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Numerical modelling of GMI effect in soft magnetic amorphous ribbons
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 1862–1863
Numerical modelling of GMI effect in soft magnetic amorphous ribbons I.Z. Rahmana,*, A. Boboca, Md. Kamruzzamana, M.A. Rahmanb a
Material and Surface Science Institute and Department of Physics, University Of Limerick, National Technological Park, Limerick, Ireland b Department Electronic and Computer Engineering, University of Limerick, National Technological Park, Limerick, Ireland
Abstract A numerical simulation model based on Machado et al. [J. Appl. Phys. 79 (1996) 6558] was developed to study the relaxation time and GMI in a series of Co-and Fe-based commercial alloys in the ribbon form as a function of excitation frequency and DC bias field. In Machado et al’s model the relaxation time was considered as constant. Based on our experimental observations, we considered the relaxation time as a function of frequency and applied field. In this paper we report on the establishment of a general expression of the relaxation time for both Fe-and Co-based alloys. r 2003 Elsevier B.V. All rights reserved. PACS: 75.50.Bb; 75.50.Kj; 75.47.m; 72.15.Lh; 75.50Tt Keywords: Soft magnetic materials; Magneto-Impedance effect; Relaxation time; Amorphous magnetic materials; Nanocrystalline materials
1. Introduction
2. Overview of the MI effect
The research on GMI began with phenomenological models developed to understand basic aspects found in experimental data. Panina, Machado and their coworkers developed theoretical models to explain the MI effect in wires and ribbons. Panina worked with the wire shaped materials and Machado worked with ribbons. Machado et al. [1,2] presented a theoretical model for Co-rich amorphous ribbons based on the skin depth effect and on the domain wall motion due to the magnetic field and AC current, which explained the MI spectra and its frequency and field dependence. In this paper, an analysis of the experimental data and comparison with theoretical predictions from an improved model (starting from Machado’s model), was carried out to reveal the relaxation time and its behaviour with axial DC magnetic field and AC frequency.
In case of a ribbon-shaped conductor, the impedance of a sample expressed using classical electrodynamics as
*Corresponding author. Tel.: +00353+61+202423. E-mail address: [email protected] (I.Z. Rahman).
Z¼
ð1 jÞL ð2promt Þ1=2 ; 2lc
ð1Þ
where r is the resistivity, mt is the transverse permeability, l and L are the ribbon width and length, respectively. From above relationship, it can be observed that the behavior of Z with DC field and frequency is determined by (omt), which, in turn depends on the response of the domains to the longitudinal DC field and the AC current.
3. Machado’s model Machado considered two neighbouring domains that have spins opposite to each other. The motion of domain walls results from the combined effect of the external longitudinal field H and transverse field h created by an AC current. After several mathematical
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.832
ARTICLE IN PRESS I.Z. Rahman et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1862–1863
2 1=4
þ ð4pw0 otÞ
;
τ of Co67Fe4Mo1.5Si16.5B11 ribbons with frequency 2
H =53.3751 (Am-1)
1.5 µs
calculations one can find, by using the Makado model, that the impedance have the following form L 4pro 1=2 ½ð1 þ o2 t2 þ 4pw0 Þ2 Z¼ 2lc 1 þ o2 t2 ð2Þ
measurements first fit second fit
1 0.5
where t is the relaxation time and w0 is the susceptibility.
0
0
1
2
3
2
4. Further work on Machado’s model
ð4Þ
In order to simplify, the dependence of the parameters a0, a1, a2 with the applied DC magnetic field, were taken to be a0 ðHÞ ¼ c0 ; c1 a1 ðHÞ ¼ pffiffiffiffiffi; H a2 ðHÞ ¼ c2 þ c3 H ;
5
6
7
ð5Þ
where, c0, c1, c2, c3 are constants that depend only on the intrinsic properties of the materials. It was observed that parameter a0 is nearly independent of the axially applied DC field, the negative values of a1 vary inversely with the DC field and a2 increases slowly with the field. Other ribbons also showed the similar trend. The study was expanded to several ribbons and the parameters c0, c1, c2 and c3 were found for all the ribbons under test by least-squares fitting. For example, the evolution of relaxation time of Co67Fe4Mo1.5Si16.5B11 as a function of frequency for several field values is shown in Fig. 1. The MI effect, described as the ratio MI=[Z(H0)Z(Hsat)]/Z(H0)], where Z(H0) and Z(Hsat) are the real part of the impedance without and with applied DC magnetic field H has been analysed. Theoretical values
µs
8
9
10
-1
H =800.6265 (Am ) measurements first fit second fit
1 0.5 0 0
1
2
3
4 5 6 7 Frequency(MHz)
8
9
10
Fig. 1. Variation of relaxation time with frequency.
MI Ratio(%)
Experimentally values of impedance of Fe- and Cobased ribbons were used in calculating the relaxation time as a function of frequency and DC magnetic field. To make the expression for t simpler, we started from a general expression shown in Eq. (4)
2
4
1.5
Machado used the relaxation time (t) calculated from dynamical susceptibility and values of Z were calculated using Eq. (2). In a particular Co-rich ribbon the t value was considered constant for all frequencies range [1,2] to profile the frequency dependence MI behaviour. Eq. (2) can be rearranged to obtain the expression for t. " #1=2 2 o2 L4 p2 r2 1 þ 4pw0 l 4 c4 z4 t¼ þy : ð3Þ l 4 c4 z4 o2 o4 L4 p2 r2
tFIT1 ¼ a0 ea1 f þ a2 f :
1863
160 140 120 100 80 60 40 20 0 -4500
Measured Fit
-2500
-500
1500
3500
dc Magnetic Field(Hdc) in A m-1 Fig. 2. MI effect for an amorphous ribbon of nominal composition Co67Fe4Mo1.5Si16.5B11.
of the MI ratio were found to agree with our experimentally measured MI ratio (see Fig. 2). In the present study, the behaviour of the relaxation time at different field and frequency was analysed and a simplified model consisting of parameters that depend on the magnetic properties of the materials has been proposed.
Acknowledgements This work is partly supported by EU under Marie Curie Host Development Fellowship Grant Scheme.
References [1] F.L.A. Machado, et al., J. Appl. Phys. 79 (1996) 6558. [2] F.L.A. Machado, et al., Phys. Rev. B Conden. Matter 51 (1995) 3926.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 1862–1863
Numerical modelling of GMI effect in soft magnetic amorphous ribbons I.Z. Rahmana,*, A. Boboca, Md. Kamruzzamana, M.A. Rahmanb a
Material and Surface Science Institute and Department of Physics, University Of Limerick, National Technological Park, Limerick, Ireland b Department Electronic and Computer Engineering, University of Limerick, National Technological Park, Limerick, Ireland
Abstract A numerical simulation model based on Machado et al. [J. Appl. Phys. 79 (1996) 6558] was developed to study the relaxation time and GMI in a series of Co-and Fe-based commercial alloys in the ribbon form as a function of excitation frequency and DC bias field. In Machado et al’s model the relaxation time was considered as constant. Based on our experimental observations, we considered the relaxation time as a function of frequency and applied field. In this paper we report on the establishment of a general expression of the relaxation time for both Fe-and Co-based alloys. r 2003 Elsevier B.V. All rights reserved. PACS: 75.50.Bb; 75.50.Kj; 75.47.m; 72.15.Lh; 75.50Tt Keywords: Soft magnetic materials; Magneto-Impedance effect; Relaxation time; Amorphous magnetic materials; Nanocrystalline materials
1. Introduction
2. Overview of the MI effect
The research on GMI began with phenomenological models developed to understand basic aspects found in experimental data. Panina, Machado and their coworkers developed theoretical models to explain the MI effect in wires and ribbons. Panina worked with the wire shaped materials and Machado worked with ribbons. Machado et al. [1,2] presented a theoretical model for Co-rich amorphous ribbons based on the skin depth effect and on the domain wall motion due to the magnetic field and AC current, which explained the MI spectra and its frequency and field dependence. In this paper, an analysis of the experimental data and comparison with theoretical predictions from an improved model (starting from Machado’s model), was carried out to reveal the relaxation time and its behaviour with axial DC magnetic field and AC frequency.
In case of a ribbon-shaped conductor, the impedance of a sample expressed using classical electrodynamics as
*Corresponding author. Tel.: +00353+61+202423. E-mail address: [email protected] (I.Z. Rahman).
Z¼
ð1 jÞL ð2promt Þ1=2 ; 2lc
ð1Þ
where r is the resistivity, mt is the transverse permeability, l and L are the ribbon width and length, respectively. From above relationship, it can be observed that the behavior of Z with DC field and frequency is determined by (omt), which, in turn depends on the response of the domains to the longitudinal DC field and the AC current.
3. Machado’s model Machado considered two neighbouring domains that have spins opposite to each other. The motion of domain walls results from the combined effect of the external longitudinal field H and transverse field h created by an AC current. After several mathematical
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.832
ARTICLE IN PRESS I.Z. Rahman et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1862–1863
2 1=4
þ ð4pw0 otÞ
;
τ of Co67Fe4Mo1.5Si16.5B11 ribbons with frequency 2
H =53.3751 (Am-1)
1.5 µs
calculations one can find, by using the Makado model, that the impedance have the following form L 4pro 1=2 ½ð1 þ o2 t2 þ 4pw0 Þ2 Z¼ 2lc 1 þ o2 t2 ð2Þ
measurements first fit second fit
1 0.5
where t is the relaxation time and w0 is the susceptibility.
0
0
1
2
3
2
4. Further work on Machado’s model
ð4Þ
In order to simplify, the dependence of the parameters a0, a1, a2 with the applied DC magnetic field, were taken to be a0 ðHÞ ¼ c0 ; c1 a1 ðHÞ ¼ pffiffiffiffiffi; H a2 ðHÞ ¼ c2 þ c3 H ;
5
6
7
ð5Þ
where, c0, c1, c2, c3 are constants that depend only on the intrinsic properties of the materials. It was observed that parameter a0 is nearly independent of the axially applied DC field, the negative values of a1 vary inversely with the DC field and a2 increases slowly with the field. Other ribbons also showed the similar trend. The study was expanded to several ribbons and the parameters c0, c1, c2 and c3 were found for all the ribbons under test by least-squares fitting. For example, the evolution of relaxation time of Co67Fe4Mo1.5Si16.5B11 as a function of frequency for several field values is shown in Fig. 1. The MI effect, described as the ratio MI=[Z(H0)Z(Hsat)]/Z(H0)], where Z(H0) and Z(Hsat) are the real part of the impedance without and with applied DC magnetic field H has been analysed. Theoretical values
µs
8
9
10
-1
H =800.6265 (Am ) measurements first fit second fit
1 0.5 0 0
1
2
3
4 5 6 7 Frequency(MHz)
8
9
10
Fig. 1. Variation of relaxation time with frequency.
MI Ratio(%)
Experimentally values of impedance of Fe- and Cobased ribbons were used in calculating the relaxation time as a function of frequency and DC magnetic field. To make the expression for t simpler, we started from a general expression shown in Eq. (4)
2
4
1.5
Machado used the relaxation time (t) calculated from dynamical susceptibility and values of Z were calculated using Eq. (2). In a particular Co-rich ribbon the t value was considered constant for all frequencies range [1,2] to profile the frequency dependence MI behaviour. Eq. (2) can be rearranged to obtain the expression for t. " #1=2 2 o2 L4 p2 r2 1 þ 4pw0 l 4 c4 z4 t¼ þy : ð3Þ l 4 c4 z4 o2 o4 L4 p2 r2
tFIT1 ¼ a0 ea1 f þ a2 f :
1863
160 140 120 100 80 60 40 20 0 -4500
Measured Fit
-2500
-500
1500
3500
dc Magnetic Field(Hdc) in A m-1 Fig. 2. MI effect for an amorphous ribbon of nominal composition Co67Fe4Mo1.5Si16.5B11.
of the MI ratio were found to agree with our experimentally measured MI ratio (see Fig. 2). In the present study, the behaviour of the relaxation time at different field and frequency was analysed and a simplified model consisting of parameters that depend on the magnetic properties of the materials has been proposed.
Acknowledgements This work is partly supported by EU under Marie Curie Host Development Fellowship Grant Scheme.
References [1] F.L.A. Machado, et al., J. Appl. Phys. 79 (1996) 6558. [2] F.L.A. Machado, et al., Phys. Rev. B Conden. Matter 51 (1995) 3926.