- Email: [email protected]
Finite element method calculations of GMI in thin films and sandwiched structures: Size and edge effects
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 320 (2008) e4–e7 www.elsevier.com/locate/jmmm
Finite element method calculations of GMI in thin films and sandwiched structures: Size and edge effects A. Garcı´ a-Arribas, J.M. Barandiara´n, D. de Cos Departamento de Electricidad y Electro´nica, Universidad del Paı´s Vasco, Apartado 644, 48080 Bilbao, Spain Available online 9 February 2008
Abstract The impedance values of magnetic thin films and magnetic/conductor/magnetic sandwiched structures with different widths are computed using the finite element method (FEM). The giant magneto-impedance (GMI) is calculated from the difference of the impedance values obtained with high and low permeability of the magnetic material. The results depend considerably on the width of the sample, demonstrating that edge effects are decisive for the GMI performance. It is shown that, besides the usual skin effect that is responsible for GMI, an ‘‘unexpected’’ increase of the current density takes place at the lateral edge of the sample. In magnetic thin films this effect is dominant when the permeability is low. In the trilayers, it is combined with the lack of shielding of the central conductor at the edge. The resulting effects on GMI are shown to be large for both kinds of samples. The conclusions of this study are of great importance for the successful design of miniaturized GMI devices. r 2008 Elsevier B.V. All rights reserved. PACS: 75.47.m; 02.70.Dc; 75.70.–i; 85.70.Kh; 84.37.+q Keywords: Giant magneto-impedance; Thin films; Trilayers; Finite element method; Edge effects
1. Introduction The mathematical description of giant magneto-impedance (GMI) is completely classical, at least in the quasistatic regime, where the dynamics of magnetization motion is ignored. However, a complete analytical solution to the Maxwell equations can only be obtained for very simple geometries, and even in these cases some approximations are required. For instance, practical analytical calculations for ribbons and films usually assume infiniteness in length and width and, consequently, edge effects are missed. Alternatively, numerical methods can be used to obtain the solution for an arbitrary sample shape. The finite element method (FEM) has been commonly used to perform GMI calculations, mostly in wires and samples with circular geometry [1], but also sometimes in films and sandwiched structures [2,3]. In this work, we analyze the importance of edge and size effects in GMI thin films and trilayers by studying the exact Corresponding author. Tel.: +34 946015307; fax: +34 946013500.
E-mail address: [email protected] (A. Garcı´ a-Arribas). 0304-8853/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.02.005
solutions provided by a two-dimensional (2D) FEM. This study only considers the effects of different sample geometries, without reference to any particular magnetization process. Besides, we neglect any dynamical effect, although we consider frequencies up to 10 GHz, where ferromagnetic resonance (FMR) plays a fundamental role [4]. In any case, the phenomena that this investigation reveals, which are somehow unexpected, will take place even in the dynamic regime, and must be considered for the successful design of thin-film GMI devices.
2. FEM calculations Two different types of samples have been studied: homogeneous magnetic thin films and sandwiched structures (magnetic/conductor/magnetic), with widths w ¼ N, 1000, 500, 200, 100, 50, 25, and 5 mm. The thickness of the films was set to 500 nm. The sandwich structures were 250 nm thick for each of the magnetic layers and 500 nm for the conductor. The samples are considered infinite in length. This justifies the use of a 2D calculation.
ARTICLE IN PRESS A. Garcı´a-Arribas et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e4–e7
3.1. Homogeneous thin film The impedance of a film at a frequency f, with infinite width and thickness h, can be calculated analytically to give pffi pffi Z ¼ Rdc jy coth jy , (1) pffiffiffi where Rdc is the dc resistance, y ¼ 2ðh=dÞ, and d ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=ðsmoÞ is the penetration depth, with o ¼ 2pf. To simulate infiniteness in FEM, periodic boundary conditions are used, imposing the solution to be the same at both sides of the simulated region (Fig. 1). FEM yields exactly the same impedance values as that of Eq. (1) for all frequencies. When the width is finite, the air surrounding the sample must also be simulated and adequate boundary conditions have to be imposed at the limits of the problem domain. Fig. 2 shows the results obtained for a 100-mm-wide sample at 10 GHz. In (a), for high permeability, the skin effect is seen along y-direction (surface of the sample), and along x-direction at the lateral edge. In (b), for low permeability, a noticeable increase of the current density is observed in the x-direction. The absolute value of the impedance, as a function of the frequency, is presented in Fig. 3 for all the geometries. For mr ¼ 1000, Fig. 3(a) shows the rapid increase of the impedance at about 600 MHz (where d equals h/2) as a
Fig. 1. FEM calculation at 10 GHz for a film of infinite width and thickness h ¼ 500 nm, with mr ¼ 1000. Symmetry allows to simulate only half of the thickness (h/2). Using periodic boundary conditions only a finite portion needs to be simulated. Solid horizontal lines represent the magnetic field. The current density is represented by color grades. The skin effect in the y-direction (thickness of the sample) is clearly visible (color online).
width
5 Z/Rdc
3. Results and discussion
Fig. 2. FEM calculation at 10 GHz for a 100-mm-wide film. The figure shows only the edge of the film (see the displayed dimensions and note that the complete sample spans from x ¼ 50 mm to 50 mm). Solid lines represent the magnetic field. The current density is displayed by color grades. In the case of mr ¼ 1000 (a), the skin effect is noticeably close to the surface in the y-direction and at the lateral edge in x-direction. In the case of mr ¼ 1 (b), there is a clear increase of the current density in the xdirection. Color scale is not the same in both cases.
infinite 1000 µm 500 µm 200 µm 100 µm 50 µm 25 µm 5 µm
3
1 1.6
µr = 1000
width infinite 1000 µm 500 µm 200 µm 100 µm 50 µm 25 µm 5 µm
1.4 Z/Rdc
The conductivities were selected as s ¼ 6.66 106 S/m (permalloy) and s ¼ 4.55 107 S/m (gold). The calculations have been performed with the FEMM package [5]. For each of the geometries, the problem is solved for 100 different frequencies, logarithmically spaced between 0 and 10 GHz. For each frequency, the impedance Z is calculated using a high value for the permeability (mr ¼ 1000) and a low one (mr ¼ 1). As the total selfinductance of an infinitely long sample diverges logarithmically, the imaginary part of the impedance is calculated using the internal self-inductance of the sample Lint, as 2pfLint. The GMI ratio is calculated as GMI ¼ 100 (ZhighZlow)/Zlow.
e5
1.2
µr = 1
1 108
109 f (Hz)
1010
Fig. 3. Absolute value of the impedance calculated for high (a) and low (b) permeability for 500-nm-thick thin films with different widths (color online).
consequence of the skin effect. The impedance of the infinite-width sample is larger than that for the others (due to the effect of edges) but, essentially, the curves are very similar for all geometries. In contrast, noticeable differences among them are found in Fig. 3(b). For mr ¼ 1, d equals h/2 at 600 GHz, so the skin effect must be insignificant at 10 GHz. However, the impedance increases considerably at lower frequencies. The reason is the increase of current density at the sample edges, shown in Fig. 2(b), which is caused by a ‘‘transversal’’ skin effect, acting in the x-direction of the sample. For w ¼ 100 mm, the penetration depth d equals w/2 at 15 MHz and the impedance of the sample starts increasing at that frequency due to the ‘‘accumulation of current’’ at the
ARTICLE IN PRESS A. Garcı´a-Arribas et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e4–e7
e6
500
GMI (%)
width infinite 1000 µm 500 µm 200 µm 100 µm 50 µm 25 µm 5 µm
300
100
1010
Fig. 4. Giant magneto-impedance ratio for thin films of different widths (color online).
edge. This effect is greater the wider the sample is, since d reaches w/2 at lower frequencies. It also occurs in the highpermeability case (in fact, d equals w/2 at 15 kHz for w ¼ 100 mm and mr ¼ 1000), but the impedance is dominated by the ‘‘conventional’’ skin effect, due to the negligible contribution of the lateral edges compared with the one of the sample surface. The consequence on GMI is observed in Fig. 4. The value of GMI is progressively reduced by the increase of the lowpermeability impedance, and therefore greater values are obtained for the narrowest sample. Edge effects are decisive and, contrary to intuition, the narrowest sample resembles more closely the behavior of the infinite-width one. 3.2. Sandwiched films Similar calculations have been performed for the case of sandwiched magnetic/conductor/magnetic structures. Fig. 5 shows the results obtained at 10 GHz for both high (a) and low (b) permeability cases. Only one quarter of the section is simulated, so the figures display the upper magnetic layer and the upper half of the conductor (and only the region close to the edge). The main features observed before are reproduced here: great skin effect in the magnetic layer when the permeability is high and a considerable increase of the current density at the edge of the sample. However, the situation is now more complicated and two main effects condition the GMI behavior. First, in the high-permeability case, the shielding effect of the magnetic layers expels the current from the central conductor. For the infinite-width sample, it vanishes completely, as in core–shell cylindrical geometries, where the conductor is completely shielded [1]. However, in sandwiched samples, the lateral edges of the conductor are unshielded and the current density increases greatly there. This affects more intensely the narrower samples, since the unshielded fraction is greater and a greater amount of current is allowed through the conductor. Second, when the permeability is low, the ‘‘accumulation’’ of current at the edges produces the increase of the impedance, as in the previous case of thin films. This effect is greater for wider samples, as discussed before. The balance of both effects on
Fig. 5. FEM calculation at 10 GHz for a 100-mm-wide sandwich. The upper magnetic layer and half of the central conductor are shown, and only the zone close to the edge. (a) mr ¼ 1000; (b) mr ¼ 1. In both cases the increase of current density at the edge of the conductor can be observed (color online).
2000
4000
width infinite 1000 µm 500 µm 200 µm 100 µm 50 µm 25 µm 5 µm
1500 1000
3000 2000
500
1000
0 107
108
109
GMI (%) (infinite)
109 f (Hz)
GMI (%) (finite-width samples)
108
0 1010
f (Hz) Fig. 6. Giant magneto-impedance ratio for sandwiched structures of different widths. The right y-axis holds for the infinite-width sample (color online).
the GMI, for the geometries considered in this study, produces the results observed in Fig. 6. In this case, the consequences of the first effect on the impedance at high permeability are more important than those of the second at low permeability, and GMI curves get more similar to the infinite one as the width increases. However, additional results show that the GMI depends greatly on the sample geometry, not only on its width but also on the thickness ratio between conductive and magnetic layers. In any case, edge effects cause great differences between samples with different widths. 4. Conclusions The finite element method has been used to determine the impedance and GMI ratio of thin films and sandwiched structures with different widths. The results reveal the importance of edge effects. We show that the validity of the ‘‘infinite width’’ approximation is limited, because it neglects the ‘‘transversal’’ skin effect that takes place at the edge. In magnetic thin films this effect is dominant when the permeability is low. In sandwiched films, it is combined with the lack of shielding of the central conductor at the edge. In both cases, the consequences on the GMI are important, and they have been ignored so far
ARTICLE IN PRESS A. Garcı´a-Arribas et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e4–e7
in GMI analysis of this kind of samples. Our results, although only consider the influence of the geometry on the GMI, can help identify the discrepancies between theory and experiment. Besides, we demonstrate that the geometry of the sample must be carefully chosen to improve device performance. Acknowledgments This work was supported by the Basque and Spanish Governments under projects ACTIMAT and MAT200506806.
e7
References [1] J.P. Sinnecker, K.R. Pirota, M. Knobel, L. Kraus, J. Magn. Magn. Mater. 249 (2002) 16. [2] G.V. Kurlyandskaya, J.L. Mun˜oz, J.M. Barandiara´n, A. Garcı´ aArribas, A.V. Svalov, V.O. Vas’kovskiy, J. Magn. Magn. Mater. 242–245 (2002) 291. [3] S. Yabukami, H. Mawatari, N. Horikoshi, Y. Murayama, T. Ozawa, K. Ishiyama, K.I. Arai, J. Magn. Magn. Mater. 290–291 (2005) 1318. [4] J.M. Barandiara´n, A. Garcı´ a-Arribas, D. de Cos, J. Appl. Phys. 99 (2006) 103904. [5] D.C. Meeker, Finite Element Method Magnetics, Version 3.4.2 (02Dec2005 Build), /http://femm.foster-miller.netS.
Journal of Magnetism and Magnetic Materials 320 (2008) e4–e7 www.elsevier.com/locate/jmmm
Finite element method calculations of GMI in thin films and sandwiched structures: Size and edge effects A. Garcı´ a-Arribas, J.M. Barandiara´n, D. de Cos Departamento de Electricidad y Electro´nica, Universidad del Paı´s Vasco, Apartado 644, 48080 Bilbao, Spain Available online 9 February 2008
Abstract The impedance values of magnetic thin films and magnetic/conductor/magnetic sandwiched structures with different widths are computed using the finite element method (FEM). The giant magneto-impedance (GMI) is calculated from the difference of the impedance values obtained with high and low permeability of the magnetic material. The results depend considerably on the width of the sample, demonstrating that edge effects are decisive for the GMI performance. It is shown that, besides the usual skin effect that is responsible for GMI, an ‘‘unexpected’’ increase of the current density takes place at the lateral edge of the sample. In magnetic thin films this effect is dominant when the permeability is low. In the trilayers, it is combined with the lack of shielding of the central conductor at the edge. The resulting effects on GMI are shown to be large for both kinds of samples. The conclusions of this study are of great importance for the successful design of miniaturized GMI devices. r 2008 Elsevier B.V. All rights reserved. PACS: 75.47.m; 02.70.Dc; 75.70.–i; 85.70.Kh; 84.37.+q Keywords: Giant magneto-impedance; Thin films; Trilayers; Finite element method; Edge effects
1. Introduction The mathematical description of giant magneto-impedance (GMI) is completely classical, at least in the quasistatic regime, where the dynamics of magnetization motion is ignored. However, a complete analytical solution to the Maxwell equations can only be obtained for very simple geometries, and even in these cases some approximations are required. For instance, practical analytical calculations for ribbons and films usually assume infiniteness in length and width and, consequently, edge effects are missed. Alternatively, numerical methods can be used to obtain the solution for an arbitrary sample shape. The finite element method (FEM) has been commonly used to perform GMI calculations, mostly in wires and samples with circular geometry [1], but also sometimes in films and sandwiched structures [2,3]. In this work, we analyze the importance of edge and size effects in GMI thin films and trilayers by studying the exact Corresponding author. Tel.: +34 946015307; fax: +34 946013500.
E-mail address: [email protected] (A. Garcı´ a-Arribas). 0304-8853/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.02.005
solutions provided by a two-dimensional (2D) FEM. This study only considers the effects of different sample geometries, without reference to any particular magnetization process. Besides, we neglect any dynamical effect, although we consider frequencies up to 10 GHz, where ferromagnetic resonance (FMR) plays a fundamental role [4]. In any case, the phenomena that this investigation reveals, which are somehow unexpected, will take place even in the dynamic regime, and must be considered for the successful design of thin-film GMI devices.
2. FEM calculations Two different types of samples have been studied: homogeneous magnetic thin films and sandwiched structures (magnetic/conductor/magnetic), with widths w ¼ N, 1000, 500, 200, 100, 50, 25, and 5 mm. The thickness of the films was set to 500 nm. The sandwich structures were 250 nm thick for each of the magnetic layers and 500 nm for the conductor. The samples are considered infinite in length. This justifies the use of a 2D calculation.
ARTICLE IN PRESS A. Garcı´a-Arribas et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e4–e7
3.1. Homogeneous thin film The impedance of a film at a frequency f, with infinite width and thickness h, can be calculated analytically to give pffi pffi Z ¼ Rdc jy coth jy , (1) pffiffiffi where Rdc is the dc resistance, y ¼ 2ðh=dÞ, and d ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=ðsmoÞ is the penetration depth, with o ¼ 2pf. To simulate infiniteness in FEM, periodic boundary conditions are used, imposing the solution to be the same at both sides of the simulated region (Fig. 1). FEM yields exactly the same impedance values as that of Eq. (1) for all frequencies. When the width is finite, the air surrounding the sample must also be simulated and adequate boundary conditions have to be imposed at the limits of the problem domain. Fig. 2 shows the results obtained for a 100-mm-wide sample at 10 GHz. In (a), for high permeability, the skin effect is seen along y-direction (surface of the sample), and along x-direction at the lateral edge. In (b), for low permeability, a noticeable increase of the current density is observed in the x-direction. The absolute value of the impedance, as a function of the frequency, is presented in Fig. 3 for all the geometries. For mr ¼ 1000, Fig. 3(a) shows the rapid increase of the impedance at about 600 MHz (where d equals h/2) as a
Fig. 1. FEM calculation at 10 GHz for a film of infinite width and thickness h ¼ 500 nm, with mr ¼ 1000. Symmetry allows to simulate only half of the thickness (h/2). Using periodic boundary conditions only a finite portion needs to be simulated. Solid horizontal lines represent the magnetic field. The current density is represented by color grades. The skin effect in the y-direction (thickness of the sample) is clearly visible (color online).
width
5 Z/Rdc
3. Results and discussion
Fig. 2. FEM calculation at 10 GHz for a 100-mm-wide film. The figure shows only the edge of the film (see the displayed dimensions and note that the complete sample spans from x ¼ 50 mm to 50 mm). Solid lines represent the magnetic field. The current density is displayed by color grades. In the case of mr ¼ 1000 (a), the skin effect is noticeably close to the surface in the y-direction and at the lateral edge in x-direction. In the case of mr ¼ 1 (b), there is a clear increase of the current density in the xdirection. Color scale is not the same in both cases.
infinite 1000 µm 500 µm 200 µm 100 µm 50 µm 25 µm 5 µm
3
1 1.6
µr = 1000
width infinite 1000 µm 500 µm 200 µm 100 µm 50 µm 25 µm 5 µm
1.4 Z/Rdc
The conductivities were selected as s ¼ 6.66 106 S/m (permalloy) and s ¼ 4.55 107 S/m (gold). The calculations have been performed with the FEMM package [5]. For each of the geometries, the problem is solved for 100 different frequencies, logarithmically spaced between 0 and 10 GHz. For each frequency, the impedance Z is calculated using a high value for the permeability (mr ¼ 1000) and a low one (mr ¼ 1). As the total selfinductance of an infinitely long sample diverges logarithmically, the imaginary part of the impedance is calculated using the internal self-inductance of the sample Lint, as 2pfLint. The GMI ratio is calculated as GMI ¼ 100 (ZhighZlow)/Zlow.
e5
1.2
µr = 1
1 108
109 f (Hz)
1010
Fig. 3. Absolute value of the impedance calculated for high (a) and low (b) permeability for 500-nm-thick thin films with different widths (color online).
consequence of the skin effect. The impedance of the infinite-width sample is larger than that for the others (due to the effect of edges) but, essentially, the curves are very similar for all geometries. In contrast, noticeable differences among them are found in Fig. 3(b). For mr ¼ 1, d equals h/2 at 600 GHz, so the skin effect must be insignificant at 10 GHz. However, the impedance increases considerably at lower frequencies. The reason is the increase of current density at the sample edges, shown in Fig. 2(b), which is caused by a ‘‘transversal’’ skin effect, acting in the x-direction of the sample. For w ¼ 100 mm, the penetration depth d equals w/2 at 15 MHz and the impedance of the sample starts increasing at that frequency due to the ‘‘accumulation of current’’ at the
ARTICLE IN PRESS A. Garcı´a-Arribas et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e4–e7
e6
500
GMI (%)
width infinite 1000 µm 500 µm 200 µm 100 µm 50 µm 25 µm 5 µm
300
100
1010
Fig. 4. Giant magneto-impedance ratio for thin films of different widths (color online).
edge. This effect is greater the wider the sample is, since d reaches w/2 at lower frequencies. It also occurs in the highpermeability case (in fact, d equals w/2 at 15 kHz for w ¼ 100 mm and mr ¼ 1000), but the impedance is dominated by the ‘‘conventional’’ skin effect, due to the negligible contribution of the lateral edges compared with the one of the sample surface. The consequence on GMI is observed in Fig. 4. The value of GMI is progressively reduced by the increase of the lowpermeability impedance, and therefore greater values are obtained for the narrowest sample. Edge effects are decisive and, contrary to intuition, the narrowest sample resembles more closely the behavior of the infinite-width one. 3.2. Sandwiched films Similar calculations have been performed for the case of sandwiched magnetic/conductor/magnetic structures. Fig. 5 shows the results obtained at 10 GHz for both high (a) and low (b) permeability cases. Only one quarter of the section is simulated, so the figures display the upper magnetic layer and the upper half of the conductor (and only the region close to the edge). The main features observed before are reproduced here: great skin effect in the magnetic layer when the permeability is high and a considerable increase of the current density at the edge of the sample. However, the situation is now more complicated and two main effects condition the GMI behavior. First, in the high-permeability case, the shielding effect of the magnetic layers expels the current from the central conductor. For the infinite-width sample, it vanishes completely, as in core–shell cylindrical geometries, where the conductor is completely shielded [1]. However, in sandwiched samples, the lateral edges of the conductor are unshielded and the current density increases greatly there. This affects more intensely the narrower samples, since the unshielded fraction is greater and a greater amount of current is allowed through the conductor. Second, when the permeability is low, the ‘‘accumulation’’ of current at the edges produces the increase of the impedance, as in the previous case of thin films. This effect is greater for wider samples, as discussed before. The balance of both effects on
Fig. 5. FEM calculation at 10 GHz for a 100-mm-wide sandwich. The upper magnetic layer and half of the central conductor are shown, and only the zone close to the edge. (a) mr ¼ 1000; (b) mr ¼ 1. In both cases the increase of current density at the edge of the conductor can be observed (color online).
2000
4000
width infinite 1000 µm 500 µm 200 µm 100 µm 50 µm 25 µm 5 µm
1500 1000
3000 2000
500
1000
0 107
108
109
GMI (%) (infinite)
109 f (Hz)
GMI (%) (finite-width samples)
108
0 1010
f (Hz) Fig. 6. Giant magneto-impedance ratio for sandwiched structures of different widths. The right y-axis holds for the infinite-width sample (color online).
the GMI, for the geometries considered in this study, produces the results observed in Fig. 6. In this case, the consequences of the first effect on the impedance at high permeability are more important than those of the second at low permeability, and GMI curves get more similar to the infinite one as the width increases. However, additional results show that the GMI depends greatly on the sample geometry, not only on its width but also on the thickness ratio between conductive and magnetic layers. In any case, edge effects cause great differences between samples with different widths. 4. Conclusions The finite element method has been used to determine the impedance and GMI ratio of thin films and sandwiched structures with different widths. The results reveal the importance of edge effects. We show that the validity of the ‘‘infinite width’’ approximation is limited, because it neglects the ‘‘transversal’’ skin effect that takes place at the edge. In magnetic thin films this effect is dominant when the permeability is low. In sandwiched films, it is combined with the lack of shielding of the central conductor at the edge. In both cases, the consequences on the GMI are important, and they have been ignored so far
ARTICLE IN PRESS A. Garcı´a-Arribas et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e4–e7
in GMI analysis of this kind of samples. Our results, although only consider the influence of the geometry on the GMI, can help identify the discrepancies between theory and experiment. Besides, we demonstrate that the geometry of the sample must be carefully chosen to improve device performance. Acknowledgments This work was supported by the Basque and Spanish Governments under projects ACTIMAT and MAT200506806.
e7
References [1] J.P. Sinnecker, K.R. Pirota, M. Knobel, L. Kraus, J. Magn. Magn. Mater. 249 (2002) 16. [2] G.V. Kurlyandskaya, J.L. Mun˜oz, J.M. Barandiara´n, A. Garcı´ aArribas, A.V. Svalov, V.O. Vas’kovskiy, J. Magn. Magn. Mater. 242–245 (2002) 291. [3] S. Yabukami, H. Mawatari, N. Horikoshi, Y. Murayama, T. Ozawa, K. Ishiyama, K.I. Arai, J. Magn. Magn. Mater. 290–291 (2005) 1318. [4] J.M. Barandiara´n, A. Garcı´ a-Arribas, D. de Cos, J. Appl. Phys. 99 (2006) 103904. [5] D.C. Meeker, Finite Element Method Magnetics, Version 3.4.2 (02Dec2005 Build), /http://femm.foster-miller.netS.