Sensors and Actuators A 119 (2005) 384–389

Skin-effect and circumferential permeability in micro-wiresutilized in GMI-sensors

Henryk K. Lachowicza,∗, Karin L. Garciab, Marek Kuzminskia,Arcady Zhukovc,d, Manuel Vazquezb

a Institute of Physics, Polish Academy of Sciences, Al. Lotnik´ow 32/46, 02-668 Warszawa, Polandb Instituto de Ciencias de Materiales, CSIC, 28049 Cantoblanco, Madrid, Spain

c Dpto. Fısica de Materiales, Fac. Qu´ımica, UPV/EHU, Apdo. 1072, 20080 San Sebasti´an, Spaind “TAMag Iberica” S.L., Parque Tecnol´ogico de Miram´on, Paseo Mikeltegi 52, 1aPlanta, 20009 San Sebasti´an, Spain

Received 10 August 2004; received in revised form 8 October 2004; accepted 9 October 2004Available online 18 November 2004

Abstract

e nominalc modelw changes int ability, isp ntities arec ta obtainedf agreementw©

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The penetration depth of the skin-effect and circumferential permeability have been calculated for a magnetic micro-wire of thomposition Co67Fe3.85Ni1.45Mo1.7Si14.5B11.5 displaying large giant magnetoimpedance (GMI) effect. For these calculations a simpleas applied in which a rough assumption was made that the changes of the real component of the impedance are due only to

he effective cross-section of the wire for the AC-current. The evolution of both, the penetration depth and circumferential permeresented as a function of the applied DC-axial field at various frequencies of the AC-current, flowing along the wire. These quaalculated using the experimental data of the real component of the micro-wire impedance. A comparison of the experimental daor the imaginary component of the impedance with those calculated by the model, shows that the latter gives only a qualitativeith the measured dependencies.2004 Elsevier B.V. All rights reserved.

eywords: Amorphous micro-wires; Giant magnetoimpedance; Skin-effect; Penetration depth; Circumferential permeability

. Introduction

The giant magnetoimpedance (GMI) effect[1–3] mani-ests itself by huge changes in the impedance of an elec-rically conducting magnetic element, usually in a form of

magnetically soft thin ribbon, tiny wire, thin film or filmtructures, submitted to a simultaneous action of an externalongitudinal DC-magnetic field and a transverse or circularin the case of wires) AC-field generated by an AC-current ofhe R.F.-frequency flowing through the magnetic conductor.uring the recent years, numerous results of intense studiesn the GMI-phenomenon, both of the basic and applied na-

ure, were reported. Owing to them, the physical background

∗ Corresponding author. Tel.: +48 22 843 5212; fax: +48 22 843 0926.E-mail address:henryk.lachowicz@ifpan.edu.pl (H.K. Lachowicz).

of the GMI-phenomenon is now generally recognizede.g.[4]). Moreover, this effect is actually widely utilizeda numerous sensitive sensors of various physical quan(see e.g.[5]). Magnetic element most frequently used in thsensors is an amorphous wire of very small diameter (frto around 100�m). The composition of these wires is simto those of metallic glasses, Co- or Fe-rich alloys. Sinceeffect is mainly observed in nearly-zero magnetostrictiveterials, for application in sensors utilizing this effect, usuCo-rich wires are used. Fe-rich wires are assigned in geto sensors based on magnetoelastic phenomena, althoucently GMI effect has been also observed in Fe-rich wafter special heat treatment[6]. Recently, thin micro-wirecoated with glass (ordinary Pyrex-type), produced usingso-called Taylor–Ulitovsky method[7,8] are often utilized insensors. Since the thermal expansion coefficient of the

924-4247/$ – see front matter © 2004 Elsevier B.V. All rights reserved.oi:10.1016/j.sna.2004.10.017

H.K. Lachowicz et al. / Sensors and Actuators A 119 (2005) 384–389 385

coating significantly differs from that of the metallic core,special effort is needed to achieve the GMI effect in this fam-ily of wires [9].

In a range of a relatively high frequency of the AC-current(0.1–100 MHz), the GMI can roughly be described in theframe of the well-known standard skin-effect. The latter man-ifests itself in that the applied current effectively flows in anouter part of the cross-section of the magnetic element. Thepenetration depth,δ, which gives a measure of this effect,is described by a commonly known expression derived un-der assumption that the dimension of the conductor in thedirection of magnetic field penetration is much larger thanthe penetration depth (in the case of a wire this conditionexpresses asδ � r, wherer is the wire-radius)

δ = (2ρ/ωµ0µ�)1/2, (1)

where ρ is the electrical resistivity of a magnetic ele-ment,ω the angular frequency of a driving current (ω = 2πf;f frequency), µ0 permeablity of the free space equals4� × 10−7 H/m, andµФ the relative circumferential perme-ability in the case of a wire or transverse one in the case of aplanar geometry of a magnetic element.

However, it should be noted that since the wire is fer-romagnetic,µФ depends on the frequency, the intensity ofthe circular AC-field (generated by the AC-current flowinga biasD tem-p d thed lex[

rs ofm ngtho nglyv nsid-e (ort gov-e en-c

y isn ng ar-r itsD mple,r rmedf att hasec areaw skin-eg

S

w thg

as af

(whereRDC is the DC-resistance of the wire andRAC thereal component of the impedance measured at a given fre-quency as a function of the axially applied DC-field) can beobtained solving a quadratic equation which appears whentaking a ratio ofRDC/RAC calculated considering the appro-priate cross-sections. The penetration depth estimated thisway is given as follows:

δ = r[1 − (1 − RDC/RAC)1/2]. (3)

It should, however, be mentioned that Eq.(3) is only mean-ingful when the penetration depth,δ, is smaller than the wireradius,r, or what is equivalentRAC >RDC (in fact the pen-etration depth should be infinitively large at the frequencyf = 0). In an opposite case, an analysis should be carried outon the grounds of the conventional eddy-current model (seee.g. [13]). However, it seems that the use of Eq.(3) for asimple analysis of the data presented in this work, is suffi-ciently justifiable if only the results of the analysis have to beof help for a qualitative interpretation the GMI-response ofthe magnetic wire-element and possible use of this elementin sensors.

It is worth to notice that similar estimation as that givenabove has been reported when the magnetic element is in theform of a ribbon[14].

The values of the penetration depth estimated on theg cu-l arq

µ

ptha ax-i ngt datao hei o-w MI-r

2

ren-t uredv lass-c itionom per-t od,h eg m-i andv lys es oft s in-

long the wire), the magnitude and orientation of an axialC-field, as well as also on the mechanical strain, anderature, thus, it has in general the tensor character anetailed analysis of the GMI is in fact much more comp

10].Since the transverse permeability can vary by orde

agnitudes with a DC magnetic field applied along the lef the magnetic element, the penetration depth also stroaries, resulting in huge changes in the impedance. Coring the above, it is obvious that the circumferential

ransverse) permeability is the key parameter whichrns the GMI-effect in the range of high enough frequies.

A direct measurement of circumferential permeabilitot an easy task and requires some special measuriangement (see e.g.[11]). However, this permeability andC-field dependence can be estimated using a very si

ough approximation. Such estimation have been perfoor magnetic wires by Knobel et al.[12]. They assumed thhe changes in the real component of the impedance (in-pomponent) are due only to changes in the effectivehere the AC-current flows as a consequence of theffect. Then, the cross-section of this outer shell,Seff, can beiven as follows:

eff = π[r2 − (r − δ)2], (2)

herer is the radius of the wire andδ the penetration depiven by Eq.(1).

Hence, a simple estimate of the penetration depthunction of the experimentally obtainable ratioRDC/RAC,

rounds of Eq.(3) for a given wire can then be used to calate its effective circular permeability,µФ (treated as a scaluantity), converting Eq.(1) to the form

� = 2ρ/ωµ0δ2. (4)

Knowledge of the evolution of both, the penetration dend the effective circumferential permeability with the

al DC-field and frequency of the AC-current flowing alohe wire, calculated making use of the experimentalf the DC-resistance,RDC, and the real component of t

mpedance,RAC(HDC), obtained for a given magnetic micrire, can be helpful to understand and interpret the G

esponse of this element.

. Experimental

Calculations of the penetration depth and circumfeial permeability have been carried out using the measalues of real component of the impedance in a goated amorphous micro-wire of the nominal composf Co67Fe3.85Ni1.45Mo1.7Si14.5B11.5, exhibiting vanishingagnetostriction and, therefore very soft magnetic pro

ies. The wire, fabricated using Taylor–Ulitovsky methad the core-diameter of 22.4�m and the thickness of thlass coating of only 0.2�m. Owing to the appropriate che

cal composition providing very soft magnetic propertiesanishing magnetostriction[15], as well as the extrememall thickness of the glass-coating, magnetic propertihe wire were not deteriorated because of low stresse

386 H.K. Lachowicz et al. / Sensors and Actuators A 119 (2005) 384–389

troduced by this coating resulting from different thermalexpansion coefficients of glass overlay and the wire mate-rial.

The calculated resistivity of the used micro-wire was equalρ = 1.78� m (the relatively large value of this quantityshows that the wire is in fact in an amorphous state).

The auto-balancing bridge (Agilent 4285A) with fourcoaxial leads (four-terminal pair configuration) was usedto measure two components of the impedance,Z = R +jX (j = √−1), in the frequency range of 1–30 MHz of anAC-current of 1 mA (rms). The four terminal method and theelectronic cable compensation function of the 4285A bridgeallowed to eliminate practically the errors introduced by 1 mlong cables and to keep constant the level (auto level func-tion) of the sinusoidal current flowing along the sample. Themicro-wire was fixed by a silver conducting paint to the cur-rent and voltage pick-up contacts (the latter placed 30 mmaway) in a home-made holder. These four contacts are con-nected by very short wires to the inner contacts of the BNCterminals of the holder (the shields of BNC connected to-gether). A computer controlled setup generating DC-currentthrough the Helmholtz coils, enables to measure precisely themagneto-impedance [Z(H)] hysteresis loop in an axial staticmagnetic field of its intensity up to 6.4 kA/m.

3

po-n iousf ire-s ationd -c n-da tion,f lyo sn( ft edw

easei pen-e thinw theg etra-t f thewf ationdn thec

eanr es icro-

Fig. 1. Real (a) and imaginary (b) components of impedance as a functionof axial DC-field at various frequencies of AC-current of 1 mA.

wire as those in the case of the penetration depth. These cal-culation were performed with the help of Eq.(4), using thevalues of the computed penetration depth and the micro-wireresistivity (ρ = 1.78� m). In order to have a better insightinto the parts of these characteristics where their changes arethe greatest, they are plotted inFig. 4b within a very narrow

Fig. 2. Calculated penetration depth vs. axial DC-field at various frequenciesof AC-current.

. Results and discussion

Fig. 1(a and b) shows both, the real and imaginary coments of the impedance plotted vs. axial DC-field at var

requencies of the 1 mA AC-current flowing along the wample. The axial DC-field dependencies of the penetrepth calculated by means of Eq.(3) for the same frequenies are shown inFig. 2. Fig. 3shows the frequency depeence of the minimum of the penetration depth (atHDC = 0)s well as the best fit of this dependence by a power func

n, which givesn=−0.40. This exponent differs noticeabf the −0.5 value expected from Eq.(1). This, however, iot surprising since circumferential permeability,µФ, in Eq.1) changes with the frequency (seeFig. 4). The value ohe DC-resistance,RDC = 135.5 , measured for the studiire-sample, was used in these calculations.As expected, the skin-effect became stronger with incr

n frequency and a maximum of this effect (the smallesttration depth) occurs in the range of the DC-field wihich the changes of the magnetization of the wire werereatest. With an increase of the DC-field, also the pen

ion depth increases monotonically towards a value oire-radius (no skin-effect) as it is observed inFig. 2. The

requency dependence of the minimum of the penetrepth (maximum of the skin-effect) shown inFig. 3, reflectsot only its relation with frequency but also variations ofircumferential permeability with this quantity.

Fig. 4a shows the axial DC-field dependences of the melative circumferential permeability,µ�, calculated for thame frequencies of the AC-current flowing along the m

H.K. Lachowicz et al. / Sensors and Actuators A 119 (2005) 384–389 387

Fig. 3. Fitted dependence of minimum of penetration depth (solid line) vs.frequency of AC-current.

range of the DC-field. It is worth to mention that the curveobtained at 1 MHz display too large values ofµ� in the rangeof high DC-field (around 4000 at 6.4 kA/m). This is clearlyan unphysical result which most probably appears because

Fado

at this frequency the conditionδ � r is not fulfilled and thanEq.(1), and consequently Eq.(4), should not be used to cal-culate the permeability. An analysis of the experimental datashould in this case be carried on with the help of the con-ventional eddy-current model[13,16], as it was mentionedabove.

As it can be expected, the greatest value of the circum-ferential permeability is achieved at the lowest frequency.Huge changes of this permeability occur in the range of theDC-field within which the magnetization of the wire changesvery steeply (at the field corresponding with the coercivity ofthe magnetization characteristics). Outside this range, the cir-cumferential permeability holds practically a constant valuewhich becomes smaller as the frequency is higher. Sincethen the wire is saturated magnetically, it can be expectedthat the permeability is due to the rotation of the total wire-magnetization caused by a circumferential field generated bythe AC-current flowing along the sample (for the used inten-sity of 1 mA, this field at the outer surface of the wire equalsaround∼80 A/m).

If δ � r (or RAC RDC), then the imaginary componentof the wire-impedance,X, can be expressed as[16]

X = ωµ�δl/4πr, (5)

wherel is the length of a wire; the other symbols have thes

hew andcft mm rod-u aryc on-s mea-s fre-qt ex-p stud-i d inF efi f thei isfi uali-t , nots C-

ig. 4. (a) Calculated dependencies of mean relative circumferential perme-bility vs. axial DC-field at various frequencies of AC-current; (b) the sameependencies as inFig. 4(a) but in narrow range of axial DC-field plotted inrder to have better insight.

fi ec -i ctsw wasn stri-b lext um-f auseo real

ame meaning as before.It is seen from Eq.(5) that the imaginary component of t

ire impedance is proportional to the penetration depthircumferential permeability. The conditionδ � r is fulfilledor the studied micro-wire at the frequenciesf ≥ 10 MHz (inhe range of the DC-field within which the wire is far froagnetic saturation). One then may expect that the pct µ�δ should be proportional to the measured imaginomponent of the impedance of the wire-specimen. Cidering the above, a comparison of this componentured at the frequency 10 MHz (as well as at higheruencies) with the calculated product,δµ�, should show

o what extent the introduced model approximates theerimental data. Such a comparison obtained for the

ed micro-wire measured at 10 and 30 MHz is presenteigs. 5 and 6(the values of the productδµ� shown in thesgures were reduced in relation to the maximum value omaginary component,X). As it can be easily noticed in thgure, the calculated dependencies approximate only qatively those obtained experimentally. This is, howeverurprising since, in particular, in the range of higher Deld the conditionRAC � RDC is not fulfilled and then thalculated values of the productδµ� are incorrect showng relatively large errors. Additionally, a number of effehich most probably can occur in a magnetic elementeglected in this rough approximation. The current diution within the wire is, in reality, much more comphan that assumed in the model. Additionally, the circerential permeability could not be homogeneous becf the “magnetic” heterogeneity usually existing in the

388 H.K. Lachowicz et al. / Sensors and Actuators A 119 (2005) 384–389

Fig. 5. Measured imaginary component of impedance (Xred) and calcu-lated product of penetration depth and circumferential permeability (δµФ)red

vs. axial DC-field at 10 MHz (values of the product� shown in thesefigures were reduced in relation to the maximum value of the imaginarycomponent,X).

Fig. 6. The same dependencies as inFig. 5but obtained at 30 MHz.

material. Besides of the fact that the conventional electrody-namic approach does not[1–5] take into account the tensorcharacter of magnetic permeabilityµ� [10], considering thisquantity as a scalar (the case of almost paramagnetic con-ductor), the existing magnetic domains are also neglectedin this approach. Also the thickness of the wire can displaycertain variations. Further, the micro-eddy currents gener-ated by the moving domain walls should also be taken intoaccount[17], particularly at relatively low frequencies. Con-sidering all the above, it seems obvious that the model usedhere is much oversimplified, and therefore, cannot well ap-proximate the dependencies obtained experimentally. How-ever, the advantage of the used model is that both the self-consistent quantities governing the GMI-effect, the penetra-tion depth and the circumferential permeability, can easily beestimated.

4. Conclusions

It is shown that the DC-field dependencies of the penetra-tion depth and the circumferential permeability in magneticmicro-wires, the key-quantities responsible for the GMI-effect, can be determined using the experimental data of thereal and imaginary components of the impedance and a sim-ple model based on the assumption that the changes in thereal component are only due to changes in the effective wirecross-section within which the AC-current flows along thewire.

A performed comparison of the experimental dependen-cies with those calculated from the model showed that onlyqualitative agreement can be achieved. However, it wasshown that determination is possible of both key-quantities,which are difficult to be determined otherwise. It demon-strates well how these quantities vary with the external DC-field as well as with the frequency of an AC-current flowingalong the wire.

Acknowledgements

The study was conducted being partially supported withinthe European Community Program ICA1-CT-2000-70018(Center of Excellence CELDIS in the Institute of Physics,P

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olish Academy of Sciences).

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Biographies

H.K. Lachowicz graduated in 1958 from the Department of Electronicsof the Warsaw University of Technology. He obtained PhD degree in1968 from the Institute of Fundamental Technological Research of thePolish Academy of Sciences, and DSc degree in 1983 from the WarsawUniversity of Technology. Present employment: Institute of Physics, Pol-ish Academy of Sciences at the position of half-time emeritus professor.Current fields of interest: magnetic materials and phenomena appeared inthem as well as their technological utilization. He has published 165 tech-nical papers most of them in the journals of the world-wide renown aswell as some chapters devoted to magnetic memories in various encyclo-pedias and monographs. He has presented a number of invited papers atthe well-known international conferences, symposia, schools, etc. Mem-ber of a number of international conference committees and editor ofa number of the conference proceedings. He was supervising 10 PhDstudents.

M. Ku zminski graduated in 1986 from the Department of TechnicalPhysics of the Warsaw University of Technology. In 1997 he receivedPhD degree in physics from the Institute of Physics, Polish Academyof Sciences Present employment: Institute of Physics, Polish Academyof Sciences. Current fields of interest: magnetic amorphous and nano-c , giantm around5 ls.

K p oft e

received her BS degree from the National University of Mexico UNAMin 2001. She is presently a PhD student in the Universidad Complutensede Madrid and belongs to the Magnetic Materials Group of the MaterialsScience Institute of Madrid ICM-CSIC. Her research interests includemagnetic properties of micro and nano structured magnetic materials forpure resarch and/or sensors applications

A.P. Zhukov graduated in 1980 from the Physics & Chemistry Depart-ment of the Moscow Steel and Alloys Institute. In 1988 received PhDdegree from the Institute of Solid State Physics of the Russian Academyof Sciences Present employment: as a contracted researcher at the Depart-ment of the Materials Physics of the University of the Basque Country inSan Sebastian. Current fields of interest: amorphous ferromagnetic ma-terials, in particular micro-wires, giant magneto-impedance, giant mag-netoresistance, magnetoelastic sensors, nanocrystalline materials. He haspublished more than 150 reviewed papers in the international journals de-voted to magnetic materials as well as some chapters in the monographsand encyclopedias editing also a number of the conference proceedings.He has presented a number of invited talks at the international magneticconferences.

M. V azquez graduated in 1974 from the Complutense University inMadrid. In 1980, he received PhD degree from the same University.Present employment: professor of research at the Instituto de Cienciade Materiales de Madrid, CSIC (Institute of Materials Science of theSpanish Superior Council for Research). Current fields of interest: softmagnetic materials: micro-wires (bistability, magnetoimpedance) and sen-sor applications; arrays of magnetic nanowires and nanoparticles. He haspublished around 380 technical papers (among them 40 reviews) beinga wnero nted an nces,s oreignp

rystalline materials, magnetization processes, domain structuresagneto-impedance, ferromagnetic resonance. He has published0 technical papers most of them in renowned international journa

.L. Garcia is an active colaborator to the Magnetic Materials Grouhe Institute of Materials Research IIM-UNAM Mexico since 1998. Sh

lso the co-author or co-editor of 5 monographs as well as an of 17 patents related to various magnetic sensors. He has preseumber of invited lectures at various international meetings, conferechool, etc. He was supervising 14 PhD students and a number of fostdoctoral visitors.