20 September 1999

Ž .Physics Letters A 260 1999 381–385www.elsevier.nlrlocaterphysleta

Concentration dependence of giant magnetoresistance in magneticgranular composites

Chen Xu b,), Zhen-ya Li a,b

a ( )CCAST World Laboratory , P.O. Box 8730, Beijing 100080, Chinab Department of Physics, Suzhou UniÕersity, Suzhou, 215006, China 1

Received 12 April 1999; accepted 4 August 1999Communicated by A.R. Bishop

Abstract

Ž .We combine effective medium theory EMT with the two-channel conducting model to study the magnetic granularŽ .concentration dependence of a giant magnetoresistance GMR in magnetic granular composites. The composite is composed

Ž . Ž .of small magnetic granules such as Co embedded in an immiscible nonmagnetic metallic matrix such as Ag . We present amodel for the composite in which the magnetic metallic granules are spherical in shape and have a distribution in sizes, andin which there are different contributions of superparamagnetic and ferromagnetic granules to conductance. The calculatedresult about the concentration dependence of GMR is in agreement with the experimental data. q 1999 Published by ElsevierScience B.V. All rights reserved.

1. Introduction

Recently, the magnetic granular concentration de-pendence of GMR in granular composites such asCo–Ag, Ni–Ag, Co–Cu granular films, has been

w xresearched 1–5 . It was shown that at a certaintemperature the GMR effect increased as the mag-netic granular concentration increased and afterreaching a maximum the GMR effect decreased withthe further increase of the concentration. There was alittle theoretical work to study this phenomenon.Some theoretical work successfully explained the

w xdependence of GMR on the granular size 6–9 , anda few papers discussed the magnetic granular con-

) Corresponding author. Fax: q86-512-511259; e-mail:zyli@suda.edu.cn

1 Mailing address.

w xcentration dependence of the GMR 6 , but theircalculation failed to explain the experimental resultsw x1,4 . How to understand the concentration depen-dence of the GMR in these magnetic composites isthe goal of this paper.

In this paper, we will present a phenomenologicaltheory to study the concentration dependence ofGMR in Co–Ag granular. Firstly, based on theexperimental data which have been reported in Ref.w x5 , we assume that the composite contains smallmagnetic granules embedded randomly in an immis-cible nonmagnetic metallic matrix and the granules

Žhave spherical shapes for example, experimentsshowed the Co granules were spherical in shapew x.16 . These magnetic granules have a distribution insizes. Secondly, for small magnetic granules, if thetemperature is higher than the average blocking tem-perature of the composite, most of these magnetic

0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 99 00550-2

( )C. Xu, Z.-y. LirPhysics Letters A 260 1999 381–385382

Ž .granules become superparamagnetic SPM . Whilethe temperature is lower than the average blockingtemperature, most of these granules become ferro-

Ž .magnetic and hence non-SPM and others remainstill SPM. We should distinguish the SPM fromnon-SPM granules, because they have a different

w xcontribution to the GMR 10–12 . Thirdly, we sup-pose that the conduction electrons in the composite

Ž . Ž .are scattered 1 in the nonmagnetic matrix, 2 at theinterface between nonmagnetic matrix and magnetic

Ž .granular, and 3 within the magnetic spherical gran-ules.

2. Theory

Now, we assume the measure temperature is lowerthan the blocking temperature of the composite andthe concentration of magnetic granules is p. Thenwe take a phenomenological parameter x to de-s

scribe the ratio of the SPM granule concentration tomagnetic granule concentration. So the ferromag-

Ž .netic granules concentration is p 1yx . It is gener-s

ally thought that the GMR is caused by heteroge-neous magnetic structures. When a magnetic field Bis applied to the magnetic granular composite, themagnetic moments of non-SPM granules m aren

aligned and saturated, and the magnetic moments ofŽ .SPM granules m have angles u of intersections

with applied field B. We adopt two-channel conduct-Ž .ing model and effective medium theory EMT to

study the magnetic granule concentration depen-w xdence of GMR 9 . For a non-SPM ferromagnetic

granule, the spin direction of conduction electronsŽ . Žmay be parallel say ‘‘spin-up’’ or anti-parallel say

.‘‘spin-down’’ to the direction of the magnetic mo-ment of the granule. For a spin up electron, thecontribution to its resistance will be high due toscattering events and then to its conductance will be

Ž .low L , while for a spin-down electron, the contri-Ž . qbution to its conductance will be high H . Let s

and sy indicate the conductivities for spin-up andspin-down electrons respectively. Using the signs ‘n’Ž . Ž .or ‘s’ for non-SPM or SPM granules, we describethe conductivities of non-SPM and SPM granules forspin-up or spin-down electron as s " and s " ,n s

respectively.We assume that the magnetic moments of non-

SPM ferromagnetic granules will be quickly aligned

and saturated in the presence of an applied field BŽ .such as Bs5.0 T . Then, for a given concentrationof magnetic granules, the average conductivities² ":s can be expressed byn

² q: qs ss , 1Ž .n nL

² y: ys ss , 2Ž .n nH

where sq and sy are independent of B.nL nH" Ž .But for the s , the magnetic moments m B ofs s

SPM particles are dependent on B. We presume the" Žvalues of s change linearly with cosu where u iss

.the angle of intersection between m and B accord-sw xing to the proposal of Gittleman et al. 13 . Therefore

we get1q qs s C m B 1qcosuŽ . Ž .s s s2

1 yq C m B 1ycosu , 3Ž . Ž . Ž .s s2

1y ys s C m B 1qcosuŽ . Ž .s s s2

1 qq C m B 1ycosu , 4Ž . Ž . Ž .s s2

where C " are two coefficients. After considerings

the distribution of SPM moments, we get the thermal" w xaverage of s as follows 12 :s

1 1q² :s s s qs I B q s ys I B ,Ž . Ž . Ž . Ž .s sL sH 1 sL sH 22 2

5Ž .1 1y² :s s s qs I B q s ys I B ,Ž . Ž . Ž . Ž .s sL sH 1 sH sL 22 2

6Ž .where

1w xI B s ln sinha ra , 7Ž . Ž .1 m m

am

aL a 1 2 L aŽ . Ž .mmI B sy q 1y da ,Ž . H2 ž /a a a0m m

8Ž .m Bm

a s , 9Ž .m k TB

Ž .and s and s are two coefficients, L a is thesL sH

Langevin function, m is the largest magnetic mo-m

ment of the Co granule which is unblocked SPMparticle at temperature T.

As the magnetic granules are randomly embeddedwith concentration p in the nonmagnetic matrix, we

Ž . w xuse effective medium theory EMT 14,15 to getthe average conductivities sq and sy for spin-upe e

( )C. Xu, Z.-y. LirPhysics Letters A 260 1999 381–385 383

and spin-down conduction electrons, respectively.Then, the average conductivity s " can be obtainede

from following equations:

² q: q ² q: qs ys s yss e n epx qp 1yxŽ .s sq q q q² : ² :s q2s s q2ss e n e

s ysqh e

q 1yp s0, 10Ž . Ž .qs q2sh e

² y: y ² y: ys ys s yss e n epx qp 1yxŽ .s sy y y y² : ² :s q2s s q2ss e n e

s ysyh e

q 1yp s0, 11Ž . Ž .ys q2sh e

where s is the conductivity of the Ag matrix.h

Being used the two-channel conducting model, thecomposite’s effective conductivity s aree

s ssqqsy. 12Ž .e e e

Ž .We have the magnetoresistance MR of the com-posite according to the definition of MR as follows:

sy1 B ysy1 0Ž . Ž .eMRs . 13Ž .y1s BŽ .e

In order to compare with the experimental data ofmagnetic granules concentration dependence of GMR

w xin Ref. 5 , we consider two cases: measure tempera-tures at Ts4.2 K and Ts300 K with the sameapplied field Bs5.0 T.

Ž . Ž .1 Ts4.2 K Bs5.0 T . The blocking tempera-ture of Co–Ag composite sample is about 20 K atlow concentration of Co granules according to Ref.w x5 . And when the Co concentration increases, theblocking temperature will also increase. Because theblocking temperature is much higher than the mea-sure temperature, we can assume that the non-SPMgranule concentration is much larger than SPM gran-ules. The phenomenological ratio x of SPM granules

concentration to whole Co concentration is small andits value decreases when the Co concentration in-creases. Because of the number of the Co granules isdifferent with the change of concentration, we as-

Ž .sume the phenomenological parameter g or g , ton s

describe the probability and degree of scattering onŽthe interface between nonmagnetic and non-SPM or

. Ž .SPM as well as within the non-SPM or SPM² ":granules. Then, the average conductivities s ofn

non-SPM granules which is in majority can be

rewritten as

² q: qs sg s , 14Ž .n n L

² y: ys sg s . 15Ž .n n H

² ":It is implied that the average conductivities sn

are changed with the increase of Co concentration.As the Co concentration is low, the magnetic gran-ules are separated and the scattering number willincrease when the number of magnetic granules in-creases. However, when the Co concentration isabout 20%, the small magnetic granules begin tointerconnect and fold up, thus the effective surfacefor scattering decreases. We can assume g is maxi-n

mum as 1.0 for the largest limit of separated granulesw xat concentration 19% 5 according to the experimen-

Ž . Ž . Ž .tal data. Putting the Eqs. 14 and 15 into Eqs. 10Ž . ² ":and 11 respectively, we will obtain s whene

parameters are given. Now, we select the values ofŽ .conductivities as 4.16 for Ag and 1.0 arbitrary units

for Co, 0.01 for sq and 10.0 for sy. For SPML H

granules, we neglect the influence of g and take ss sL

as 0.1and s as 2.0 because of the very small SPMsH

concentration and assume the maximum moment ms

about 4.5m as the sizes of SPM granules are inB

nanometers. The two phenomenological parametersx and g will be determined by a nonlinear least-s n

w xsquare fit to the GMR data in Ref. 5 .In Table 1, we show the values of g and x forn s

various Co concentrations. The selection of g andn

x is reasonable for above assumptions. Namely, thes

value of g increases first then decreases later as Con

concentration increases and x decreases as Co con-s

centration increases. The curve of GMR vs concen-Žtration at Ts4.2 K is shown in Fig. 1 the filled dots

.are experimental data . The theoretical result is inw xagreement with the experimental report in Ref. 5 .

Table 1Values of the parameters at 50 K

Ž .Vol. % T s4.2 K T s300K

g x gn s s

7.14 0.6640 0.070 0.525018.83 0.9480 0.060 0.656023.70 0.7250 0.058 0.572035.23 0.4340 0.040 0.411549.03 0.3746 0.020 0.3608

( )C. Xu, Z.-y. LirPhysics Letters A 260 1999 381–385384

Fig. 1. Magnetic Co concentration dependence of the GMR at twodifferent temperatures. The symbols represent the experiment data

w xof Ref. 5 and the curves give the calculated values.

Ž . Ž .2 Ts300 K Bs5.0 T . The average blockingŽ .temperature 20 K of the composite sample is much

lower than the measure temperature 300 K at therange of 0 to 50% Co concentration according to the

w xexperimental results in Ref. 5 . So we can assumethe Co–Ag composite containing only SPM gran-ules. The influence of parameter g can not bes

neglected because in this case, all the magneticgranules are superparamagnetic. And we can predictthat g is smaller than g at the same Co concentra-s n

tion because a granule’s magnetic moment is smallerat SPM state than in non-SPM state. Namely, non-SPM states are more effective on spin-dependentconductance, so g should be larger than g . Thus,n s

we should substitute g s and g s for s ands sL s sH sLŽ . Ž . ² ":s in Eqs. 3 and 4 . Then s can be rewrittensH s

by1q² :s s g s qs I BŽ . Ž .s s sL sH 12

1q g s ys I B , 16Ž . Ž . Ž .s sL sH 22

1y² :s s g s qs I BŽ . Ž .s s sL sH 12

1q g s ys I B . 17Ž . Ž . Ž .s sH sL 22

Now we take s as 0.01 and s as 10.0, andsL sH

assume the maximum moment m as 744m , whichs B

is much larger than the value of m in case 1 at lows

temperature because the maximum size of the Cogranule is much larger than that in case 1. Themaximum moment is larger and correspondingly

higher s and lower s . The value of parameter gsH sL s² ":is given in Table 1 and s can be obtained bye

² q: q qs ys s yss e h ep q 1yp s0, 18Ž . Ž .q q q² :s q2s s q2ss e h e

² y: y ys ys s yss e h ep q 1yp s0. 19Ž . Ž .y y y² :s q2s s q2ss e h e

The theoretical curve is in good agreement withŽ .experimental data open dots in Fig. 1.

3. Discussion and conclusion

Ž .We have studied the magnetic granule Co con-centration dependence of GMR by a phenomenologi-cal method. In order to study the magnetoresistanceof the composite which contains fine magnetic gran-ules, we should distinguish different contribution tospin-dependent conductance between superparamag-netic and ferromagnetic granules. The blocking tem-perature is important to determine whether the Cogranules lie in SPM state or in mixed non-SPM andSPM state. And we can only give a phenomenologi-cal parameter x to describe the concentration ofs

SPM and non-SPM state. When given the magneticCo concentration, the parameters g reflect then, s

spin-dependent scattering at the magnetic-nonmag-netic interface and within the magnetic granules. Thesurface spin-dependent scattering is more importantthan body scattering within magnetic granules.Therefore g and g increase at first with the in-n s

crease of Co concentration then begin to decrease atthe concentration about 20%. We studied the GMReffect at two measure temperatures 4.2 K and 300 Kwhen the magnetic Co concentration was changed.Using our method, we predict that when measuretemperatures are between 4.2 K and 300 K, the peaksof GMR will also be between the two peaks inFig. 1. The main reason is that the non-SPM granuleconcentration will decrease when temperature in-creases from 4.2 K to 300 K, which lead to smaller gn

and larger x .s

In summary, we combined effective medium the-ory with two-channel conducting model to calculatethe magnetic Co concentration dependence of theGMR. The result is in good agreement with experi-

( )C. Xu, Z.-y. LirPhysics Letters A 260 1999 381–385 385

ment, this shows the model presented and the theo-retical method are essentially applicable for magneticgranular composites.

Acknowledgements

This work was supported by the National NaturalScience Foundation of China under Grant 19774042.

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