Theory of giant magneto-impedance effect in amorphous wires with different types of magnetic...

*Corresponding author. Fax: #7 (095) 334 5776; e-mail:usov@fly.triniti.troitsk.ru.

Journal of Magnetism and Magnetic Materials 185 (1998) 159—173

Theory of giant magneto-impedance effect in amorphous wireswith different types of magnetic anisotropy

N.A. Usov!,*, A.S. Antonov", A.N. Lagar’kov"

! Troitsk Institute for Innovation and Fusion Research, 142092, Troitsk, Moscow region, Russia" Scientific Center for Applied Problems in Electrodynamics RAN, 127412, Moscow, Russia

Received 20 November 1997

Abstract

The effect of various types of magnetic anisotropy on the giant magneto-impedance spectra of amorphous wires withlow magnetostriction is studied theoretically. For the first of the model considered the easy anisotropy axis is supposed tobe parallel to the wire axis, whereas for the second model it has circumferential direction. In case of a wire with axial typeof magnetic anisotropy the transverse magnetic permeability is obtained as a function of external magnetic fieldamplitude H

0and other magnetic parameters of the wire. A strong dependence of this quantity on H

0is shown to explain

the giant magneto-impedance (GMI) effect in this type of amorphous wire. Besides, the amplitude of the GMI effectdepends substantially on the value of a phenomenological damping parameter. In case of a wire with circumferentialanisotropy the classical expression [Landau and Lifshitz, Electrodynamics of Continuous Media, 2nd ed, Pergamon,New York, 1984] for the wire impedance is not valid in the range of external magnetic field 0)H

!, where H

!is the

anisotropy field. In this range of external magnetic field the wire impedance becomes a tensor. It has both longitudinaland transverse components that can be measured experimentally [Antonov et al., IEEE Trans. Magn. 33 (1997) 3367]. Incase of a wire with circumferential anisotropy the peak of the longitudinal component of wire impedance as the functionof H

0corresponds to the anisotropy field, whereas in case of axial anisotropy it is found at zero magnetic field. The results

obtained show that the good magnetic softness of amorphous wire is one of the most important conditions to observe theGMI effect; neither any domain structure nor circumferential anisotropy are strictly necessary. ( 1998 Elsevier ScienceB.V. All rights reserved.

PACS: 75.50.Kj; 72.15.Gd

Keywords: Magneto-impedance; Amorphous wires; Magnetization processes

1. Introduction

About 70 years ago the impedance Z of a thinferromagnetic wire was found [3] to depend toa great extent on an external magnetic field appliedparallel to the wire axis. The phenomenon was

attributed [4] to a nonlinear dependence of mag-netic induction of ferromagnetic wire on the ex-ternal magnetic field. More recently, very largechanges in the high-frequency impedance havebeen observed in Co-rich amorphous wires [5,6]with nearly vanishing magnetostriction under theapplication of external uniform magnetic field withsmall amplitude. The phenomenon was called thegiant magneto-impedance (GMI) effect. The greatsensitivity of the effect to very low values of externalmagnetic field makes it very promising for techno-logical applications [7].

It is widely accepted [6—11] that the main char-acteristics of GMI effect in amorphous wires andribbons can be understood in the frame of classicalelectrodynamics [1]. It is shown that the experi-mental results [6,9] can be described qualitativelyby means of usual expression [1] for the high-frequency impedance of metallic wire (see alsoEq. (16) further) taking into account that in case offerromagnetic wire the skin depth depends also ona transverse magnetic permeability. The latterquantity is considered [6,9] as a fitting parameter.A theoretical approach [8,10,11] is based on thecalculation of AC complex impedance of a fer-romagnetic wire with a domain structure. The aver-aging of the permeability tensor of the wire over thedomains with different magnetization simplifies thecalculation of the wire impedance greatly. It shouldbe noted, however, that the amorphous wire withvery low magnetostriction may not possess anydomain structure at all [12].

In the present paper, we make an attempt todevelop a consistent theoretical description of theGMI effect in Co-rich amorphous wire. We con-sider simplest possible models for initial magneti-zation distribution in the wire. At first we take intoaccount that the GMI effect is observed in the wireswith very low magnetostriction constant. Thus, theanisotropy field of amorphous wire is very smallcompared to the saturation magnetization, so thatone can expect that any domain structure along thelength of a sufficiently perfect wire is energeticallyunfavorable and can hardly be stable [12]. Thissuggestion is supported also by the fact thatfor wires with vanishing magnetostriction the so-called magnetic after effect [13] turns out to be verysmall.

Neglecting the domain structure of amorphouswire means that we take into account only themagnetization rotation processes. On the otherhand, it is well-known [7,14,15] that the magneti-zation distribution is non-uniform throughouta cross-section of a Co-rich wire. Namely, there isusually an inner core uniformly magnetized alongthe wire axis and an outer shell with circumferentialmagnetization. It can be shown [16] that the radiusof the inner core of Co-rich wire is determined bythe distribution of the residual quenching stressesthroughout the wire volume. Besides, it can easilybe changed by means of a proper heat treatment ofthe wire in an external uniform magnetic field or ina circumferential magnetic field of a DC currentflowing through the wire [17]. Thus, it seems rea-sonable to consider two limiting cases for initialmagnetization distribution of Co-rich amorphouswire. If the inner core radius is close to the wireradius, one can neglect the outer shell in the firstapproximation. Then, neglecting also the effect ofdemagnetizing field near the wire ends, one canconsider a sufficiently long piece of amorphous wireuniformly magnetized along the wire axis. In theopposite limit, when the inner core radius is smallwith respect to the wire radius, one can consider thewire to be magnetized circumferentially. For sim-plicity, for both models we suppose also that theanisotropy constants do not depend on the coordi-nate along the wire radius.

The aim of the present paper is to compare theGMI spectra for wires with axial and circumferen-tial types of magnetic anisotropy. At first we calcu-late a permeability tensor of amorphous wiresolving the linearized Landau—Lifshitz—Gilbertequation [18,19]. In case of wire with axial anisot-ropy, a transverse magnetic permeability is ob-tained as a function of external magnetic fieldamplitude H

0and other magnetic parameters of

the wire. Due to a large ratio of M4/H

!A1, the

transverse magnetic permeability turns out to bevery high in zero external magnetic field. On theother hand, an external magnetic field appliedalong the wire axis increases the magnetic ‘hard-ness’ of the wire. Then the magnetic permeability ofthe wire decreases and both real and imaginarycomponents of the wire impedance drop rapidlywith increasing H

160 N.A. Usov et al. / Journal of Magnetism and Magnetic Materials 185 (1998) 159—173

In case of a wire with circumferential anisotropythe classical expression [1] for the wire impedanceis not valid in the range of external magnetic field0)H

!, where the structure of the permeabil-

ity tensor is more complicated. As a result, oneneeds to analyze Maxwell equations separatelywithin the wire and in the outer space. Then, theboundary conditions at the surface of the wire mustbe used to determine a correct solution. It is shownin Section 3 that in case of wire with circumferen-tial anisotropy the wire impedance is, generallyspeaking, a tensor. It has both longitudinal andtransverse components which can be measured ex-perimentally [2].

Comparing the results of calculations of wireimpedance for two models considered, one canprove that the giant magneto-impedance spectraare directly determined by the type of magneticanisotropy of amorphous wire. For example, incase of a wire with circumferential anisotropy, thepeak of the real component of wire impedance asa function of external magnetic field corresponds tothe anisotropy field, whereas in case of axial anisot-ropy the maximal value of this component takesplace at zero magnetic field. The important con-clusion is also that the good magnetic softness ofamorphous ferromagnetic wire is one of the mostimportant conditions to observe the GMI effect.Neither any domain structure nor circumferentialanisotropy are strictly necessary.

In Section 2, the permeability tensors for wireswith axial and circumferential types of magneticanisotropy have been obtained. The calculation ofwire impedance for both cases of magnetic anisot-ropy is carried out in Section 3. The discussion ofthe results obtained is presented in Section 4.

2. Permeability tensor

To obtain a permeability tensor of an amorph-ous wire we use the well-known procedure [19].Let a be the unit magnetization vector of amorphouswire which is governed by the Landau—Lifshitz—Gilbert (LLG) equation [18,19],

t"!c[a, H

%&]#iCa,

tD, (1)

where c is the gyromagnetic ratio and i is thephenomenological damping parameter. Neglectingspatial dispersion, the effective magnetic field of theamorphous wire is given by [18]

0. (2)

Here w!

is the density of a magnetic anisotropyenergy, M

4is the saturation magnetization, H

the total magnetic field which is a sum of both themagnetic field of the alternating current flowingthrough the wire and that of the magnetic chargesarising due to magnetization perturbation. The lastterm in Eq. (2) is an external uniform magnetic fieldthat is assumed to be parallel to the wire axis.

2.1. Uniaxial anisotropy

At first, let us consider the case of uniaxial mag-netic anisotropy with an energy density

1[1!(e

z) a)2], (3)

where ezis the unit vector of the cylindrical coordi-

nates (r, u, z) parallel to the wire axis and K1

is thecorresponding anisotropy constant. In this paper,we do not take into account the demagnetizingfields existing near the ends of the amorphous wire[7] assuming the wire to be sufficiently long. Then,in the absence of the AC current the unperturbedvalues of the unit magnetization vector and effec-tive magnetic field in the wire are given by

a(0)"(0, 0, 1), H0%&"(0, 0, H

!), (4)

where H!"2K

4is the anisotropy field. It is

easy to see that the vectors (4) satisfy stationaryLLG equation

[a(0), H(0)%&

]"0. (5)

Now let an alternating current with a frequencyu is flowing through the sample. Then the per-turbations of the magnetization and the effectivemagnetic field are given by

a(r, t)"a(0)#a(1) exp(!iut),

H%&(r, t)"H(0)

%&#H(1)

%&exp(!iut).

N.A. Usov et al. / Journal of Magnetism and Magnetic Materials 185 (1998) 159—173 161

In view of Eq. (5), the latter satisfy the linearizedLLG equation

!iua(1)"!c[a(1), H(0)%&

!c[a(0), H(1)%&

]!iui[a(0), a(1)]. (6)

Due to normalizing condition a(0) ) a(1)"0 [18],the perturbation of the unit magnetization vectortakes a form

a(1)"(a(1)o , a(1)r , 0). (7)

Then, using Eqs. (2), (3) and (7) it can be shown thatthe perturbation of the effective magnetic field isdetermined by the total alternating magnetic fieldonly, H(1)

%&"H(1)

.. Denoting, for simplicity, H(1)

one can obtain from Eqs. (6) and (7) a susceptibilitytensor sL which couples the perturbations of themagnetization and the total alternating magneticfield, a(1)"sL h. Having derived the susceptibilitytensor, the permeability tensor of the amorphouswire can be easily calculated in cylindrical coordi-nates as

kL "IK#4pM4sL "A

k !ika

0 0 1B , (8a)

where IK is the unit tensor and the permeabilitytensor components are given by

k"(l#u

m)l!u2

l2!u2, k

l2!u2. (8b)

Here we put

um"4pM

4c, l"u

K!iui,

0. (8c)

2.2. Circumferential anisotropy

In case of wire with circumferential type of mag-netic anisotropy the anisotropy energy density isassumed to be

1[1!(er ) a)2], (9)

where er is the unit vector of the cylindrical coordi-nates (r, u, z) pointing in circumferential direction.

One can obtain the permeability tensor for themodel (9) by analogy with the previous calcu-lations. It turns out that in the given case thepermeability tensor has different expressions de-pending on the amplitude of an external uniformmagnetic field applied along the wire axis. It is easyto see that in the magnetic field range 0)H

the unit magnetization vector components aregiven by

"cos h"u

, a(0)r "sin h"$J1!(a(0)z

the sign of the circumferential component beingdetermined by the direction of the unit magneti-zation vector rotation with respect to the wire axis.

Solving the linearized LLG Eq. (6), one canprove that the permeability tensor of the amorph-ous wire in the aforementioned interval of externalmagnetic field is given by

kL (1)"Ak(1)o !ik(1)

aik(1)

bik(1)

ak(1)r k(1)

c!ik(1)

bk(1)c

B , (11a)

where the permeability tensor components can becalculated as

k(1)o "1#u

Hcos h)

k(1)r "1#u

Kcos2 h

k(1)z"1#

Ksin2 h

, k(1)a

umu cos hX2

k(1)b"

umu sin hX2

, k(1)c"!

Ksin h cos hX2

In Eq. (11b) we define

K!iui, X2

Hcos h)!u2.

On the other hand, in case of H0*H

!, when the

components of the unit magnetization vector are

given by a(0)z"1, a(0)r "0, the permeability tensor

takes the form

kL (2)"Ak(2)o !ik(2)

ik(2)a

k(2)r 0

0 0 1B, (12a)

k(2)o "1#u

k(2)r "1#u

, k(2)a"

. (12b)

Besides, in the given case we put

H!iui, X2

K)!u2. (12c)

Eqs. (11a), (11b), (12a) and (12b) represent the per-meability tensor of an amorphous wire with azi-muthal type of magnetic anisotropy in differentintervals of external magnetic field.

3. Wire impedance

Maxwell equations for the perturbations of elec-tric and magnetic fields in an amorphous wire aregiven by

rot e"ik0kL h, rot h"!ik

0ee, (13)

where permeability tensor is given by Eqs. (8a) and(8b) or Eqs. (11a), (11b), (12a) and (12b) dependingon the type of the wire magnetic anisotropy. Herewe neglect the displacement current since the wireconductivity p is usually large enough [5,6] anddenote

e"4ppi/u, k0"u/c.

In this paper, we consider the propagation of thelowest electromagnetic mode in the amorphouswire, so that the perturbations of the electric andmagnetic fields do not depend on the u coordinate[20,21]. We also assume the wire to be infinitelylong, so that the z-dependence of the perturbationsis described by the exponential factor exp(ibz). Fur-ther, we omit this exponential factor everywhere. Itshould also be noted that in case of not very highfrequencies, one can neglect the longitudinal wave

number putting b"0 in all relations concerningthe domain o)a, where a is the wire radius [21].Actually, it can be proved that this value is smallwith respect to the transverse wave numbers withinthe amorphous wire.

3.1. Uniaxial anisotropy

At first, let us consider the GMI effect inamorphous wire with axial type of magnetic anisot-ropy. We are interested in the solution of MaxwellEq. (13) with non-zero e

z-component. It satisfies

the following set of equations:

bo"kho!ikahr"0, !

aho#khr),

o(ohr)"!ik

z, (14)

where bo is the o-component of the magnetic induc-tion. Eliminating the magnetic field componentsfrom the first two equations in Eq. (14) one canobtain for the e

z-component a Bessel-type equation

which has a solution

zJ0(jo), j2"k2

kM. (15a)

Here and further Ji(x) are the Bessel functions of

the first kind whereas the skin-depth d0

and theeffective transverse wire permeability k

Mare given

J2pup, k

(l#um)2!u2

l(l#um)!u2

We see therefore that the structure of the electro-magnetic field propagating in the amorphous wirewith uniaxial type of magnetic anisotropy is analo-gous to the usual case of non-ferromagnetic wire[1,20,21]. Thus, the wire impedance in the givencase can be calculated by means of the well-knownrelation [1]

jaJ0(ja)

2J1(ja)

, (16)

where RDC

"1/pa2p is the DC resistance of the unitlength of the amorphous wire.

3.2. Circumferential anisotropy: Case of0)H

It follows from Eqs. (11a), (11b), (12a) and (12b)that in case of wire with circumferential type ofmagnetic anisotropy, the form of Maxwell equa-tions within the amorphous wire depends on thevalue of the external magnetic field amplitude. Atfirst let us consider the interval of magnetic fields0)H

!, where the wire permeability tensor is

given by Eqs. (11a) and (11b). Then, within the wire,in the domain o(a, Maxwell Eq. (13) take theform

bo"k(1)o ho!ik(1)a

hr#ik(1)b

0(ik(1)

aho#k(1)r hr#k(1)

o(oer)"ik

0(!ik(1)

bho#k(1)

chr#k(1)

o(ohr)"!ik

z. (17)

Eliminating ho, er and ez

components of theelectric and magnetic fields from Eq. (17), one canobtain for the hr and h

zcomponents the following

set of equations:

o(ohr)D"!k2

0e(krrhr#krzhz),

oD"!k20e(k

zrhr#kzzhz). (18a)

Here we denote

krr"k(1)r !

(k(1)a

k(1)o"1#kJ cos2 h,

kzz"k(1)

(k(1)b

k(1)o"1#kJ sin2 h,

kzr"krz"k(1)

k(1)o"!kJ sin h cos h. (18b)

It is interesting to note that the electrodynamics ofthe amorphous wire is evidently determined by theparameter kJ only. The latter is given by the relation

kJ "(u

Hcos h)!u2

. (19)

If the hr and hz

components of the magnetic fieldhave been obtained from Eqs. (18a) and (18b), theelectric field components can be calculated bymeans of equations

(ohr). (20)

3.2.1. Approximate solutionAn approximate solution of the set of Eqs. (18a)

and (18b) can be easily obtained in the limit of highenough frequency. Then, the effective skin depth issmall with respect to the wire radius and the coeffi-cients on the right-hand sides of Eqs. (18a) and(18b) are large enough. Assuming

zexp[ij(o!a)], hr"Hr exp[ij(o!a)],

and taking into account only the leading terms withrespect to j on the left-hand sides of Eqs. (18a) and(18b), one can obtain for this quantity the followingequation:

j4!k20e(krr#k

zz)j2#(k2

0e)2(krrkzz

!k2rz)"0.

Using the relations (18b) it is easy to see that thesolutions of Eq. (22a) are given by

j21"k2

0e, j2

0e(1#kJ ). (22b)

The transverse wave numbers (22b) correspond totwo different electromagnetic modes propagatingin amorphous wire with circumferential type ofmagnetic anisotropy in the range of magnetic field0)H

!; the parameter kJ plays the role of the

transverse wire permeability in the given case.It should be noted that in view of Eq. (21), in the

domain o)a, it is necessary to use the solutions(22b) satisfying the conditions Im j

1,2(0. From

Eq. (18a) it can also be shown that the amplitudes

of magnetic field components for the first and thesecond modes (22b) are related by the equations

hrB1"cos hsin h

hrB2"!

sin hcos h

. (22c)

Therefore, in the limit of high enough frequency,the general solution of the Eqs. (18a), (18b) and (20)has a form

hz"A cos h exp[ij

1(o!a)]

!B sin h exp[ij2(o!a)],

hr"A sin h exp[ij1(o!a)]

#B cos h exp[ij2(o!a)],

k0esin h exp[ij

1(o!a)]

k0ecos h exp[ij

2(o!a)],

er"Aj1

k0ecos h exp[ij

1(o!a)]

k0esin h exp[ij

2(o!a)], (23)

where A and B are the arbitrary mode amplitudes.As we shall see further, unlike the usual case of

non-ferromagnetic wire, to calculate the wire impe-dance in the case considered it is necessary toobtain the amplitude ratio A/B. In accordance withthe general procedure [20,21] this ratio can bedetermined by means of the boundary conditions atthe surface of an amorphous wire. Outside the wire,in the region of o'a, the general solution of theMaxwell Eq. (13) is an arbitrary linear combinationof the modes of electric

"CH(2)0

(po), hr,0"!Cik

(po), (24a)

and magnetic types [21]

"DH(2)0

(po), er,0"Dik

(po). (24b)

Here C and D are the mode amplitudes, H(2)i

(x)(i"1, 2) are the Bessel functions of the third kindand p2"k2

0!b2. Note, the longitudinal wave

number b cannot be omitted in the domain o'ato get a regular behavior of the electromagneticfield of the wire in the limit of oPR [20,21].

Taking into account the boundary conditions atthe surface of the amorphous wire at o"a one canobtain the following set of linear equations for thecoefficients A, B, C and D:

A cos h!B sin h"DH(2)0

A sin h#B cos h"!Cik

k0esin h!B

k0ecos h"CH(2)

0(pa),

k0ecos h!B

k0esin h"D

1(pa). (25)

Next, equating the determinant of this linear set ofequations to zero, one arrives at the following char-acteristic equation for the longitudinal wave num-ber b:

G(p),pH(2)

ik0H(2)

1#j1j2

k20e2$SA1!

k20e2B

2k0e B

2sin2 2h

k0ecos2 h#

k0esin2 hB

It should be noted, however, that the mode ratiosA/B, as well as C/D, do not depend on the longitu-dinal wave number in the given quasistatic approx-imation at all. Actually, by means of Eqs. (25) and(26) these values can be explicitly evaluated as thefunctions of transverse wave numbers (22b),

AG(p)!j2

k0eB cos h

AG(p)!j1

k0eB sin h

, (27a)

G(p)!Cj1

k0esin2 h#

k0ecos2 hD

G(p)j2!j

sin h cos h. (27b)

Taking into account Eq. (22b) it is easy to see thatin the rather broad frequency range the undimen-sional parameters

k0eK&K

JeK@1, Kj2

k0eK&KS

kJe K@1 (28)

are very small. Thus, one can evaluate the solutionsof the characteristic Eq. (26) for two different elec-tromagnetic waves existing in the amorphous wirewith circumferential type of magnetic anisotropy asthe following series with respect to the small par-ameters (28):

G`(p)+

k0ecos2 h#

k0esin2 h

] C1!Aj1!j

2k0e B

2sin2 h cos2 h#2D ,

G~(p)+C

k0esin2 h#

k0ecos2 hD

]C1#Aj1!j

2k0e B

2sin2 h cos2 h#2D .

As a result, the amplitude ratios of Eqs. (27a) and(27b) can be expressed as the corresponding seriestoo. Namely, for two different electromagneticmodes within the wire we have the relations

+!C1#AAj1

k20e2BD

cos hsin h

2Aj1!j

2k0e B

2sin 2h#2, (30a)

+C1#AAj1

k20e2BD

sin hcos h

2Aj1!j

2k0e B

2sin 2h#2, (30b)

whereas for the same electromagnetic modes out-side the wire we obtain

sin h cos h#2 , (31a)

sin h cos h#2 . (31b)

It is easy to see from Eqs. (31a) and (31b) that themode designated by a plus sign in Eqs. (29)—(31) isclose to the magnetic type excitation (24b) since forthis mode the corresponding amplitude ratio out-side the wire is very high, D(D/C)

`DA1. On the other

hand, for the mode designated by a minus sign inEqs. (29)—(31) the same amplitude ratio turns out tobe very small, D(D/C)

~D@1. Therefore, outside the

wire this mode is close to the electric type excitation(24a).

Let us define the longitudinal component of wireimpedance by means of the usual relation [21]

2ez(a)

achr(a), (32a)

where Iz

is the total current flowing through thewire cross-section and e

z(a) and hr(a) are the values

of the electric and magnetic field components at thewire surface.

Using Eq. (23), one can obtain the reduced longi-tudinal component of the amorphous wire impe-dance from the following relation:

sin h#Bj2

cos hA sin h#B cos h

. (32b)

It follows from Eqs. (30a), (30b) and (32b) that thisquantity is finite for the mode (29b) only. Neglect-ing the small terms in Eq. (30b), one can finallyobtain for this mode a simple relation

1sin2 h#j

2cos2 h). (32c)

It is important to note that for the same mode(29b) the transverse component of the wire impe-dance has non-zero value in the range of magnetic

field 0)H0)H

!, too. The latter can be defined by

means of the relation

Zrz"er(a)

2er(a)

achr(a). (33a)

Using Eq. (23), one can obtain for the reducedtransverse component of the amorphous wire impe-dance a relation

cos h!Bj2

sin hA sin h#B cos h

. (33b)

It is easy to see from Eqs. (30a), (30b) and (33b) thatthe transverse component of wire impedance is alsofinite for the mode (29b) only. Neglecting the smallterms in Eq. (29b), one can obtain for this mode anexpression

1) sin h cos h . (33c)

3.2.2. Exact solutionAs we shall see further, the approximate expres-

sion given by Eq. (32c) is very close to the exactresult if the skin depth in the amorphous wire issmall with respect to the wire radius. More strictly,Eq. (32c) is valid if both of the conditionsj1a, j

2a@1 are fulfilled for transverse wave num-

bers (22b). The latter satisfy in case of large enoughalternating current frequency. For lower frequen-cies at least the first of the inequalities violates. Asa result, Eq. (32c) leads to a wrong predictionZ

zz+0 in the limit of H

0P0, instead of the exact

result Zzz"R

DCwhich holds at H

0"0 for not

very high frequencies. Therefore, in the interval ofexternal magnetic fields H

!the approximate

solution (23) ceases to be valid, so one needs in theexact solution for Eqs. (18a) and (18b).

It can be shown that the exact solution ofEqs. (18a) and (18b) having regular behavior ato"0 is the arbitrarily linear combination of evenand odd types of solutions

hr"Aº(1)r (o)#Bº(2)r (o),

hz"Aº(1)

z(o)#Bº(2)

z(o). (34a)

The latter can be determined by means of equations

º(1,2)r "+n

n, º(1,2)r "+

n. (34b)

In case of odd solution the series (34b) are over oddn. For n"1 one can put a

1"1, b

1"0, whereas

the coefficients with n"2k#1, k*1, can be cal-culated by means of the recursion formulas

k20ea2

n2!1(krran~2

#krzbn~2),

k20ea2

n2(krzan~2

#kzzbn~2

). (34c)

On the other hand, in case of even solution we puta0"0; b

0"1 for n"0; the coefficients with

n"2k, k*1, can be obtained with the help of thesame Eq. (34c). Having derived the magnetic fieldcomponents in accordance with Eqs. (34a), (34b)and (34c), one can obtain the electric field compo-nents by means of Eq. (20).

As we have seen in the previous subsection, tocalculate the wire impedance it is necessary to ob-tain the amplitude ratio A/B in the Eq. (34a). Thiscan be done by means of the same procedure asgiven in the previous subsection. Taking into ac-count the boundary conditions at the surface of thewire, one can obtain the linear set of equationswhich determine the corresponding amplitude ra-tios A/B and C/D in Eqs. (34a) and (24a). By anal-ogy with the previous calculations it can be shownthat these ratios do not depend on the longitudinalwave number in the quasistatic approximationused. It can be shown that the A/B ratio is deter-mined by means of an equation

(Aº(1)r #Bº(2)r )(Aº(1)z#Bº(2)

(k0e)2

(Aº{(1)r #Bº{(2)r )(Aº{(1)z

#Bº{(2)z

where the corresponding field amplitudes at thewire surface are designated as

º(1,2)r "º(1,2)r (a), º(1,2)z

"º(1,2)z

º@(1,2)r "C1

o(oº(1,2)r )Do/a

º@(1,2)z

oº(1,2)

z Do/a

. (35b)

It can be shown that the right-hand side ofEq. (35a) is proportional to a very small parameter

Dj2/(k0e)2D&D1/eD@1 and can be omitted. Thus,

we see that in the range of magnetic fields0)H

!, there are two independent electro-

magnetic modes in amorphous wire with circum-ferential type of magnetic anisotropy. For the firstmode the approximate relation holds

Aº(1)r #Bº(2)r +0. (36a)

It can be shown that for this mode the ratioD(D/C)DA1, so the structure of the mode outside thewire is close to that of magnetic type exitation (24b).For the second mode, a similar relation satisfies

Aº(1)z#Bº(2)

z+0. (36b)

Besides, for this mode the ratio D(D/C)D@1. Thus,outside the wire this mode is close to the electrictype excitation (24a).

Based on Eqs. (34a), (34b) and (34c), one canobtain an exact expression for the longitudinalcomponent of wire impedance, instead of approx-imate Eq. (32b), namely

Aº{(1)r #Bº{(2)rAº(1)r #Bº(2)r

. (37a)

It follows from Eqs. (36a) and (36b) that the longi-tudinal component of wire impedance is finite forthe mode (36b) only. It is given by an equation

º(1)z

(a) C1

do(oº(2)r )Do/a

!º(2)z

do(oº(1)r Do/a

º(1)z

(a)º(2)r (a)!º(2)r (a)º(1)r (a).

As we shall see further, unlike Eq. (32c), Eq. (37b)leads to correct behavior of longitudinal compon-ent of wire impedance in the limit of H

For transverse component of wire impedance,instead of approximate Eq. (33c), one can obtain anexact expression

Aº@z(1)#Bº@

Aº(1)r #Bº(2)r. (38a)

Again, the transverse component of wire impe-dance is finite for the mode (36b) only. For thismode it is given by an equation

º(2)z

(a)Cdº(1)

zdo Do/a

!º(1)z

(a)Cdº(2)

zdo Do/a

º(1)z

(a)º(2)r (a)!º(2)z

(a)º(1)r (a).

3.3. Circumferential anisotropy: Case of H0*H

If the amplitude of external magnetic field ex-ceeds the anisotropy field, H

!, the structure of

the permeability tensor, as given by Eq. (12a), sim-plifies greatly. In fact, it is analogous to the struc-ture of the permeability tensor for wire withuniaxial anisotropy (8a), (8b). As a result, it is easyto see that in given case, the wire impedance can becalculated by means of the same Eq. (16) with thedifference that the effective transverse wire per-meability (15b) substitutes for

kJ @"k(2)r !

(k(2)a

k(2)o"

K#m)(u

m#m)!u2

K#m)m!u2

where we denote m"uH!u

K!iui.

4. Results and discussion

4.1. Axial anisotropy

It is easy to see that in case of wire with axial typeof magnetic anisotropy the behavior of the trans-verse wire permeability (15b) is responsible for theGMI effect. Actually, the dependence of wire impe-dance both on the amplitude of external magneticfield and the alternating current frequency is main-ly determined by this quantity. To estimate thetransverse wire permeability we assume here thefollowing values of the magnetic parametersof an amorphous wire: saturation magnetizationM

4"640 G [6], anisotropy field H

!)0.5 Oe

and phenomenological damping parameter i"10~2—10~1. As a result one can estimate the fre-quencies u

m+1011 s~1 and u

K)107 s~1. Another

frequencies are assumed to vary in the intervals

Fig. 1. Effective magnetic permeability of amorphous wire withaxial type of magnetic anisotropy at different values of anisot-ropy field: (1) H

!"0.2 Oe; (2) H

!"0.1 Oe; (3) H

!"0.01 Oe.

Fig. 2. Real and imaginary components of wire impedance asfunctions of external magnetic field in case of wire with axialtype of magnetic anisotropy at different values of alternatingcurrent frequencies: (1) f"105 Hz; (2) f"5]105 Hz; (3) f"106 Hz.

uH"0—109 s~1 and u"0—107 s~1, respectively.

Thus, we see that the um

value is usually muchlarger than the other frequencies, so that the trans-verse wire permeability (15b) can be estimated as

. (40)

Therefore, this quantity can be of the order of103—104 at small values of u

H, but it decreases

rapidly with increasing of the external magneticfield amplitude. The field dependence of the trans-verse wire permeability (15b) is shown in Fig. 1 atdifferent values of wire anisotropy field.

Fig. 2 shows the field dependence of the real andimaginary components of wire impedance at differ-ent values of current frequency, whereas Fig. 3shows the same quantities as the functions of fre-quency at different values of anisotropy field. Thesecalculations are carried out by means of Eqs. (15a),(15b) and (16). The values of the magnetic par-ameters are chosen close to that used in the experi-ments [6,9] except for the H

!and i values which

are not strictly known. One can see from Figs. 2and 3 that the experimental results [6,9] are quali-tatively in accordance with the behavior of theimpedance of wire with axial type of magnetic an-isotropy.

It is important to note that the amplitude of theGMI effect depends strongly on the anisotropyfield value. For example, it can be seen in Fig. 3that in amorphous wire with H

!"10 Oe the am-

plitude of the GMI effect is negligible up to a ratherhigh alternating current frequency f"106 Hz.Therefore, the good magnetic softness of amorph-ous wire is one of the most important conditionsto observe the GMI effect. The anisotropy fieldof amorphous wire can be obtained from thehysteresis loop measurements [7]. Then there isa possibility to estimate the value of the pheno-menological damping parameter from the experi-ment [6,9].

On the basis of Eqs. (15a), (15b) and (16) one canexplain the GMI effect in amorphous wire withaxial type of magnetic anisotropy as follows. Onecan see from Eqs. (15a) and (15b) that the actualskin depth in the amorphous wire is given by

d"d0/Jk

M, the transverse wire permeability being

approximately given by Eq. (40). At f"106 Hz andresistivity o"100 l) cm [6], one obtains d

0.05 cm which exceeds considerably the wire radiusa"6]10~3 cm. On the other hand, if one takesinto account the k~1@2

Mmultiplier, the radius-to-

skin depth ratio will change noticeably as a func-tion of H

0. For example, at M

4"640 G and

H!"0.5 Oe the ratio a/d is about 15 at H

while it decreases to a/d+3 in case of H0"10 Oe.

The change of the radius-to-skin depth ratio leads

Fig. 3. Real and imaginary components of wire impedance asfunctions of alternating current frequency in case of wire withaxial type of magnetic anisotropy at different values of aniso-tropy field: (1) H

!"0 Oe; (2) H

!"1 Oe; (3) H

!"10 Oe.

Fig. 4. Effective magnetic permeability of amorphous wire withcircumferential type of magnetic anisotropy: (1) H

!"0.5 Oe,

i"0.1; (2) H!"1.0 Oe; i"0.2.

evidently to the corresponding variation of the wireimpedance. In fact, this explanation of GMI effect isclose to that given in Ref. [6] and in the followingpapers [7]. It should be noted however that here wecalculate the transverse wire permeability (15b) ex-plicitly as a function of the magnetic parameters ofamorphous wire. This makes it possible to checkthe model assumed and to obtain the correspond-ing magnetic parameters of amorphous wire fromthe experiment.

4.2. Circumferential anisotropy

Let us consider now the case of amorphous wirewith circumferential type of magnetic anisotropy. Itis believed [7,14,15] that this type of magneticanisotropy exists in the outer shell of Co-richamorphous wire with negative magnetostriction.Thus, it is important to consider this case theo-retically and to compare the results with the cor-responding calculations for a wire with axialanisotropy.

First of all it should be stressed that in case ofamorphous wire with circumferential anisotropy,the usual expression (16) for the wire impedance,generally speaking, is not valid. As we have seen inSection 3.2, in the range of external magnetic field0)H

!, the structure of the electromagnetic

wave propagating along the amorphous wire, asgiven approximately by means of Eqs. (23), (24a)

and (24b), becomes more complicated than in theprevious case, Section 3.1. As a result, the wireimpedance in this range of external magnetic fieldbecomes a tensor. It has both longitudinal compon-ent Z

zz, Eq. (32c) or Eq. (37b), and the transverse

one Zrz, (see Eq. (33c) or Eq. (38b)). On the otherhand, if the amplitude of external magnetic fieldH

!, the structure of the electromagnetic wave

simplifies and the wire impedance has the longitu-dinal component only. The latter is again given byEq. (16) with the difference that the transverse per-meability of the wire must be calculated by meansof Eq. (39). It should be noted that the existence ofthe transverse component of the impedance of Co-rich amorphous wire has been recently confirmedin the experiment [2].

Fig. 4 shows the real and imaginary componentsof magnetic permeability of amorphous wire withcircumferential anisotropy at different values bothof the anisotropy field and the damping parameter.The curves in Fig. 4 are plotted in accordance withEq. (19) in the region of H

!and with Eq. (39)

in case of H0'H

!, respectively. Note that the

peaks in the imaginary component of the magneticpermeability of the wire as well as the gaps in thereal one correspond to the value of anisotropy field.The magnitude of the peaks depends strongly onthe value of damping parameter. The field depend-ence of the real and imaginary components of thewire impedance is shown in Fig. 5a and b. Thecurves labeled 1 and 2 in Fig. 5 correspond to theapproximate, Eq. (32c), and exact, Eq. (37b), ex-pressions for the longitudinal component of wire

Fig. 5. (a) Real and imaginary components of the longitudinalcomponent of wire impedance as functions of external magneticfield in case of wire with circumferential type of magnetic aniso-tropy. Magnetic parameters of the wire are given byH

!"0.2 Oe, i"0.1, f"5]105 Hz. Curves 1 and 2 indicate

approximate (high-frequency limit) and exact solutions, respec-tively. (b) Same as in Fig. 5a but with different values ofthe magnetic parameters of the wire H

!"0.5 Oe; i"0.2;

f"106 Hz.

impedance, respectively. The difference between theapproximate and exact solutions exists for the realcomponent of the wire impedance only. It can beproved that for imaginary component of Z

zzas well

as for both components of the Zrz the approximateand exact solutions are very close to each other. Itshould be noted also that for exact solution,Eq. (37b), the real component of wire impedance atzero external magnetic field turns out to be equalto the DC resistance of the amorphous wire,Z

zz(0)"R

Comparing Figs. 2 and 5 one can see that themagneto-impedance spectra are directly deter-mined by the type of magnetic anisotropy ofamorphous wire. For example, in case of wire with

circumferential anisotropy, the peak of the realcomponent of wire impedance as a function ofexternal magnetic field corresponds to the anisot-ropy field, whereas in case of axial anisotropy themaximal value of this component corresponds tozero magnetic field.

Bearing in mind the simple models considered inthis paper (see Eqs. (3) and (9)), one can prove thatneither any domain structure nor circumferentialanisotropy are strictly necessary to observe theGMI effect. Nevertheless, to describe the magneto-impedance spectra of real amorphous wires in de-tail one needs to take into account the actual distri-bution of easy anisotropy axes throughout the wirecross-section. It is well known [7,14,15] that thereis a uniformly magnetized core near the center ofCo-rich wire where the easy anisotropy axis hasprobably an axial direction. On the other hand, inthe outer shell of Co-rich wire the easy anisotropyaxis points in circumferential direction. Thus, it isinteresting to study the electrodynamics of wirehaving axial type of magnetic anisotropy in theinner core and circumferential one in the outer shell(Fig. 6). One can expect that the refined model willbe able to avoid some limitations peculiar to thesimplest models considered above. For example,though the model of wire with purely circumferen-tial anisotropy describes qualitatively the mag-neto-impedance spectra of Co-rich amorphouswire, it leads simultaneously to the conclusion thatthe real component of wire impedance at zero ex-ternal magnetic field coincides with the DC resist-ance of the wire, Z

zz(0)"R

DC. This fact contradicts

the existing experimental data [7,8,22] which showthat usually Z

zz(0)AR

DC. On the basis of Eqs. (18a)

and (18b) it can be proved that in case of H0@H

!the actual skin depth in a wire with circumferentialanisotropy turns out to be much larger than thewire radius. In such a case one can expect that inthe domain of H

!the influence of the uniform-

ly magnetized core on the impedance of Co-richamorphous wire may be very important. We planto consider this effect in a separate paper.

It seems important also to take into accountpossible radial dependences of the anisotropy con-stants in the core and shell regions of an amorph-ous wire. To take this fact into account one cansubdivide the wire into several layers with slightly

Fig. 6. Field dependence of the real and imaginary componentsof the transverse component of wire impedance in case of wirewith circumferential type of magnetic anisotropy. Magneticparameters of wire are given by (a) H

!"0.2 Oe, i"0.1,

f"5]105 Hz; (b) H!"0.5 Oe, i"0.2, f"106 Hz.

different anisotropy constants. This evidently willlead to a certain averaging of the magneto-impe-dance spectra of amorphous wire. As a result onecan expect the broadening of the peaks in the realcomponent of the wire impedance simultaneouslywith the disappearing of the sharp gaps in theimaginary component. A similar model can prob-ably be able to take into account the influence oftensile stress on the field response of impedance inlow magnetostriction amorphous wires. Recently[22,23], this effect has been extensively studied ex-perimentally.

As we have seen above, the model of wire withcircumferential type of magnetic anisotropy de-scribes at least the most important features of theGMI effect in Co-rich amorphous wire with nearlyvanishing magnetostriction. Besides, we expect thattaking into account the actual distribution of theeasy anisotropy axes along the wire radius one willbe able to fit the experimental data for varioustypes of amorphous wires more successfully. Notethat we do not consider in this paper the possibleinfluence of any type of domain structure on theGMI characteristics of amorphous wire since webelieve that in case of wire with very low magnetos-triction any domain structure along the wire lengthis energetically unfavorable and can hardly bestable [12]. This suggestion is supported also bythe remarkable fact that for wires with vanishingmagnetostriction the so-called magnetic aftereffect

[13] turns out to be very small. The latter is sup-posed to be related to the relaxation of the domainstructure of amorphous wire after a sharp change ofthe amplitude of external magnetic field. It seems tobe observed in the wires with not very small satura-tion magnetostriction constant.

Acknowledgements

The authors would like to express their appreci-ation to I.T. Iakubov for helpful discussions. Thiswork was supported by Russian Foundation forBasic Research (Grant 96-02-16346).

References

[1] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continu-ous Media, 2nd ed., Pergamon, New York, 1984.

[2] A.S. Antonov, I.T. Iakubov, A.N. Lagar’kov, IEEE Trans.Magn. 33 (1997) 3367.

[3] E.P. Harrison, G.L. Turney, H. Rowe, H. Gollop, Proc.Roy. Soc. 157 (1937) 651.

[4] G.S. Gorelic, Izv. Acad. Nauk Ser. Fiz. 8 (1944) 172.[5] K. Mohri, T. Kohzawa, K. Kawashima, H. Yoshida, L.V.

Panina, IEEE Trans. Magn. 28 (1992) 3150.[6] R.S. Beach, A.E. Berkowitz, Appl. Phys. Lett. 64 (1994)

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Magn. Mater. 174 (1997) 127.[13] M. Knobel, M.L. Sartorelli, J.P. Sinnecker, Phys. Rev. B 55

(1997) 3362.[14] K. Mohri, F.B. Humphrey, K. Kawashima, K. Kimura, M.

Mizutani, IEEE Trans. Magn. 26 (1990) 1789.[15] P.T. Squire, D. Atkinson, M.R.J. Gibbs, S. Atalay, J.

Magn. Magn. Mater. 132 (1994) 10.[16] A. Antonov, A. Dykhne, A. Lagar’kov, N. Usov, Physica

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Phys. 76 (1994) 6204.[18] W.F. Brown Jr., Micromagnetics, Interscience, New York,

1963.[19] A.I. Akhiezer, V.G. Bar’yakhtar, S.V. Peletminskii, Spin

Waves, Wiley, New York, 1968.

[20] A. Sommerfeld, Electrodynamik, Akademische Verlag,Leipzig, 1949.

[21] V.V. Nikolskii, Electrodynamics and Radiowave Propaga-tion, Nauka, Moscow, 1978 (in Russian).

[22] M. Knobel, M.L. Sanchez, J. Velazquez, M. Vazquez,J Phys.: Condens. Matter 7 (1995) L115.

[23] M. Knobel, M. Vazquez, M.L. Sanchez, A. Hernando, J.Magn. Magn. Mater. 169 (1997) 89.

Theory of giant magneto-impedance effect in amorphous wires with different types of magnetic...

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