- Email: [email protected]
Au(0 0 1) multilayers
Journal of Magnetism and Magnetic Materials 240 (2002) 79–82
Micromagnetic investigation of sub-1 0 0-nm magnetic domains in atomically stacked Fe(0 0 1)/Au(0 0 1) multilayers a, . M. Kohler *, J. Zwecka, G. Bayreuthera, P. Fischerb, G. Schutz . b, G. Denbeauxc, D. Attwoodc Universitat . Regensburg, Institut f. Exp. u. Angew. Physik, D-93040 Regensburg, Germany Universitat Lehrst. f. Exp. Phys. IV, Am Hubland, D-97074 Wurzburg, Germany . Wurzburg, . . c Lawrence Berkeley National Laboratory, MSXO, 1 Cyclotron Road, Berkeley, CA 94720, USA a
b
Abstract Continuous films and nanostructures of atomically stacked epitaxial Fe(0 0 1)/Au(0 0 1) multilayers have been studied by soft X-ray and Lorentz microscopy as well as micromagnetic simulations. Domain imaging shows about 65 nm wide magnetic stripe domains, in which the magnetization is oriented perpendicular to the film plane. These results are confirmed by micromagnetic simulations, which also yield additional information about the internal structure of domains and walls. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Ordered alloy; AnisotropyFuniaxial; Micromagnetism; Labyrinth pattern
Properties of ferromagnetic materials are governed to a large extent by their magnetic microstructure. However, no magnetic imaging method can yield complete information about the three dimensional magnetization distribution in a given sample. This restriction can, as demonstrated in Ref. [1], be overcome to a certain degree by a combination of complementary observation methods. In a previous work [2], continuous films of atomically stacked [Fe(0 0 1)/Au(0 0 1)]n monocrystalline multilayers have been studied by magnetic transmission X-ray microscopy (MTXM) [3] and Lorentz transmission electron microscopy (LTEM). These techniques complement each other very well, since for normal incidence MTXM is sensitive to the perpendicular magnetization, while Lorentz microscopy yields the inplane components of the magnetic flux. However, for a unique interpretation the model structures derived from experiment have to be compared with micromagnetic simulations, which can also yield additional information
*Corresponding author. Fax: +49-941-943-4544. E-mail address: [email protected]. burg.de (M. Kohler).
about the internal structure of domain walls not directly accessible by any imaging method. In the present work domain observations in Fe(0 0 1)/ Au(0 0 1) multilayers are extended to nanostructured films and compared to micromagnetic simulations based on measured magnetic material parameters. Films were grown by molecular beam epitaxy (MBE) at room temperature maintaining a pressure of o4.0 1010 Torr. Deposition was monitored by reflection high-energy electron diffraction (RHEED) and quartz microbalances calibrated by X-ray fluorescence analysis (RFA) and RHEED oscillations. First, a n+GaAs(0 0 1) substrate was cleaned in situ by simultaneous heating to 5801C and sputter cleaning with 500 eV Ar+-ions [4]. Then a 6 monolayers (ML) thick Fe(0 0 1) seed layer was deposited at a rate of 1.5 ML/min, followed by 20 nm of Au(0 0 1) at 2 ML/min serving as a buffer layer and later as membrane and etch stop layer in the membrane fabrication process. During the next step the multilayer system was grown by alternating the deposition of Fe(0 0 1) and Au(0 0 1) layers, each at 1.5 ML/min. Finally, the samples were covered by a 15 ML Au(0 0 1) cap layer to protect them from corrosion.
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 7 5 0 - 8
80
M. Kohler et al. / Journal of Magnetism and Magnetic Materials 240 (2002) 79–82 .
Nanostructures were prepared by first covering parts of the sample with a 40 nm Ti mask structured by a standard e-beam lithography and lift-off process. Then the sample was subjected to argon-ion etching with Auger electron depth-monitoring, which served to protect the Au membrane layer by indicating when the etching process must be stopped, i.e. when the Fe signal vanishes. X-ray and electron transparent samples were produced by etching a window into the GaAs substrates leaving the remaining film as a monocrystalline membrane. A new technique was developed to obtain windows with small lateral dimensions, thus ensuring their mechanical stability: laser-induced etching [5] was used to create a hole with high aspect ratio through the rear side of the n+-GaAs(0 0 1) substrate up to the Au(0 0 1) layer. For this purpose the sample was immersed in dilute nitric acid and a 15 mW green–blue Ar+ ion laser beam was focused to a spot of about 20 mm diameter onto the rear side of the substrate, thus locally activating the etch process. The front side of the sample was protected by clear wax which was later removed by carefully rinsing in high-purity acetone. Windows with diameters of about 40 mm could be obtained in this way. The single crystalline structure of the multilayer was verified by electron- and X-ray diffraction (XRD). Magnetization curves were measured by alternating gradient magnetometry (AGM), polar magneto-optic Kerr effect (MOKE), and magnetic anisotropy constants were determined by torque magnetometry. The saturation magnetization and exchange stiffness were derived from spin–wave excitations measured in a SQUID magnetometer. Magnetic domains were imaged with a Philips CM-30 TEM (modified for Lorentz microscopy), and the transmission soft X-ray microscope located at the Advanced Light Source, Lawrence Berkeley National Laboratory in Berkeley, CA. For [1.0 ML Fe/1.0 ML Au]n samples, electron- and X-ray diffraction (XRD) did not show any deviation from the monocrystalline L10 structure. The average vertical Fe/Au lattice spacing d(0 0 2) as determined by ( in good agreement with Ref. [6]. XRD is 1.92A, Investigations of the multilayer surface by in situ scanning tunneling microscope (STM) show a smallscale roughness caused by monoatomic steps separating 3–10 nm wide atomically flat terraces, while on a larger scale the sample appears sufficiently smooth (over a 1 mm2 scan area the typical z-range is 2 nm). Torque magnetometer measurements indicate a strong uniaxial anisotropy with an easy axis perpendicular to the film plane. Samples with slightly larger single layer thicknesses d up to 1.5 ML show a somewhat weaker anisotropy, but in all other respects behave very similarly to those with d ¼ 1:0 ML. According to Ref. [6], the uniaxial anisotropy and the vertical lattice
spacing show an oscillatory behavior with a period of 1 ML. Both have a maximum for d ¼ 1:0 ML and reach a minimum for d ¼ 1:5 ML. The measured anisotropy constants of second and fourth order are in the range Ku1 ¼ ð771Þ 105 J/m3 and Ku2 ¼ ð4:570:5Þ 104 J/m3. The saturation magnetization and exchange stiffness constant as derived from SQUID measurements are JS ¼ 1:1370:1 T (1.25 T for ( (assuming isotropic d ¼ 1:5 ML) and AE3:271 meV/A exchange interaction). Thus the Q-factor (=Ku =Kd ) for our samples is 1.470.3, which means that, since Q is close to 1, numerical micromagnetic methods are very well applicable in the present case [7], while on the other hand common domain theoretical methods (e.g. Ref. [8]), which usually either assume Q51 or Qb1, are likely to yield wrong results. Since Q is larger than 1, one does neither expect ‘‘weak’’ (or dense) stripe domains, which occur in the case of small anistropies often along with a tilted anisotropy axis, nor ‘‘strong’’ stripe domains for which the tilted axis is a prerequisite. On the other hand, the value of Q is smaller than in typical high-anisotropy perpendicular films (‘‘bubble films’’, where 1=Q can often be treated as a small quantity), thus making this system an interesting intermediate case. Fig. 1 shows a labyrinth domain pattern observed by MTXM. Since the sample was observed under perpendicular incidence of X-rays, the contrast indicates variations in the perpendicular magnetization component averaged through the sample thickness. The inplane components are best imaged by LTEM. A similar pattern can be observed by Fresnel mode Lorentz
Fig. 1. [1.5 ML Fe/1.5 ML Au]110 sample observed by MTXM at the Fe L3-edge. Domain width: 6575 nm.
M. Kohler et al. / Journal of Magnetism and Magnetic Materials 240 (2002) 79–82 .
81
Fig. 2. [1.15 ML Fe/1.0 ML Au]130 sample observed by Fresnel mode Lorentz microscopy. Beam incidence 351 from normal. Fig. 4. Cross-section through a simulated wall (perpendicular to film plane and wall). Q ¼ 1:5; finite element size 1 nm.
Fig. 3. [1.5 ML Fe/1.5 ML Au]110 nanostructures observed by MTXM at the Fe L3-edge. The thickness of the structures is approximately 38 nm. QE1:1:
microscopy in a tilted sample [2]. The image vanishes, however, for normal beam incidence, which means that no variation in the averaged in-plane magnetization components can be detected. Since almost no remanence was detected in in-plane AGM measurements, one can conclude that the bulk of the magnetic domains is oriented perpendicular to the film plane and that the observed domains can be classified as band domains characteristic for high-Q films. Despite the somewhat different anisotropy and film thicknesses of the studied samples of Figs. 1 and 2 the domain width is very similar in all cases: 6575 nm. MTXM images of nanostructures are displayed in Fig. 3. It is remarkable how little the domain pattern seems to be disturbed by the particle edges. In order to understand this behavior, micromagnetic simulations were carried out using the Scheinfein LLG Micromagnetics Simulator. Fig. 4 shows the cross-
section through a simulated domain wall. This twisted wall structure is typical for bubble films: in the center of the film, the wall is of a regular Bloch type with a singularity at its very center, where the magnetization lies in the film plane, parallel to the wall. At the top and bottom surface of the film, however, the wall gets twisted towards a Ne! el wall by stray fields: the magnetization is forced completely into the film plane above and below the center line. Obviously, there are no well-defined closure domains, but rather a continuous flux closure configuration, which is about 30 nm wide at the film surfaces. Its in-plane magnetization components exactly cancel out when averaged over the sample thickness and thus are invisible in perpendicular incidence LTEM, as is the singularity in the center of the wall, which is below 8 nm wide and has too little volume to be detected. Fig. 5 shows the different magnetization components in the center plane of a simulated 256 nm 256 nm 40 nm particle. The shape and width of the resulting band domains agree well with our experimental results. There is some variation in the domain width, which may be attributed to the fact that only a small particle, not a continuous film, has been (and can be) simulated. Looking at the in-plane components in the center of the walls one immediately notices discontinuities, where the magnetization direction relative to the wall reverses over a very small length. These can be identified as Bloch lines, which reduce the extra stray field energy of walls that intercept the particle
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M. Kohler et al. / Journal of Magnetism and Magnetic Materials 240 (2002) 79–82 .
walls that intersect particle boundaries and thus why the domain pattern is disturbed so little at the edges of the nanostructures. Financial support by the Volkswagen-Stiftung is gratefully acknowledged.
References
Fig. 5. Simulation of a 256 nm 256 nm 40 nm platelet, Q ¼ 1:5; finite element size 2 nm. Cross-section through the middle of the platelet (z ¼ 20 nm). The resulting domain width is 55715 nm. (a) Explanation of geometry and color scale, (b) mz (normal component of magnetization) (c) my (vertical component), (d) mx (horizontal component).
edges because the longitudinal net flux carried by the walls is reduced. The formation of Bloch lines help to explain why there is almost no energetic penalty for
[1] R. Gomez, T. Luu, A. Pak, K. Kirk, J. Chapman, J. Appl. Phys. 85 (8) (1999) 6163–6165. . [2] M. Kohler, T. Eimuller, . T. Schweinb.ock, J. Zweck, P. Fischer, G. Schutz, . P. Guttmann, G. Schmahl, G. Bayreuther, J. Appl. Phys. 87 (9) (2000) 6481–6483. [3] P. Fischer, T. Eimuller, . G. Schutz, . P. Guttmann, G. . Schmahl, M. Kohler, G. Bayreuther, J. Magn. Soc. Japan 23 (suppl. S1) (1999) 205–208. . . [4] M. Zolfl, M. Brockmann, M. Kohler, S. Kreuzer, T. . Schweinbock, S. Miethaner, F. Bensch, G. Bayreuther, J. Magn. Magn. Mater. 175 (1997) 16–22. [5] D. Podlesnik, H. Gilgen, R. Osgood Jr., Appl. Phys. Lett. 48 (1986) 496. [6] K. Takanashi, S. Mitani, K. Himi, H. Fujimori, Appl. Phys. Lett. 72 (1998) 737. [7] A. Hubert, R. Sch.afer, Magnetic Domains, Springer, Berlin, 1998, p. 327. [8] C. Kooy, U. Enz, Philips Res. Rep. 15 (1960) 7–29.
Micromagnetic investigation of sub-1 0 0-nm magnetic domains in atomically stacked Fe(0 0 1)/Au(0 0 1) multilayers a, . M. Kohler *, J. Zwecka, G. Bayreuthera, P. Fischerb, G. Schutz . b, G. Denbeauxc, D. Attwoodc Universitat . Regensburg, Institut f. Exp. u. Angew. Physik, D-93040 Regensburg, Germany Universitat Lehrst. f. Exp. Phys. IV, Am Hubland, D-97074 Wurzburg, Germany . Wurzburg, . . c Lawrence Berkeley National Laboratory, MSXO, 1 Cyclotron Road, Berkeley, CA 94720, USA a
b
Abstract Continuous films and nanostructures of atomically stacked epitaxial Fe(0 0 1)/Au(0 0 1) multilayers have been studied by soft X-ray and Lorentz microscopy as well as micromagnetic simulations. Domain imaging shows about 65 nm wide magnetic stripe domains, in which the magnetization is oriented perpendicular to the film plane. These results are confirmed by micromagnetic simulations, which also yield additional information about the internal structure of domains and walls. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Ordered alloy; AnisotropyFuniaxial; Micromagnetism; Labyrinth pattern
Properties of ferromagnetic materials are governed to a large extent by their magnetic microstructure. However, no magnetic imaging method can yield complete information about the three dimensional magnetization distribution in a given sample. This restriction can, as demonstrated in Ref. [1], be overcome to a certain degree by a combination of complementary observation methods. In a previous work [2], continuous films of atomically stacked [Fe(0 0 1)/Au(0 0 1)]n monocrystalline multilayers have been studied by magnetic transmission X-ray microscopy (MTXM) [3] and Lorentz transmission electron microscopy (LTEM). These techniques complement each other very well, since for normal incidence MTXM is sensitive to the perpendicular magnetization, while Lorentz microscopy yields the inplane components of the magnetic flux. However, for a unique interpretation the model structures derived from experiment have to be compared with micromagnetic simulations, which can also yield additional information
*Corresponding author. Fax: +49-941-943-4544. E-mail address: [email protected]. burg.de (M. Kohler).
about the internal structure of domain walls not directly accessible by any imaging method. In the present work domain observations in Fe(0 0 1)/ Au(0 0 1) multilayers are extended to nanostructured films and compared to micromagnetic simulations based on measured magnetic material parameters. Films were grown by molecular beam epitaxy (MBE) at room temperature maintaining a pressure of o4.0 1010 Torr. Deposition was monitored by reflection high-energy electron diffraction (RHEED) and quartz microbalances calibrated by X-ray fluorescence analysis (RFA) and RHEED oscillations. First, a n+GaAs(0 0 1) substrate was cleaned in situ by simultaneous heating to 5801C and sputter cleaning with 500 eV Ar+-ions [4]. Then a 6 monolayers (ML) thick Fe(0 0 1) seed layer was deposited at a rate of 1.5 ML/min, followed by 20 nm of Au(0 0 1) at 2 ML/min serving as a buffer layer and later as membrane and etch stop layer in the membrane fabrication process. During the next step the multilayer system was grown by alternating the deposition of Fe(0 0 1) and Au(0 0 1) layers, each at 1.5 ML/min. Finally, the samples were covered by a 15 ML Au(0 0 1) cap layer to protect them from corrosion.
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 7 5 0 - 8
80
M. Kohler et al. / Journal of Magnetism and Magnetic Materials 240 (2002) 79–82 .
Nanostructures were prepared by first covering parts of the sample with a 40 nm Ti mask structured by a standard e-beam lithography and lift-off process. Then the sample was subjected to argon-ion etching with Auger electron depth-monitoring, which served to protect the Au membrane layer by indicating when the etching process must be stopped, i.e. when the Fe signal vanishes. X-ray and electron transparent samples were produced by etching a window into the GaAs substrates leaving the remaining film as a monocrystalline membrane. A new technique was developed to obtain windows with small lateral dimensions, thus ensuring their mechanical stability: laser-induced etching [5] was used to create a hole with high aspect ratio through the rear side of the n+-GaAs(0 0 1) substrate up to the Au(0 0 1) layer. For this purpose the sample was immersed in dilute nitric acid and a 15 mW green–blue Ar+ ion laser beam was focused to a spot of about 20 mm diameter onto the rear side of the substrate, thus locally activating the etch process. The front side of the sample was protected by clear wax which was later removed by carefully rinsing in high-purity acetone. Windows with diameters of about 40 mm could be obtained in this way. The single crystalline structure of the multilayer was verified by electron- and X-ray diffraction (XRD). Magnetization curves were measured by alternating gradient magnetometry (AGM), polar magneto-optic Kerr effect (MOKE), and magnetic anisotropy constants were determined by torque magnetometry. The saturation magnetization and exchange stiffness were derived from spin–wave excitations measured in a SQUID magnetometer. Magnetic domains were imaged with a Philips CM-30 TEM (modified for Lorentz microscopy), and the transmission soft X-ray microscope located at the Advanced Light Source, Lawrence Berkeley National Laboratory in Berkeley, CA. For [1.0 ML Fe/1.0 ML Au]n samples, electron- and X-ray diffraction (XRD) did not show any deviation from the monocrystalline L10 structure. The average vertical Fe/Au lattice spacing d(0 0 2) as determined by ( in good agreement with Ref. [6]. XRD is 1.92A, Investigations of the multilayer surface by in situ scanning tunneling microscope (STM) show a smallscale roughness caused by monoatomic steps separating 3–10 nm wide atomically flat terraces, while on a larger scale the sample appears sufficiently smooth (over a 1 mm2 scan area the typical z-range is 2 nm). Torque magnetometer measurements indicate a strong uniaxial anisotropy with an easy axis perpendicular to the film plane. Samples with slightly larger single layer thicknesses d up to 1.5 ML show a somewhat weaker anisotropy, but in all other respects behave very similarly to those with d ¼ 1:0 ML. According to Ref. [6], the uniaxial anisotropy and the vertical lattice
spacing show an oscillatory behavior with a period of 1 ML. Both have a maximum for d ¼ 1:0 ML and reach a minimum for d ¼ 1:5 ML. The measured anisotropy constants of second and fourth order are in the range Ku1 ¼ ð771Þ 105 J/m3 and Ku2 ¼ ð4:570:5Þ 104 J/m3. The saturation magnetization and exchange stiffness constant as derived from SQUID measurements are JS ¼ 1:1370:1 T (1.25 T for ( (assuming isotropic d ¼ 1:5 ML) and AE3:271 meV/A exchange interaction). Thus the Q-factor (=Ku =Kd ) for our samples is 1.470.3, which means that, since Q is close to 1, numerical micromagnetic methods are very well applicable in the present case [7], while on the other hand common domain theoretical methods (e.g. Ref. [8]), which usually either assume Q51 or Qb1, are likely to yield wrong results. Since Q is larger than 1, one does neither expect ‘‘weak’’ (or dense) stripe domains, which occur in the case of small anistropies often along with a tilted anisotropy axis, nor ‘‘strong’’ stripe domains for which the tilted axis is a prerequisite. On the other hand, the value of Q is smaller than in typical high-anisotropy perpendicular films (‘‘bubble films’’, where 1=Q can often be treated as a small quantity), thus making this system an interesting intermediate case. Fig. 1 shows a labyrinth domain pattern observed by MTXM. Since the sample was observed under perpendicular incidence of X-rays, the contrast indicates variations in the perpendicular magnetization component averaged through the sample thickness. The inplane components are best imaged by LTEM. A similar pattern can be observed by Fresnel mode Lorentz
Fig. 1. [1.5 ML Fe/1.5 ML Au]110 sample observed by MTXM at the Fe L3-edge. Domain width: 6575 nm.
M. Kohler et al. / Journal of Magnetism and Magnetic Materials 240 (2002) 79–82 .
81
Fig. 2. [1.15 ML Fe/1.0 ML Au]130 sample observed by Fresnel mode Lorentz microscopy. Beam incidence 351 from normal. Fig. 4. Cross-section through a simulated wall (perpendicular to film plane and wall). Q ¼ 1:5; finite element size 1 nm.
Fig. 3. [1.5 ML Fe/1.5 ML Au]110 nanostructures observed by MTXM at the Fe L3-edge. The thickness of the structures is approximately 38 nm. QE1:1:
microscopy in a tilted sample [2]. The image vanishes, however, for normal beam incidence, which means that no variation in the averaged in-plane magnetization components can be detected. Since almost no remanence was detected in in-plane AGM measurements, one can conclude that the bulk of the magnetic domains is oriented perpendicular to the film plane and that the observed domains can be classified as band domains characteristic for high-Q films. Despite the somewhat different anisotropy and film thicknesses of the studied samples of Figs. 1 and 2 the domain width is very similar in all cases: 6575 nm. MTXM images of nanostructures are displayed in Fig. 3. It is remarkable how little the domain pattern seems to be disturbed by the particle edges. In order to understand this behavior, micromagnetic simulations were carried out using the Scheinfein LLG Micromagnetics Simulator. Fig. 4 shows the cross-
section through a simulated domain wall. This twisted wall structure is typical for bubble films: in the center of the film, the wall is of a regular Bloch type with a singularity at its very center, where the magnetization lies in the film plane, parallel to the wall. At the top and bottom surface of the film, however, the wall gets twisted towards a Ne! el wall by stray fields: the magnetization is forced completely into the film plane above and below the center line. Obviously, there are no well-defined closure domains, but rather a continuous flux closure configuration, which is about 30 nm wide at the film surfaces. Its in-plane magnetization components exactly cancel out when averaged over the sample thickness and thus are invisible in perpendicular incidence LTEM, as is the singularity in the center of the wall, which is below 8 nm wide and has too little volume to be detected. Fig. 5 shows the different magnetization components in the center plane of a simulated 256 nm 256 nm 40 nm particle. The shape and width of the resulting band domains agree well with our experimental results. There is some variation in the domain width, which may be attributed to the fact that only a small particle, not a continuous film, has been (and can be) simulated. Looking at the in-plane components in the center of the walls one immediately notices discontinuities, where the magnetization direction relative to the wall reverses over a very small length. These can be identified as Bloch lines, which reduce the extra stray field energy of walls that intercept the particle
82
M. Kohler et al. / Journal of Magnetism and Magnetic Materials 240 (2002) 79–82 .
walls that intersect particle boundaries and thus why the domain pattern is disturbed so little at the edges of the nanostructures. Financial support by the Volkswagen-Stiftung is gratefully acknowledged.
References
Fig. 5. Simulation of a 256 nm 256 nm 40 nm platelet, Q ¼ 1:5; finite element size 2 nm. Cross-section through the middle of the platelet (z ¼ 20 nm). The resulting domain width is 55715 nm. (a) Explanation of geometry and color scale, (b) mz (normal component of magnetization) (c) my (vertical component), (d) mx (horizontal component).
edges because the longitudinal net flux carried by the walls is reduced. The formation of Bloch lines help to explain why there is almost no energetic penalty for
[1] R. Gomez, T. Luu, A. Pak, K. Kirk, J. Chapman, J. Appl. Phys. 85 (8) (1999) 6163–6165. . [2] M. Kohler, T. Eimuller, . T. Schweinb.ock, J. Zweck, P. Fischer, G. Schutz, . P. Guttmann, G. Schmahl, G. Bayreuther, J. Appl. Phys. 87 (9) (2000) 6481–6483. [3] P. Fischer, T. Eimuller, . G. Schutz, . P. Guttmann, G. . Schmahl, M. Kohler, G. Bayreuther, J. Magn. Soc. Japan 23 (suppl. S1) (1999) 205–208. . . [4] M. Zolfl, M. Brockmann, M. Kohler, S. Kreuzer, T. . Schweinbock, S. Miethaner, F. Bensch, G. Bayreuther, J. Magn. Magn. Mater. 175 (1997) 16–22. [5] D. Podlesnik, H. Gilgen, R. Osgood Jr., Appl. Phys. Lett. 48 (1986) 496. [6] K. Takanashi, S. Mitani, K. Himi, H. Fujimori, Appl. Phys. Lett. 72 (1998) 737. [7] A. Hubert, R. Sch.afer, Magnetic Domains, Springer, Berlin, 1998, p. 327. [8] C. Kooy, U. Enz, Philips Res. Rep. 15 (1960) 7–29.