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Modeling of the directional dependence of the hysteresis in nanostructured thin films by ion irradiation
ARTICLE IN PRESS
Physica B 372 (2006) 33–36 www.elsevier.com/locate/physb
Modeling of the directional dependence of the hysteresis in nanostructured thin films by ion irradiation H. Hausera,, D. Bajalana, P.L. Fulmeka,b a
b
Institute of Sensor and Actuator Systems, Vienna University of Technology, Austria Christian Doppler Laboratory for Automotive Measurement Research, Gusshausstrasse 25–29/366, A-1040 Vienna, Austria
Abstract An energetic hysteresis model is applied to the prediction of the directional dependence of the hysteresis in thin film nanostructures, using the inner demagnetizing factor and the anisotropy dependence of the model parameters. The elongated ferromagnetic nanoparticles have been formed by heavy ion irradiation of paramagnetic YCo2 thin films, thus affecting both anisotropy and spontaneous magnetization. r 2005 Elsevier B.V. All rights reserved. Keywords: Nanoparticles; Ion irradiation; YCo2 thin film; Hysteresis model; Anisotropy
1. Introduction The magnetic properties of an ultra thin multilayer can be patterned by controlled ion beam irradiation [1]. The basic step in this technique is to control the changes in the magnetic properties induced by the irradiation process. It would be very useful to be able to predict the changes of the magnetic properties due to other physical quantities such as applied and residual stresses, fatigue, temperature, or irradiation. Moreover, engineering applications increasingly require the integration of a model into system design software and therefore need fast computation and efficient parameter identification strategies. Reliable magnetic modeling of the behavior of ferromagnetic materials has great impact on the field of magnetism [2–6]. In most cases the models that are proposed concentrate on a limited subset of the magnetization processes. For example, the widely used Preisach model focuses only on irreversible switching processes of an assembly of single domain switching elements called hysterons. The Stoner–Wohlfarth model focuses only on the rotational magnetization processes (both reversible and irreversible) in non-interacting single domain particles. The Corresponding author. Tel.: +43 1 58801 36614; fax: +43 1 58801 36695. E-mail address: [email protected] (H. Hauser).
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.011
models developed by Nee´l and others concentrate on domain wall dynamics. The stochastic process models of Bertotti are similarly directed towards the discontinuous motion of domain walls. The micromagnetic models based on the Landau–Lifshitz–Gilbert equation consider the reorientation of a small number of spins, by 10–15 orders of magnitude less than the numbers of spins which comprise a typical single domain in a magnetic material. In a former work the energetic model (EM) [9] was used to describe the dependence of the hysteresis on the spontaneous magnetization [10]. The model is now expanded to uniaxial anisotropy and applied to predict the directional dependence of the hysteresis in nanostructured thin films by heavy ion irradiation. The consideration of the internal demagnetizing factor and the law of approach to saturation lead to equations which allow anisotropy to be explicitly incorporated into the model parameters. 2. Experiments An assembly of ferromagnetic amorphous nanoparticles has been prepared by heavy ion irradiation of paramagnetic YCo2 thin films [7,8]. Several irradiation experiments carried out on YCo2 samples have shown that fluences on the order 1012 U ions=cm2 causes changes in magnetic
ARTICLE IN PRESS 34
H. Hauser et al. / Physica B 372 (2006) 33–36
properties of the samples. Important changes are reported to take place after the irradiation: a change of spontaneous magnetization, coercivity and initial susceptibility [8], and a distinct change of the anisotropy perpendicular to the film plane is observed [7]. 3. Magnetization curve at strong fields The Law of Approach (LA) to saturation [11–13] is used here as a reference to which the EM may be compared. The LA describes the dependence of magnetization on the applied field HbH c near to the saturation magnetization M s according to a b c M ¼ Ms 1 2 3 . (1) H H H Using the reduced magnetization m ¼ M=M s and the demagnetizing field H d ¼ N d M (where N d is a geometric demagnetizing factor), and neglecting the a=H term and the H n terms with n42 in the LA and considering the difference between the inner and external fields (H i ¼ H ext þ N d mM s ) we find for 0pmo1 rffiffiffiffiffiffiffiffiffiffiffiffi b þ N d M s m, (2) H¼ 1m where the parameter b ¼ H 2k
(3)
is related to the fictive anisotropy field H k ¼ 2K u =m0 M s (anisotropy constant K u ) and is a microscopic constant considering the crystal geometry; 0:02 in iron, for example Ref. [13]. Relating the parameters to anisotropy is considered as follows. At a certain field H a , which is a fraction n of H k , we find ma ¼ 1 n2 from Eq. (2). Without considering N d —because it does not affect the parameter relations with anisotropy—we then compare the differential susceptibility w0a ¼ dM=dH at M a with the prediction of the EM. This suggests that M a ðH a Þ is approximately the same in LA and EM, which should be satisfied for H c 5H a oH k . The reversible field contribution in the EM is [9] H r ¼ h½ðð1 þ mÞ1þm ð1 mÞ1m Þg=2 1 þ M s mN e
(4)
resulting from an ordering process with statistical domain behavior, where g and h are model parameters. The saturation field is defined as H s 2g h, where the magnetization changes that are determined by a probability function of the distribution of the domain volumes over the easy directions, e.g. domain wall displacements, are finished. It turns out that materials with larger H s exhibit stronger reversible impediments of domain wall displacements due to inner stray fields, for example. This will also affect the slope of the ideal magnetization curve close to the origin (or the slope of the hysteresis curve close to coercivity).
Using the loss parameter k and the pinning site distribution parameter q, the fictitious irreversible field is q i k h 1 k exp jm mo j , (5) Hl ¼ m0 M s k without considering the field dependence of the magnetization reversal speed, where the (old) value mo is the reduced magnetization at the last reversal point. The reversal function k ¼ 1 for the initial magnetization and k ¼ 2 for the major loop branches [9]. The hysteresis equation is finally H ¼ H r þ H l . We distinguish now between two types of demagnetizing factors. The effective demagnetizing factor is Ne ¼ Nd þ Ni
(6)
and N i is the inner demagnetizing factor, e.g., due to the magnetostatic stray fields within the microstructure of grains or particles (1=N e is the slope of the anhysteretic curve in the origin). Until now no distinction has been made between N e and N d . The differentiation with respect to m at m ¼ ma ¼ 1 n2 of Eq. (2) dH Hk ¼ þ N dM s dm 2n3 and (4)
(7)
2 dH gh 2 n2 2 ¼ ½ð2 n2 Þ2n n2 n g=2 ln þ ðN i þ N d ÞM s dm 2 n2 (8)
leads to f h 2g h þ f N N i M s ¼ H k , where f h and f N are functions of n and ; if 2g b1 we may find the linear relationships 2g h ¼ ð2g hÞc þ ch H k
(9)
and N i ¼ N c þ cN
Hk , Ms
(10)
where ð2g hÞc and N c are the values of 2g h and N i independent on H k . The microscopic constants ch and cN describe the relations of the saturation field H s 2g h of Eq. (4), where all domain processes are finished— considering H s bH c —and of N i with H k , respectively. A numerical evaluation for typical values of g and h shows a linear dependence of the reciprocal of the differential susceptibility 1=w0a versus 2g h for 2g b1 if ma is close to 1, or is small, respectively, and indicating ð2g hÞc 0. 4. Application to uniaxial anisotropy of particulate media Consider for example a two dimensional particle assembly with a distinct long axis orientation of the (interacting) particles in the x direction. The inner demagnetizing matrix of the assembly is N i ¼ ðN x ; N y ; N z Þ
(11)
ARTICLE IN PRESS H. Hauser et al. / Physica B 372 (2006) 33–36
35
with N x oN y 5N z . Then the inner shape anisotropy energy density is ws ¼
m0 M 2s ðN x cos2 j þ N y sin2 jÞ, 2
(12)
corresponding to the magnetocrystalline anisotropy energy wu ¼ K 0 þ K u sin2 j, where K 0 corresponds to N x m0 M 2s =2 and K u corresponds to ðN y N x Þm0 M 2s =2 for an angle j between M and the easy x axis in the xy plane. The losses of interacting switching particles are assumed to be proportional to the elementary hysteresis of the single particles and to their interactions, as is also the case in the Preisach models. Therefore, the anisotropy energy of long ellipsoid shaped particles (axes a; b; c with N a þ N b þ N c ¼ 1 and abb ¼ c) is m0 M 2s ðN b N a Þ=2 and with the adaptive relation cd ¼ ð1 3N a Þ=4 it follows that k ¼ cd m0 M 2s .
Fig. 1. Calculated hysteresis of YCo2 thin films at T ¼ 10 K perpendicular (solid –) and parallel (dashed - -) to the film plane after U ion irradiation with 1012 ions/cm2 ; measurements perpendicular () and parallel (D) [7] (H m ¼ 160 kA/m).
(13)
To identify the parameters k and q (which are depending on N i ) we may use H c and relation [9] q kq exp . Ni ¼ (14) 2 2 m0 M s Using Eqs. (14), (12), and (13) we find the directional dependencies q¼
N x cos2 j þ N y sin2 j q exp 2 cd
(15)
(this transcendent equation has to be solved numerically). Former considerations have shown that g ¼ cg Dwg =m0 M 2s is proportional to the energy difference Dwg which has to be overcome during a magnetization reversal. This could be used to identify g and h independently. With K 0 as a direction independent term we find cg g¼ ðK 0 þ K u cos2 jÞ, (16) m0 M 2s
Fig. 2. Calculated hysteresis of YCo2 thin films at T ¼ 10 K perpendicular (solid –) and parallel (dashed - -) to the film plane after U ion irradiation with 5 1012 ions/cm2 ; measurements perpendicular () and parallel (D) [7] (H m ¼ 160 kA/m).
and furthermore
N i;p=2 ¼ N i;0
h ¼ Ms
ch cd N d . ðcr þ 1Þ½expðg ln 2Þ 1
(17)
The identification can be done with a measurement of small j in order to obtain the approximations eq 51 and 2g b1, e.g. at j ¼ 0 yielding g0 , q0 and j1 ¼ p=2 yielding g1 , q1 . Then the inner demagnetizing factors of the particle assembly are q q (18) N x ¼ q0 cd exp 0 ; N y ¼ q1 cd exp 1 , 2 2 and we can identify cg ¼
m0 M 2s ðg0 g1 Þ Ku
(19)
and K 0 ¼ m0 M 2s
g1 . cg
(20)
The exact identification of the parameters and the microscopic constants is described in Ref. [9]. With N i;0 ¼ N i at j ¼ 0 and N i;p=2 ¼ N i at j ¼ p=2 we find K0 þ Ku K0
(21)
and we can calculate N d from N d ¼ N e;0
K0 ðN e;p=2 N e;0 Þ Ku
(22)
as N d ¼ 3:2 107 . Figs. 1 and 2 show the calculated hysteresis loops using the relations above for the fluences of 1012 and 5 1012 U ions=cm2 . The hysteresis loop of j ¼ p=4 was then calculated without any new parameter identification. Some measurement points from Ref. [7] are drawn for comparison. 5. Conclusions Ion beam modification of magnetic layers may be the possible future of ultra-high density magnetic recording media. After ion irradiation of YCo2 thin films with different fluence values, the measured magnetization curves
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H. Hauser et al. / Physica B 372 (2006) 33–36
clearly show a perpendicular shape anisotropy [7]. As the EM parameters are related to anisotropy it is possible to calculate the direction dependence of these magnetization curves. References [1] J. Ferre, C. Chappert, H. Bernas, J.-P. Jamet, P. Meyer, O. Kaitasov, S. Lemerle, V. Mathet, F. Rousseaux, H. Launois, J. Magn. Magn. Mater. 198 (1999) 191. [2] G. Bertotti, Hysteresis in Magnetism, Academic Press, London, 1998. [3] E. Della Torre, Magnetic Hysteresis, IEEE Press, Piscataway, 1999. [4] A. Iva´nyi, Hysteresis Models in Electromagnetic Computation, Akade´miai Kiado´, Budapest, 1997.
[5] A. Iva´nyi, Magnetic Field Computation with R-Functions, Akade´miai Kiado´, Budapest, 1998. [6] I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer, New York, 1991. [7] D. Givord, J.P. Nozie´res, M. Ghidini, B. Gervais, Y. Otani, J. Magn. Magn. Mater. 148 (1995) 253. [8] M. Solzi, M. Ghidini, G. Asti, Magn. Nanostructures 4 (2002) 123. [9] H. Hauser, J. Appl. Phys. 96 (2004) 2753. [10] D. Bajalan, H. Hauser, P.L. Fulmek, Physica B 343 (2004) 384. [11] N.S. Akulov, Z. Phys. 69 (1931) 822. [12] L. Nee´l, J. Phys. Rad. 9 (1948) 193. [13] S. Chikazumi, Physics of Magnetism, Wiley, New York, 1964, pp. 274–280.
Physica B 372 (2006) 33–36 www.elsevier.com/locate/physb
Modeling of the directional dependence of the hysteresis in nanostructured thin films by ion irradiation H. Hausera,, D. Bajalana, P.L. Fulmeka,b a
b
Institute of Sensor and Actuator Systems, Vienna University of Technology, Austria Christian Doppler Laboratory for Automotive Measurement Research, Gusshausstrasse 25–29/366, A-1040 Vienna, Austria
Abstract An energetic hysteresis model is applied to the prediction of the directional dependence of the hysteresis in thin film nanostructures, using the inner demagnetizing factor and the anisotropy dependence of the model parameters. The elongated ferromagnetic nanoparticles have been formed by heavy ion irradiation of paramagnetic YCo2 thin films, thus affecting both anisotropy and spontaneous magnetization. r 2005 Elsevier B.V. All rights reserved. Keywords: Nanoparticles; Ion irradiation; YCo2 thin film; Hysteresis model; Anisotropy
1. Introduction The magnetic properties of an ultra thin multilayer can be patterned by controlled ion beam irradiation [1]. The basic step in this technique is to control the changes in the magnetic properties induced by the irradiation process. It would be very useful to be able to predict the changes of the magnetic properties due to other physical quantities such as applied and residual stresses, fatigue, temperature, or irradiation. Moreover, engineering applications increasingly require the integration of a model into system design software and therefore need fast computation and efficient parameter identification strategies. Reliable magnetic modeling of the behavior of ferromagnetic materials has great impact on the field of magnetism [2–6]. In most cases the models that are proposed concentrate on a limited subset of the magnetization processes. For example, the widely used Preisach model focuses only on irreversible switching processes of an assembly of single domain switching elements called hysterons. The Stoner–Wohlfarth model focuses only on the rotational magnetization processes (both reversible and irreversible) in non-interacting single domain particles. The Corresponding author. Tel.: +43 1 58801 36614; fax: +43 1 58801 36695. E-mail address: [email protected] (H. Hauser).
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.011
models developed by Nee´l and others concentrate on domain wall dynamics. The stochastic process models of Bertotti are similarly directed towards the discontinuous motion of domain walls. The micromagnetic models based on the Landau–Lifshitz–Gilbert equation consider the reorientation of a small number of spins, by 10–15 orders of magnitude less than the numbers of spins which comprise a typical single domain in a magnetic material. In a former work the energetic model (EM) [9] was used to describe the dependence of the hysteresis on the spontaneous magnetization [10]. The model is now expanded to uniaxial anisotropy and applied to predict the directional dependence of the hysteresis in nanostructured thin films by heavy ion irradiation. The consideration of the internal demagnetizing factor and the law of approach to saturation lead to equations which allow anisotropy to be explicitly incorporated into the model parameters. 2. Experiments An assembly of ferromagnetic amorphous nanoparticles has been prepared by heavy ion irradiation of paramagnetic YCo2 thin films [7,8]. Several irradiation experiments carried out on YCo2 samples have shown that fluences on the order 1012 U ions=cm2 causes changes in magnetic
ARTICLE IN PRESS 34
H. Hauser et al. / Physica B 372 (2006) 33–36
properties of the samples. Important changes are reported to take place after the irradiation: a change of spontaneous magnetization, coercivity and initial susceptibility [8], and a distinct change of the anisotropy perpendicular to the film plane is observed [7]. 3. Magnetization curve at strong fields The Law of Approach (LA) to saturation [11–13] is used here as a reference to which the EM may be compared. The LA describes the dependence of magnetization on the applied field HbH c near to the saturation magnetization M s according to a b c M ¼ Ms 1 2 3 . (1) H H H Using the reduced magnetization m ¼ M=M s and the demagnetizing field H d ¼ N d M (where N d is a geometric demagnetizing factor), and neglecting the a=H term and the H n terms with n42 in the LA and considering the difference between the inner and external fields (H i ¼ H ext þ N d mM s ) we find for 0pmo1 rffiffiffiffiffiffiffiffiffiffiffiffi b þ N d M s m, (2) H¼ 1m where the parameter b ¼ H 2k
(3)
is related to the fictive anisotropy field H k ¼ 2K u =m0 M s (anisotropy constant K u ) and is a microscopic constant considering the crystal geometry; 0:02 in iron, for example Ref. [13]. Relating the parameters to anisotropy is considered as follows. At a certain field H a , which is a fraction n of H k , we find ma ¼ 1 n2 from Eq. (2). Without considering N d —because it does not affect the parameter relations with anisotropy—we then compare the differential susceptibility w0a ¼ dM=dH at M a with the prediction of the EM. This suggests that M a ðH a Þ is approximately the same in LA and EM, which should be satisfied for H c 5H a oH k . The reversible field contribution in the EM is [9] H r ¼ h½ðð1 þ mÞ1þm ð1 mÞ1m Þg=2 1 þ M s mN e
(4)
resulting from an ordering process with statistical domain behavior, where g and h are model parameters. The saturation field is defined as H s 2g h, where the magnetization changes that are determined by a probability function of the distribution of the domain volumes over the easy directions, e.g. domain wall displacements, are finished. It turns out that materials with larger H s exhibit stronger reversible impediments of domain wall displacements due to inner stray fields, for example. This will also affect the slope of the ideal magnetization curve close to the origin (or the slope of the hysteresis curve close to coercivity).
Using the loss parameter k and the pinning site distribution parameter q, the fictitious irreversible field is q i k h 1 k exp jm mo j , (5) Hl ¼ m0 M s k without considering the field dependence of the magnetization reversal speed, where the (old) value mo is the reduced magnetization at the last reversal point. The reversal function k ¼ 1 for the initial magnetization and k ¼ 2 for the major loop branches [9]. The hysteresis equation is finally H ¼ H r þ H l . We distinguish now between two types of demagnetizing factors. The effective demagnetizing factor is Ne ¼ Nd þ Ni
(6)
and N i is the inner demagnetizing factor, e.g., due to the magnetostatic stray fields within the microstructure of grains or particles (1=N e is the slope of the anhysteretic curve in the origin). Until now no distinction has been made between N e and N d . The differentiation with respect to m at m ¼ ma ¼ 1 n2 of Eq. (2) dH Hk ¼ þ N dM s dm 2n3 and (4)
(7)
2 dH gh 2 n2 2 ¼ ½ð2 n2 Þ2n n2 n g=2 ln þ ðN i þ N d ÞM s dm 2 n2 (8)
leads to f h 2g h þ f N N i M s ¼ H k , where f h and f N are functions of n and ; if 2g b1 we may find the linear relationships 2g h ¼ ð2g hÞc þ ch H k
(9)
and N i ¼ N c þ cN
Hk , Ms
(10)
where ð2g hÞc and N c are the values of 2g h and N i independent on H k . The microscopic constants ch and cN describe the relations of the saturation field H s 2g h of Eq. (4), where all domain processes are finished— considering H s bH c —and of N i with H k , respectively. A numerical evaluation for typical values of g and h shows a linear dependence of the reciprocal of the differential susceptibility 1=w0a versus 2g h for 2g b1 if ma is close to 1, or is small, respectively, and indicating ð2g hÞc 0. 4. Application to uniaxial anisotropy of particulate media Consider for example a two dimensional particle assembly with a distinct long axis orientation of the (interacting) particles in the x direction. The inner demagnetizing matrix of the assembly is N i ¼ ðN x ; N y ; N z Þ
(11)
ARTICLE IN PRESS H. Hauser et al. / Physica B 372 (2006) 33–36
35
with N x oN y 5N z . Then the inner shape anisotropy energy density is ws ¼
m0 M 2s ðN x cos2 j þ N y sin2 jÞ, 2
(12)
corresponding to the magnetocrystalline anisotropy energy wu ¼ K 0 þ K u sin2 j, where K 0 corresponds to N x m0 M 2s =2 and K u corresponds to ðN y N x Þm0 M 2s =2 for an angle j between M and the easy x axis in the xy plane. The losses of interacting switching particles are assumed to be proportional to the elementary hysteresis of the single particles and to their interactions, as is also the case in the Preisach models. Therefore, the anisotropy energy of long ellipsoid shaped particles (axes a; b; c with N a þ N b þ N c ¼ 1 and abb ¼ c) is m0 M 2s ðN b N a Þ=2 and with the adaptive relation cd ¼ ð1 3N a Þ=4 it follows that k ¼ cd m0 M 2s .
Fig. 1. Calculated hysteresis of YCo2 thin films at T ¼ 10 K perpendicular (solid –) and parallel (dashed - -) to the film plane after U ion irradiation with 1012 ions/cm2 ; measurements perpendicular () and parallel (D) [7] (H m ¼ 160 kA/m).
(13)
To identify the parameters k and q (which are depending on N i ) we may use H c and relation [9] q kq exp . Ni ¼ (14) 2 2 m0 M s Using Eqs. (14), (12), and (13) we find the directional dependencies q¼
N x cos2 j þ N y sin2 j q exp 2 cd
(15)
(this transcendent equation has to be solved numerically). Former considerations have shown that g ¼ cg Dwg =m0 M 2s is proportional to the energy difference Dwg which has to be overcome during a magnetization reversal. This could be used to identify g and h independently. With K 0 as a direction independent term we find cg g¼ ðK 0 þ K u cos2 jÞ, (16) m0 M 2s
Fig. 2. Calculated hysteresis of YCo2 thin films at T ¼ 10 K perpendicular (solid –) and parallel (dashed - -) to the film plane after U ion irradiation with 5 1012 ions/cm2 ; measurements perpendicular () and parallel (D) [7] (H m ¼ 160 kA/m).
and furthermore
N i;p=2 ¼ N i;0
h ¼ Ms
ch cd N d . ðcr þ 1Þ½expðg ln 2Þ 1
(17)
The identification can be done with a measurement of small j in order to obtain the approximations eq 51 and 2g b1, e.g. at j ¼ 0 yielding g0 , q0 and j1 ¼ p=2 yielding g1 , q1 . Then the inner demagnetizing factors of the particle assembly are q q (18) N x ¼ q0 cd exp 0 ; N y ¼ q1 cd exp 1 , 2 2 and we can identify cg ¼
m0 M 2s ðg0 g1 Þ Ku
(19)
and K 0 ¼ m0 M 2s
g1 . cg
(20)
The exact identification of the parameters and the microscopic constants is described in Ref. [9]. With N i;0 ¼ N i at j ¼ 0 and N i;p=2 ¼ N i at j ¼ p=2 we find K0 þ Ku K0
(21)
and we can calculate N d from N d ¼ N e;0
K0 ðN e;p=2 N e;0 Þ Ku
(22)
as N d ¼ 3:2 107 . Figs. 1 and 2 show the calculated hysteresis loops using the relations above for the fluences of 1012 and 5 1012 U ions=cm2 . The hysteresis loop of j ¼ p=4 was then calculated without any new parameter identification. Some measurement points from Ref. [7] are drawn for comparison. 5. Conclusions Ion beam modification of magnetic layers may be the possible future of ultra-high density magnetic recording media. After ion irradiation of YCo2 thin films with different fluence values, the measured magnetization curves
ARTICLE IN PRESS 36
H. Hauser et al. / Physica B 372 (2006) 33–36
clearly show a perpendicular shape anisotropy [7]. As the EM parameters are related to anisotropy it is possible to calculate the direction dependence of these magnetization curves. References [1] J. Ferre, C. Chappert, H. Bernas, J.-P. Jamet, P. Meyer, O. Kaitasov, S. Lemerle, V. Mathet, F. Rousseaux, H. Launois, J. Magn. Magn. Mater. 198 (1999) 191. [2] G. Bertotti, Hysteresis in Magnetism, Academic Press, London, 1998. [3] E. Della Torre, Magnetic Hysteresis, IEEE Press, Piscataway, 1999. [4] A. Iva´nyi, Hysteresis Models in Electromagnetic Computation, Akade´miai Kiado´, Budapest, 1997.
[5] A. Iva´nyi, Magnetic Field Computation with R-Functions, Akade´miai Kiado´, Budapest, 1998. [6] I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer, New York, 1991. [7] D. Givord, J.P. Nozie´res, M. Ghidini, B. Gervais, Y. Otani, J. Magn. Magn. Mater. 148 (1995) 253. [8] M. Solzi, M. Ghidini, G. Asti, Magn. Nanostructures 4 (2002) 123. [9] H. Hauser, J. Appl. Phys. 96 (2004) 2753. [10] D. Bajalan, H. Hauser, P.L. Fulmek, Physica B 343 (2004) 384. [11] N.S. Akulov, Z. Phys. 69 (1931) 822. [12] L. Nee´l, J. Phys. Rad. 9 (1948) 193. [13] S. Chikazumi, Physics of Magnetism, Wiley, New York, 1964, pp. 274–280.