Effects of bias field and driving current on the

J. Phys. D: Appl. Phys. 28 (1995) 2404-2410. Printed in the UK

Effects of bias field and driving current on the equivalent circuit response of magnetoimpedance in amorphous wires R Valenzuelat, M Knobell, M Vazquez and A Hernando lnstituto de Magnetismo'Aplicado, RENFE-UCM, and lnstituto de Ciencia de Materiales, CSlC PO Box 155, 28230 Las Rozas, Madrid, Spain Received 4 May 1995, in final form 11 AugusJ.1995 Abstract. Magnetoimpedance in as-cast, non-magnetostrictive CoFeBSi amorphous ferromagnetic wires, submitted to AC electric current, i,,,,$, in the 0.1-20 mA range and frequencies between 100 Hz and 100 kHz, is analysed in terms of equivalent circuits. The effects of the bias longitudinal field, Hdc, up to 3600 A m-l are also investigated. It is shown that the equivalent circuit representing the wire frequency behaviour can be approximated by a series R,L, arrangement, in series with a parallel LPRp arm. L, and Lp inductor elements are associated with the rotational and domain wall contributions to circumferential permeablity. respectively. Rp is related to wall damping and Rs accounts for all the resistances in the circuit (the wire itself, contacts and so on). The circumferential permeability associated with domain wall movements exhibits a maximum for i,ms = 5 mA (that Is, a circumferential field HAs = 12 A m-'), similar to the classical behaviour of wall permeability. The increase in bias field has the effect of strongly decreasing the Lp value; for Hdc = 3600 A m-', the series circuit along accounts for the frequency response of the wire. The assoclation of the circuit elements with basic magnetization processes is discussed. Results are interpreted in terms of the influence of both fields (DC bias, Hdc.and AC circumferential, HLSsfields) on the inner-core-outer-shellmagnetic structure of the wire.

1. Introduction The impedance response of amorphous materials in the shape of ribbons [1,2] and wires [2-6], when subjected to a longitudinal AC current has shown to be extremely sensitive to DC longitudinal fields. These phenomena are known as giant magnetoimpedance (GMI) and are receiving considerable interest, since they can lead to original applications in field sensors such s.a miniature, sensitive heads for magnetic recording. Recent results have shown that GMI is a classical electrodynamics effect [2,3], depending essentially on the interaction between the circular field generated by the current and the transverse magnetization of the material. In ribbons, transverse domains can be produced by appropriate field annealings. However, GMI effects are stronger i n Co-based amorphous wires obtained by the in-rotatingwater technology [7]. Changes i n impedance as high as 600% Oe-' have recently been reported 141 in FeCoBSi

t On sabbatical leave from'lhe

National University of Mexico. $ Pemmenl address: lnstituto de Fisica Gleb Wataghin, Universidade Estadual de Campinas (UNICAMP), PO Box 6165, Campinas 13083-970. SP Brazil.

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wires at 90 W z . These wires exhibit very low, negative magnetostriction resulting in a domain structure [SI which can be schematically described in terms of an inner core with axially magnetized domains and an outer shell where domains are circumferentially oriented, figure 1 , Deviations from this simple model indicating that some magnetic moments are neither strictly axial not circumferential have recently been observed 191. The inner core cannot be considered as a single domain, but rather must be considered as a multidomain structure, figure I . Also, the boundary between core and shell is not as well defined as i n highly magnetostrictive wires. The study of GMI i s further complicated because its strongest effects occur at high frequencies, at which skindepth effects become significant. Several approximations have been proposed [5,10,1 I ] to account for these effects, with reasonable agreement when compared to experimental results. An exact calculation of the eddy-current-influence is, however, extremely dificult, since i t depends on (and affects) the detailed microscopic domain structure and its dynamics. Complex impedance formalisms have recently been used to investigate [12-161 the frequency response of

Effects of bias field in amorphous wires (iii) association of equivalent circuit elements with physical parameters of the sample.

I I I I I Figure 1. ( a ) A schematic representation of low, negativemagnetostriction, as-cast amorphous wires, with an axially magnetized inner core and an outer shell with circular domains. (b)A cross sectional view.

magnetic materials. This approach (also known as impedance spectroscopy [17]), combined with the use of equivalent circuits, allows a resolution of the magnetization processes. In some cases a direct association of equivalent circuit elements with physical parameters of the material has been established [15]. In a preliminary report [6], we have applied for the first time this analysis technique to the frequency response of amorphous wires. We have shown that magnetization processes in these materials can be clearly associated with specific equivalent circuits. In this article, we apply impedance spectroscopy methodology to analyse the frequency response of ascast CoFeBSi wires in the 100 Hz to 100 kHz frequency range, at AC currents in the range 0.1-20 mA (RMS) and DC fields of 0,240,800 and 3600 A m-'. We show that (i) to a good approximation, the rotational contribution to the circumferential permeability can be separated from the total permeability; (ii) the frequency response of these wires can be interpreted in terms of a combination of simple equivalent circuits, namely a series R L circuit for the rotational part of the circumferential permeability and a parallel R L arrangement for the domain wall contribution to the circumferential permeability; and (iii) GMI phenomena can be schematically interpreted on the basis of changes in domain structure produced both by the DC bias field and by the AC circumferential field. 2. Equivalent circuits: basic concepts

The equivalent circuit methodology, also known as impedance spectroscopy was originally developed [ 171 to study electrical properties of dielectric materials. It can be summarized in terms of three main steps: (i) measurement of the material's response as a function of frequency of the exciting field in the widest frequency range; (ii) modelling of the material's response by means of an equivalent circuit; and

The total polarization response of any given material depends on the contribution of all of the polarization processes. Since each polarization mechanism possesses a different time-constant, measurements in a wide frequency range allow the resolution (separation) of each polarization process contribution. At the lowest frequencies, all the mechanisms are active and contribute to the total polarization. As the frequency of the excitation signal increases, polarization mechanisms with large time constant become unable to follow the field and the total polarization shows a decrease (a 'dispersion') at a given frequency, = l/%, with WO the angular frequency and to the time constant. Only processes with very small time constants remain active at high frequencies. A plot of polarization response as a function of frequency appears therefore as a succession of decreasing steps, sometimes with a characteristic shape (relaxation, resonance) between them. The use of an equivalent circuit formed by ideal elements, (step (ii)), is extremely useful to represent the sample's behaviour. In the general case, however, any given frequency behaviour can be modelled by more than one equivalent circuit if only mathematical (or numerical) simulation is involved. The 'good' equivalent circuit (namely the one with a physical meaning) has to be validated by step (iii), in which a direct association of circuit elements with physical parameters of the sample is established. In the general case, there is no foolproof procedure to find the pertinent equivalent circuit. However, simple circuits (such as parallel and series arrangements) produce characteristic point distributions on plots (complex plane, spectroscopic plots and so on) of impedance, admittance, permittivity or permeability which, with some experience and practice, can be easily recognized. When dealing with magnetic systems, it is more instructive to use the complex inductance formalism, L* = L' - jL", directly obtained from impedance, Z*, by the transformation L' = (-j/w)Z* (1) where w is the angular frequency and j the basis of imaginary numbers. This transformation leads to an exchange i n the real-imaginary components, since the real part of the inductance, L', depends on the imaginary component of the impedance, Z"; conversely, L" is obtained from 2'. Therefore, phenomena associated with magnetization (permeabilities) depend on L', whereas the dissipative effects should be related to L". Of course. skindepth phenomena are expected to affect both components of permeability. Spin rotation at frequencies well below the natural spin resonance can be represented [I31 by a series R L circuit; L is related to the rotational permeability, p,,,, = K L , where K is the geometrical constant. The complex plot, namely the plot of the imaginary part as a function of the real part of permeability, p" versus p', shows a characteristic vertical line (a 'spike'), figure 2.(a). In the frequency range of most impedance analysers (from some millihertz to tens of megahertz), no dispersion is observed, since spin resonance is usually in the gigahertz range. 2405

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Figure 2. Ideal complex conductance plots of (a) a series RL circuit and (b) a parallel RL circuit. The values of the corresponding inductors, as extracted from plots, are shown, as is the direction of increasing frequency. Initial permeability processes (reversible magnetization at low fields,produced by domain wall bulging [181) in amorphous ribbons have been shown [13-151 to be conveniently represented by a parallel R L circuit, which exhibits a semicircle in the complex permeability plot representation, figure 2(b). In this case, there is a domain wall relaxation at a frequency ox= 2xfXgiven by w, = R J L .

kindly provided by Unitika Ltd, Japan. Frequency measurements were performed on samples 8 cm long. After cleaning with a soft acid solution, current leads were attached to sample ends and voltage leads were attached 6 cm apart To improve electrical continuity, Ag paint was used on contacts. Wire resistivity [2] (1.35 pS2 m) accounted for most of the DC resistance; typically, total lead contact resistance was about 0.9 Q. The value of the saturation magnetostriction constant of these wiures is I191 -0.4 x 10-7. Real. Z', and imaginary, Z", component of the complex impedance were measured in the frequency range 100 Hz to 100 kHz by means OF a Stanford Research Systems SR 850 dual phase lock-in amplifier. The AC current Rowing through the wire, ,,,i was varied i n the range 0.1-20 mA, which leads to circumferential fields, H$m,y, in the range 0.25-50.9 A m-' (calculated on the surface of the wire and neglecting skin effects). DC magnetic fields, Hdc, of 0,240,800 and 3600 A m-I were applied along the longitudinal axis of the wire by a Helmholtz system of coils. All the measurements were performed at room temperature, with the axis of the wire aligned perpendicular to the earth's field.

(2) 4. Experimental results

The combination of equivalent circuits with the equation of motion for domain walls led [I51 to the association of equivalent circuit elements with wall parameters: particularly, it has been shown that R is related to the inverse of the viscous damping factor, ,9. and L (related to permeability) depends on the inverse of the domain wall restoring force, a. Wall bulging dynamics is thus controlled by wall rigidity (proportional to the restoring force a ) and damping (depending on the term B ) . The relaxation frequency can be therefore written as w, = uf.6. In the case of domain wall resonance, the circuit includes a capacitor which is associated with the wall effective mass, m. Among the benefits of representing a material's behaviour with the aid of an equivalent circuit is the possibility to model the material response under a variety of conditions. The effort to find a physically sound circuit gives new insights into the material capability. Impedance spectroscopy is particularly useful in the investigation of magnetization processes, since it can lead to their resolution, as well as to the evaluation of physical parameters of domain walls from simple impedance data analysis. This methodology has been applied to a variety of magnetic materials, such as polycrystalline soft [ 121 and hard [13] ferrites, amorphous ferromagnetic ribbons [13-15], and amorphous ferromagnetic wires [6.16]. Impedance spectroscopy can be expected to become a valuable characterization technique for soft ferromagnetic materials in the near future. 3. Experimental

We used as-cast amorphous ferromagnetic wires obtained by the in-rotating water quenching technique [7] of nominal composition Cos~,lFe4,4B1sSilz~ and 125 p m in diameter, 2406

The frequency behaviour of real and imaginary components of impedance showed the features previously reported in other papers [2,3,6, IO]. Asdiscussed above, magnetic permeabilities are better investigated by using complex inductance formalisms, which can be obtained from impedance through transformation (1). The results of this transformation from impedance data are shown in figure 3. Whereas Hdc = 0 results show a behaviour which cannot be associated with a simple equivalent circuit, the H d c = 3600 A m - ' are distributed in a way (a 'spike') that reminds one of a series R L circuit. If such a circuit is used for the Hdc = 0 condition, which we call the series circuit R,L,, then Rs accounts for the DC resistance in the circuit. On the other hand, we can assume that, since this wire is a soft ferromagnet, the longitudinal field results in saturation and therefore LI is associated with the rotational contribution to the circumferential permeability, &. We can also assume that this rotational permeability can be subtracted from the total inductive response of the wire to obtain the domain wall contribution, to the circumferential permeability. When this operation, which can be expressed as = L'WdC = 0) - L;,=,,, (3)

&,,

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(4) L&it = L"(Hdc = 0) - L&d,,mo) is performed for each frequency, a dramatic change is observed in the complex L" plot, figure 4 . Data points are now distributed with a clear tendency to form a semicircle. The upper frequency limit of our system does not allow one to observe the whole locus of the curve for high frequencies; with the higher frequency ranges available in other materials [13-151, near completion of the semicircle has been obtained. In the present case, completion of the semicircle could be expected, although some deviations

Effects of bias field in amorphous wires

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from this behaviour could appear as a consequence of the skin-depth effect, which is especially strong at high frequencies. Spectroscopic Lk,,, and LL,,, plots, figure 5, clearly show relaxation features; the real component (associated with domain wall permeability) decreases as the imaginary part (associated with dissipative processes) goes through

a maximum. Such a frequency response has been widely observed [12-161 to be typical of complex permeability associated with domain wall movements. In terms of equivalent circuits it appears that the frequency response of these wires can be approximated by a series R,L, circuit, in series with a parallel R p L p arm. The application of a strong bias field simply eliminates the parallel arm. The values of the equivalent circuit elements can be determined as follows. R, can be determined from DC resistance measurements; L,, associated with the rotational permeability of the wire, is found as the intersection of the 'spike' with the real L' axis in the complex plot for the Hdc = 3600 A m-' experiment, figure 3. L p , related with the circumferential wall permeability, is simply the semicircle's diameter in figure 4 (for Hdc = 0). R, is not a real resistor; this component has been found [14,15] to be associated with the viscous damping of domain walls. It can be determined from the relaxation condition, equation (2), once the relaxation frequency is evaluated, for instance, from log LL,,,-log j plots, figure 5. where it appears as a maximum. The observed values of equivalent circuit elements and relaxation frequency are listed in table I . The effects of intermediate values of Hdc can be investigated by performing a subtraction operation similar to that for equations (3) and (4). We use measurements at Hdc = 240 and 800 A m-] (instead of Hdc = 0), and substract, for each frequency, the points obtained at 2407

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The tendency to form a semicircle is again observed, with two main differences: the semicircle diameters are strongly decreased by the field, and they appear even more incomplete than in the Hdc = 0 case. In terms of equivalent circuits, this indicates a decrease in L,, which is just the semicircle's diameter, whereas Rp probably remains unchanged. The latter can be inferred from the fact that semicircles are here far from completion, which means that the upper frequency limit is lower than the relaxation frequency; the ratio R,/L, has increased. The effects of AC current amplitude are not so strong as those of bias field. For Hdc = 3600 A m-I, the impedance response of the wire was virtually independent of the current amplitude value. This is consistent with the assumption that, under these conditions. only the rotational part of the circumferential permeability (which is nearly independent of the circumferential field value) is observed. The wall contribution for each current was obtained as before, by subtracting the high-field measurements. Once again, semicircles are observed in the complex plot, figure 6. The low-frequency limit of wall permeability can be calculated from L, (the semicircle's diameter) by the expression I201 ,-I.

&,, = 2.857

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where I is the wire's length. When plotted as a function of the circumferential field, a maximum is observed or H$,,,r = 12.7 A m-' (0.16 Oe) (corresponding to ,,i = 5 mA, figure 7, followed by a hyperbolic decrease, as H?,$ increases. These plots are typical [ 14,211 of domain wall permeabili!y in soR ferromagnetics. 5. Discussion

We first discuss the effects of circumferential field generated by the AC current at zero bias field. We use the cross section of the model in figure 1, showing the outer-shell domain structure, figure 8(a). When irmE reaches a maximum, domains with spins parallel to the field direction increase their volume by wall displacements at the expense of domains with opposite orientation. Except by the circular geometry, this process should be the same than 2408

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H+rms ("1 Figure 7. The influence of the circumferential field on the wall permeability (lines are shown only to guide the eye, and do not represent any mathematical fit). in any domain structure with 180" Bloch walls, submitted to alternate fields in the easy axis direction. The field behaviour of the low-frequency, quasi-state permeability is comparable to the typical wall permeability of soft ferromagnets, for which a maximum, followed by a hyperbolic decrease (constant)H,%,!) is commonly observed [14,21]. By recalling the basic definition of permeability as the slope in B = H plots and the general features of a typical magnetization curve, it is clear that a p versus H plot should exhibit a constant behaviour at low fields (initial permeability), an increase as field increases (unpinning and initiation of wall propagation), a maximum (for the H value at which the magnetization curve begins to curve toward saturation), and then a hyperbolic decrease (as saturation is approached). The only difference with our ease is that, at small fields, no constant permeability range appears, which can be attributed to the fact that the propagation field, namely the threshold field needed to initiate wall displacement is smaller than our smallest applied field (on the wire surface), H,m,, = 0.254 A m-' (3 mOe). The field for the maximum permeability, = 12.7 A m-] (160 mOe) is of the same order of magnitude than coercivity values observed 1221 in these wires, as also observed in wall processes. The frequency behaviour has also been found to be consistent with this interpretation. The application of the bias field leads to a strong decrease in wall permeability. The schematic domain model can provide a qualitative explanation of these results, figure S(b). The decrease in permeability has been ascribed [22,23] to a rotational mechanism, whereby circumferential domain spins are reoriented toward the wire axis under the action of HA