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Analysis of switching times of inhomogeneous magnetization processes in thin platelets
ARTICLE IN PRESS
Physica B 372 (2006) 273–276 www.elsevier.com/locate/physb
Analysis of switching times of inhomogeneous magnetization processes in thin platelets D. Goll, G. Schu¨tz, H. Kronmu¨ller Max-Planck-Institut fu¨r Metallforschung, Heisenbergstr. 3, D-70569 Stuttgart, Germany
Abstract High-density magnetic recording requires large switching fields and short sub-ns switching times of single-domain particles. Switching times are shown to depend on sample size and geometry, the magnetic material parameters and type and strength of applied fields. Numerical solutions of the Landau–Lifshitz–Gilbert equation show a minimum of the switching times as a function of the damping parameter. With increasing magnetic field an increase of switching times is observed for small damping constants and a decrease for large ones. r 2005 Published by Elsevier B.V. PACS: 75.40.Mg; 75.50.Bb; 75.60.Jk Keywords: Numerical micromagnetism; Thin platelets; Switching times
1. Introduction The basic problems of high-density magnetic recording are the tailoring of desired magnetic properties and the thermostability of the written information on a long time scale. Concerning the magnetic properties large magnetocrystalline uniaxial anisotropies allow the use of nanodots, i.e., large recording densities and guarantee thermostability at finite temperatures. Large switching fields ðm0 H c 40:1 TÞ may be overcome by high writing temperatures or the use of bilayers with a metamagnetic transition from an antiferromagnetic to a soft ferromagnetic layer [1–4]. Another important property is the so-called switching time which describes the time interval required to reverse the magnetization by 90 and which should be in the sub-ns region ðo0:1 nsÞ. It turns out that the switching time depends on the magnetic material parameters as the spontaneous polarization, J s ¼ m0 M s , the magnetocrystalline anisotropy constant, K 1 , the exchange constant, A, and the damping constant, a. But also the particles shape and Corresponding author. Tel.: +49 711 689 1910, +49 711 689 1911; fax: +49 711 689 1912. E-mail address: [email protected] (H. Kronmu¨ller).
0921-4526/$ - see front matter r 2005 Published by Elsevier B.V. doi:10.1016/j.physb.2005.10.065
dimensions as well as the strength of the applied magnetic field play an important role. It is the aim of this paper to determine the conditions under which minimum switching times may be realized for hard or middle-hard magnetic materials. 2. Micromagnetic equations The micromagnetic background for the investigation of the dynamics of magnetization processes is based on the socalled Landau–Lifshitz–Gilbert (LLG) equation [5–7] dj 1 a ¼ ½j heff ½j ½j heff , (1) 2 dt 1þa 1 þ a2 with j ¼ J=J s , heff ¼ Heff =H K , H K ¼ 2K 1 =J s and t ¼ tgH K . Here, t denotes the time, g the gyromagnetic ratio and H eff ¼ ð1=J s Þq=qj where ¼ A þ K þ S þ H is the sum of exchange, magnetocrystalline, stray field and Zeeman energy. Analytical solutions of the LLG equation in general are determined for homogeneous precession or periodic spin wave-type excitations. The switching of magnetization in spherical or platelet-like particles cannot be adequately described by these processes [8]. Therefore, numerical methods have to be applied. In particular the finite element method (FEM) has been found to be rather
ARTICLE IN PRESS D. Goll et al. / Physica B 372 (2006) 273–276
274
effective to investigate the dynamics of magnetization processes [5–7]. This method starts from a discretization of the total volume of the particle into tetrahedral elements by Delaunay triangulation followed by refinements of the grid in regions where the magnetization varies its orientation on length scales comparable to the smallest exchange length [9]. At the nodes of the finite elements the orientation of J and the potential U are calculated by solving the LLG equation where the stray field follows from Brown’s functional by maximizing the scalar potential U [10]. Within the tetrahedral elements J and U are linearly interpolated [9]. Discretization of time starts from a splitting of the LLG equation into a precessional and a relaxation term. Within the framework of spherical coordinates—azimuthal and polar angles j and y—the changes Dj and Dy after a timestep t to t þ d t are given by Dj ¼ jheff j
Dt ; 1 þ a2
Dy ¼ jm heff j
aDt . 1 þ a2
(2)
In the following we neglect thermal fluctuations which means that we determine the upper bound of the switching times at T ¼ 0 K.
calculations the magnetic field instantaneously applied corresponds to m0 H K ¼ 5:34 T for Nd2 Fe14 B and to m0 H K ¼ 0:31 T for Co. The corresponding distributions of Js at t ¼ ts are shown in Fig. 1 for Nd2 Fe14 B platelets with in-plane and out-of-plane anisotropy axis and aspect ratios 0.02 and 0.5. Starting from the homogeneous state the reversal process of J s starts in all cases at the corners of the platelets as a consequence of the enhanced stray fields at the corners of the platelets. The switching process proceeds by the displacement of Ne´el-type domain walls moving from the edges into the interior of the platelet leading finally to the fully reversed magnetic state. Due to the misalignment of the initial state the reversal starts for small aspect ratios at two diagonal corners where the demagnetizing field of the surface charges exerts the strongest torque on J s . Therefore, the distribution of J s reveals a diagonal asymmetry. In the case of large aspect ratios the situation is different. Due to the larger magnetic stray fields at the corners the reversal process starts at all four corners of the square block. 4. Switching times of inhomogeneous magnetization processes
3. Inhomogeneous switching processes In the following we consider the magnetization process in square platelets of edge length 12.5 nm and varying thickness with aspect ratios 0.02 (ultrathin film) to 1 (cubic particle). The materials investigated are Nd2 Fe14 B and Co both at room temperature with the following material parameters: Nd2 Fe14 B: K 1 ¼ 4:3 MJ=m3 , J s ¼ 1:61 T, A ¼ 7:7 pJ=m; Co: K 1 ¼ 0:4 MJ=m3 , J s ¼ 1:8 T, A ¼ 13 pJ=m. The corresponding exchange lengths are for Nd2 Fe14 B l K ¼ 1:3 nm and ! s ¼ 2:7 nm and for Co l K ¼ 5:7 nm and l s ¼ 3:2 nm (l K ¼ ðA=K 1 Þ1=2 , l s ¼ ð2m0 A=J 2s Þ1=2 ). In Fig. 1, the distribution of magnetization is shown for the switching time, ts , where the average component hJ x i in field direction becomes zero. The platelets with an easy direction parallel to the x-axis originally are magnetized homogeneously at the beginning ðt ¼ 0Þ with a misalignment angle of 5 with respect to the easy direction in order to guarantee a finite magnetic torque if the magnetic field is applied antiparallel to the easy direction. For these
The switching time, ts , is defined as the time elapsed until the magnetization component in field direction becomes zero for the first time after applying an inverse magnetic field with respect to the original direction of magnetization. In general, two extreme cases have to be distinguished. In case of large damping constants, a ¼ 1, the reversal of J s takes place monotonously with a continuously decreasing J x component approaching J x ¼ 0 at the well-defined switching time, ts . In contrast, at small damping parameters, a51, the reversal of magnetization takes place by oscillating, known as the so-called ringing, intersecting J x ¼ 0 many times, however, with decreasing amplitude for a40. In Fig. 2 switching times as a function of a are represented for Nd2 Fe14 B-based square platelets with inplane and out-of-plane anisotropy direction for two aspect ratios p ¼ 0:02 and 0.5. In all cases there exists a damping constant, amin , for which the switching time shows a minimum. For values a4amin the switching process
Fig. 1. Magnetization distribution for Nd2 Fe14 B films for the switching times at hJ x i ¼ 0. From left to right: in-plane anisotropy/p ¼ 0:02, in-plane anisotropy/p ¼ 0:5, out-of-plane-anisotropy/p ¼ 0:02, out-of-plane-anisotropy/p ¼ 0:5.
ARTICLE IN PRESS D. Goll et al. / Physica B 372 (2006) 273–276
Nd2Fe14B (in plane)
275
Nd2Fe14B (out of plane)
ts [ps]
p = 0.02
p = 0.5
p = 0.5 p = 0.02
HHk Fig. 5. Field dependence of ts of Co for p ¼ 0:02 and a ¼ 0:02 and 1. Fig. 2. Switching times ts as a function of a for Nd2 Fe14 B and aspect ratios p ¼ 0:02 and 0.5.
0.8 αmin
α
0.6
tsmin min
ts
0.4
(a)
αmin
in-plane out-of-plane 0.0
Fig. 3. Minimum switching times tmin and amin as a function of p for s Nd2 Fe14 B with in-plane and out-of-plane easy direction.
250 (b) 200
ts [ps]
p = 0.5 150
100 p = 0.02 50 (c)
Co (in-plane) 0
0
0.2
0.4
0.6
0.8
Fig. 6. J x ðtÞ of Co in the three stages of the field dependence.
1
Fig. 4. Switching times ts as a function of a for Co and aspect ratios p ¼ 0:02 and 0.5.
corresponds to a monotonous aperiodic relaxation process with switching times increasing with increasing a. For values aoamin the switching time again increases with decreasing a because now the reversal process is governed by the precessional process with (many) precessions before
reversal takes place which leads to a delay in time. The min minimum reversal time, tmin show characteristic s , and a dependencies on the parameter p (Fig. 3). Fig. 4 presents the switching times for square Co platelets with in-plane anisotropy direction as a function of a. In contrast to the results obtained for Nd2 Fe14 B, the ts ðaÞ-relation does not show a minimum for p ¼ 0:02. The switching time increases monotonously with increasing a showing its minimum
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value for a ! 0. With increasing parameters p the switching time changes its character revealing for p ¼ 0:5 a minimum of ts at around a ¼ 0:6. The missing minimum of ts for middle-hard (and also soft) magnetic materials as Co in the case of small damping parameters and small aspect ratios is due to the ringing process which starts for Co right after and for Nd2 Fe14 B already before reaching the switching time. 5. Field dependence of switching times All preceding results were obtained for applied fields H K ¼ 2K 1 =J s . There exists however a well-defined dependence of ts for instantaneously applied constant fields of varying strengths. As for large applied fields (HX1:5H K ), the switching process takes place by (quasi-)homogeneous rotation, in the following the LLG equation is solved analytically. Since Co shows the smallest values of the switching time for thin films with small damping constant (see Fig. 4), the field dependence of ts has been determined for p ¼ 0:02 and a ¼ 0:02 and 1, respectively. The obtained results are represented in Fig. 5. Altogether for small avalues we may distinguish three stages of ts as a function of H. In stage I up to H=H K ¼ 2 the switching time ts decreases. In stage II ts increases with increasing field and approaches a maximum value at H=H K 4. For even larger fields (stage III) ts decreases monotonously over the whole field range. It is of interest to note that Nd2 Fe14 B shows a qualitatively similar field dependence of ts as Co. The existence of three field stages for small a-values is explained by the different types of ringing modes as shown in Fig. 6.
6. Conclusions In the case of Nd2 Fe14 B the reversal process starts at the corners and further proceeds by the displacement of a domain wall from the corners into the center of the platelet. Due to the larger exchange length of Co here no domain walls are formed for thin films and reversal takes place by quasihomogeneous rotation. In NdFeB and Co switching times are obtained in the sub-ns range with minimum values of ts for a-values around 0.5. In the case of Co, the minimum ts is found for small p-values and very small avalues, whereas in the case of NdFeB, below the minimum ts increases for decreasing a-values. The dependence of ts on the applied magnetic field is characterized by three stages for small a-values. With increasing field in stage I, ts decreases up to H=H K ¼ 2, then increases up to a maximum at around H=H K ¼ 4 and then again decreases smoothly. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
D. Weller, A. Moser, IEEE Trans. Magn. 35 (1999) 4423. S. Cumpson, P. Hidding, IEEE Trans. Magn. 36 (2000) 2271. A. Lyberatos, K.Y. Guslienko, J. Appl. Phys. 94 (2003) 1119. K.Y. Guslienko, O. Chubykalo-Fesenko, O. Myrasov, R. Chantrell, D. Weller, Phys. Rev. B 70 (2004) 104405. T.L. Gilbert, Phys. Rev. 100 (1955) 1243. T. Leineweber, H. Kronmu¨ller, J. Magn. Magn. Mater. 192 (1999) 575; Physica B 275 (2000) 5. J. Fidler, T. Schrefl, W. Scholz, D. Suess, V.D. Tsiantos, Physica B 306 (2001) 216. R. Kikuchi, J. Appl. Phys. 27 (1956) 1352. R. Hertel, H. Kronmu¨ller, J. Magn. Magn. Mater. 238 (2002) 185. W.F. Brown Jr., Magnetostatic Principles in Ferromagnetism, NorthHolland, Amsterdam, 1962.
Physica B 372 (2006) 273–276 www.elsevier.com/locate/physb
Analysis of switching times of inhomogeneous magnetization processes in thin platelets D. Goll, G. Schu¨tz, H. Kronmu¨ller Max-Planck-Institut fu¨r Metallforschung, Heisenbergstr. 3, D-70569 Stuttgart, Germany
Abstract High-density magnetic recording requires large switching fields and short sub-ns switching times of single-domain particles. Switching times are shown to depend on sample size and geometry, the magnetic material parameters and type and strength of applied fields. Numerical solutions of the Landau–Lifshitz–Gilbert equation show a minimum of the switching times as a function of the damping parameter. With increasing magnetic field an increase of switching times is observed for small damping constants and a decrease for large ones. r 2005 Published by Elsevier B.V. PACS: 75.40.Mg; 75.50.Bb; 75.60.Jk Keywords: Numerical micromagnetism; Thin platelets; Switching times
1. Introduction The basic problems of high-density magnetic recording are the tailoring of desired magnetic properties and the thermostability of the written information on a long time scale. Concerning the magnetic properties large magnetocrystalline uniaxial anisotropies allow the use of nanodots, i.e., large recording densities and guarantee thermostability at finite temperatures. Large switching fields ðm0 H c 40:1 TÞ may be overcome by high writing temperatures or the use of bilayers with a metamagnetic transition from an antiferromagnetic to a soft ferromagnetic layer [1–4]. Another important property is the so-called switching time which describes the time interval required to reverse the magnetization by 90 and which should be in the sub-ns region ðo0:1 nsÞ. It turns out that the switching time depends on the magnetic material parameters as the spontaneous polarization, J s ¼ m0 M s , the magnetocrystalline anisotropy constant, K 1 , the exchange constant, A, and the damping constant, a. But also the particles shape and Corresponding author. Tel.: +49 711 689 1910, +49 711 689 1911; fax: +49 711 689 1912. E-mail address: [email protected] (H. Kronmu¨ller).
0921-4526/$ - see front matter r 2005 Published by Elsevier B.V. doi:10.1016/j.physb.2005.10.065
dimensions as well as the strength of the applied magnetic field play an important role. It is the aim of this paper to determine the conditions under which minimum switching times may be realized for hard or middle-hard magnetic materials. 2. Micromagnetic equations The micromagnetic background for the investigation of the dynamics of magnetization processes is based on the socalled Landau–Lifshitz–Gilbert (LLG) equation [5–7] dj 1 a ¼ ½j heff ½j ½j heff , (1) 2 dt 1þa 1 þ a2 with j ¼ J=J s , heff ¼ Heff =H K , H K ¼ 2K 1 =J s and t ¼ tgH K . Here, t denotes the time, g the gyromagnetic ratio and H eff ¼ ð1=J s Þq=qj where ¼ A þ K þ S þ H is the sum of exchange, magnetocrystalline, stray field and Zeeman energy. Analytical solutions of the LLG equation in general are determined for homogeneous precession or periodic spin wave-type excitations. The switching of magnetization in spherical or platelet-like particles cannot be adequately described by these processes [8]. Therefore, numerical methods have to be applied. In particular the finite element method (FEM) has been found to be rather
ARTICLE IN PRESS D. Goll et al. / Physica B 372 (2006) 273–276
274
effective to investigate the dynamics of magnetization processes [5–7]. This method starts from a discretization of the total volume of the particle into tetrahedral elements by Delaunay triangulation followed by refinements of the grid in regions where the magnetization varies its orientation on length scales comparable to the smallest exchange length [9]. At the nodes of the finite elements the orientation of J and the potential U are calculated by solving the LLG equation where the stray field follows from Brown’s functional by maximizing the scalar potential U [10]. Within the tetrahedral elements J and U are linearly interpolated [9]. Discretization of time starts from a splitting of the LLG equation into a precessional and a relaxation term. Within the framework of spherical coordinates—azimuthal and polar angles j and y—the changes Dj and Dy after a timestep t to t þ d t are given by Dj ¼ jheff j
Dt ; 1 þ a2
Dy ¼ jm heff j
aDt . 1 þ a2
(2)
In the following we neglect thermal fluctuations which means that we determine the upper bound of the switching times at T ¼ 0 K.
calculations the magnetic field instantaneously applied corresponds to m0 H K ¼ 5:34 T for Nd2 Fe14 B and to m0 H K ¼ 0:31 T for Co. The corresponding distributions of Js at t ¼ ts are shown in Fig. 1 for Nd2 Fe14 B platelets with in-plane and out-of-plane anisotropy axis and aspect ratios 0.02 and 0.5. Starting from the homogeneous state the reversal process of J s starts in all cases at the corners of the platelets as a consequence of the enhanced stray fields at the corners of the platelets. The switching process proceeds by the displacement of Ne´el-type domain walls moving from the edges into the interior of the platelet leading finally to the fully reversed magnetic state. Due to the misalignment of the initial state the reversal starts for small aspect ratios at two diagonal corners where the demagnetizing field of the surface charges exerts the strongest torque on J s . Therefore, the distribution of J s reveals a diagonal asymmetry. In the case of large aspect ratios the situation is different. Due to the larger magnetic stray fields at the corners the reversal process starts at all four corners of the square block. 4. Switching times of inhomogeneous magnetization processes
3. Inhomogeneous switching processes In the following we consider the magnetization process in square platelets of edge length 12.5 nm and varying thickness with aspect ratios 0.02 (ultrathin film) to 1 (cubic particle). The materials investigated are Nd2 Fe14 B and Co both at room temperature with the following material parameters: Nd2 Fe14 B: K 1 ¼ 4:3 MJ=m3 , J s ¼ 1:61 T, A ¼ 7:7 pJ=m; Co: K 1 ¼ 0:4 MJ=m3 , J s ¼ 1:8 T, A ¼ 13 pJ=m. The corresponding exchange lengths are for Nd2 Fe14 B l K ¼ 1:3 nm and ! s ¼ 2:7 nm and for Co l K ¼ 5:7 nm and l s ¼ 3:2 nm (l K ¼ ðA=K 1 Þ1=2 , l s ¼ ð2m0 A=J 2s Þ1=2 ). In Fig. 1, the distribution of magnetization is shown for the switching time, ts , where the average component hJ x i in field direction becomes zero. The platelets with an easy direction parallel to the x-axis originally are magnetized homogeneously at the beginning ðt ¼ 0Þ with a misalignment angle of 5 with respect to the easy direction in order to guarantee a finite magnetic torque if the magnetic field is applied antiparallel to the easy direction. For these
The switching time, ts , is defined as the time elapsed until the magnetization component in field direction becomes zero for the first time after applying an inverse magnetic field with respect to the original direction of magnetization. In general, two extreme cases have to be distinguished. In case of large damping constants, a ¼ 1, the reversal of J s takes place monotonously with a continuously decreasing J x component approaching J x ¼ 0 at the well-defined switching time, ts . In contrast, at small damping parameters, a51, the reversal of magnetization takes place by oscillating, known as the so-called ringing, intersecting J x ¼ 0 many times, however, with decreasing amplitude for a40. In Fig. 2 switching times as a function of a are represented for Nd2 Fe14 B-based square platelets with inplane and out-of-plane anisotropy direction for two aspect ratios p ¼ 0:02 and 0.5. In all cases there exists a damping constant, amin , for which the switching time shows a minimum. For values a4amin the switching process
Fig. 1. Magnetization distribution for Nd2 Fe14 B films for the switching times at hJ x i ¼ 0. From left to right: in-plane anisotropy/p ¼ 0:02, in-plane anisotropy/p ¼ 0:5, out-of-plane-anisotropy/p ¼ 0:02, out-of-plane-anisotropy/p ¼ 0:5.
ARTICLE IN PRESS D. Goll et al. / Physica B 372 (2006) 273–276
Nd2Fe14B (in plane)
275
Nd2Fe14B (out of plane)
ts [ps]
p = 0.02
p = 0.5
p = 0.5 p = 0.02
HHk Fig. 5. Field dependence of ts of Co for p ¼ 0:02 and a ¼ 0:02 and 1. Fig. 2. Switching times ts as a function of a for Nd2 Fe14 B and aspect ratios p ¼ 0:02 and 0.5.
0.8 αmin
α
0.6
tsmin min
ts
0.4
(a)
αmin
in-plane out-of-plane 0.0
Fig. 3. Minimum switching times tmin and amin as a function of p for s Nd2 Fe14 B with in-plane and out-of-plane easy direction.
250 (b) 200
ts [ps]
p = 0.5 150
100 p = 0.02 50 (c)
Co (in-plane) 0
0
0.2
0.4
0.6
0.8
Fig. 6. J x ðtÞ of Co in the three stages of the field dependence.
1
Fig. 4. Switching times ts as a function of a for Co and aspect ratios p ¼ 0:02 and 0.5.
corresponds to a monotonous aperiodic relaxation process with switching times increasing with increasing a. For values aoamin the switching time again increases with decreasing a because now the reversal process is governed by the precessional process with (many) precessions before
reversal takes place which leads to a delay in time. The min minimum reversal time, tmin show characteristic s , and a dependencies on the parameter p (Fig. 3). Fig. 4 presents the switching times for square Co platelets with in-plane anisotropy direction as a function of a. In contrast to the results obtained for Nd2 Fe14 B, the ts ðaÞ-relation does not show a minimum for p ¼ 0:02. The switching time increases monotonously with increasing a showing its minimum
ARTICLE IN PRESS 276
D. Goll et al. / Physica B 372 (2006) 273–276
value for a ! 0. With increasing parameters p the switching time changes its character revealing for p ¼ 0:5 a minimum of ts at around a ¼ 0:6. The missing minimum of ts for middle-hard (and also soft) magnetic materials as Co in the case of small damping parameters and small aspect ratios is due to the ringing process which starts for Co right after and for Nd2 Fe14 B already before reaching the switching time. 5. Field dependence of switching times All preceding results were obtained for applied fields H K ¼ 2K 1 =J s . There exists however a well-defined dependence of ts for instantaneously applied constant fields of varying strengths. As for large applied fields (HX1:5H K ), the switching process takes place by (quasi-)homogeneous rotation, in the following the LLG equation is solved analytically. Since Co shows the smallest values of the switching time for thin films with small damping constant (see Fig. 4), the field dependence of ts has been determined for p ¼ 0:02 and a ¼ 0:02 and 1, respectively. The obtained results are represented in Fig. 5. Altogether for small avalues we may distinguish three stages of ts as a function of H. In stage I up to H=H K ¼ 2 the switching time ts decreases. In stage II ts increases with increasing field and approaches a maximum value at H=H K 4. For even larger fields (stage III) ts decreases monotonously over the whole field range. It is of interest to note that Nd2 Fe14 B shows a qualitatively similar field dependence of ts as Co. The existence of three field stages for small a-values is explained by the different types of ringing modes as shown in Fig. 6.
6. Conclusions In the case of Nd2 Fe14 B the reversal process starts at the corners and further proceeds by the displacement of a domain wall from the corners into the center of the platelet. Due to the larger exchange length of Co here no domain walls are formed for thin films and reversal takes place by quasihomogeneous rotation. In NdFeB and Co switching times are obtained in the sub-ns range with minimum values of ts for a-values around 0.5. In the case of Co, the minimum ts is found for small p-values and very small avalues, whereas in the case of NdFeB, below the minimum ts increases for decreasing a-values. The dependence of ts on the applied magnetic field is characterized by three stages for small a-values. With increasing field in stage I, ts decreases up to H=H K ¼ 2, then increases up to a maximum at around H=H K ¼ 4 and then again decreases smoothly. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
D. Weller, A. Moser, IEEE Trans. Magn. 35 (1999) 4423. S. Cumpson, P. Hidding, IEEE Trans. Magn. 36 (2000) 2271. A. Lyberatos, K.Y. Guslienko, J. Appl. Phys. 94 (2003) 1119. K.Y. Guslienko, O. Chubykalo-Fesenko, O. Myrasov, R. Chantrell, D. Weller, Phys. Rev. B 70 (2004) 104405. T.L. Gilbert, Phys. Rev. 100 (1955) 1243. T. Leineweber, H. Kronmu¨ller, J. Magn. Magn. Mater. 192 (1999) 575; Physica B 275 (2000) 5. J. Fidler, T. Schrefl, W. Scholz, D. Suess, V.D. Tsiantos, Physica B 306 (2001) 216. R. Kikuchi, J. Appl. Phys. 27 (1956) 1352. R. Hertel, H. Kronmu¨ller, J. Magn. Magn. Mater. 238 (2002) 185. W.F. Brown Jr., Magnetostatic Principles in Ferromagnetism, NorthHolland, Amsterdam, 1962.