- Email: [email protected]
On the equivalent circuit approximation of magnetoimpedance in amorphous wires
cm _-
February
__ l!!iB
1998
Materials Letters 34 (1998) IO-13
ELSEWIER
On the equivalent circuit approximation of magnetoimpedance amorphous wires K.L. Garcia, R. Valenzuela Institute.for Materials Research,
National University
in
*
ofMexico, P.O. Box 70-360, Mexico D.F. 04510, Mexico
Received 30 April 1997; accepted 6 May 1997
Abstract The magnetoimpedance response of CoFeBSi wires was measured at frequencies equal, below and above the relaxation frequency. We show that the behavior of the real and imaginary parts of inductance, as a function of the current amplitude is clearly consistent with modeling of the wire response by means of a parallel RL equivalent circuit. 0 1998 Elsevier Science B.V. PACS: 75.5O.Kj; 75.50.Q; 75.70.K~ Ke?;words: Magnetoimpedance; Amorphous
wire; CoFeBSi alloy; Equivalent
1. Introduction Magnetoimpedance (MI) consists in the variations of the impedance response of a ferromagnetic material (submitted to an ac current of small amplitude), when a dc magnetic field is applied. This extremely sensitive response leads to original applications on magnetic field sensors technology. MI can be fully explained by classical electromagnetism. Recent results has shown [1,2] that ac current flowing through the material produces a circumferential or perimetral (depending of the cross section of the sample) ac magnetic field which inter-
* Corresponding author. [email protected].
Fax:
+52-5-6161371;
e-mail:
mon-
circuits
acts with domains and domain walls. This coupling results in an additional contribution to the total impedance of the sample. When a dc magnetic field is applied, it leads to a damping of the domain wall movement; if it is large enough to saturate the sample, only rotational magnetization processes remain and the inductive coupling decreases significantly. In the case of samples with small cross section, the combination of high frequencies and small resistivity results in a limited penetration of the magnetic field, and therefore in an additional impedance contribution associated with eddy currents. CoFeBSi amorphous wires with very low and negative magnetostriction exhibits a strong MI effect. This is related with their magnetic structure, which can be described [3] as formed by two main parts:
00167-577X/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. PI/ SO167-577X(97)00131-6
an inner
core,
with domains
oriented
close
[4]
K.L. Garci;?, R. Valenzuela /Materials Letters 34 (1998) IO-13
to the longitudinal direction of the wire, and an outer shell with circumferentially oriented domains. Since the magnetic field produced by the ac current has also a circumferential geometry, the most efficient coupling occurs with the outer’s shell domains. In this paper, we carried out complex inductance measurements on these wires; at some selected frequencies, as a function of the ac current amplitude. We show that these experimental results are in very good agreement with an RL parallel equivalent circuit approximation, which was previously investigated [6] as a function of frequency.
2. Experimental techniques
We used small pieces (- 4 cm long) of as-cast corn position wires of amorphous kindly provided by Uni(Co 0.094Fe0.06)72.5B15Si12.5’ tika, Japan, prepared by the in-rotating-water technique [6]. Impedance measurements were carried out by using a system which includes an HP 4192A Impedance Analyzer in the 5 Hz- 13 MHz frequency range at a current amplitude, i,,, of 0.4 mA (RMS). Some measurements were done by varying the ac current amplitude between 0.22 and 9.65 mA (RMS) at fixed frequencies of 1 kHz, 65 kHz and 5 MHz. In order to apply a longitudinal dc magnetic field, HdC, we used a 200-turn, 10 cm long solenoid; magnetic fields up to 6400 A/m (80 Oe) could be applied to the samples. We have shown [5] that the complex inductance formalism, L* = L’ + jC, where L is the real part of inductance, j is J-1, and E’ is the imaginary part of inductance, is more convenient to investigate magnetic phenomena. L* can be derived directly from complex impedance data, 2 * , as: L* = (-j/w)Z*
II
3. Experimental results and discussion We first carried out measurements of the imaginary part of inductance as a function of frequency at L, = 0.4 mA. It can be seen that L!’ goes through a maximum in the 5 Hz-13 MHz frequency range (inset Fig. 1). We have previously shown [5,7] that this dispersion has a relaxation character, and can be directly associated with the domain wall relaxation. The relaxation frequency, o, (or ,f,, if measured in Hz), can then be directly determined from the maximum in C(w). The observed value for relaxation frequency was f,= 65 kHz. Since this frequency is a critical one, we have carried out measurements at f =f, (65 kI-Iz), f -=c f,(1 kHz), and f s-f, (5 MHz). At 65 kHz, L! and L!’ show similar values and also a maximum at the same current amplitude Fig. 1. At low frequencies (as compared with f,), L!and L!’ are considerably different both in value and behavior as a function of current amplitude. Fig. 2 shows that C exhibits a larger value than c’ and also a maximum, while c’ seems rather insensitive to i rms. At (5 MHz), both L! and L!’ have small values quite independent of i,,,, Fig. 3.
25
-
20 e t -1 15
(1)
where w is the angular frequency ( w = 2n-f,,,,,). We have also shown [5] that by subtracting the impedance values obtained under a saturating dc field measurement (HdC 2 6400 A/m) to the zero field measurement run, it is possible to separate the domain wall contribution from total inductance. In the following, we consider only these differences.
101
0
/
I
I
2
4
6
8
10
i,,, WV
Fig. 1. Variations of the C and L!’ as a function of i,,, at f = 65 Hz. In the inset, the variations of the imaginary part of inductance, L!‘, as a function of frequency at i,,, = 0.4 mA are shown.
12
K.L. Gar&,
.
R. Valenzuela /Materials
. .
. .
.
. . 0
oIm0
’ 2
1
I
4
6
8
10
i,,,(W Fig. 2. L’ and c’ as a function of i,,
at
f
=
1 Wz.
Letters 34 (19981 lo-13
R, is related to the inverse of the domain wall damping parameter 171. When the experiment is carried out with a strong dc field applied longitudinally, the wire is magnetically saturated and therefore the parallel arm, which represents the domain wall contributions is eliminated. The difference (L” jHdc=o - ( L* jHdc=6400A , m, done point-to-point for each measurement run, thus represents only the domain wall contribution. In the equivalent circuit representation, this difference leads to the elimination of the R, L, series arm; experimental results are therefore associated with the domain wall contribution, represented only by the R, L, parallel arrangement. The complex inductance of an R, L, parallel circuit can be easily derived as: L’ = R; L,/(
R; + w’L?p)
c’ = wR, L;/( We have shown [5] that the impedance response of an amorphous wire submitted to an ac current of small amplitude can be modeled to a good approximation by means of a simple equivalent circuit, formed by a R,L, series section, in series with a R,L, parallel arm. These circuit elements are associated with physical parameters of the sample; R, is the wire dc resistance, L, is the inductance associated with the rotational permeability of the wire; L, is associated with the domain wall permeability, and
(2)
R; + w*L;)
(3)
At the relaxation frequency o, = R,/L,. If this condition is substituted in Eqs. (2) and (3) it leads to: L’ = c’ = L,/2 This expression is in good agreement with results in Fig. 1, where L and Z!’ not only have similar values but also a similar behavior. At low frequencies, Eqs. (2) and (3) can be written: L = R; L,,/R;
= L,
(4)
L!’ = wR, Li/Rfy = wL~/R,
= small
(5)
5MHz
.
3--
l
l
l
l
.
.
..
This approximation, as results in Fig. 2, effectively shows a small value for L!’ when compared
.
l
5.0
. .
0.0
0
2
4
6
8
i,,,(W Fig. 3. C and I?’ as a function of i,,,
at f = 5 MHz.
10
I
I
5
10
.
5
L'(W)
Fig. 4. Spectroscopic plot of real, L’, and imaginary, inductance at i,,, = 0.4 mA.
C, parts of
K.L. Car&,
R. Valenzuela/Materials
with the real part of inductance C more important, the latter exhibits a maximum; we have shown [5] that L’ is proportional (by the corresponding geometrical constant) to the circumferential permeability, and therefore, should show such a maximum followed by a hyperbolic decrease as the field produced by the ac current increases. A comparison of L! at f= 65 kHz and L’ at f = 1 kHz leads to a ratio =: 0.5, as expected from Eq. (4) and Eq. (5). The resulting approximation for the high frequency case are: L! = Ri L,/o’L’, L” = wR, L~/o’L’p
= Ri/m2L, = R,/w
= small = small
(6)
Lerterr 34 (1998) IO-13
13
ties of dielectric materials. In this case, it has been interpreted in terms of a relaxing system with a large distribution of time-constants. The combination of such a distributed impedance response results in a deformed semicircle. In our case, the observed deformed semicircle can be explained by considering that domains in the wire’s outer shell have a variety of widths, which leads to a distributed inductance response. As a conclusion. we have shown that the complex inductance behavior as a function of’ the ac current amplitude, at constant frequency, is in good agreement with the modeling of magnetoimpedance by means of simple equivalent circuits.
(7)
Experimental results are in good agreement, since at 5 MHz, Fig. 4, both L’ and L!’ are quite small and insensitive to i,,,. Since frequency is clearly higher (f~.[,.,> than the domain wall relaxation frequency, domain walls are unable to follow the field excitations, and the dominant magnetization process is spin rotation. This mechanism results in a quite small contribution to impedance as compared to domain wall magnetization processes [5]. As has been shown the proposed circuit approximation is good, but some clear deviations are observed. The main deviation from the ideal behavior of a R,L, parallel circuit is that in the relaxation frequency range, experimental data shows a value of L!’ smaller than the one expected from the circuit. This can also be observed by considering the L!L’ complex plane, Fig. 4. On this representation, the locus of data from an ideal R,L, parallel circuit is a semicircle. This kind of deviation has been widely observed [8] when investigating the electrical proper-
Acknowledgements K.L.G. thanks CONACyT-Mexico, ship under grant 3101P-A9607.
for a fellow-
References [I] R.S. Beach, A.E. Berkowitz, J. Appl. Phya. 76 (1994) 6209. [2] L.V. Panina, K. Mohri, Appl. Phys. Lett. 65 (1994) 1189. [3] M. Takajo, T. Yamasaki, F.B. Humphrey. IEEE. Trans. 29 (1993) 3484. [4] M. VBzquez, A. Hernando, Appl. Phys. 29 (1996) 939. [5] R. Valenzuela, M. Knobel, M. Vizquez, A. Hernando, J. Phys. D: Appl. Phys. 28 (1995) 2404. [6] Y. Waseda. S. Ueno, M. Hagiwara, K.T. Austen, Prog. Mater. Sci. 34 (1990) 149. [7] Aguilar-SahagGn, P. Quintana, E. Amano, R. Valenzuela, J.T.S. Irvine, J. Appl. Phys. 75 (1994) 7000. [8] R.J. MacDonald, Impedance Spectroscopy, vol. 34, Wiley-Interscience, New York. 1987.
February
__ l!!iB
1998
Materials Letters 34 (1998) IO-13
ELSEWIER
On the equivalent circuit approximation of magnetoimpedance amorphous wires K.L. Garcia, R. Valenzuela Institute.for Materials Research,
National University
in
*
ofMexico, P.O. Box 70-360, Mexico D.F. 04510, Mexico
Received 30 April 1997; accepted 6 May 1997
Abstract The magnetoimpedance response of CoFeBSi wires was measured at frequencies equal, below and above the relaxation frequency. We show that the behavior of the real and imaginary parts of inductance, as a function of the current amplitude is clearly consistent with modeling of the wire response by means of a parallel RL equivalent circuit. 0 1998 Elsevier Science B.V. PACS: 75.5O.Kj; 75.50.Q; 75.70.K~ Ke?;words: Magnetoimpedance; Amorphous
wire; CoFeBSi alloy; Equivalent
1. Introduction Magnetoimpedance (MI) consists in the variations of the impedance response of a ferromagnetic material (submitted to an ac current of small amplitude), when a dc magnetic field is applied. This extremely sensitive response leads to original applications on magnetic field sensors technology. MI can be fully explained by classical electromagnetism. Recent results has shown [1,2] that ac current flowing through the material produces a circumferential or perimetral (depending of the cross section of the sample) ac magnetic field which inter-
* Corresponding author. [email protected].
Fax:
+52-5-6161371;
e-mail:
mon-
circuits
acts with domains and domain walls. This coupling results in an additional contribution to the total impedance of the sample. When a dc magnetic field is applied, it leads to a damping of the domain wall movement; if it is large enough to saturate the sample, only rotational magnetization processes remain and the inductive coupling decreases significantly. In the case of samples with small cross section, the combination of high frequencies and small resistivity results in a limited penetration of the magnetic field, and therefore in an additional impedance contribution associated with eddy currents. CoFeBSi amorphous wires with very low and negative magnetostriction exhibits a strong MI effect. This is related with their magnetic structure, which can be described [3] as formed by two main parts:
00167-577X/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. PI/ SO167-577X(97)00131-6
an inner
core,
with domains
oriented
close
[4]
K.L. Garci;?, R. Valenzuela /Materials Letters 34 (1998) IO-13
to the longitudinal direction of the wire, and an outer shell with circumferentially oriented domains. Since the magnetic field produced by the ac current has also a circumferential geometry, the most efficient coupling occurs with the outer’s shell domains. In this paper, we carried out complex inductance measurements on these wires; at some selected frequencies, as a function of the ac current amplitude. We show that these experimental results are in very good agreement with an RL parallel equivalent circuit approximation, which was previously investigated [6] as a function of frequency.
2. Experimental techniques
We used small pieces (- 4 cm long) of as-cast corn position wires of amorphous kindly provided by Uni(Co 0.094Fe0.06)72.5B15Si12.5’ tika, Japan, prepared by the in-rotating-water technique [6]. Impedance measurements were carried out by using a system which includes an HP 4192A Impedance Analyzer in the 5 Hz- 13 MHz frequency range at a current amplitude, i,,, of 0.4 mA (RMS). Some measurements were done by varying the ac current amplitude between 0.22 and 9.65 mA (RMS) at fixed frequencies of 1 kHz, 65 kHz and 5 MHz. In order to apply a longitudinal dc magnetic field, HdC, we used a 200-turn, 10 cm long solenoid; magnetic fields up to 6400 A/m (80 Oe) could be applied to the samples. We have shown [5] that the complex inductance formalism, L* = L’ + jC, where L is the real part of inductance, j is J-1, and E’ is the imaginary part of inductance, is more convenient to investigate magnetic phenomena. L* can be derived directly from complex impedance data, 2 * , as: L* = (-j/w)Z*
II
3. Experimental results and discussion We first carried out measurements of the imaginary part of inductance as a function of frequency at L, = 0.4 mA. It can be seen that L!’ goes through a maximum in the 5 Hz-13 MHz frequency range (inset Fig. 1). We have previously shown [5,7] that this dispersion has a relaxation character, and can be directly associated with the domain wall relaxation. The relaxation frequency, o, (or ,f,, if measured in Hz), can then be directly determined from the maximum in C(w). The observed value for relaxation frequency was f,= 65 kHz. Since this frequency is a critical one, we have carried out measurements at f =f, (65 kI-Iz), f -=c f,(1 kHz), and f s-f, (5 MHz). At 65 kHz, L! and L!’ show similar values and also a maximum at the same current amplitude Fig. 1. At low frequencies (as compared with f,), L!and L!’ are considerably different both in value and behavior as a function of current amplitude. Fig. 2 shows that C exhibits a larger value than c’ and also a maximum, while c’ seems rather insensitive to i rms. At (5 MHz), both L! and L!’ have small values quite independent of i,,,, Fig. 3.
25
-
20 e t -1 15
(1)
where w is the angular frequency ( w = 2n-f,,,,,). We have also shown [5] that by subtracting the impedance values obtained under a saturating dc field measurement (HdC 2 6400 A/m) to the zero field measurement run, it is possible to separate the domain wall contribution from total inductance. In the following, we consider only these differences.
101
0
/
I
I
2
4
6
8
10
i,,, WV
Fig. 1. Variations of the C and L!’ as a function of i,,, at f = 65 Hz. In the inset, the variations of the imaginary part of inductance, L!‘, as a function of frequency at i,,, = 0.4 mA are shown.
12
K.L. Gar&,
.
R. Valenzuela /Materials
. .
. .
.
. . 0
oIm0
’ 2
1
I
4
6
8
10
i,,,(W Fig. 2. L’ and c’ as a function of i,,
at
f
=
1 Wz.
Letters 34 (19981 lo-13
R, is related to the inverse of the domain wall damping parameter 171. When the experiment is carried out with a strong dc field applied longitudinally, the wire is magnetically saturated and therefore the parallel arm, which represents the domain wall contributions is eliminated. The difference (L” jHdc=o - ( L* jHdc=6400A , m, done point-to-point for each measurement run, thus represents only the domain wall contribution. In the equivalent circuit representation, this difference leads to the elimination of the R, L, series arm; experimental results are therefore associated with the domain wall contribution, represented only by the R, L, parallel arrangement. The complex inductance of an R, L, parallel circuit can be easily derived as: L’ = R; L,/(
R; + w’L?p)
c’ = wR, L;/( We have shown [5] that the impedance response of an amorphous wire submitted to an ac current of small amplitude can be modeled to a good approximation by means of a simple equivalent circuit, formed by a R,L, series section, in series with a R,L, parallel arm. These circuit elements are associated with physical parameters of the sample; R, is the wire dc resistance, L, is the inductance associated with the rotational permeability of the wire; L, is associated with the domain wall permeability, and
(2)
R; + w*L;)
(3)
At the relaxation frequency o, = R,/L,. If this condition is substituted in Eqs. (2) and (3) it leads to: L’ = c’ = L,/2 This expression is in good agreement with results in Fig. 1, where L and Z!’ not only have similar values but also a similar behavior. At low frequencies, Eqs. (2) and (3) can be written: L = R; L,,/R;
= L,
(4)
L!’ = wR, Li/Rfy = wL~/R,
= small
(5)
5MHz
.
3--
l
l
l
l
.
.
..
This approximation, as results in Fig. 2, effectively shows a small value for L!’ when compared
.
l
5.0
. .
0.0
0
2
4
6
8
i,,,(W Fig. 3. C and I?’ as a function of i,,,
at f = 5 MHz.
10
I
I
5
10
.
5
L'(W)
Fig. 4. Spectroscopic plot of real, L’, and imaginary, inductance at i,,, = 0.4 mA.
C, parts of
K.L. Car&,
R. Valenzuela/Materials
with the real part of inductance C more important, the latter exhibits a maximum; we have shown [5] that L’ is proportional (by the corresponding geometrical constant) to the circumferential permeability, and therefore, should show such a maximum followed by a hyperbolic decrease as the field produced by the ac current increases. A comparison of L! at f= 65 kHz and L’ at f = 1 kHz leads to a ratio =: 0.5, as expected from Eq. (4) and Eq. (5). The resulting approximation for the high frequency case are: L! = Ri L,/o’L’, L” = wR, L~/o’L’p
= Ri/m2L, = R,/w
= small = small
(6)
Lerterr 34 (1998) IO-13
13
ties of dielectric materials. In this case, it has been interpreted in terms of a relaxing system with a large distribution of time-constants. The combination of such a distributed impedance response results in a deformed semicircle. In our case, the observed deformed semicircle can be explained by considering that domains in the wire’s outer shell have a variety of widths, which leads to a distributed inductance response. As a conclusion. we have shown that the complex inductance behavior as a function of’ the ac current amplitude, at constant frequency, is in good agreement with the modeling of magnetoimpedance by means of simple equivalent circuits.
(7)
Experimental results are in good agreement, since at 5 MHz, Fig. 4, both L’ and L!’ are quite small and insensitive to i,,,. Since frequency is clearly higher (f~.[,.,> than the domain wall relaxation frequency, domain walls are unable to follow the field excitations, and the dominant magnetization process is spin rotation. This mechanism results in a quite small contribution to impedance as compared to domain wall magnetization processes [5]. As has been shown the proposed circuit approximation is good, but some clear deviations are observed. The main deviation from the ideal behavior of a R,L, parallel circuit is that in the relaxation frequency range, experimental data shows a value of L!’ smaller than the one expected from the circuit. This can also be observed by considering the L!L’ complex plane, Fig. 4. On this representation, the locus of data from an ideal R,L, parallel circuit is a semicircle. This kind of deviation has been widely observed [8] when investigating the electrical proper-
Acknowledgements K.L.G. thanks CONACyT-Mexico, ship under grant 3101P-A9607.
for a fellow-
References [I] R.S. Beach, A.E. Berkowitz, J. Appl. Phya. 76 (1994) 6209. [2] L.V. Panina, K. Mohri, Appl. Phys. Lett. 65 (1994) 1189. [3] M. Takajo, T. Yamasaki, F.B. Humphrey. IEEE. Trans. 29 (1993) 3484. [4] M. VBzquez, A. Hernando, Appl. Phys. 29 (1996) 939. [5] R. Valenzuela, M. Knobel, M. Vizquez, A. Hernando, J. Phys. D: Appl. Phys. 28 (1995) 2404. [6] Y. Waseda. S. Ueno, M. Hagiwara, K.T. Austen, Prog. Mater. Sci. 34 (1990) 149. [7] Aguilar-SahagGn, P. Quintana, E. Amano, R. Valenzuela, J.T.S. Irvine, J. Appl. Phys. 75 (1994) 7000. [8] R.J. MacDonald, Impedance Spectroscopy, vol. 34, Wiley-Interscience, New York. 1987.