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A novel type of domain walls with two-dimensional magnetization distribution in magnetic triaxial films
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 298 (2006) 1–6 www.elsevier.com/locate/jmmm
A novel type of domain walls with two-dimensional magnetization distribution in magnetic triaxial films L.G. Korzunina, B.N. Filippova, F.A. Kassan-Oglya,, I.A. Chaikovskyb a
Institute of Metal Physics, Ural Division,Russian Academy of Sciences, ul. S.Kovalevskoi 18, Ekaterinburg 620219, Russian Federation b Ben-Gurion University of the Negev, POB 653, Beer-Sheva 84105, Israel Received 15 December 2004; received in revised form 28 January 2005 Available online 24 March 2005
Abstract On the basis of numerical minimization of total energy in magnetic triaxial ferromagnetic films with a surface of a (1 1 0)-type, we investigated two-dimensional structures of domain walls within a rigorous micromagnetic approach that takes into account all the main interactions including the dipole–dipole one. Novel two-vortex and three-vortex domain wall structures are established to exist. The profiles of domain wall structures and their stability regions are studied. r 2005 Elsevier B.V. All rights reserved. PACS: 05.45.a; 87.17.Aa; 75.70.Kw Keywords: A. Magnetic films and multilayers; D. Spin dynamics
1. Introduction It is theoretically [1,2] and experimentally [3–5] shown that in the magnetic uniaxial films of a certain thickness (more than 40 nm in Permalloy films) the equilibrium magnetization distribution in interdomain regions (domain walls) is asymmetric and vortex like. The walls with such internal structures are called asymmetric Bloch Corresponding author. Tel.: +7 343 3783528;
fax: +7 343 3745244. E-mail address: [email protected] (F.A. Kassan-Ogly).
walls, but for our purposes, it is better to call them one-vortex Bloch walls (1VB). Walls with similar structure were also found in magnetic triaxial films [3], whose surface coincides with a crystallographic plane of a (1 0 0) type (we will call them hereafter (1 0 0)-films). In this paper we show that similar walls should also exist in (1 1 0)-films. We also show that in such films novel types of two-vortex and three-vortex asymmetric walls (2VA and 3VA) can appear, whose existence is related to supplementary lowering of the anisotropy energy. The stability regions of such walls are found that depend on the thickness and magnetic parameters of the films. The studies are carried out within the
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.02.064
ARTICLE IN PRESS L.G. Korzunin et al. / Journal of Magnetism and Magnetic Materials 298 (2006) 1–6
2
framework of a model of two-dimensional magnetization M distribution using the numerical minimization of total energy that ensures the magnetization distribution in the films. We give the study results of domain wall profiles (the dependence of magnetization components averaged over the film thickness on the coordinate along the film surface) and show that the profiles allows one to experimentally reveal the walls of new types.
consists of the densities of the exchange energy wex ; the energy of magnetic anisotropy wa ; and the energy of magnetization wm in the magnetostatic field HðmÞ : wa ¼ K½m2x m2z þ m2y m2z þ 14ðm2y m2x Þ2 for ð1 1 0Þ-films. " 2 # qm 2 qm þ wex ¼ A , qx qy
ð2Þ
(3)
2. Statement of the problem
wm ¼ 12ðM HðmÞ Þ,
Let us consider magnetic triaxial films with surfaces of a (1 1 0) type ((1 1 0)-films). And let one of the easy axes (EMA) be parallel to the film surface and the other EMAs be tilted by the angles 45 1 and 135 1 to the surface (see Fig. 1). Let the 180 1 domain wall (DW) be in the V region in the shape of a parallelepiped elongated along the z-axis (the computation region). The DW separates the domains magnetized along and against the [0 0 1] direction. And let the orientation of M in the V region vary such that M ¼ Mðx; yÞ (two-dimensional model of magnetization distribution). The equilibrium distribution of M is obtained by the numerical minimization of the energy per unit length along the z-axis: Z Z W¼ w dx dz, (1)
where m ¼ M=M s ; M s is the saturation magnetization, A is the exchange parameter, and K is the anisotropy constant. The following boundary conditions are fulfilled at the boundaries of the V region:
D
where D is the cross-section of V with a plane z ¼ const: The D region has the (a b) dimensions, where b is the film thickness. The energy density w
Fig. 1. The geometry of problem.
(4)
mz jx¼a=2 ¼ 1; mx jx¼a=2 ¼ 0; myjx¼a=2 ¼ 0; qmy qmx j j ¼ 0; ¼ 0. ð5Þ qy y¼b=2 qy y¼b=2 The minimization of (1) is carried out numerically and it allows one to obtain the equilibrium configurations of a domain wall and its energy W 0 : The method of numerical minimization is described in a series of papers (see, for example, Refs. [1,6,7]). We used the network method with the maximal number of cells 150 50; and assumed the mesh size to be on the order of absolute single-domain size, that is approximately equal to 10 nm for iron. As for the size of the a computation region, we carried out numerical computations for the ratio a=b 2 ð1 6Þ at various fixed film thicknesses and established that the wall energy decreases with ratio a=b increasing. The wall energy difference between a=b ¼ 3 and a=bX4 does not exceed 0.2%. The films with b 2 ð40; 500Þ nm thickness are considered. The parameters A ¼ 2 106 erg=cm (exchange parameter), K ¼ 5:2 105 erg=cm3 (constant of cubic anisotropy), M s ¼ 1740 G (saturation magnetization) characteristic of iron were used as basal parameters. The criterion for termination of computations was chosen as that suggested in Ref. [8].
ARTICLE IN PRESS L.G. Korzunin et al. / Journal of Magnetism and Magnetic Materials 298 (2006) 1–6
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3. Results and discussion The numerical computations were carried out in a wide range of the saturation induction Bs ¼ 4pM s and the anisotropy field H a ¼ 2K=M s : The computation results can be displayed as projections of the m vectors onto the plane perpendicular to easy axis (the xy plane). The projections of m onto the plane are depicted by arrows of various lengths. Fig. 2 shows the typical asymmetric walls characteristic of the (1 1 0)-films. The dashed curves correspond to the mz ¼ const level lines (see Ref. [1]). At the middle line the change of the mz component sign takes place. It is seen that the projection of m onto the xy plane has a vortex-like character and one, two, and three vortex-like patterns are seen in Figs. 2a–c. The walls with the 2a–2c structures are the 1VB, 2VA, and 3VA, respectively. Below we show that different walls will be stable in the films with different thickness. The 1VB walls are stable in (1 1 0)-films with the thickness bpbc : For the films with basal parameters the critical thickness bc is approximately equal to 200 nm, and it varies with variation of magnetic parameters. At greater thickness, the 1VB walls become unstable and their structure rearranges into the structure of three-vortex walls (Fig. 2c). Nothing of the kind could be observed both in magnetic uniaxial and triaxial (1 0 0)-films at any studied film thickness. The existence of the 3VA structures in (1 1 0)-films is related to a possible decrease of the anisotropy energy, when there appear segments of walls with M orientations close to EMA directions tilted by the angles 451 and 1351 to the surface of the film. The appearance of such segments is impossible in thinner films because of the strong growth of exchange energy due to the formation of three vortices. Since the vortex character of a wall is not always pronounced, it is convenient to draw the surfaces of the mðx; yÞ components. The mz ¼ mz ðx; yÞ surfaces turn out to be similar to those in classical Bloch walls. However, in our case one, two, or three extrema appear on the mz ¼ mz ðx; yÞ surfaces, but they are not always pronounced. The mx ¼ mx ðx; yÞ and my ¼ my ðx; yÞ are much more representative. Fig. 3
Fig. 2. Typical asymmetric walls in (1 1 0)-films of 200 nm (a,b) and 250 nm (c) thickness. Films with basal parameters.
shows, as an example, the my ¼ my ðx; yÞ surfaces that clearly distinguish the M distribution in the 1VB and 2VA walls.
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L.G. Korzunin et al. / Journal of Magnetism and Magnetic Materials 298 (2006) 1–6
central line is S-shaped. The other feature of the 2b-type walls that distinguishes them from the ANWs is related to a well-pronounced character of two vortices, while the vortices are actually absent in the ANWs. From comparison of Figs. 2a and c it is seen that the 3VA wall could be considered as two superimposed 2VA walls with the mutual central vortex. Fig. 4 shows an example of the wall energy dependence on the film thickness, and quite a peculiar feature can be seen in it, namely, at the film thickness above 250 nm; the 2AB walls become stable. Nothing of the kind can be observed both in uniaxial and (1 0 0)-magnetic triaxial films. The available studies (see, for example, Ref. [2]) show that the ANWs are metastable in these films. The result obtained for basal films is valid in a wide range of the saturation induction, i.e. 0.5–2.2 T. Fig. 5 shows the dependence of DW energy on the anisotropy field H a for the films with b ¼ 300 nm:
Fig. 3. The my ðx; yÞ surfaces for 1VB (a) and 3VA (b) walls in (1 0 0)-films of 200 nm (a) and 250 nm (b) thickness. Films with basal parameters.
The 2VA walls (Fig. 2b) have features similar to the asymmetric Ne´el walls (ANW) predicted in [2] in uniaxial magnetic films. In particular, both the wall types are asymmetric relative to the xz surface and to the yz surface. The tangents on two nearsurface portions of the central lines of the ANWs and 2VA walls are like-tilted to the film surface (the dmx =dx derivatives have like signs), but the ANWs and 2VA walls are not quite identical. In the ANW, the dmx =dx derivatives have like signs all along the central line, whereas in the 2VA wall the sign changes with the depth of the film, and the
Fig. 4. The dependence of the 1VB and 3VA walls energy (open circles) and the 2VA (solid circles) in (1 1 0)-films on the film thickness. Films with basal parameters. The 1VB walls exist approximately up to b ¼ 200 nm: Circles are the numerical computation results; the curves are guides to the eye.
ARTICLE IN PRESS L.G. Korzunin et al. / Journal of Magnetism and Magnetic Materials 298 (2006) 1–6
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Fig. 5. The dependence of the 1VB and 3VA walls energy (solid circles) and the 2VA (open circles) on the field H a : Films with A and Bs basal parameters and b ¼ 300 nm: Circles are the numerical computation results; the curves are guides to the eye.
A wide range of H a where the 2VA walls are stable is seen in the figure, whereas at relatively small H a the 1VB walls are stable. Since the transition from 1VB to 2VA walls occurs gradually, in a narrow region to the left of the curves intersection point (Fig. 5) the 3VA walls can be stable. However, in this case their three-vortex structure is pronounced only slightly. The computations show that the stable 3VA wall also exists in a narrow region near H a ¼ 260 Oe in the films with b ¼ 300 nm: The 3VA walls are also stable at high H a ðX2000 OeÞ: However, in this case the distortion of their structure takes place, which calls for special investigations. In closing let us bring out the profiles of domain walls: Z 1 b mi mi ðxÞ ¼ mi dy; i ¼ ðx; y; zÞ. (6) b 0 An example of such profiles is shown in Fig. 6 for the 2VA and 3VA walls. They are seen to differ from one another, and from comparison with the data of [4] it also
Fig. 6. The profiles for 2VA (a) and 3VA (b) walls in the films with basal parameters; b ¼ 200 nm (a), b ¼ 250 nm (b).
follows that they differ from the 1VB wall profiles. This suggests that the described novel types of walls could be experimentally detected by the wall profile studies by electron microscopy, as in Ref. [4].
4. Conclusions In magnetic triaxial films with a surface of a (1 1 0)-type the novel types of asymmetric domain walls were shown to exist: two-vortex walls with an S-shaped central line, and asymmetric three-vortex walls. The structures of these walls substantially differ from the structure of classical Bloch and Ne´el walls and also from asymmetric Bloch [1] and Ne´el [2] walls.
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In the (1 1 0)-films the asymmetric Bloch walls of a type predicted in [1] can also exist. They are stable in the relatively thin films (less than 200 nm in iron films). In the thicker films, the two- and three-vortex asymmetric walls appear to be stable. The stability of these walls is related to that their structure is favorable for lowering the anisotropy energy under the condition of minimum total energy. The qualitative difference of the one- , two- and three-vortex asymmetric wall profiles is established, which gives a possibility to experimentally discover the new types of walls by using a conventional electron microscopy technique. Acknowledgements This work was partially supported by the project of the Department of Physical Sciences of Russian
Academy of Sciences and the Russian Foundation for Basic Research (Project no. 03-02-16185).
References [1] A.E. LaBonte, J. Appl. Phys. 40 (1969) 2450. [2] A. Hubert, Phys. Stat. Solidi.(b) 32 (1969) 519. [3] S. Tsukahara, H. Kavakatsu, J. Phys. Soc. Japan 32 (1972) 1993. [4] J.N. Chapman, G.R. Morisson, J.P. Jakubovics, R.A. Taylor, J. Magn. Magn. Mat. 49 (1985) 277. [5] M.R. Scheinfein, J. Unguris, R.J. Celotta, et al., Phys. Rev. Lett. 63 (1989) 668. [6] B.N. Filippov, L.G. Korzunin, F.A. Kassan-Ogly, Phys. Rev. B 64 (2001) 104412. [7] B.N. Filippov, L.G. Korzunin, JETP 121 (1993) 372 (in Russian). [8] A. Aharoni, J. Appl. Phys. 39 (1968) 861.
Journal of Magnetism and Magnetic Materials 298 (2006) 1–6 www.elsevier.com/locate/jmmm
A novel type of domain walls with two-dimensional magnetization distribution in magnetic triaxial films L.G. Korzunina, B.N. Filippova, F.A. Kassan-Oglya,, I.A. Chaikovskyb a
Institute of Metal Physics, Ural Division,Russian Academy of Sciences, ul. S.Kovalevskoi 18, Ekaterinburg 620219, Russian Federation b Ben-Gurion University of the Negev, POB 653, Beer-Sheva 84105, Israel Received 15 December 2004; received in revised form 28 January 2005 Available online 24 March 2005
Abstract On the basis of numerical minimization of total energy in magnetic triaxial ferromagnetic films with a surface of a (1 1 0)-type, we investigated two-dimensional structures of domain walls within a rigorous micromagnetic approach that takes into account all the main interactions including the dipole–dipole one. Novel two-vortex and three-vortex domain wall structures are established to exist. The profiles of domain wall structures and their stability regions are studied. r 2005 Elsevier B.V. All rights reserved. PACS: 05.45.a; 87.17.Aa; 75.70.Kw Keywords: A. Magnetic films and multilayers; D. Spin dynamics
1. Introduction It is theoretically [1,2] and experimentally [3–5] shown that in the magnetic uniaxial films of a certain thickness (more than 40 nm in Permalloy films) the equilibrium magnetization distribution in interdomain regions (domain walls) is asymmetric and vortex like. The walls with such internal structures are called asymmetric Bloch Corresponding author. Tel.: +7 343 3783528;
fax: +7 343 3745244. E-mail address: [email protected] (F.A. Kassan-Ogly).
walls, but for our purposes, it is better to call them one-vortex Bloch walls (1VB). Walls with similar structure were also found in magnetic triaxial films [3], whose surface coincides with a crystallographic plane of a (1 0 0) type (we will call them hereafter (1 0 0)-films). In this paper we show that similar walls should also exist in (1 1 0)-films. We also show that in such films novel types of two-vortex and three-vortex asymmetric walls (2VA and 3VA) can appear, whose existence is related to supplementary lowering of the anisotropy energy. The stability regions of such walls are found that depend on the thickness and magnetic parameters of the films. The studies are carried out within the
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.02.064
ARTICLE IN PRESS L.G. Korzunin et al. / Journal of Magnetism and Magnetic Materials 298 (2006) 1–6
2
framework of a model of two-dimensional magnetization M distribution using the numerical minimization of total energy that ensures the magnetization distribution in the films. We give the study results of domain wall profiles (the dependence of magnetization components averaged over the film thickness on the coordinate along the film surface) and show that the profiles allows one to experimentally reveal the walls of new types.
consists of the densities of the exchange energy wex ; the energy of magnetic anisotropy wa ; and the energy of magnetization wm in the magnetostatic field HðmÞ : wa ¼ K½m2x m2z þ m2y m2z þ 14ðm2y m2x Þ2 for ð1 1 0Þ-films. " 2 # qm 2 qm þ wex ¼ A , qx qy
ð2Þ
(3)
2. Statement of the problem
wm ¼ 12ðM HðmÞ Þ,
Let us consider magnetic triaxial films with surfaces of a (1 1 0) type ((1 1 0)-films). And let one of the easy axes (EMA) be parallel to the film surface and the other EMAs be tilted by the angles 45 1 and 135 1 to the surface (see Fig. 1). Let the 180 1 domain wall (DW) be in the V region in the shape of a parallelepiped elongated along the z-axis (the computation region). The DW separates the domains magnetized along and against the [0 0 1] direction. And let the orientation of M in the V region vary such that M ¼ Mðx; yÞ (two-dimensional model of magnetization distribution). The equilibrium distribution of M is obtained by the numerical minimization of the energy per unit length along the z-axis: Z Z W¼ w dx dz, (1)
where m ¼ M=M s ; M s is the saturation magnetization, A is the exchange parameter, and K is the anisotropy constant. The following boundary conditions are fulfilled at the boundaries of the V region:
D
where D is the cross-section of V with a plane z ¼ const: The D region has the (a b) dimensions, where b is the film thickness. The energy density w
Fig. 1. The geometry of problem.
(4)
mz jx¼a=2 ¼ 1; mx jx¼a=2 ¼ 0; myjx¼a=2 ¼ 0; qmy qmx j j ¼ 0; ¼ 0. ð5Þ qy y¼b=2 qy y¼b=2 The minimization of (1) is carried out numerically and it allows one to obtain the equilibrium configurations of a domain wall and its energy W 0 : The method of numerical minimization is described in a series of papers (see, for example, Refs. [1,6,7]). We used the network method with the maximal number of cells 150 50; and assumed the mesh size to be on the order of absolute single-domain size, that is approximately equal to 10 nm for iron. As for the size of the a computation region, we carried out numerical computations for the ratio a=b 2 ð1 6Þ at various fixed film thicknesses and established that the wall energy decreases with ratio a=b increasing. The wall energy difference between a=b ¼ 3 and a=bX4 does not exceed 0.2%. The films with b 2 ð40; 500Þ nm thickness are considered. The parameters A ¼ 2 106 erg=cm (exchange parameter), K ¼ 5:2 105 erg=cm3 (constant of cubic anisotropy), M s ¼ 1740 G (saturation magnetization) characteristic of iron were used as basal parameters. The criterion for termination of computations was chosen as that suggested in Ref. [8].
ARTICLE IN PRESS L.G. Korzunin et al. / Journal of Magnetism and Magnetic Materials 298 (2006) 1–6
3
3. Results and discussion The numerical computations were carried out in a wide range of the saturation induction Bs ¼ 4pM s and the anisotropy field H a ¼ 2K=M s : The computation results can be displayed as projections of the m vectors onto the plane perpendicular to easy axis (the xy plane). The projections of m onto the plane are depicted by arrows of various lengths. Fig. 2 shows the typical asymmetric walls characteristic of the (1 1 0)-films. The dashed curves correspond to the mz ¼ const level lines (see Ref. [1]). At the middle line the change of the mz component sign takes place. It is seen that the projection of m onto the xy plane has a vortex-like character and one, two, and three vortex-like patterns are seen in Figs. 2a–c. The walls with the 2a–2c structures are the 1VB, 2VA, and 3VA, respectively. Below we show that different walls will be stable in the films with different thickness. The 1VB walls are stable in (1 1 0)-films with the thickness bpbc : For the films with basal parameters the critical thickness bc is approximately equal to 200 nm, and it varies with variation of magnetic parameters. At greater thickness, the 1VB walls become unstable and their structure rearranges into the structure of three-vortex walls (Fig. 2c). Nothing of the kind could be observed both in magnetic uniaxial and triaxial (1 0 0)-films at any studied film thickness. The existence of the 3VA structures in (1 1 0)-films is related to a possible decrease of the anisotropy energy, when there appear segments of walls with M orientations close to EMA directions tilted by the angles 451 and 1351 to the surface of the film. The appearance of such segments is impossible in thinner films because of the strong growth of exchange energy due to the formation of three vortices. Since the vortex character of a wall is not always pronounced, it is convenient to draw the surfaces of the mðx; yÞ components. The mz ¼ mz ðx; yÞ surfaces turn out to be similar to those in classical Bloch walls. However, in our case one, two, or three extrema appear on the mz ¼ mz ðx; yÞ surfaces, but they are not always pronounced. The mx ¼ mx ðx; yÞ and my ¼ my ðx; yÞ are much more representative. Fig. 3
Fig. 2. Typical asymmetric walls in (1 1 0)-films of 200 nm (a,b) and 250 nm (c) thickness. Films with basal parameters.
shows, as an example, the my ¼ my ðx; yÞ surfaces that clearly distinguish the M distribution in the 1VB and 2VA walls.
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central line is S-shaped. The other feature of the 2b-type walls that distinguishes them from the ANWs is related to a well-pronounced character of two vortices, while the vortices are actually absent in the ANWs. From comparison of Figs. 2a and c it is seen that the 3VA wall could be considered as two superimposed 2VA walls with the mutual central vortex. Fig. 4 shows an example of the wall energy dependence on the film thickness, and quite a peculiar feature can be seen in it, namely, at the film thickness above 250 nm; the 2AB walls become stable. Nothing of the kind can be observed both in uniaxial and (1 0 0)-magnetic triaxial films. The available studies (see, for example, Ref. [2]) show that the ANWs are metastable in these films. The result obtained for basal films is valid in a wide range of the saturation induction, i.e. 0.5–2.2 T. Fig. 5 shows the dependence of DW energy on the anisotropy field H a for the films with b ¼ 300 nm:
Fig. 3. The my ðx; yÞ surfaces for 1VB (a) and 3VA (b) walls in (1 0 0)-films of 200 nm (a) and 250 nm (b) thickness. Films with basal parameters.
The 2VA walls (Fig. 2b) have features similar to the asymmetric Ne´el walls (ANW) predicted in [2] in uniaxial magnetic films. In particular, both the wall types are asymmetric relative to the xz surface and to the yz surface. The tangents on two nearsurface portions of the central lines of the ANWs and 2VA walls are like-tilted to the film surface (the dmx =dx derivatives have like signs), but the ANWs and 2VA walls are not quite identical. In the ANW, the dmx =dx derivatives have like signs all along the central line, whereas in the 2VA wall the sign changes with the depth of the film, and the
Fig. 4. The dependence of the 1VB and 3VA walls energy (open circles) and the 2VA (solid circles) in (1 1 0)-films on the film thickness. Films with basal parameters. The 1VB walls exist approximately up to b ¼ 200 nm: Circles are the numerical computation results; the curves are guides to the eye.
ARTICLE IN PRESS L.G. Korzunin et al. / Journal of Magnetism and Magnetic Materials 298 (2006) 1–6
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Fig. 5. The dependence of the 1VB and 3VA walls energy (solid circles) and the 2VA (open circles) on the field H a : Films with A and Bs basal parameters and b ¼ 300 nm: Circles are the numerical computation results; the curves are guides to the eye.
A wide range of H a where the 2VA walls are stable is seen in the figure, whereas at relatively small H a the 1VB walls are stable. Since the transition from 1VB to 2VA walls occurs gradually, in a narrow region to the left of the curves intersection point (Fig. 5) the 3VA walls can be stable. However, in this case their three-vortex structure is pronounced only slightly. The computations show that the stable 3VA wall also exists in a narrow region near H a ¼ 260 Oe in the films with b ¼ 300 nm: The 3VA walls are also stable at high H a ðX2000 OeÞ: However, in this case the distortion of their structure takes place, which calls for special investigations. In closing let us bring out the profiles of domain walls: Z 1 b mi mi ðxÞ ¼ mi dy; i ¼ ðx; y; zÞ. (6) b 0 An example of such profiles is shown in Fig. 6 for the 2VA and 3VA walls. They are seen to differ from one another, and from comparison with the data of [4] it also
Fig. 6. The profiles for 2VA (a) and 3VA (b) walls in the films with basal parameters; b ¼ 200 nm (a), b ¼ 250 nm (b).
follows that they differ from the 1VB wall profiles. This suggests that the described novel types of walls could be experimentally detected by the wall profile studies by electron microscopy, as in Ref. [4].
4. Conclusions In magnetic triaxial films with a surface of a (1 1 0)-type the novel types of asymmetric domain walls were shown to exist: two-vortex walls with an S-shaped central line, and asymmetric three-vortex walls. The structures of these walls substantially differ from the structure of classical Bloch and Ne´el walls and also from asymmetric Bloch [1] and Ne´el [2] walls.
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In the (1 1 0)-films the asymmetric Bloch walls of a type predicted in [1] can also exist. They are stable in the relatively thin films (less than 200 nm in iron films). In the thicker films, the two- and three-vortex asymmetric walls appear to be stable. The stability of these walls is related to that their structure is favorable for lowering the anisotropy energy under the condition of minimum total energy. The qualitative difference of the one- , two- and three-vortex asymmetric wall profiles is established, which gives a possibility to experimentally discover the new types of walls by using a conventional electron microscopy technique. Acknowledgements This work was partially supported by the project of the Department of Physical Sciences of Russian
Academy of Sciences and the Russian Foundation for Basic Research (Project no. 03-02-16185).
References [1] A.E. LaBonte, J. Appl. Phys. 40 (1969) 2450. [2] A. Hubert, Phys. Stat. Solidi.(b) 32 (1969) 519. [3] S. Tsukahara, H. Kavakatsu, J. Phys. Soc. Japan 32 (1972) 1993. [4] J.N. Chapman, G.R. Morisson, J.P. Jakubovics, R.A. Taylor, J. Magn. Magn. Mat. 49 (1985) 277. [5] M.R. Scheinfein, J. Unguris, R.J. Celotta, et al., Phys. Rev. Lett. 63 (1989) 668. [6] B.N. Filippov, L.G. Korzunin, F.A. Kassan-Ogly, Phys. Rev. B 64 (2001) 104412. [7] B.N. Filippov, L.G. Korzunin, JETP 121 (1993) 372 (in Russian). [8] A. Aharoni, J. Appl. Phys. 39 (1968) 861.