Domain structure screening of a local magnetic inhomogeneity

M.L. Akimov, P.A. Polyakov, N.N. Usmanov

Moscow State University, Leninskie gory, Moscow, 119992 Russia

A magnetic domain ordering is known to occur in ferromagnetic materials and to promote decrease of the sample’s magnetostatic energy [1]. This phenomenon can be considered as screening external and intrinsic magnetic fields by the magnetic sample. In this paper, an efficiency of screening a magnetic field of a cylinder-shaped magnetic inhomogeneity by a stripe domain structure is analyzed.

We used a magnetic film of composition with orientation (210) to obtain a static domain configuration experimentally. The parameters of the selected sample were the following: is the thickness of the film, is the inclination of the easy direction, is the saturation magnetization, is the dimensionless Hilbert damping parameter determined from the FMR line width, is the field of the orthorhombic anisotropy.

A photo of an obtained domain structure is presented at fig. 1. The width of a stripe domain (the dark one at fig. 1) containing a bubble domain equals 16 μm. The width of adjacent stripe domains (light ones) equals 10 μm. The mean radius of the bubble domain is 6.75 μm.

Let us consider a stripe domain of width in presence of a cylinder-shaped inhomogeneity of radius R on a side of it. The origin of coordinates is in the center of the inhomogeneity. The stripe domain is located along the x coordinate axis in an infinite film of thickness h. The z coordinate axis is directed perpendicularly to the film’s plane, and the y coordinate axis is directed perpendicularly to the stripe domain walls. A magnetostatic stray field of the inhomogeneity distorts the stripe domain’s shape and leads to a dependence of its width on x coordinate.

Assume that functions and determine the domain walls’ curves. After calculating variational derivatives of a magnetostatic energy functional and , we yield a system of integral equations with respect to functions and .

The equations can be linearized for comparatively small deformations of domain walls.Expressing functions and in terms in the range of integration, we yield a system of linear integral equations of a convolution kind which can be solved with a Fourier transformation method.

Fig. 1. A bubble domain in a stripe domain structure.

After Fourier transformations, we obtain the following expressions for distortion shapes of domain walls of the stripe domain [2]:

, (1)

(2)

(3)

(4)

, (5)

, (6)

, (7)

(8)

(9)

Fig. 2. Domain walls computed by formulas (1)–(9).

Based on expressions (1)–(9), we plotted theoretical curves which describe the distortion shape of domain walls of a stripe domain in the presence of a cylinder-shaped magnetic inhomogeneity of radius R = 6.75 μm on a side of it (fig. 2).

The theoretical computation of a maximum domain wall bend by formulas (1)–(9) (fig. 2) for parameters which fit the experimental data (w = 10 μm is the width of the stripe domain, h = 13 μm is the film’s thickness, R = 6.75 μm is the mean radius of the bubble domain, c = 8 μm is the distance between the center of the bubble domain and the nearest stripe domain wall) gives values 5.44 μm and 0.71 μm. The maximum stripe domain wall bend values obtained experimentally for the above parameters are 3.9 μm and 1.1 μm. Therefore, the values calculated by formulas (1)–(9) conform the experimentally obtained stripe domain wall bend values.

It follows from the obtained results and graphs shown at fig. 1 and 2 that the field of the cylinder-shaped inhomogeneity influences significantly upon the nearest domain wall only. The next one curves much smaller. Physically, it means that the magnetic field of the inhomogeneity is almost totally screened by a magnetic charge induced by the curvature of the nearest domain wall (see fig. 1). This phenomenon was also observed experimentally in [3]–[5]. Thus the presented research shows that a stripe domain in a magnetic film can effectively screen a magnetostatic field of a magnetic inhomogeneity with a slight distortion of the domain’s shape.

References

[1] L. D. Landau and E. M. Lifshitz. Electrodynamics of continuous media. Pergamon, Oxford, 1984.

[2] M. L. Akimov and P. A. Polyakov. Kvazilokalniy kharakter vliyaniya polya magnitnoy neodnorodnosti na polosovuyu domennuyu strukturu (Quasilocal character of influence of a field of a magnetic inhomogeneity on a stripe domain structure), Vestnik MGU. Ser. 3. Fizika. Astronomiya. (Bulletin of the Moscow University. Ser. 3. Physics, Astronomy), 2004, № 2, p. 47-50 (in Russian).

[3] M. L. Akimov, Yu. V. Boltasova and P. A. Polyakov. The Effect of a Pointlike Asymmetric Laser-Induced Action on a Magnetic Film Medium. Radiotekhnika i Elektronika, 46 (2001), p. 504 (in Russian) [Translation: J. Comm. Tech. Electronics, 46 (2001) p. 469].

[4] M. L. Akimov, P. A. Polyakov and N. N. Usmanov. A Mixed Domain Structure in Ferrite–Garnet Films. ZhETF, 121 (2002), 347 (in Russian) [Translation: J. Exp. Theor. Phys., 94(2) (2002), p. 293].

[5] A. S. Logginov, A. V. Nikolaev, E. P. Nikolaeva and V. N. Onishchuk. Modification of the Domain Wall Structure and Generation of Submicron Magnetic Formations by Local Optical Irradiation. ZhETF, 117 (2000), 571 (in Russian) [Translation: J. Exp. Theor. Phys., 90 (2000), p. 499