- Email: [email protected]
Mechanisms responsible for magnetostriction in heterogeneous magnetic systems
Journal of Magnetism and Magnetic Materials 191 (1999) 199 — 202
Mechanisms responsible for magnetostriction in heterogeneous magnetic systems H. Szymczak *, R. Z uberek , J. Gonzalez Institute of Physics, Polish Academy of Sciences, Al. Lotniko& w 32, 02-668 Warsaw, Poland Departamento de Fisica de Materiales, Facultad de Quimicas, Universidad del Pais Vasco, Paseo Manuel de Lardizabal 3, 20009 San Sebastian, Spain Received 6 June 1999; received in revised form 8 August 1998
Abstract Experimental and theoretical research on magnetoelastic properties of heterogeneous magnetic materials is reviewed. Particular attention is devoted to granular and nanocrystalline systems. It is shown that localisation of magnetoelastic strains at the interface leads to the dependence of effective magnetostriction on the size of magnetic inhomogeneities (radius of grains, thickness of magnetic sublayers, etc.). Several phenomenological models of magnetostriction are presented and analysed in detail. 1999 Elsevier Science B.V. All rights reserved. PACS: 75.80; 75.70; 75.70.c; 75.50K Keywords: Heterogeneous magnetic systems; Nanocrystalline materials; Magnetostriction; Surface magnetostriction; Magnetoelastic properties
Heterogenous magnetic materials, which include nanoscale magnetic films and multilayers, various granular systems, as well as various magnetic composites have been a subject of interest in recent years. The motivation for the studies of heterogenous nanoscale systems is twofold. First, one is interested in the novel overall properties of these materials which are determined by their high density of topological defects. The amount of the disorder in nanometre-scale systems is so high that the interfacial atoms having neither short range order nor long range order constitute in fact a new phase of condensed matter. A second reason to study
* Corresponding author. Tel.: #48-22-843-68-71; fax: #4822-843-26-09; e-mail: [email protected].
these systems is related to their interesting potential applications in many branches of high technology. Nanoscale materials offer a new opportunity for tailoring metallic, magnetic and semiconductor materials. Progress in this field is made mainly due to a tremendous improvement in the production and characterisation of nanoscale systems. Among various heterogenous systems magnetic nanometre-scale materials have been receiving the greatest interest. Nanoscale magnetic multilayers are used in erasable magneto-optic and perpendicular recording (see Ref. [1] for details and references). The discovery of giant magnetoresistance [2] has opened the possibility to apply these types of materials in magnetic devices such as magnetoresistive heads and sensors. Recently very good soft magnetic materials with high initial magnetic
0304-8853/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 3 3 5 - 7
200
H. Szymczak et al. / Journal of Magnetism and Magnetic Materials 191 (1999) 199 — 202
permeability and low coercivity have been found (see Ref. [3] for details and references). These materials consist of nanosize magnetic granules embedded in an amorphous matrix. All of these materials exhibit unusual properties which ordinary solids do not have, such as the reduction of saturation magnetisation (nanocrystalline iron [4] and nickel [5]), surface magnetic anisotropy (see Ref. [6] and references therein for thin films and Refs. [7,8] for nanoparticles), surface magnetostrictive strains (see Ref. [9] and references therein for thin films and Refs. [10—12] for nanoparticles), magnetic anisotropy oscillations [13] and many others. Explanation of the phenomena in these magnetic nanometre-scale systems has generally relied on the specific properties (due to the reduced symmetry) of atoms at the interfaces. It was Ne´el [14] who first predicted that an ideal surface (or interface) could exhibit this special type of behaviour.The above-mentioned excellent soft magnetic properties of nanocrystalline materials are attributed to the absence (or very small value) of magnetostriction. The magnetostriction is also responsible for magneto-optic and perpendicular recording applications of magnetic multilayers. Generally, magnetostriction in heterogenous magnetic systems is expected to play a particularly important role by its influence on various (intrinsic and extrinsic) properties. The importance of this problem results from the fact [15] that all nanometre-scale metallic systems are in a state of stress. This is mainly due to differences in the thermal expansion coefficients of the contacting materials and also due to differences in their lattice parameters. We are going to sum up in this review the results of experimental and theoretical investigations of magnetostriction in heterogenous magnetic systems. The review will be focused on the granular systems (mainly nanocrystalline alloys) because only little attention has been paid in the literature to the mechanisms responsible for magnetostriction in these materials. Magnetoelastic energy, simply related to the magnetostriction constants, can be written in the following form in the coordinate axes connected to the local anisotropy. º " B aGaGeG , KC IJKL I J KL
(1)
Fig. 1. Effective magnetostriction of Fe Zr B Cu nanocrys talline alloys versus crystalline fraction p. Solid line is the fitting curve (according to Eq. (3)).
where aG are the direction cosines of the local I magnetisation and eG are the components of the KL local strain tensor. It has been shown, both experimentally and theoretically (see Ref. [9] for references) that the magnetoelastic tensor [B] in nanoscale magnetic multilayers consists of two parts: [B]"[B] #2t\[B] (2) where [B] describes bulk (volume) magnetoelastic coupling and [B] describes surface magnetoelastic coupling (t is the thickness of magnetic layers). Recently, the experimental dependencies of the magnetostriction constant on the volume fraction of the crystalline phase, measured both in FeCuNbSiB and in FeZrBCu nanocrystalline alloys, have also been interpreted in terms of surface and bulk contributions to the effective magnetostriction [10,12,16]. As an example the experimental results of the effective saturation magnetostriction j of Fe Zr B Cu nanocrystalline alloy as a function of the volume fraction of the crystalline phase p are shown in Fig. 1. The solid line in Fig. 1 describes the j(p) proposed in Refs. [10,12,16]: 3j j(p)"pj#(1!p)(j #kp)#p , (3) R where j and j are the magnetostriction constants of the grains and amorphous matrix,
H. Szymczak et al. / Journal of Magnetism and Magnetic Materials 191 (1999) 199 — 202
respectively; j is the magnetostriction constant which describes the interface contribution; R is the effective radius of the grains. The calculations have been performed for j and j known from Ref. [10]. The surface magnetostriction was found to be j"10.9;10\ nm. The important feature of the results presented is the different signs of j and j which lead to relatively low values of j. It can be seen from Eq. (3) that the behaviour of j(p) is more complicated than a simple superposition of crystalline and amorphous contributions (as it is usually assumed [3]) because of the important role of the surface effects. The existence of the surface contribution to the effective magnetostriction of nanocrystalline alloys has been confirmed theoretically in terms of the dipolar model [17,18]. It has been shown that, due to limited radius of nanoparticles, additional magnetoelastic stresses are localised at the interface. This localisation effect leads to the R\ term in effective magnetostriction of the system in accordance with experimental data (see Eq. (3)). The pure magnetic dipolar interactions between grains are not expected to contribute to the effective magnetostriction of nanocrystalline alloys. But recently [19,20] it has been shown that RKKY exchange is responsible for interactions between small metallic spheres. Since RKKY exchange has, generally, an anisotropic (pseudodipolar) part [21] one should expect that the intergrain interactions have, in addition to the symmetric RKKY term, a pseudodipolar contribution. Such a term has been observed in Gd—Co amorphous films [22]. The anisotropic RKKY interactions in nanometer-scale systems explain, in a natural way, the oscillations of magnetic anisotropy in nanoscale magnetic films [13] and should contribute to magnetostriction in heterogenous magnetic systems. Another interesting case has been considered in Ref. [23]. They have studied the effective magnetostriction of a system consisting of highly magnetostrictive spherical particles embedded in a nonmagnetostrictive (or low-magnetostrictive) matrix. Both components of such composite are elastically and magnetoelastically isotropic. The effective magnetostriction of the composite can be
201
written as p j"j (1.15!1.26p!0.21p ), 1!1.25p
(4)
where p is the Poisson’s ratio for low-magnetostrictive component. Further development of this problem, based on Green’s function method was given recently by Ce-Wen Nan [24]. The strong suppression of local magneto-crystalline anisotropy in nanocrystalline magnetic alloys has been observed experimentally and confirmed theoretically in terms of the random anisotropy model [3]. The same model describes magnetostriction in amorphous alloys [25,26]. This model can easily be adapted to the case of granular systems. The energy density º of magnetic grains embedded in an amorphous magnetic matrix can be written in the following way: º"K sin h !j1M 2 M cos *h , (5) G
G G G G where K is the uniaxial anisotropy of grains, M is G their magnetisation, M is the magnetisation of
the amorphous matrix, h is the angle between the G local magnetization and the uniaxial anisotropy of the grain, and *h is the angle between the magnetG isation M and the exchange field H "j1M2. The G # magnetoelastic energy written in coordinate axes connected to the grains can be written as º " B aae , (6)
IJKL I J KL IJKL where a are the direction cosines of magnetisation I in grains and e are the components of the strain KL tensor in grains. After averaging º (taking into
account the conditions resulting from the minimisation of º) the anisotropy of B will be supIJKL pressed (similarly to suppression of K [3]), and the effective magnetoelastic tensor will have isotropic properties and will be determined by one component B which is a linear combination of local magnetoelastic tensor components B [25,26]: GHIJ B (H, ¹)" A (H, ¹)B . GHIJ GHIJ GHIJ
(7)
202
H. Szymczak et al. / Journal of Magnetism and Magnetic Materials 191 (1999) 199 — 202
The coefficients A are calculated according to the GHIJ procedure given in Ref. [25]. In conclusion, the magnetoelastic interactions as well as other magnetic properties of heterogenous magnetic systems are very complex and need further theoretical and experimental studies. At the moment only phenomenological models can be used in order to describe their properties. It seems that computer simulations will be essential in the description of magnetic properties of various heterogenous magnetic systems. This work was supported in part by the State Committee for Scientific Research under Grant No. 2 P03B 031 09.
References [1] J.A.C. Bland, B. Heinrich (Eds.), Ultrathin Magnetic Structures, Springer, Berlin, 1994. [2] M.N. Babich, J.M. Broto, A. Fert, F. Nguyen Van Dan, F. Petroff, P. Etienne, G. Creuzet, A. Fridrich, J. Chazelass, Phys. Rev. Lett. 61 (1988) 2472. [3] G. Herzer, in: K.H.J. Buschow (Ed.), Handbook of Magnetic Materials, Vol. 10, Elsevier, Amsterdam, 1997. [4] R. Biringer, U. Herr, H. Gleiter, Trans. Jpn. Inst. Met. 27 (1986) 43. [5] K.Y. Mulyukov, S.B. Khapizov, R.Z. Valiev, Phys. Stat. Sol. (a) 133 (1992) 447. [6] W.J.M. de Jonge, P.J.H. Bloemen, F.J.A. den Broeder, in: J.A.C. Bland, B. Heinrich (Eds.), Ultrathin Magnetic Structures, Springer, Berlin, 1994, p. 65. [7] F. Bodker, S. Morup, S. Linderoth, Phys. Rev. Lett. 72 (1994) 282.
[8] B.J. Hickey, M.A. Hawson, D. Greig, N. Wiser, Phys. Rev. B 53 (1996) 32. [9] H. Szymczak, J. Appl. Phys. 81 (1997) 5411. [10] A. S¨lawska-Waniewska, R. Z uberek, J. Gonzalez, H. Szymczak, Rapidly Quenched and Metastable Materials (Suppl.) Elsevier, Amsterdam, 1997, pp. 220—223. [11] P. Garcia-Tello, J.M. Blanco, R. Z uberek, N. Murillo, J. Gonzalez, A. S¨lawska-Waniewska, J.M. Gonzalez, IEEE Trans. Magn. MAG-33 (1997) 3919. [12] A. S¨lawska-Waniewska, A. Roig, E. Molins, J.M. Greneche, R. Z uberek, J. Appl. Phys. 81 (1997) 4652. [13] C.H. Back, W. Weber, Ch. Wursch, A. Bishof, D. Pescia, R. Allenspach, J. Appl. Phys. 81 (1997) 5054. [14] L. Ne´el, J. Phys. Radium 15 (1954) 225. [15] J.A. Thornton, D.W. Hoffman, Thin Solid Films 171 (1989) 5. [16] A. S¨lawska-Waniewska, R. Z uberek, P. Nowicki, J. Magn. Magn. Mater. 157—158 (1996) 147. [17] R. Z uberek, T. Szumiata, A. S¨lawska-Waniewska, H. Szymczak, in: R. Rivas, M.A. Lopez-Quintela (Eds.), Noncrystalline and Nanoscale Materials, World Scientific, Singapore, 1998, pp. 505—510. [18] T. Szumiata, R. Z uberek, J. Gonzalez, A. S¨lawskaWaniewska, H. Szymczak, J. Magn. Magn. Mater. 191 (1998) this Proceedings. [19] G.M. Genkin, M.V. Sapoznikov, Appl. Phys. Lett. 64 (1994) 794. [20] P. Vargas, D. Altbir, J. Magn. Magn. Mater. 167 (1997) 161. [21] Yu.P. Irkhin, V.P. Karpenko, Fiz. Tverd. Tela 13 (1971) 3586. [22] K. Twardowski, H.K. Lachowicz, M. Gutowski, H. Szymczak, Phys. Stat. Sol. (a) 63 (1981) 103. [23] J.F. Herbst, T.W. Capehart, F.E. Pinkerton, Appl. Phys. Lett. 70 (1997) 3041. [24] Ce-Wen Nan, Appl. Phys. Lett. 72 (1998) 2897. [25] H. Szymczak, R. Z uberek, J. Phys. F 12 (1982) 1841. [26] H. Szymczak, R. Z uberek, J. Magn. Magn. Mater. 35 (1983) 149.
Mechanisms responsible for magnetostriction in heterogeneous magnetic systems H. Szymczak *, R. Z uberek , J. Gonzalez Institute of Physics, Polish Academy of Sciences, Al. Lotniko& w 32, 02-668 Warsaw, Poland Departamento de Fisica de Materiales, Facultad de Quimicas, Universidad del Pais Vasco, Paseo Manuel de Lardizabal 3, 20009 San Sebastian, Spain Received 6 June 1999; received in revised form 8 August 1998
Abstract Experimental and theoretical research on magnetoelastic properties of heterogeneous magnetic materials is reviewed. Particular attention is devoted to granular and nanocrystalline systems. It is shown that localisation of magnetoelastic strains at the interface leads to the dependence of effective magnetostriction on the size of magnetic inhomogeneities (radius of grains, thickness of magnetic sublayers, etc.). Several phenomenological models of magnetostriction are presented and analysed in detail. 1999 Elsevier Science B.V. All rights reserved. PACS: 75.80; 75.70; 75.70.c; 75.50K Keywords: Heterogeneous magnetic systems; Nanocrystalline materials; Magnetostriction; Surface magnetostriction; Magnetoelastic properties
Heterogenous magnetic materials, which include nanoscale magnetic films and multilayers, various granular systems, as well as various magnetic composites have been a subject of interest in recent years. The motivation for the studies of heterogenous nanoscale systems is twofold. First, one is interested in the novel overall properties of these materials which are determined by their high density of topological defects. The amount of the disorder in nanometre-scale systems is so high that the interfacial atoms having neither short range order nor long range order constitute in fact a new phase of condensed matter. A second reason to study
* Corresponding author. Tel.: #48-22-843-68-71; fax: #4822-843-26-09; e-mail: [email protected].
these systems is related to their interesting potential applications in many branches of high technology. Nanoscale materials offer a new opportunity for tailoring metallic, magnetic and semiconductor materials. Progress in this field is made mainly due to a tremendous improvement in the production and characterisation of nanoscale systems. Among various heterogenous systems magnetic nanometre-scale materials have been receiving the greatest interest. Nanoscale magnetic multilayers are used in erasable magneto-optic and perpendicular recording (see Ref. [1] for details and references). The discovery of giant magnetoresistance [2] has opened the possibility to apply these types of materials in magnetic devices such as magnetoresistive heads and sensors. Recently very good soft magnetic materials with high initial magnetic
0304-8853/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 3 3 5 - 7
200
H. Szymczak et al. / Journal of Magnetism and Magnetic Materials 191 (1999) 199 — 202
permeability and low coercivity have been found (see Ref. [3] for details and references). These materials consist of nanosize magnetic granules embedded in an amorphous matrix. All of these materials exhibit unusual properties which ordinary solids do not have, such as the reduction of saturation magnetisation (nanocrystalline iron [4] and nickel [5]), surface magnetic anisotropy (see Ref. [6] and references therein for thin films and Refs. [7,8] for nanoparticles), surface magnetostrictive strains (see Ref. [9] and references therein for thin films and Refs. [10—12] for nanoparticles), magnetic anisotropy oscillations [13] and many others. Explanation of the phenomena in these magnetic nanometre-scale systems has generally relied on the specific properties (due to the reduced symmetry) of atoms at the interfaces. It was Ne´el [14] who first predicted that an ideal surface (or interface) could exhibit this special type of behaviour.The above-mentioned excellent soft magnetic properties of nanocrystalline materials are attributed to the absence (or very small value) of magnetostriction. The magnetostriction is also responsible for magneto-optic and perpendicular recording applications of magnetic multilayers. Generally, magnetostriction in heterogenous magnetic systems is expected to play a particularly important role by its influence on various (intrinsic and extrinsic) properties. The importance of this problem results from the fact [15] that all nanometre-scale metallic systems are in a state of stress. This is mainly due to differences in the thermal expansion coefficients of the contacting materials and also due to differences in their lattice parameters. We are going to sum up in this review the results of experimental and theoretical investigations of magnetostriction in heterogenous magnetic systems. The review will be focused on the granular systems (mainly nanocrystalline alloys) because only little attention has been paid in the literature to the mechanisms responsible for magnetostriction in these materials. Magnetoelastic energy, simply related to the magnetostriction constants, can be written in the following form in the coordinate axes connected to the local anisotropy. º " B aGaGeG , KC IJKL I J KL
(1)
Fig. 1. Effective magnetostriction of Fe Zr B Cu nanocrys talline alloys versus crystalline fraction p. Solid line is the fitting curve (according to Eq. (3)).
where aG are the direction cosines of the local I magnetisation and eG are the components of the KL local strain tensor. It has been shown, both experimentally and theoretically (see Ref. [9] for references) that the magnetoelastic tensor [B] in nanoscale magnetic multilayers consists of two parts: [B]"[B] #2t\[B] (2) where [B] describes bulk (volume) magnetoelastic coupling and [B] describes surface magnetoelastic coupling (t is the thickness of magnetic layers). Recently, the experimental dependencies of the magnetostriction constant on the volume fraction of the crystalline phase, measured both in FeCuNbSiB and in FeZrBCu nanocrystalline alloys, have also been interpreted in terms of surface and bulk contributions to the effective magnetostriction [10,12,16]. As an example the experimental results of the effective saturation magnetostriction j of Fe Zr B Cu nanocrystalline alloy as a function of the volume fraction of the crystalline phase p are shown in Fig. 1. The solid line in Fig. 1 describes the j(p) proposed in Refs. [10,12,16]: 3j j(p)"pj#(1!p)(j #kp)#p , (3) R where j and j are the magnetostriction constants of the grains and amorphous matrix,
H. Szymczak et al. / Journal of Magnetism and Magnetic Materials 191 (1999) 199 — 202
respectively; j is the magnetostriction constant which describes the interface contribution; R is the effective radius of the grains. The calculations have been performed for j and j known from Ref. [10]. The surface magnetostriction was found to be j"10.9;10\ nm. The important feature of the results presented is the different signs of j and j which lead to relatively low values of j. It can be seen from Eq. (3) that the behaviour of j(p) is more complicated than a simple superposition of crystalline and amorphous contributions (as it is usually assumed [3]) because of the important role of the surface effects. The existence of the surface contribution to the effective magnetostriction of nanocrystalline alloys has been confirmed theoretically in terms of the dipolar model [17,18]. It has been shown that, due to limited radius of nanoparticles, additional magnetoelastic stresses are localised at the interface. This localisation effect leads to the R\ term in effective magnetostriction of the system in accordance with experimental data (see Eq. (3)). The pure magnetic dipolar interactions between grains are not expected to contribute to the effective magnetostriction of nanocrystalline alloys. But recently [19,20] it has been shown that RKKY exchange is responsible for interactions between small metallic spheres. Since RKKY exchange has, generally, an anisotropic (pseudodipolar) part [21] one should expect that the intergrain interactions have, in addition to the symmetric RKKY term, a pseudodipolar contribution. Such a term has been observed in Gd—Co amorphous films [22]. The anisotropic RKKY interactions in nanometer-scale systems explain, in a natural way, the oscillations of magnetic anisotropy in nanoscale magnetic films [13] and should contribute to magnetostriction in heterogenous magnetic systems. Another interesting case has been considered in Ref. [23]. They have studied the effective magnetostriction of a system consisting of highly magnetostrictive spherical particles embedded in a nonmagnetostrictive (or low-magnetostrictive) matrix. Both components of such composite are elastically and magnetoelastically isotropic. The effective magnetostriction of the composite can be
201
written as p j"j (1.15!1.26p!0.21p ), 1!1.25p
(4)
where p is the Poisson’s ratio for low-magnetostrictive component. Further development of this problem, based on Green’s function method was given recently by Ce-Wen Nan [24]. The strong suppression of local magneto-crystalline anisotropy in nanocrystalline magnetic alloys has been observed experimentally and confirmed theoretically in terms of the random anisotropy model [3]. The same model describes magnetostriction in amorphous alloys [25,26]. This model can easily be adapted to the case of granular systems. The energy density º of magnetic grains embedded in an amorphous magnetic matrix can be written in the following way: º"K sin h !j1M 2 M cos *h , (5) G
G G G G where K is the uniaxial anisotropy of grains, M is G their magnetisation, M is the magnetisation of
the amorphous matrix, h is the angle between the G local magnetization and the uniaxial anisotropy of the grain, and *h is the angle between the magnetG isation M and the exchange field H "j1M2. The G # magnetoelastic energy written in coordinate axes connected to the grains can be written as º " B aae , (6)
IJKL I J KL IJKL where a are the direction cosines of magnetisation I in grains and e are the components of the strain KL tensor in grains. After averaging º (taking into
account the conditions resulting from the minimisation of º) the anisotropy of B will be supIJKL pressed (similarly to suppression of K [3]), and the effective magnetoelastic tensor will have isotropic properties and will be determined by one component B which is a linear combination of local magnetoelastic tensor components B [25,26]: GHIJ B (H, ¹)" A (H, ¹)B . GHIJ GHIJ GHIJ
(7)
202
H. Szymczak et al. / Journal of Magnetism and Magnetic Materials 191 (1999) 199 — 202
The coefficients A are calculated according to the GHIJ procedure given in Ref. [25]. In conclusion, the magnetoelastic interactions as well as other magnetic properties of heterogenous magnetic systems are very complex and need further theoretical and experimental studies. At the moment only phenomenological models can be used in order to describe their properties. It seems that computer simulations will be essential in the description of magnetic properties of various heterogenous magnetic systems. This work was supported in part by the State Committee for Scientific Research under Grant No. 2 P03B 031 09.
References [1] J.A.C. Bland, B. Heinrich (Eds.), Ultrathin Magnetic Structures, Springer, Berlin, 1994. [2] M.N. Babich, J.M. Broto, A. Fert, F. Nguyen Van Dan, F. Petroff, P. Etienne, G. Creuzet, A. Fridrich, J. Chazelass, Phys. Rev. Lett. 61 (1988) 2472. [3] G. Herzer, in: K.H.J. Buschow (Ed.), Handbook of Magnetic Materials, Vol. 10, Elsevier, Amsterdam, 1997. [4] R. Biringer, U. Herr, H. Gleiter, Trans. Jpn. Inst. Met. 27 (1986) 43. [5] K.Y. Mulyukov, S.B. Khapizov, R.Z. Valiev, Phys. Stat. Sol. (a) 133 (1992) 447. [6] W.J.M. de Jonge, P.J.H. Bloemen, F.J.A. den Broeder, in: J.A.C. Bland, B. Heinrich (Eds.), Ultrathin Magnetic Structures, Springer, Berlin, 1994, p. 65. [7] F. Bodker, S. Morup, S. Linderoth, Phys. Rev. Lett. 72 (1994) 282.
[8] B.J. Hickey, M.A. Hawson, D. Greig, N. Wiser, Phys. Rev. B 53 (1996) 32. [9] H. Szymczak, J. Appl. Phys. 81 (1997) 5411. [10] A. S¨lawska-Waniewska, R. Z uberek, J. Gonzalez, H. Szymczak, Rapidly Quenched and Metastable Materials (Suppl.) Elsevier, Amsterdam, 1997, pp. 220—223. [11] P. Garcia-Tello, J.M. Blanco, R. Z uberek, N. Murillo, J. Gonzalez, A. S¨lawska-Waniewska, J.M. Gonzalez, IEEE Trans. Magn. MAG-33 (1997) 3919. [12] A. S¨lawska-Waniewska, A. Roig, E. Molins, J.M. Greneche, R. Z uberek, J. Appl. Phys. 81 (1997) 4652. [13] C.H. Back, W. Weber, Ch. Wursch, A. Bishof, D. Pescia, R. Allenspach, J. Appl. Phys. 81 (1997) 5054. [14] L. Ne´el, J. Phys. Radium 15 (1954) 225. [15] J.A. Thornton, D.W. Hoffman, Thin Solid Films 171 (1989) 5. [16] A. S¨lawska-Waniewska, R. Z uberek, P. Nowicki, J. Magn. Magn. Mater. 157—158 (1996) 147. [17] R. Z uberek, T. Szumiata, A. S¨lawska-Waniewska, H. Szymczak, in: R. Rivas, M.A. Lopez-Quintela (Eds.), Noncrystalline and Nanoscale Materials, World Scientific, Singapore, 1998, pp. 505—510. [18] T. Szumiata, R. Z uberek, J. Gonzalez, A. S¨lawskaWaniewska, H. Szymczak, J. Magn. Magn. Mater. 191 (1998) this Proceedings. [19] G.M. Genkin, M.V. Sapoznikov, Appl. Phys. Lett. 64 (1994) 794. [20] P. Vargas, D. Altbir, J. Magn. Magn. Mater. 167 (1997) 161. [21] Yu.P. Irkhin, V.P. Karpenko, Fiz. Tverd. Tela 13 (1971) 3586. [22] K. Twardowski, H.K. Lachowicz, M. Gutowski, H. Szymczak, Phys. Stat. Sol. (a) 63 (1981) 103. [23] J.F. Herbst, T.W. Capehart, F.E. Pinkerton, Appl. Phys. Lett. 70 (1997) 3041. [24] Ce-Wen Nan, Appl. Phys. Lett. 72 (1998) 2897. [25] H. Szymczak, R. Z uberek, J. Phys. F 12 (1982) 1841. [26] H. Szymczak, R. Z uberek, J. Magn. Magn. Mater. 35 (1983) 149.