REVISTA ~IEXICA~A I>E FISICA.f.f S('I'I,E.\IL\:T() 1, 7 \ 77

l\lagnctorcsistancc in granul:u' matcrials amI sizc cffccts

D. Altbir and P, Var!!asDeparwmell!() de' F(üc{/ (h'fI'e'rsidlll/ de .)'(/Il!i(/,~o de Chile

Casi{{a }07 .)'(///li(/go-2 Chile

Recihido el Xde diciemhre de I\)\J7: accpl;ldo el IXdc khlWO de It)l)X

SEI'TIE~lBRE 199X

Cll1l~iderin~ dipol:lr intcrar.:lions hetween lll,lgnelic grains. lhe Cfll.'CIllr clusler Síl.l' disl ríhll,illll" in tlw lIla~nclorcsi~raJlce nf ~r:ll1l1larIllclallic•.y",ems is ,maly/cd. In paniculnr. (he redu('cd magnctorcsisl,lncl' ,1l11Jma~nl'II;r;lIillll are ••\Ildi\.'d rOl'diffcrClll sy••lcrns. and il is shll\\'n Ih<llwhcn a h["(ladgrain size distribulíon is considered. strong clfecb are Ilhscnc(l, 110\\l'\ l'I. the diptlll'-tllPI 1le-••IlI111 rangl' l'tnrCI,ltil In al ;rero lietddocs nOIcllallgc. when a c1usler si"-t.~dislrihlllion is consitlercd Thcrefnrc lile redllccd lll;¡~nl'lnrcsi ...I;IlKC\',tlUl' al tero field i...nol afkl'lcu.

A'{'\'II'(ml.,': flysteresis: granular sylcms: Illcgneloresístencl'

Se anaJi;ra el efeclo dc la distrihurlón de lamano de lo", nírnulo.'" 1.'11la llIagl1ctnre",islcnrl<l lk ...í",tl'llI<l~grallulare~ IIlcl:llicos. nmsidcranduuna inleracción dipolnr enlre los granos Illagnélil'o~, En particular. ",e l'....tudían la l1la~nl'lOl'CSi",II'I1Ci¡1rcdurlda y la magllctizacilÍll paradiversos sislemns y se muestra que, si se nmsidera ulla dislrihul'ÍlÍn de tamaño and),1 de lo.'"gr,ullls. "'c ohtiene un efeclo apreciahlc. Sinemhargo. la correlación dipolo-dipolo a l'Cll"loalcance. a campo cero. no camhia al comiderar lll1;1dislrihul'Íon de tamaflO dc los granos.Consecuenlemenle. la magnetorcsislencia reducida a campo cero no es aft:clada.

1)(',I'CI"i/1/{m'J: liistéresis: sistemas granulares; magnetOfe",istcflcia

mes: 75.50.Pa: 75AO.Mg: 75.60.-J

l. Intmductilln

\1a~llcli(' ('oupling helwecn low dimensional syslellls h;:hheen a suhjcct 01' inlerest in the lasl few ycars. hccausc 01rhcir inlcrcsting hasic propcrtics. In particular. granular sys-lcms consisling uf single dOlllain c1uslers embedded in tl nOI1-Illagnctic mClallic rnalrix had generatcd considerahle intereslhccausc Ihey are eas)' to manufacture in ribbon fmm and e\~hibil an isolmpic giant magnelorcsislanee (Cit\-,]f{)dh'L'r [11.C\'l.'1Ial room temperalure.

In Illis \\'ork we cOllsider systCllls Iikc Co-Cu (JI' CO-,Agallo)'s. I~lhricated hy Illeltspun lcchnique or low~energy clllS-IL'r be<lm deposition (LECBD). Thcse SYSlcI11shavc a COI11-plex mieroscopic struclUre, wherc the small clusters (each of:=::::: ]()Q() eo atollls) hchavc like a single magnelic dumain.whercas lhe hig ferromagnctie grains have Illany lIlagneticdomains, Then. ami assuming Ihat the grains do nOl interacl,rhc magnelic hehaviour, i.e. Ihe magnelizatioll 01' the syslelll\'s. external magnclic neld, (hysleresis curve) is a cOlllbina-lion nI' a L:lIlgcvin fUllclion and Ihe hyslercsis curves nf lheslllalt fcrnllllagnclie gr¡¡ins.

Al! exisling lheorics which do not cOl1sidcr any Illagncticil11eraclion hetwcen Ihe clusters, aSSlIllle a dependcllce nI' therractional magneloresislancc t5.H / Ron the square 01' Ihe rc-dIH.:ed magneti/alion JI/'\I,., where .\1" is the saturated mag-Ilcli/ation. I1owl'ver. cxperimcntal reslllts sllow clear devi;:¡~liolls fmm Ihis prcdictcd hehaviour. prO\'iJing evidcllcc fOImagnetic corrctalions helween Ihe cluslers. Thcse magneliccorrelaliolls arise fwm a long range illleraclion alllong Ihcdll'ilCrs and l\\'() lllechanislTls have been prorosed to accourll

rOl' lhis coupl ing: Rudcrrllan- Kiul'l- Kasllya- Yosida (RKK y)alhl dip(II:11' inlcraclions. II()\\-'cvcr. \Ve llave recclltly delllol1-slralcl! Ihal rOl' ctll •.•lcrs \..-tlOse •.•i/es amI spalial distribulion •.•arc con si sIc ni wilh cxperimental dala. thc dipolar inleraclionis lilc uOlllinanl olle 121.

In lhis work we use a simpk model lo study the Illagnc-toresistann' 01' ~ranlJtar syslclIls, whcre only single JIlaglleticdOlllain clu:-'lcrs are l'onsidered in ortler lo study lhe dkclS 01'111l'grain ...i;re di:-.lriblllion, \\'ilhin Ihis model. \vc Shtl\V lhallhe cluS!L'r .SilCdi ...lribulioll cOlltrihlllcs lo Ihe llatlcning (Ir Illl'curve rOl' lhc rL'duL'I.:dlIlat:nclore.sislancc 1'('r.HlS reduced Illag-1IL'li;ralion and lo allcnlJanCCI11Cll1 nI' the hystcresis curve.

2, i\lodl'l alllll'aiculatiollS

\VL'considl'l" a systelll nI' N single doma in c1l1stl'l"s elllbeddedin a 110n lIlagnelic Illclallic malri.\. each cluster repn.:senlcl!by l'lTeclive Illagnclic dipoles (;/ ¡:q.

In Ih~sl' sySIL'lI1s. llll' f;i\lR elfecl is Ihollghl to he re.!alL'd to lhc ",pin-dcpendenl seatteri ng 01"condllction cleclrons<lnLl lhe reoriL'lllalioll (JI' Ihe rna~llclic 1ll0lllCIlIs hv an l'.\lcrnalIllagnclic lil'ld J7 [ l. .~d.For ¡j = O,. lhe lolal n;agnelil' 1110-II1L'llls /// or IhL' L'luslcrs "re oricllted in sllch a way lhal theJlL'l lIlagllcli;ration of thc syslem is zcro and the rcsislancc inhOI!l lhc IIp- and down-spin chanllels is higll. Ih)we\'cr. lhealigl111lL'1l1(Ir 11ll' I!lOIllCnls by Ihe magnelic licld !Cads lo aSholl-circuil L'lh'L"l in Dne ul lhe spin-chanllcls, wilh a signif-icalll redllclioll in lhe lolal rcsislance U (Ir lhc systclIl. It isclear lllal ti milT(lscopic rlll'ory uf transpon \\'ould nccessar-ily l'l)Jlsider Ihc 'pin-lkpcndenl scaltcring Illcrhanislll. and

11 ALTBII{ ANO P.VARGAS

Using Eq, (2), the reduced magnelorcsistance can he wril-

ILn as:

fvloreo\'er. \Ve kllow lhat ('ti. It)) 11I=0 is very c10se 10 I.ero,Ihcll we approxilllatc lhe reduced magnelizalion as

(5)\

~L £') - L ¡I} . /7 ,- l.:t) )=1

\vhcre Eu is lhe c1assical dipolar pair interaclinl1 oetweenclusters 11; and 11../.

\Ve pcrforJllL'd our calclllaliolls in systems consisting 01"Cn cluslers arrangcLl in a squared lauice with a lattice param-eter 01' 70 Á, \\.'hich c(lrrespullds 10 a measurcd mean graindistance in eo-eu systcms 111. Thc position of Ihe clusters,each olle represented hy an ellective magnetic moment 131.is initially Incalcd al the sites 01' a square latticc. Howevcr,oecausc the spalial arrangement 01' the cluslers in ordinarygranular matcrials arc distorted, \Ve have randomly displacedthe positiolls uf Ihe Illomeols wilh respect to the lallice sites,aecording lo a Gaussian prohabilily distribulion with zcru ay.erage,

Figure I shm\'s lhe positions uf the c1uslcrs in a 2D sys-[em of 441 c1usll'rS, wherc a Gaussiall prob.abilily distrihulionis used for (he positinlls, wilh I() A standard deviation in eachspace direL:lioll.

\Ve also inlroduce ;J (jaussian distrihulion for the sile nf(he grains, Fig, 2. is an hislogralll 01"Ibe size Llistrihution ofthe Co clusters. The vallle 1 mcans lhat the cluster has 750atoms, amI Ihcll a Illagneli/.a(ion of 7;j0 I/-B, assulTling lhatlile magneti/.alioll 01"every alOlIl of en is LO Jiu, as slaled byN.H, Duc ano T.K. Oanh 17],

Here we invesligale lhe lIIagnetic field dependellce uf lhemagneloresistance and Illagnelizalion 01"granular malerials.\Ve u,e a slandard i\tul1le CarIo al!!tlrilhm (lO!; thermalilafionsteps and lO,1 stl'PS rOl' the l'nsel;lhle average). Our I:alcula-lions ha ve hecn perfonncd fOl"lcmperatures T ,.....GI"':, wherehu T is "mall as l'illllpawd ti) tlll.' inlcraction cnergy t)f the sys-telll. \Ve cOllsidl'r compact arrays of magnetic c1usters, rcp-rcscllted by lIlagnetic 1ll0Illents 11" find the equilihrum COI1-figuralion of (he variolls systellls, and calculalc lheir reducetlmagnetorcsislalH.:e and rnagnetization, as a function of an ex-lemally applicd magnetic ticld 17. The reduced magnetore-siSlancc i.s calclllated evalllaling Eq. (..1-) for diffcrl"..llt values01' lhe magllL'lic lield,

JI is in gl'lIl.'1"aldirticult toCS[illlalC lhe real distrihutiun 01"grain si/cs ¡¡Ild also Jlerccntagcs 01"single magnclic domains.Howe\'eL rel:L'1l1repons on lhe manllfactering of Co-Ag gran-ular lilms prepared hy low-energy cluster hearn deposilion [61:,how that grain si/e Llislrihulions can oc very narrow (::::::1nm) around Ihe 1ll1'¡1Ilgrain diaIlll'ter (3 nm). Thereforc, andin ordcr 10 sludy the e1le<.:tof (hl' grain distribulion, we <':00-sider a systelll iJf single magnelic dUlllain Co clusters emoed-ded in a IlIclallic lllatrix (Cu 01' Ag), Clusler sizc elTecls haveheen t1isclIsscd hefore. in non interacting: systcll1s. hy Ferrariel a/. [81. In this work \Ve assullle Ihal the c1usters interaetonly throllgh Iheir t1ipolar liclds, The elfeet of the RKKY in-Icractiolls are Ilcgligihle 121 at the cluster sizes <':oIlsidered (::::::I ()()() Cn alOms),

Givcll fhe IHJ,itioll of lhe clusters in space, lhe encrgy oflhe syslem can he writlcll as

(4 )

(2)

(1 )

(ji, . ¡I}) 11

(/1.i . ¡7j)m=1

1¿,¡ldL.I¡I'¡/11 =

(1" '¡lj)",=1 - (/" .¡I})H(/Ii .llj)m=1 - (/Ii 'jij}m=O

(:'>I!) _)_a Tt'fl

( :'>;!) '''/

:'>I!/ fI = ,,(¡I, .¡Ij)o = (¡I~.¡I})Po - f,,(/I¡ 'I/j)o

wherc (/7;. ¡7j)o = (/7; '11))11=0.I[ i" con\"l'nil'llt to deline the rcduced magnctization /JI

nf lhe systclll, goillg from zero 10 olle when al! the maglll'tic1ll0lllents are paralll'L as:

This magniludc le¡¡Lls us lo introLluce Ihe so-calletl reducetllllagnewresistancc [1 J, dellneLl as:

(:'>J!) (:'>fI/fI)-(6.I!/fI)",=, (.1)

R n'" = (6.R/ R)",=o - (6.fI/ R)oo=1

This Iasl cqualioll gives a relalion hetwecn the retlucetl mag-neloresistance and the magnelization uf lhe clusters.

Ihcn Ihc shape anisolropy has to be induLleLl. Howevcr, inlhis work our allclllioll is cenlered UIl Ihe dipular inleraclionhel\Vl'cn the grains. ami hmv it is alTeclcd whcn a grain sill'di'ilrihution is l'onsidercLl. anJ then we do not include shapeanisíllropy and any spin.dependent scaltering mechanisl11.

The ¡¡e Id dl'l1l'lldenec nI" the alloy resislivilY (1 is wel! dc-"LTihl'd hy a silllpll' Illodcl proposcd hy Gittleman ('[l/l. 1.1].This Illodel (,:ollsiders onl1' the relalive orielllation hetwcclllile magnetic IlHllJlcnls nI' the cluslcrs, ignoring scatteringwithin Ihe graills. Acconting lo lhis model, the alJoy rcsis-livily call he obtainctl as:

whl'rc {/(I and 1; arl' constants, ami ¡i, ami ,7.1 are lhc lolalIllagllctic 1ll01llcnls 01"duster i and j, respectively. Herc it isa""ulllcd Ihal the lotal l11agnetic moments nI' the clusters ,7¡h•.l\"(: Ihe same magnitllde.

\\'e will assllme nmv that Ihe systcm is composed hy Ne!tl"ters of ditTercllt siles, each (lnc characterized hy a IOtallIlagllclic 1lI0111ent ¡7, 01' any magnitudc. \Ve can generalil.ellll.' e.\pressiun for Ihe resistivity if \ve adlllit that (/7; , 17./)call hc t1ilfcrent from zero when 1I = O, as expccleLl in Iheprl''ienCe (lf correlation among the magnetic Illumenls of lilel'IU:-,tl'IS.This means that .1R / R = ~p/ fJ can be \\TiIICn ,1'

R{'l', MI'x, rú . .w SI (199Xl 7J-77

MAGNETORESISTANCE IN GRANULAR MATERIAL.S ANIl SIZE EFFECTS 75

100 ,o ~~ ~ \-_.06 1:: "(~) 06 I "

R "" 04 :J~'O ~' ,, ''''02 ~~-:;.:..::--/ '••...~-

0.0 - -_.~-_. ~-- ~- . '-6 -2 O , 4 ti

H(KOe)

>- O

80

-1 00L~~~~.L~~~~.L~~~~..L~~~~..J-100 -50 O 50 100

X

FICjURE l. ShCmalic illustratioll of a 2D syslcm 01'441 lIlagncticsingle domains uf Co c1usters. Thc x anu y coordinares are in units01 10 Á.

10 /;;7.- ]os t Im 00 /1O"10 ~~/-~~.. ~- -~-

" -, O o11(K<k)

PI(;URE). Redllccd magnctorcsistance (top hall) and rcduccJmagnctil.;ltlOIl (llottolll 11::111)as a functinn nI' applieJ cxtcrnalmag-lldie licld. rOl" a 2D sySIClll01'441 elllstcrs.

in Figs. 1 ami 2. The eOlllinuous hne iJlustrales (he rcsuhsohlained considering (he size dislribution discussed ahoye.ami lhe dashed Iinc dcpiclS results ohlained consideringaH the grains willl Ihe same sil.c and Illagnclil.<llion (750¡lB each ol1e).

\Ve SCe almosl no dilTcrence hetwcen thcse (wo SYStClllS,prohahly hccausc of lhe small dcviation of thc sizc dislri-hlllion wilh respccl lo lhe average valuc. Correlalion alllonglIlagnelic grains does not change when Ihis clusler size dis-trihution is considered. ami then. lhc magnetorcsislance doesnol change. as illustrcd in Fig. 3.

We a!so consider a syslelll 01" 100 cluslcrs distrihulcd ina dislorted 2D lattice. Here we usc a uniform distributionfor the lllagnclic 1Il0tllenls. ranging fmm O lO 2, which iscOllsiL!erahly larger lhan (he distrihulion considered for lhesyslelll illustraled in Pigs. I and 2. It is important lo nOliceIhat. when a t,;luster sLl.e distrihutioJl is considerctl ror stllallsystems « --lOO cIuslers) in a distorted lattice. Ihere are notslill prcsclll the spalial sYllll11clries lhat are presentcd whcnalt lhe grains have Ihe salllC sizc and the lanice is nol dis-Imled. Then. the responsc of these systcl11s is sensitive tothe direclioll 01' (he applil'd lIlagnetic field. Figure 4 despicIsthe rcsults 01' Ihe magnetil.<llion (Iower pannel) and reducedlllagnelorcsislance (upper pannel) as a funclion of lhe ex-lernal Illagllctic field. rOl' (lne particular direction of Ihe ap-plied magnetic field. The conlinuous line shows lhe resultsohtaincd cOllsidcring lllat all grains have lhe same magni-tude and lIlagncli¡alion. whercas lhe dotled line illustrates there ...•ults ohlained considering the size distribulion dcscrihcdahove. Sil.e clfects are evidenced in the rcduccd magnelorc-sistance curve. and also little cffecl is observed in the hystcre-siso The hysleresis curve 01' lhe disordcred sysletll saturales athigher nelds. indicaling thal sume grains. lhe higger ones.!leed higher l¡elds lo he aligned. Thc rnagneliza(ion jump inIh(: high l¡eld rcgiun is a conscqllencc 01" the parlicular dis-

'6\..val "e

06oo.

-'O

20

Thc rcsislancc 01' a S)'SICIll. as a transport propcrty. dc-pcnds on lhe mean free path.A orthe clectrons in lhe materia!.TIH:n, lhe average 01' ¡ti' ¡tj in Eq. (1), and (hen in Eq. (4), istakcn over pairs 01' c1ustcrs scparalcd hy distances n01 1TI1Ichlargcr fhan A hy a large alllounl [1 l. A defincs lhe eXlension01"lhe correlation hctwcen rcgions in our systclll and its illl1u-cnec on lhe Illagnelic hehavior of granular systcms has hceneXlcllsively invcstigated hy Vargas et al. 19J. For thesc calcu-lations we use A :::::150 Á. a valuc experilllentally acceptcdfor eo-eu granular syslems [1¡.

Figure J ¡lIustrales the rcduccd magnclorcsislance I'.\'.

magnclic field (upper pannel) and lhe rcduccd magnetizalionI'S. magnetk field (Iower pannel) for the 20 systelll t1cscrihcd

60

FIGURE 2. Histogram of the size dislribulion of the systcm of 441magnctic single domains. The value onc means tha! the cluster has750 atolTIs ano ,1 magnetizatían of 750 JtB.

R('I~ ¡\J('X. Fú. .uSI (1998)73-77

710 D, ALTHIR A;"¡f) p, VAR.GAS

2015

•'f ''''.~ .1-.i

1 5

05

00O

Sizc 1,0

10

Fl(illlU 5. Thrcc diqrihuli(lllS 1\11'Ihc grain sin ..•.in ;1 sy"lcrn 01'20 du .•.lL.r.•..TIIl' \'~II11eolle corre",polld_slo a clU.sler wilh;l rnagneli.¡;Ilioll uf 750 1111.Tlle l"OlllíIllIOIl'"lilH.'rcpn:scnts a sy.sICl1lwith allclustcrs 01 l!le sallle .•.i!l', Thc doued JíllC represellls ;l díslrihutioll\\illl lwice ;ISbrge dc\'ialíolls lo thal illustralcJ with Ihe dot-dashcJ

10 linc,

o 5II(KOc)

-510

04

06

° -00

'°f -05' l'm 00

05

1010 -5 ° 5

II(KOc)

F[(illRE 6 RcduceJ \la!,!llctnrcsislant'C (top hnlf) and rcduccdllla~l1Cli¡alioll (hol1oTllh;lIfl " .•.a fllllctioll nI' external applicJ ticld,fOIa ~D syslelll of 20 Clll.•.lcrs wilh Il1rL'cditTcrcnt sizc distrihutions .a....SIlOWllin Fig. 5. Thc inse! illlllL' lowcr pannel illuslrate sehcmal-¡e,lIly Ihe sy.•.IClll as well íl'" Illl' din.:L:lion01' Ihe applicd magnctirlidd.

Illagnl'torcsistancc rol' low lIlagnelic lields, when a grainsilc dislrihutioll i'i cOllsideretl. For a hroaucr dislribulion. a~trol1gcr ellcct is ohservcd. HO\\'cvcr. at higher licios. the rc-dUl'l'd magnclorcsislance is cnhanccd hccause the magnctit.a-lion sall/mles al hig:hcr tlcld". as shown in (he lower panel.

\Vhcl1 alll11agnctic chl"itcrs llave lhe "iame sizc. (he corrc-I;ltion. i.l' .. ¡Ill' a\'crag:l's in El]. (4), considcred over a lenglhsc;de oj" ,\. shou]d nol he dillt.renl Irolll lhe corrclatioll oh-lainl'd when a I.:lustcr sit.e dislrihution is considercd, ir il isproperly avcraged, Howevcr. a real systelll has only olle grain

1:11;1In -~ Redun:d ma!!lll'(orl'~i~[alll'C ((or hall) and rcdul'cJ111,1\.!llCll/,lll(lll (ho[[tllll hall) as a fllllCllOIl or applicd cxtcrnallield.Ipl;t 21> \~\ICJl1 01 lOO single m<lf!l1clic uOfllains.

ItllIL'l! "p:llial po"iliollS nI" clusters. \Vh~1l ¡\Vn clusters are\l'I~ rlllsl' ;llId thL'ir Jlla~Ill'li/ali()lls are hig cllough. ¡!len;1 \'l'I) hi~11 l1Iaglll'lic l¡cld is lll'cdcd lo ¡¡I¡gll IhcllI 10 lheIll'Jd, Tlll' JlI;¡~IlL'lil.ation jUlllps suddcnly frorn olle Cljuilih-l'iUlll \;t!I!L'ltlll1l' llL',\( cquilihriulIl \"aluc al a highcr.ficld. Thisrl''>L.lllbk•....11lL'BarkhauSCll cllcc!. Tlll' jUlllps in Ihe magm:li-1;llloll dlr\L' han';¡ hig ellcct in lhe magnctorcsistancc curn'.kL";llhL' Il1l' ...hl\r1 rangc corn:lalioll (on.'r a region 01' radius\) 111lile dipol-.' IlHllllcnl is slrongly dependclll 01' lhe rel-dli\'l' tlliclllalilln 01' lhe ncighhoring magnctic oipolcs. In ahi1.-'~l'r SYSICIIl. l!lis jumps \vill pruhahly disappcar. occauscIhe pmhahility Ihal a givcn sizc dis[rihution populates lIJanynl'i).!l1horing siles wilh !ligh lIlagnclic lllo111cnls is 10\\1.Thcn,hl'll' []¡l' jlllllP" inlhc curvc" cvidclln.' linite sile cllecls,

TII oh'\.'I\\' ...,[rong sile clTccts, wc considcr l\Vo chains(\1 1\.'Jl grains l'ach. in \Vich lhc dusll'rs are 7U Á arpan.dlld IlIl' di ....l¡lIlÚ' het\Vecn chains is X5 A. This sySlCIll was.dl\.';¡d~ pl'Opo,ed ;¡" an experimenlal prolotype for ohtain-IIl~ ;111111lpml;ltlt cnhanlTlllellt 01' lile magnclorcsistancc d-Icd 1::] \\'hl'll an exlerna] magnctic Iield is applicd parallcl (o111l'dlains. \Vc .stl1dieo 1\••..0 dillen:nl sizc dislrihl1lions 01' theI1l;11.-'-llclicgrains. which are illustralco in Fig, 5,

rllc sy •.•tCIIl is schcmalically shown in lhe inset of Fig. ()(1I1\\l'l' pallllL'I). The eXlernallllagnL'lic lield is applieJ pcr-Pl'lldicular lo lile dlains. as dcspicll'O in Ihc inset. The C011-lilllllJU....lilll' illu'arales rcsults for a SYSICIll01' 20 c1uslers. all01 l!ll' .•.anll' SilL', Thc dOI-dashcd linc depicls rcsults whell a,,111;111di,pl'rsion in the grain si/l~ is cOllsidcreo (dol-dashel!lilll' in I:ig. 51 and Ihe dOllcd line illuslrales resulls whell ;]hlo;ldl'l dislll'r~i{\ll is considcrco (tlollco li!lc in Fig. 5). Inllll'''l' l"<LiruJaliolls \Ve llscd an t'!cl'lronil' 1l1l'~1l1frec path 01\ = t:~.~.A. \Vl' can l'Iearly nolice a deCl"l'<tsl' of lile rcducl'd

I 41 ,

(~)10

1OSR "" 06

04Oc00-6 -4 -, O ,

H(KOc)

10 .,'--'.......... ;/..........05

h00 /m-#.:

-O 5 /'

-1 O /-6 -4 ~2 O 2 4

H(KOc)

6

6

R¡'\'. Akr_ Fú . .w Sil ItJlJX) 73-77

~1AGNET()RESISTANCE IN GRANUI.AR ~1ATERIALS AND SIZE EFFECTS 77

size distributioll, ami we shuulJ scc a behaviour likc the olledepicled in Fig. 6, The crfeL:t is slrnnger whcn the systcm issrnall ¡¡Ild lile grain si/.e distrihution is very hrnat!. 17m smallsy'stClIls, tlle correlation al short dislanccs depends slrDnglyOlllhc size 01" Ihe consitlcretl c1ustcr. For larger syslCI1lS. 11m\'.ever. si/.e elTects 011 Ihe short rangc correlalion averages outanJ the cffecti\'c magnelic correlalion is similar lo Ihe OIlCohtainctI when aH the grains have Ihe same size.

3. Cnnclllsinns

In cOllclusion, \\lC have included si/e efTccts in our mode!for Ihc magnctoresistance and magnclizatioll 01' granular sys-lel1ls. Thc lllain cfrcct 01' cOllsidcring a hroad dislrihlltion inlhe grain size oí' a disorderetl systel1l is a hroadcning 01' IhercducctI magncloresistance curve, and almOSI no rctlucliol1nI' ils maximulll value. For an ortlereJ 2D system \Ve h.l\'c aC(HllraClion 01' the reduced magnctoresislance as a funclioll ()Icxlcrnalmagnctic tield. This is a cOllsequcllcc ortlle !lC\\' bc-havior 01' thc rcJuccd magnelization in lhe 10\\' f1eld region.

1. 1'. Allia. 1\1. Knohcl. P. Tibcrto. ami F Vinai. PhYJ. I<el'. n 52(]995) 15398.

.} D. t\lIbir. J. (J'Alhuqucrqlll: e Castro. ano P. Vargas. I'hy.\ I«'\'./l5"¡ (199(1) R682~,

:3. D. Ahhir. P. Vargas. J. o'Alhuqucrquc e Castro. ami U. Rall.""ys. Rl'I'. n 57 (1998) 136().t..

.1. J.L Giuleman. Y. Goldstcin. anO S Bozowski. Ph\'.\" ReJ'. nI)(1972) 3(¡()9.

Broader si/.c distrioutiolls givc rise lo hmaJcr hysteresis hc-havior. ill Ihe Sl'nsc lhal a highcr Ileld is llcedcd to reach thesaturatiull \'<lltI\.'.Also, \v!lcn \Ve compare Ollr rcslllts to the{mes ohtained IDI rol' ditlercllt vallles {jI" .\. \Ve nOlicc that oursimple model for lhe magnelorcsislancc in Illclallic granularlIlillerials is far more scnsilivc lo the cleclronie mcan free paththall lo Ihe grain size distrihlltion. Finally. we also concluJeIhal lhe ma.\illlllll1 vallle 01' Ihe reJuceJ magneloresislanccat tcro lield. relalctl lo the "lIlliform si/.c" case, remain UIl-changed as a cOllsequcnce of lhe invariancc 01' lhe shOrl ranget1ipole-dipole correlation al zcro IIcld \\'!len a size distrihulionis C'onsiden:d.

Ackllowlcdglllcnts

\Ve ackllowlcdgc DI'. Ulrich RatT for crilical reading of lhemanusnipt. Financial support fmm DICYT of lhe Universi-dad de Santiago lindel' granl nUlIlhcr O-l973 [AD is grall'ful1yacknowlcdgeJ.

d. \1.R. I'arker, J.A. Barnard. 1). Scalc. anll A. Wakins, J. Appl.1'lIr.l" 7.1 (199)) 5512.

¡-i. F. Part'll[ ('/111., 1'11\,.1 Net'. n 55 (1 ()c)71 ~Mn.l. 1':.11. Dut:. and T.K. <bnh . .1. I'hy.\".: (owl. lHat1l'r 9 (IY97)

15K5.

S. E.F Fl'lTari. l.es da Silv~l, amI ¡,\1.Knobd. I'/¡YJ. Rl,t, n 56( 19(7) hOX()

!l. P. Varga~, D. t\Jthi •..J. tl'Alhuqucrquc e Ca~tro. and U. Raff. 1.Phy.I',: ('(JIId. MlIt1l'1" l) (1997) 99~ l.

R('l'. ,\h.l. Fú . ..w SI (199g) 7J-77