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Magnetization processes in ferromagnetic nanostructures with competing anisotropies
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 290–291 (2005) 772–775 www.elsevier.com/locate/jmmm
Magnetization processes in ferromagnetic nanostructures with competing anisotropies U.K. Ro¨Xlera,, S.V. Bukhtiyarovab, I.V. Zhikharevb,c, A.N. Bogdanova,c a
Leibniz Institute for Solid State and Materials Research Dresden, Postfach 270116, D-01171 Dresden, Germany b Lugansk State Pedagogical University, Obronna 2, 91011 Lugansk, Ukraine c Donetsk Institute for Physics and Technology, R. Luxemburg 72, 83114 Donetsk, Ukraine Available online 15 December 2004
Abstract A micromagnetic framework based on the phenomenological theory for induced magnetic interactions is developed to analyse states in ferromagnetic nanolayers under competing intrinsic and (uniaxial, unidirectional) surface-induced anisotropies. Magnetic phase diagrams in components of applied fields show complex systems of critical lines and lines of first-order transitions corresponding to two-phase and multiphase coexistence regions. The role of these first-order transitions for multidomain states and limiting cases of the corresponding magnetization processes are discussed. The inhomogeneous distribution of induced magnetic anisotropies across the thickness of a layer may occasion noncollinear magnetic states. r 2004 Elsevier B.V. All rights reserved. Keywords: Magnetization—field dependent; Thin films—magnetic; Anisotropy—surface
1. Introduction In bulk materials with competing magnetic anisotropies, the magnetization processes are characterized by different types of reorientation transitions accompanied by complex evolution of metastable and multidomain states (see Refs. [1,2] and further references in Ref. [3]). In nanoscale magnetic layers and particles, strong surface/interface-induced magnetic anisotropies arise in addition to the usual intrinsic (magnetocrystalline) contributions. These anisotropies may own lower symmetry and be stronger than those of the magnetic material of the layer. Hence, surfaces or couplings to suitable substrates may induce uniaxial anisotropy, and the coupling to an antiferromagnet may yield unidirecCorresponding author. Tel.: +49 351 4659 542; fax: +49 351 4659 537. E-mail address: [email protected] (U.K. Ro¨Xler).
tional anisotropy leading to exchange bias [4]. The interplay between these different types of anisotropies causes remarkable phenomena such as shifted and double-shifted hysteresis loops and multidomain states [4–7]. These effects are commonly described by including the heuristic Ne´el ansatz for surface anisotropies into the micromagnetic energy functional [5]. This model is valid only in the limit of sufficiently thick films. It is inadequate when finite widths of magnetic defect regions due to surface/interface effects come into play [8]. In a more realistic phenomenological theory, the surface/ interface-induced magnetic interactions should be treated as internal spatially inhomogeneous variables [8,9]. This approach explains general features observed in magnetic nanostructures and is consistent with microscopic mechanisms for surface effects on magnetic interactions [10]. In this contribution, we investigate the field-driven evolution of magnetic states in a magnetic layer with
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.402
ARTICLE IN PRESS U.K. Ro¨X ler et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 772–775
773
competing induced and intrinsic anisotropies. We discuss their magnetization processes for exemplary cases of a cubic ferromagnetic layer with different induced uniaxial or unidirectional anisotropies.
layers (d5d), it tends to a constant value Bi : For systems with induced uniaxial anisotropy a ¼ 2; we denote the angles between a and m (H) by y (c). Then, for the two co-planar cases discussed above, the homogeneous part of the energy (1) yields the same potential for y:
2. Phenomenological theory
F ¼ sgnðKÞ cos4 y þ ð1 kÞ cos2 y
We consider a layer with thickness d magnetically homogeneous, infinite in x and y directions, and confined by planar surfaces at z ¼ d=2: Its magnetic energy within a phenomenological approach [8,9] is Z d=2 " X 2 qm 1 Wm ¼ A M H M Hd qx 2 i d=2 i # Kðm4x þ m4y þ m4z Þ bðzÞðm aÞa dz,
ð1Þ
where m ¼ M=jMj is the normalized value of the magnetization vector M; A is the exchange constant, and H and Hd are the external and demagnetizing fields, respectively. The last term describes an induced anisotropy, unidirectional or unaxial for a ¼ 1; 2; respectively, with direction or axis given by the fixed unity vector a and with inhomogeneous strengths bðzÞ through the layer. To simplify discussion, we restrict the model to cases with coplanar geometry, i.e. the (easy) axis a; effective field and the magnetization vector m share a common plane: Case (i) is a layer with induced perpendicular anisotropy, i.e. a along the z-axis, and identical surface properties at the top and bottom. Then b depends on the position within the layer, b1 ðzÞ ¼ B0 coshð2z=dÞ= coshðd=dÞ; with a characteristic ‘‘penetration depth’’ d and B0 a phenomenological constant [8,9]. This induced anisotropy is confined near the surfaces for thick layers dbd; whereas it will vary only little across thin layers d5d: For coplanar biaxial configurations with fields and magnetization in the xz-plane, the intrinsic cubic anisotropy must have (1 0 0)-easy axes, i.e. K40: Case (ii) is a layer where magnetization is strictly in plane. An induced anisotropy or exchange coupling may be taken also in plane, a along the y-direction, and is restricted to the top side of the layer. Then b2 ðzÞ ¼ B0 =2½coshð2z=dÞ= coshðd=dÞ þ sinhð2z=dÞ= sinhðd=dÞ ; and the cases K40 or Ko0 describe different orientations of the four-fold intrinsic anisotropy in plane. Clearly, the two induced contributions of cases (i) and (ii) may also occur simultaneously in the same layer. The analysis has to start by considering homogeneous states. The induced anisotropies can be transformed to R d=2 effective integral parameters BðdÞ ¼ d 1 d=2 bi ðzÞ dz ¼ Bi dd 1 tanhðd=dÞ with B1;2 ¼ B0 or B0 =2 for i ¼ 1; 2 [9]. In the limit of thick layers this effective anisotropy transforms to the Ne´el ansatz B ¼ Bi d=d: For ‘‘thin’’
h cosðy cÞ
ð2Þ
with kðdÞ ¼ BðdÞ=ð2KÞ: Induced unidirectional anisotropy for a ¼ 1 only adds an effective field to Eq. (2). In potential expressions for homogeneous magnetic states, as the simplified example Eq. (2), field h ¼ HjMj=ð2KÞ has to be understood as an internal field. For (modulated) domain states, demagnetization effects can be taken into account within the phase approximation by a standard mapping between the phase diagrams written in components of the field perpendicular and along the chosen easy axis ðhx ; hjj Þ; and the corresponding external ðeÞ field ðhðeÞ x ; hjj Þ [3]. The energy (2) is mathematically equivalent to the phenomenological energy of a bulk ferromagnet, where variations of the second-order anisotropy constant with temperature or pressure cause a reorientation of the easy magnetization direction (see bibliography in Ref. [3]) and results for these systems can be used to describe the phase structure of homogeneous states in magnetic nanolayers. However, in layers with induced uniaxial anisotropies reorientation effects may occur also due to the change of the parameter k that describes the variation of the effective anisotropy with thickness.
3. (hjj ; hx ) magnetic phase diagram The magnetization processes described by Eq. (2) can be summarized in phase diagrams with metastability regions for different states and first-order lines. Metastable states: The equations for the stability limits (dF=dy ¼ 0; d2 F=dy2 ¼ 0) describe a transformation of the Stoner–Wohlfarth astroid into eight cusp lines in the interval 0oko5 (Fig. 2) [1]. This process starts in the points k ¼ 5 by a bifurcation in one pair of opposite cusps forming so-called ‘‘swallow tails’’. The internal cusps (points c1 and c2 in Fig. 2) move to the centre, and then penetrate each other (Fig. 1). In the point k ¼ 0 an astroid with eight cusps is established (Fig. 2(B)). Because Eq. (2) is invariant under transformation k ! k; y ! y þ p=2; c ! c þ p=2; the same process occurs for negative k along transformed axes. The equations for the critical lines are also invariant under this transformation and a change of sign in K. Thus, the metastable regions are the same for K40 and Ko0; but within these regions phase structure and locations of
ARTICLE IN PRESS U.K. Ro¨X ler et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 772–775
774
4
2
d1
(A)
b1
b2
(B) a
0
2 h||
b2
b3 a1
0
c2 c1
b3
-2
b4
-2
0
2
a2
b1
b4
10
(C)
-2 0
-4
-10
d2 -2
-1
0 hx
1
-10
2
0
10
Fig. 1. Phase diagram as in Fig. 2 for 54k40: Thicker lines for first-order transitions with Ko0:
d1 2
h||
b2
b3 c1
0
c2
b1
b4
-2 d2 -2
0 hx
2
Fig. 2. Phase diagrams in terms of internal fields for biaxial systems described by Eq. (2): (A) 54k40; (B) k ¼ 0; (C) k45: Thicker internal lines stand for first-order transitions in the case K40:
first-order transitions change as the energy of Eq. (2) is not invariant under this operation. First-order transitions: In Figs. 2 the lines of first-order transitions between the magnetic states are given for K40: There are four segments ai bj with critical endpoints at bj ; and the segment a1 a2 : Along the line a1 a2 the phase transition occurs between canted phases designated by (% &) with equal components along x and antiparallel along a: In points a1 and a2 transitions occur between two such canted phases and a perpendicular phase: (t - ) at a1 and (% & !) at a2 : Along lines (ai bj ) one of the canted phases coexists with a ‘‘distorted’’ phase derived from or !: At k ¼ 0; the points a1 and a2 merge and form a point where four phases coexist. For Ko0; the first-order lines between two canted phases are segments along the field
components hx and hjj (Fig. 1). The canted states transform into collinear phases at the critical end-points of these first-order transitions (c1 ; c2 ; d 1 ; d 2 ). At (0; 0) four canted phases coexist. Multidomain states and twisted phases. The first-order transitions between magnetic states are accompanied by thermodynamically stable multidomain states built from the co-existing phases [3]. Depending on the number of these phases (Figs. 2, 1) two-, three- and four-phase multidomain patterns will arise. Such multidomain states should be particularly interesting for systems with perpendicular induced anisotropy resulting in out-ofplane components of magnetization. Moreover, noncollinear distortions of the one-dimensional magnetization structures perpendicular through the layer are embodied in the general magnetic energy Eq. (1) with oblique field directions due to the inhomogeneous distribution bðzÞ: Numerical investigations show that these distortions are usually small for thin layers [11], but in the vicinity of the first-order lines twisted phases may occur [9,11]. Magnetization processes: The magnetization processes in real systems mediate between the two limiting cases of perfectly stiff and soft magnetic materials. The first socalled Stoner–Wohlfarth limit is the case of coherently rotating particles. The hysteresis loops are determined by the critical stability lines. The perfectly soft case describes an anhysteretic transformation via equilibrium multidomain states [12]. For nanoscale ferromagnetic objects, only specific cases of magnetization curves in the Stoner–Wohlfarth limit have been analysed in detail [5]. However, evidence for multidomain states and incoherent spin rotations is found in various thin layered systems, e.g. in epitaxial NiMn/Co films [6] and MnPd/ Fe bilayers [7]. In conclusion, the micromagnetic methods highlighted in this paper based on the effective energy (1) provide tools to investigate magnetization processes in nanolayers with competing anisotropies.
References [1] I.S. Edelman, Fiz. Met. Metalloved. 20 (1965) 683; A.I. Mitsek, N.P. Kolmakova, D.I. Sirota, Fiz. Met. Metalloved. 38 (1974) 35. [2] A.N. Bogdanov, I.Ya. Granovskii, Sov. Phys. Solid State 29 (1987) 1674. [3] V.G. Baryakhtar, A.N. Bogdanov, D.A. Yablonskii, Sov. Phys. Usp. 31 (1988) 810. [4] J. Nogues, I.K. Schuller, J. Magn. Magn. Mater. 192 (1999) 203. [5] C.-R. Chang, J.-S. Yang, J.C.A. Huang, C.H. Lai, J. Phys. Chem. Sol. 62 (2001) 1737. [6] C.H. Lai, et al., Phys. Rev. B 64 (2001) 094420. [7] P. Blomqvist, K.M. Krishnan, E. Girt, J. Appl. Phys. 95 (2004) 8487.
ARTICLE IN PRESS U.K. Ro¨X ler et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 772–775 [8] A.N. Bogdanov, U.K. Ro¨Xler, Phys. Rev. Lett. 86 (2001) 037203. [9] A.N. Bogdanov, U.K. Ro¨Xler, K.-H. Mu¨ller, J. Magn. Magn. Mater. 238 (2002) 155; A.N. Bogdanov, U.K. Ro¨Xler, K.-H. Mu¨ller, J. Magn. Magn. Mater. 242 (2002) 594.
775
[10] D. Weller, et al., Phys. Rev. Lett. 75 (1995) 3752; S.-K. Kim, J.B. Kortright, Phys. Rev. Lett. 86 (2001) 1347. [11] S.V. Bukhtiyarova et al., to be published [12] A. Hubert, R. Scha¨fer, Magnetic Domains, Springer, Berlin, 1998.
Journal of Magnetism and Magnetic Materials 290–291 (2005) 772–775 www.elsevier.com/locate/jmmm
Magnetization processes in ferromagnetic nanostructures with competing anisotropies U.K. Ro¨Xlera,, S.V. Bukhtiyarovab, I.V. Zhikharevb,c, A.N. Bogdanova,c a
Leibniz Institute for Solid State and Materials Research Dresden, Postfach 270116, D-01171 Dresden, Germany b Lugansk State Pedagogical University, Obronna 2, 91011 Lugansk, Ukraine c Donetsk Institute for Physics and Technology, R. Luxemburg 72, 83114 Donetsk, Ukraine Available online 15 December 2004
Abstract A micromagnetic framework based on the phenomenological theory for induced magnetic interactions is developed to analyse states in ferromagnetic nanolayers under competing intrinsic and (uniaxial, unidirectional) surface-induced anisotropies. Magnetic phase diagrams in components of applied fields show complex systems of critical lines and lines of first-order transitions corresponding to two-phase and multiphase coexistence regions. The role of these first-order transitions for multidomain states and limiting cases of the corresponding magnetization processes are discussed. The inhomogeneous distribution of induced magnetic anisotropies across the thickness of a layer may occasion noncollinear magnetic states. r 2004 Elsevier B.V. All rights reserved. Keywords: Magnetization—field dependent; Thin films—magnetic; Anisotropy—surface
1. Introduction In bulk materials with competing magnetic anisotropies, the magnetization processes are characterized by different types of reorientation transitions accompanied by complex evolution of metastable and multidomain states (see Refs. [1,2] and further references in Ref. [3]). In nanoscale magnetic layers and particles, strong surface/interface-induced magnetic anisotropies arise in addition to the usual intrinsic (magnetocrystalline) contributions. These anisotropies may own lower symmetry and be stronger than those of the magnetic material of the layer. Hence, surfaces or couplings to suitable substrates may induce uniaxial anisotropy, and the coupling to an antiferromagnet may yield unidirecCorresponding author. Tel.: +49 351 4659 542; fax: +49 351 4659 537. E-mail address: [email protected] (U.K. Ro¨Xler).
tional anisotropy leading to exchange bias [4]. The interplay between these different types of anisotropies causes remarkable phenomena such as shifted and double-shifted hysteresis loops and multidomain states [4–7]. These effects are commonly described by including the heuristic Ne´el ansatz for surface anisotropies into the micromagnetic energy functional [5]. This model is valid only in the limit of sufficiently thick films. It is inadequate when finite widths of magnetic defect regions due to surface/interface effects come into play [8]. In a more realistic phenomenological theory, the surface/ interface-induced magnetic interactions should be treated as internal spatially inhomogeneous variables [8,9]. This approach explains general features observed in magnetic nanostructures and is consistent with microscopic mechanisms for surface effects on magnetic interactions [10]. In this contribution, we investigate the field-driven evolution of magnetic states in a magnetic layer with
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.402
ARTICLE IN PRESS U.K. Ro¨X ler et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 772–775
773
competing induced and intrinsic anisotropies. We discuss their magnetization processes for exemplary cases of a cubic ferromagnetic layer with different induced uniaxial or unidirectional anisotropies.
layers (d5d), it tends to a constant value Bi : For systems with induced uniaxial anisotropy a ¼ 2; we denote the angles between a and m (H) by y (c). Then, for the two co-planar cases discussed above, the homogeneous part of the energy (1) yields the same potential for y:
2. Phenomenological theory
F ¼ sgnðKÞ cos4 y þ ð1 kÞ cos2 y
We consider a layer with thickness d magnetically homogeneous, infinite in x and y directions, and confined by planar surfaces at z ¼ d=2: Its magnetic energy within a phenomenological approach [8,9] is Z d=2 " X 2 qm 1 Wm ¼ A M H M Hd qx 2 i d=2 i # Kðm4x þ m4y þ m4z Þ bðzÞðm aÞa dz,
ð1Þ
where m ¼ M=jMj is the normalized value of the magnetization vector M; A is the exchange constant, and H and Hd are the external and demagnetizing fields, respectively. The last term describes an induced anisotropy, unidirectional or unaxial for a ¼ 1; 2; respectively, with direction or axis given by the fixed unity vector a and with inhomogeneous strengths bðzÞ through the layer. To simplify discussion, we restrict the model to cases with coplanar geometry, i.e. the (easy) axis a; effective field and the magnetization vector m share a common plane: Case (i) is a layer with induced perpendicular anisotropy, i.e. a along the z-axis, and identical surface properties at the top and bottom. Then b depends on the position within the layer, b1 ðzÞ ¼ B0 coshð2z=dÞ= coshðd=dÞ; with a characteristic ‘‘penetration depth’’ d and B0 a phenomenological constant [8,9]. This induced anisotropy is confined near the surfaces for thick layers dbd; whereas it will vary only little across thin layers d5d: For coplanar biaxial configurations with fields and magnetization in the xz-plane, the intrinsic cubic anisotropy must have (1 0 0)-easy axes, i.e. K40: Case (ii) is a layer where magnetization is strictly in plane. An induced anisotropy or exchange coupling may be taken also in plane, a along the y-direction, and is restricted to the top side of the layer. Then b2 ðzÞ ¼ B0 =2½coshð2z=dÞ= coshðd=dÞ þ sinhð2z=dÞ= sinhðd=dÞ ; and the cases K40 or Ko0 describe different orientations of the four-fold intrinsic anisotropy in plane. Clearly, the two induced contributions of cases (i) and (ii) may also occur simultaneously in the same layer. The analysis has to start by considering homogeneous states. The induced anisotropies can be transformed to R d=2 effective integral parameters BðdÞ ¼ d 1 d=2 bi ðzÞ dz ¼ Bi dd 1 tanhðd=dÞ with B1;2 ¼ B0 or B0 =2 for i ¼ 1; 2 [9]. In the limit of thick layers this effective anisotropy transforms to the Ne´el ansatz B ¼ Bi d=d: For ‘‘thin’’
h cosðy cÞ
ð2Þ
with kðdÞ ¼ BðdÞ=ð2KÞ: Induced unidirectional anisotropy for a ¼ 1 only adds an effective field to Eq. (2). In potential expressions for homogeneous magnetic states, as the simplified example Eq. (2), field h ¼ HjMj=ð2KÞ has to be understood as an internal field. For (modulated) domain states, demagnetization effects can be taken into account within the phase approximation by a standard mapping between the phase diagrams written in components of the field perpendicular and along the chosen easy axis ðhx ; hjj Þ; and the corresponding external ðeÞ field ðhðeÞ x ; hjj Þ [3]. The energy (2) is mathematically equivalent to the phenomenological energy of a bulk ferromagnet, where variations of the second-order anisotropy constant with temperature or pressure cause a reorientation of the easy magnetization direction (see bibliography in Ref. [3]) and results for these systems can be used to describe the phase structure of homogeneous states in magnetic nanolayers. However, in layers with induced uniaxial anisotropies reorientation effects may occur also due to the change of the parameter k that describes the variation of the effective anisotropy with thickness.
3. (hjj ; hx ) magnetic phase diagram The magnetization processes described by Eq. (2) can be summarized in phase diagrams with metastability regions for different states and first-order lines. Metastable states: The equations for the stability limits (dF=dy ¼ 0; d2 F=dy2 ¼ 0) describe a transformation of the Stoner–Wohlfarth astroid into eight cusp lines in the interval 0oko5 (Fig. 2) [1]. This process starts in the points k ¼ 5 by a bifurcation in one pair of opposite cusps forming so-called ‘‘swallow tails’’. The internal cusps (points c1 and c2 in Fig. 2) move to the centre, and then penetrate each other (Fig. 1). In the point k ¼ 0 an astroid with eight cusps is established (Fig. 2(B)). Because Eq. (2) is invariant under transformation k ! k; y ! y þ p=2; c ! c þ p=2; the same process occurs for negative k along transformed axes. The equations for the critical lines are also invariant under this transformation and a change of sign in K. Thus, the metastable regions are the same for K40 and Ko0; but within these regions phase structure and locations of
ARTICLE IN PRESS U.K. Ro¨X ler et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 772–775
774
4
2
d1
(A)
b1
b2
(B) a
0
2 h||
b2
b3 a1
0
c2 c1
b3
-2
b4
-2
0
2
a2
b1
b4
10
(C)
-2 0
-4
-10
d2 -2
-1
0 hx
1
-10
2
0
10
Fig. 1. Phase diagram as in Fig. 2 for 54k40: Thicker lines for first-order transitions with Ko0:
d1 2
h||
b2
b3 c1
0
c2
b1
b4
-2 d2 -2
0 hx
2
Fig. 2. Phase diagrams in terms of internal fields for biaxial systems described by Eq. (2): (A) 54k40; (B) k ¼ 0; (C) k45: Thicker internal lines stand for first-order transitions in the case K40:
first-order transitions change as the energy of Eq. (2) is not invariant under this operation. First-order transitions: In Figs. 2 the lines of first-order transitions between the magnetic states are given for K40: There are four segments ai bj with critical endpoints at bj ; and the segment a1 a2 : Along the line a1 a2 the phase transition occurs between canted phases designated by (% &) with equal components along x and antiparallel along a: In points a1 and a2 transitions occur between two such canted phases and a perpendicular phase: (t - ) at a1 and (% & !) at a2 : Along lines (ai bj ) one of the canted phases coexists with a ‘‘distorted’’ phase derived from or !: At k ¼ 0; the points a1 and a2 merge and form a point where four phases coexist. For Ko0; the first-order lines between two canted phases are segments along the field
components hx and hjj (Fig. 1). The canted states transform into collinear phases at the critical end-points of these first-order transitions (c1 ; c2 ; d 1 ; d 2 ). At (0; 0) four canted phases coexist. Multidomain states and twisted phases. The first-order transitions between magnetic states are accompanied by thermodynamically stable multidomain states built from the co-existing phases [3]. Depending on the number of these phases (Figs. 2, 1) two-, three- and four-phase multidomain patterns will arise. Such multidomain states should be particularly interesting for systems with perpendicular induced anisotropy resulting in out-ofplane components of magnetization. Moreover, noncollinear distortions of the one-dimensional magnetization structures perpendicular through the layer are embodied in the general magnetic energy Eq. (1) with oblique field directions due to the inhomogeneous distribution bðzÞ: Numerical investigations show that these distortions are usually small for thin layers [11], but in the vicinity of the first-order lines twisted phases may occur [9,11]. Magnetization processes: The magnetization processes in real systems mediate between the two limiting cases of perfectly stiff and soft magnetic materials. The first socalled Stoner–Wohlfarth limit is the case of coherently rotating particles. The hysteresis loops are determined by the critical stability lines. The perfectly soft case describes an anhysteretic transformation via equilibrium multidomain states [12]. For nanoscale ferromagnetic objects, only specific cases of magnetization curves in the Stoner–Wohlfarth limit have been analysed in detail [5]. However, evidence for multidomain states and incoherent spin rotations is found in various thin layered systems, e.g. in epitaxial NiMn/Co films [6] and MnPd/ Fe bilayers [7]. In conclusion, the micromagnetic methods highlighted in this paper based on the effective energy (1) provide tools to investigate magnetization processes in nanolayers with competing anisotropies.
References [1] I.S. Edelman, Fiz. Met. Metalloved. 20 (1965) 683; A.I. Mitsek, N.P. Kolmakova, D.I. Sirota, Fiz. Met. Metalloved. 38 (1974) 35. [2] A.N. Bogdanov, I.Ya. Granovskii, Sov. Phys. Solid State 29 (1987) 1674. [3] V.G. Baryakhtar, A.N. Bogdanov, D.A. Yablonskii, Sov. Phys. Usp. 31 (1988) 810. [4] J. Nogues, I.K. Schuller, J. Magn. Magn. Mater. 192 (1999) 203. [5] C.-R. Chang, J.-S. Yang, J.C.A. Huang, C.H. Lai, J. Phys. Chem. Sol. 62 (2001) 1737. [6] C.H. Lai, et al., Phys. Rev. B 64 (2001) 094420. [7] P. Blomqvist, K.M. Krishnan, E. Girt, J. Appl. Phys. 95 (2004) 8487.
ARTICLE IN PRESS U.K. Ro¨X ler et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 772–775 [8] A.N. Bogdanov, U.K. Ro¨Xler, Phys. Rev. Lett. 86 (2001) 037203. [9] A.N. Bogdanov, U.K. Ro¨Xler, K.-H. Mu¨ller, J. Magn. Magn. Mater. 238 (2002) 155; A.N. Bogdanov, U.K. Ro¨Xler, K.-H. Mu¨ller, J. Magn. Magn. Mater. 242 (2002) 594.
775
[10] D. Weller, et al., Phys. Rev. Lett. 75 (1995) 3752; S.-K. Kim, J.B. Kortright, Phys. Rev. Lett. 86 (2001) 1347. [11] S.V. Bukhtiyarova et al., to be published [12] A. Hubert, R. Scha¨fer, Magnetic Domains, Springer, Berlin, 1998.