Although the main interest here is the in-plane anisot-ropy, the micromagnetic model based on magnetoelastic en-ergy is not limited to only in-plane anisotropy. Actually, themaze-like stripe domains representative of perpendicular an-isotropy were also seen from the simulation results forFe65Co35 film elements with equal biaxial compressivestress, consistent with experimental observations.2

1P. Zou, W. Yu, and J. A. Bain, IEEE Trans. Magn.38, 3501~2002!.2J. Dho, Y. N. Kim, Y. S. Hwang, J. C. Kim, and N. H. Hur, Appl. Phys.Lett. 82, 1434~2003!.

3A. Hubert and R. Schafer,Magnetic Domains: The Analysis of MagneticMicrostructures~Springer, Berlin, 1998!.

4J.-G. Zhu, IEEE Trans. Magn.29, 195 ~1993!.5G. Khanna, B. M. Clemens, H. Zhou, and H. N. Bertram, IEEE Trans.Magn.37, 1468~2001!.

6C. Y. Mao, Ph.D. Dissertation, Carnegie Mellon University, Pittsburgh,PA, 2000.

7B. D. Cullity, Introduction to Magnetic Materials~Addison-Wesley, Read-ing, MA, 1972!.

FIG. 3. The magnetic domains of a Fe thin-film element under a 5 GPauniaxial horizontal tensile stress for~a! 110 texture and~b! 100 texture.

FIG. 4. The magnetic domains of a Fe65Co35 thin-film element on a Sisubstrate with a uniform strain of 0.6%.

6866 J. Appl. Phys., Vol. 95, No. 11, Part 2, 1 June 2004 Bai et al.

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