Relationship between coercivity and magnetic moment of superparamagnetic particles with dipolar interaction

Rela ionship be ween coe ci i y and magne ic momen o supe pa amagne ic pa icles wi h dipola in e ac ion V. F anco, C. F. Conde, and A. Conde Dp o. Física de la Ma e ia Condensada, ICMSE-CSIC, Uni e sidad de Se illa, P. O. Box 1065, 41080 Se illa, Spain L. F. Kiss Resea ch Ins i u e o Solid S a e Physics and Op ics, Hunga ian Academy o Sciences, P. O. Box 49, 1525 Budapes , Hunga y 共Recei ed 28 July 2005; e ised manusc ip ecei ed 8 Sep embe 2005; published 22 No embe 2005兲 The empe a u e dependence o he hys e esis loops o Nanope m- ype alloys has been s udied. In he high- empe a u e egion abo e he coe ci i y maximum, he esponse o he sys em can be modeled as ha o dipola -in e ac ion supe pa amagne ic pa icles, conside ing a mean in e ac ion ield. Special a en ion has been paid o he in luence o he pa icle size dis ibu ion on he applicabili y o he mean- ield model. The wo main e ec s o he dipola in e ac ion 共coe ci i y and dis o ion o he he mal dependence o he appa en magne ic momen 兲ha e been co ela ed. DOI: 10.1103/PhysRe B.72.174424 PACS numbe 共s兲: 75.50.T , 75.50.Kj, 75.10.N I. INTRODUCTION The magne ic cha ac e iza ion o ine pa icles is a ield o cu en in e es . No only do he p ope ies o hese pa icles de ia e om hose o hei bulk coun e pa s, which is an in e es ing subjec o s udy in undamen al physics, bu also hese pa icles a e p omising o nume ous echnological ap- plica ions, anging om da a s o age o bio echnology.1 As he cha ac e is ic size o he ma e ial dec eases, in e - pa icle in e ac ions 共exchange and/o dipola 兲ge inc easing impo ance, o igina ing nume ous scien i ic s udies.2,3 De- pending on he ype o ma e ial, i s mic os uc u e, and he ope a ing empe a u e, he ype o in e pa icle in e ac ion ha con ols i s magne ic beha io can be di e en . Fo ex- ample, cen e ing ou a en ion on magne ic eco ding media, while in he case o hin ilms he dominan in e ac ion is exchange coupling, o pa icula e media he leading ole is played by he magne os a ic in e ac ion.4A special ype o ma e ials a e he nanoc ys alline alloys o he Fineme and Nanope m amilies. In hese cases, e omagne ic nanoc ys- alline pa icles a e embedded in a esidual amo phous ma- ix wi h a ela i ely low Cu ie empe a u e 共⬃300 °C兲. P o- ided ha he c ys alline ac ion is small enough, he ele an in e ac ion be ween he nanoc ys als can be con- olled by simply changing he measu ing empe a u e, mak- ing hese sys ems pa icula ly in e es ing o analyzing he in luence o he di e en in e ac ion ypes on he magne ic cha ac e is ics o samples. The main i ue o hese sys ems is ha hei mic os uc u e emains unchanged o he di e - en in e ac ion egimes. Recen ly, he magne ic hys e esis obse ed in g anula magne ic ma e ials a empe a u es abo e he supe pa amag- ne ic blocking empe a u e o he pa icles has been modeled by in oducing a memo y e m inside he a gumen o he Lange in unc ion, ela ed o a mean dipola in e ac ion ield.5This model has been success ully applied o di e en kind o g anula ma e ials, such as Co-Cu alloys,5Co nano- pa icles embedded in a SiO2ma ix,6and Fineme - ype alloys.7 A ema kable expe imen al ac is ha o supe pa amag- ne ic pa icles wi h dipola in e ac ion, he magne ic momen calcula ed om he Lange in i ing o he anhys e e ic mag- ne iza ion cu e does no co espond o he eal magne ic momen o he pa icles. This disc epancy is e idenced by analyzing he he mal dependence o he calcula ed magne ic momen o a sys em o he mally s able pa icles, a momen wi h an appa en inc ease wi h empe a u e. This appa en ly anomalous e ec o he dipola in e ac ion be ween he pa - icles has been modeled by using an e ec i e empe a u e ela ed o he ms dipola ene gy.8The applica ion o his model o Fineme - ype alloys p o ided a i s app oach o ela e bo h e ec s o he dipola in e ac ion be ween he pa icles.7 I is impo an o men ion ha geome ical ac o s can play a ele an ole in he e ec s o he dipola in e ac ion be ween pa icles.9Speci ically, he exis ence o a g ain size dis ibu ion in he s udied sample can a ec he applicabili y o he mean- ield model. In he case o Co-Cu alloys, g ain size dis ibu ions we e aken in o accoun by conside ing ha he esponse o he sys em o he addi ional exci a ion ield was ha o an a e age momen .5In o de o check he appli- cabili y o ha assump ion o o he sys ems, we ha e s udied C - and C -Mo-subs i u ed Nanope m- ype alloys a he ea ly s ages o nanoc ys alliza ion. The p esence o Mo in he al- loy composi ion educes he Cu ie empe a u e o he amo - phous alloy close o oom empe a u e, acili a ing he ex- change decoupling be ween he nanoc ys als,10 while C has a simila e ec in he Fineme - ype alloys.7I will be shown ha he e a e cases whe e he esponse o a sys em wi h a dis ibu ion o pa icle sizes canno be ep esen ed by he esponse o an a e age sys em, al hough he mean dipola in e ac ion ield e ains i s meaning and use ulness. The s uc u e o his pape will be he ollowing. Sec ion II summa izes he expe imen al echniques used. Sec ion III p esen s he expe imen al esul s ega ding he de i i ica ion p ocess o he s udied alloys and he he mal dependence o he hys e esis loops, which will be subsequen ly modeled in Sec. IV. Ini ially, an a emp is made o model he loops wi h PHYSICAL REVIEW B 72, 174424 共2005兲 1098-0121/2005/72共17兲/174424共10兲/$23.00 ©2005 The Ame ican Physical Socie y174424-1 he use o a single a e age magne ic momen . A e ealizing ha in one case his app oach 共i.e., using an a e age memo y unc ion兲 ails, he way in which pa icle size dis ibu ions a e aken in o accoun in he hys e esis model is modi ied, esul ing in a much be e i o he loops. Sec ion V makes he ela ion be ween bo h e ec s o he dipola in e ac ion mo e p ecise. Finally, conclusions a e d awn. II. EXPERIMENT Amo phous ibbons o he nominal composi ion Fe76C 8Cu1B15 and Fe76C 4Mo4Cu1B15 we e p epa ed by single olle mel spinning. Thei amo phous cha ac e was checked by x- ay di ac ion. The de i i ica ion p ocess was s udied wi h a Pe king-Elme DSC-7 di e en ial scanning calo ime e 共DSC兲. The e olu ion o he Cu ie empe a u e o he nanos uc u ed ma e ial du ing de i i ica ion was de- e mined wi h he help o a Pe kin-Elme TGA-7 he mobal- ance applying he ield o a small pe manen magne . To ob ain he desi ed mic os uc u e, samples we e hea ed a 10 K/min up o he a ge empe a u e. The he mal depen- dence o he hys e esis loops was measu ed by a supe con- duc ing quan um in e e ence de ice magne ome e 共Quan- um Design MPMS-5S兲 om 300 up o 700 K. T ansmission elec on mic oscopy TEM obse a ions we e pe o med on a Philips CM-200 elec on mic oscope wi h an accele a ing ol age o 200 kV. III. RESULTS A. De i i ica ion p ocess The de i i ica ion p ocess o he s udied alloys is simila o ha obse ed in nanoc ys alline so magne ic ma e ials o he Nanope m amily.10–13 I akes place in wo main s ages, as e idenced in he di e en ial scanning calo ime y plo 共Fig. 1兲. The i s one co esponds o he appea ance o a bcc Fe nanophase embedded in he esidual amo phous ma ix, while he second one, a e which he sample is ully c ys- allized, is associa ed wi h he o ma ion o bo ide- ype phases and includes he ec ys alliza ion o some p eexis ing bcc Fe c ys als.14,15 The p esence o Mo in he alloy p o- duces a s abiliza ion o he amo phous phase agains nanoc- ys alliza ion, as e idenced by he displacemen o highe empe a u es o he nanoc ys alliza ion onse , as well as he s abiliza ion o he nanophase by inc easing he onse em- pe a u e o he second c ys alliza ion s age. This la e e ec p e en s he o e lapping o he wo c ys alliza ion s ages o he Mo-con aining alloy. The empe a u e e olu ion o he low- ield magne iza ion o he as-cas alloys shows 共Fig. 2兲a anishing magne iza- ion 共M兲a he Cu ie empe a u e 共TC兲o he amo phous phase, an inc ease in Massocia ed wi h he onse o nanoc- ys alliza ion, and a u he dec ease co esponding o he ec ys alliza ion p ocess and hen o he TCo he bcc Fe phase. In he case o he Mo- ee alloy, he he momagne ic cu e eco ded du ing he cooling a e ull c ys alliza ion is also plo ed, showing he Cu ie empe a u e o he inal mag- ne ic c ys alline phases: bcc Fe and bo ide phases. Fo bo h alloys, he he momagne ic scans o he p ehea ed samples indica e ha annealing he samples a mode a ed empe a- u es p oduces a sligh educ ion in he TCo he amo phous alloy due o s uc u al elaxa ion. As nanoc ys alliza ion be- gins, TCinc eases p og essi ely. The main di e ence be- ween he wo alloys is a highe TCo he amo phous phase in he case o he Mo- ee alloy. Table I con ains he peak empe a u es o he DSC exo he mal maxima, as well as he e olu ion o he Cu ie empe a u e o he esidual amo phous FIG. 1. Di e en ial scanning calo ime y cu es, aken a 10 K/min, o he as-cas Fe76C 8Cu1B15 and Fe76C 4Mo4Cu1B15 samples. FIG. 2. Tempe a u e dependence o he low- ield magne iza ion o he as-cas and p ehea ed samples 共up o he indica ed empe a- u e兲. Fo he Mo- ee alloy, he cooling p ocess o a ully c ys al- lized sample is also eco ded. FRANCO e al. PHYSICAL REVIEW B 72, 174424 共2005兲 174424-2 ma ix as a unc ion o he p e ious he mal ea men . B. Tempe a u e dependence o hys e esis Figu e 3 shows he he mal dependence o coe ci i y o he s udied nanoc ys alline samples. The empe a u e ange has been selec ed o a oid he e olu ion o he mic os uc u e o he samples du ing he measu emen p ocess. Th ee di - e en egimes can be de ec ed in he cu es. A low empe a- u es, he coe ci i y has a small alue due o he exchange coupling be ween he nanoc ys als h ough he e omagne ic amo phous ma ix, which p oduces he a e aging ou o he magne oc ys alline aniso opy.16 As he Cu ie empe a u e o he ma ix is app oached, he coe ci i y inc eases as he ma- ix canno ansmi he exchange so e icien ly.17 A highe empe a u es, a e a maximum, he coe ci i y ends o di- minish. This las egime has been explained in so magne ic nanoc ys alline ma e ials as a ansi ion o supe pa amagne ism,18 con olled by he dipola in e ac ion be ween he nanoc ys als, which causes he p og essi e de- c ease o he coe ci i y. I has been shown ha Fineme - ype alloys con aining C 共Re s. 7, 18, and 19兲o C -Mo 共Re . 20兲 show ully supe pa amagne ic beha io a empe a u es abo e he Cu ie empe a u e o he emaining amo phous ma ix. Also, o ypical 共FeSiBCuNb兲Fineme alloys, supe - pa amagne ism occu s a he ea ly s ages o nanoc ys alliza ion,21 which has been in e p e ed as a dem- ons a ion ha supe pa amagne ism is a gene al ea u e in so magne ic nanoc ys alline alloys o low enough c ys al- line ac ions. Fo Nanope m alloys, wi h a simila mic o- s uc u e o ha o Fineme - ype alloys 共in he sense ha Fe nanoc ys als—Fe,Si in he case o Fineme —a e embedded in a emaining amo phous ma ix兲, supe pa amagne ic be- ha io has been epo ed.22 The e o e, al hough in he expe i- men al da a he comple e supe pa amagne ic egime is no eached 共 he coe ci i y does no all o ze o兲, i is easonable o conside ha he men ioned coe ci i y dec ease a e he maximum is e idence o he ansi ion o supe pa amag- ne ism. As he nanoc ys alline ma e ial is me as able, i is no possible o inc ease u he he measu ing empe a u e wi h- ou al e ing he mic os uc u e o he sample, p e en ing us om obse ing he ze o coe ci i y. As he c ys alline ac ion inc eases, he maximum alue o he coe ci i y inc eases due o he s onge in e ac ion be ween he pa icles.7Also, he empe a u e a which he maximum is de ec ed shi s o highe empe a u es, in co e- la ion wi h he inc ease o he Cu ie empe a u e o he e- maining amo phous ma ix men ioned abo e. The essen ial di e ence be ween he beha io o he wo alloys is ha while a maximum in coe ci i y can be de ec ed o he Mo-con aining alloy, he Mo- ee samples show ei- he a con inuous inc ease in coe ci i y, o a pla eau. This can also be ela ed o he highe Cu ie empe a u e o he amo phous ma ix in he la e case. The e o e, he Mo- ee samples will no be adequa e o analyzing he dipola in e - ac ion be ween he nanopa icles. IV. MODELING THE HYSTERESIS LOOPS A. Using a single magne ic momen In he empe a u e ange abo e he coe ci i y maximum, he hys e esis loop o he samples can be analyzed using a mean- ield model.5I conside s he e ec o he dipola in- e ac ions be ween he supe pa amagne ic pa icles as a memo y unc ion and can be summa ized as ollows. The anhys e e ic educed magne iza ion o he sample, m,isde- ined as he hal sum o he wo b anches o he educed hys e esis loop. ⌬is de ined as he hal di e ence o he wo b anches o he educed hys e esis loop. The skele on o he loop should co espond o a Lange in unc ion 共 he ex en- sion o a dis ibu ion o pa icle sizes will be conside ed in nex sec ion兲, as his is he beha io o he pa icles in he absence o in e ac ions. Fo majo hys e esis loops, he only ones conside ed in his pape , he memo y unc ion ␦ 共m兲 depends on he anhys e e ic educed magne iza ion o he sample, m. The b anches o he majo loop can be desc ibed as m±⬇L 冉 ␮ H kT ± ␦ 共m兲 冊 ,共1兲 whe e ␮ is he magne ic momen o he pa icle 共which will be e e ed o la e as he “appa en magne ic momen ”兲,His TABLE I. Peak empe a u es o he DSC cu es o he s udied alloys, and Cu ie empe a u es o he 共 esidual兲amo phous ma ix o he samples p ehea ed up o he empe a u e indica ed in b ack- e s, de i ed om he magne iza ion cu es. Fe76C 8Cu1B15 Fe76C 4Mo4Cu1B15 Tp1共K兲715 727 Tp2共K兲771 832 Tc共am.兲共K兲关As cas 兴392 346 Tc共am.兲共K兲关703 K兴382 Tc共am.兲共K兲关705 K兴383 Tc共am.兲共K兲关708 K兴385 Tc共am.兲共K兲关715 K兴335 Tc共am.兲共K兲关720 K兴341 Tc共am.兲共K兲关730 K兴358 FIG. 3. Tempe a u e dependence o he coe ci i y o p ehea ed samples o he s udied alloys. RELATIONSHIP BETWEEN COERCIVITY AND …PHYSICAL REVIEW B 72, 174424 共2005兲 174424-3 he applied ield, Tis he measu ing empe a u e, and kis he Bol zmann cons an . The memo y unc ion can be ew i en as a unc ion o an e ec i e in e ac ion ield H0共 ela ed o he oo mean squa e o he dipola ield兲and a “cu o ” unc ion F共m兲as ollows: ␦ 共m兲= ␮ H0 kT F共m兲.共2兲 The magne ic momen o he pa icles can be calcula ed h ough a simple Lange in i ing o he anhys e e ic cu e. Meanwhile, he educed hal di e ence o he wo b anches 关⌬共m兲兴 is a measu e o how he hys e esis loop sepa a es om he supe pa amagne ic skele on. The e o e, i should be ela ed o he cu o unc ion. A e some manipula ions, de- ining m as he educed emanence o a majo loop, he cu o unc ion can be ob ained as F共m兲=⌬共m兲 3m u⬘共m兲,共3兲 whe e u⬘共m兲is he i s de i a i e o he Lange in unc ion wi h espec o i s a gumen . An al e na i e exp ession o he cu o unc ion can also be ob ained: F=3 冑3u⬘共m兲 冉 1−2mkT ␮ H−m2 冊 1/2 .共4兲 In his case, i has o be no ed ha he F共m兲cu e can be gene a ed i he magne ic momen o he pa icles is known. The di e en pa ame e s equi ed o econs uc ing he b anches o a majo loop a e ob ained as ollows. The mag- ne ic momen is de e mined om he Lange in i ing o he anhys e e ic cu e; he cu o unc ion can be ob ained ei he om he educed hal di e ence o he b anches h ough Eq. 共3兲共expe imen al F兲, o om Eq. 共4兲共p edic ed F兲once he magne ic momen is known, and inally, he in e ac ion ield is calcula ed om m =1 3 ␮ H0 kT .共5兲 The majo loop is ob ained as m±=L 冉 ␮ H kT ± ␮ H0 kT F共m兲 冊 .共6兲 As indica ed in he p e ious sec ion, supe pa amagne ism is a common ea u e o his kind o so magne ic nanoc s alline alloy a empe a u es abo e he coe ci i y maximum. Howe e , he exis ence o dipola in e ac ions be- ween he nanopa icles causes a g adual ansi ion o he supe pa amagne ic egime. Following he desc ibed p oce- du e, he hys e esis loops o he Mo-con aining sample p e- hea ed up o 730 K can be modeled wi h ema kable ag ee- men in he empe a u e ange s a ing om he coe ci i y maximum. As an example, Fig. 4 shows he i o he anhys- e e ic magne iza ion cu e, measu ed a 475 K, o he Mo- con aining alloy p ehea ed up o 730 K. In his case, a Lange in unc ion plus a linea con ibu ion we e i ed o he cu e. The eason o using his linea pa 共wi h a mag- ne iza ion e ol ing as ␹ H兲, whose alue is o he o de o 10−7 emu/Oe, is o ake in o accoun he magne ic signal no coming om he nanopa icles bu om he pa amagne ic ma ix and he sampleholde . Figu e 5 shows he expe imen- al ⌬共m兲da a oge he wi h hose ob ained om he p edic ed F, o he samples p ehea ed up o 715 and 730 K. Figu e 6 shows he expe imen al 共c osses兲and calcula ed loops o hose samples. Ca e has o be aken when using Eq. 共3兲 o ob aining he cu o unc ion 共expe imen al F兲. While o he highe an- nealing empe a u e he e is no di e ence be ween he i ed loops using he expe imen al Fand hose p o ided by Eq. 共4兲 共p edic ed F兲, o he sample annealed a he lowe empe a- u e he use o he p edic ed Fshows disc epancies be ween FIG. 4. Anhys e e ic magne iza ion cu e ob ained om he hal sum o he hys e esis loop b anches o he Fe76C 4Mo4Cu1B15 alloy p ehea ed up o 730 K and measu ed a 475 K. The line co esponds o a Lange in i . FIG. 5. ⌬共m兲o he Fe76C 4Mo4Cu1B15 samples p ehea ed up o 715 and 730 K measu ed a 475 K: expe imen al da a 共c osses兲; calcula ed om he p edic ed F关Eq. 共4兲兴 共lines兲. FRANCO e al. PHYSICAL REVIEW B 72, 174424 共2005兲 174424-4 he expe imen al and p edic ed loops. This ailu e o he p e- dic ed Fin ep oducing he expe imen al loop would be an indica ion ha ei he he model is no accu a e o ha case, o as we will show o ou s udied samples, some modi ica- ion has o be in oduced o accu a ely ep oduce he loop. The e can be se e al easons which could jus i y ha he model o supe pa amagne ic pa icles wi h dipola in e ac- ion does no wo k o a speci ic sample, e.g., he s eng h o he in e ac ions, which could o ce us o in oduce highe - o de e ms in he Taylo expansions used in he model, o he p esence o in e ac ions o nondipola o igin, which was p obably he case o he Mo- ee alloy, since he coe ci i y maximum could no be eached due o he s uc u al e olu- ion o he samples. Howe e , hese easons canno be e- sponsible o he beha io o he Mo-con aining alloy p e- hea ed up o 715 K, as in ha case bo h he Cu ie empe a u e o he emaining amo phous ma ix and he c ys alline ac ion a e smalle han hose o he sample p e- hea ed a 730 K, and his would jus i y he opposi e beha io 共a s onge ailu e o he model o he 730 K sample, which is no he case兲. Ano he possible e ec ha can be disca ded is he p es- ence o blocked pa icles in he s udied samples. I has been shown23 ha in he case o he coexis ence o blocked and in e ac ing supe pa amagne ic pa icles, he no malized ⌬共m兲 cu e 关⌬N共m兲兴 o di e en measu ing empe a u es should no o e lap, he blocked pa icles being esponsible o e- ma kable de ia ions om he mas e cu e beha io . This is ce ainly no he case, which is demons a ed in Fig. 7 whe e he o e lapping is sa is ied o he 730 K sample, while he 715 K sample shows only mino de ia ions om he mas e cu e. The e o e, we should conside he possibili y ha he be- ha io o he 715 K sample is due o a dis ibu ion o he magne ic momen s in he sample, b oade han in he case o he 730 K sample, which could be i ed wi h a single mag- ne ic momen . B. Conside ing he g ain size dis ibu ion Acco ding o he o iginal model,5 he ex ension o a dis- ibu ion o magne ic momen s should be made by jus using he mean magne ic momen in he exp essions o he memo y unc ion. The e o e, he b anches o he hys e esis loop can be ob ained as M±=兺 i Ni ␮ im±,i,共7兲 whe e Niis he densi y o pa icles o each con ibu ion, ␮ i i s magne ic momen , and m±,i=L 冉 ␮ iH kT ± ␮ iH0 kT F共m0兲 冊 ,共8兲 whe e m0indica es ha he mean magne ic momen has been used in he exp ession o F. The in e ac ion ield o his sys em o pa icles would be de ined as H0=3kTm 具 ␮ a典,共9兲 whe e 具 ␮ a典=兺pi ␮ iis he a e age magne ic momen o he pa icles, pibeing he ac ion o each con ibu ion. Acco ding o his easoning, he hal di e ence o he educed magne iza ion cu e should co espond o FIG. 6. Reduced hys e esis loops o he Fe76C 4Mo4Cu1B15 samples p ehea ed up o 715 and 730 K, measu ed a 475 K. C osses, expe imen al da a; solid lines, i using he expe imen al F; dashed lines, i using he p edic ed F. Loops we e measu ed o ields up o 2 kOe. Fo 730 K, bo h i s o e lap. FIG. 7. Expe imen al ⌬N共m兲o he Fe76C 4Mo4Cu1B15 samples p ehea ed up o 715 and 730 K, measu ed o se e al empe a u es. RELATIONSHIP BETWEEN COERCIVITY AND …PHYSICAL REVIEW B 72, 174424 共2005兲 174424-5 ⌬=具 ␮ a典H0 kT F共m0兲u⬘共m0兲.共10兲 The e o e, acco ding o he model, he ⌬共m兲cu e co e- sponds o a single magne ic momen : he a e age alue o he dis ibu ion. Figu e 8共a兲shows he i o Eq. 共10兲o he expe imen al ⌬共m兲cu e o he 715 K sample, e idencing ha al hough he e a e some imp o emen s wi h espec o Fig. 5, he e a e s ill impo an disc epancies be ween he expe imen al and p edic ed beha io o bo h high and low ields. The eason o he be e i ing compa ed o Fig. 5 is ha in he p esen case he leas -squa es i ing is applied di ec ly o he ⌬Ncu e and he e o e, he i ed cu e goes abo e and below he expe imen al da a in he di e en pa s o he cu e; in he case o Fig. 5, he i was o he magne- iza ion cu e, making he i ed ⌬共m兲cu e always below he co esponding expe imen al ⌬共m兲cu e. I is also wo h men ioning ha he nume ical p ocedu e o i ing ⌬Nuses Has he abscissa axis. The eason o his is ha m, he abscissa axis o hese igu es, depends on he p e ious de e - mina ion o he magne ic momen o he pa icles, hei num- be densi y, and he linea con ibu ion p e iously men- ioned. The e o e, mino di e ences in he de e mina ion o hese pa ame e s due o he di e en quali y o he i ings p oduce he sligh modi ica ions in he shape o he ⌬共m兲 cu e, as e idenced in he igu es. To educe he emaining disc epancies in he i ing o ⌬, le us conside ha he o igin o he hys e esis in his mean- ield model is he in e ac ion ield H0, which should depend on he a e age magne ic momen o he pa icles h ough Eq. 共9兲and is a measu e o he dipola ield s eng h p oduced by he es o he sys em 共i s physical meaning does no change wi h espec o he single-magne ic-momen model兲. The con ibu ion o each componen o he educed magne iza ion loop can be ew i en om Eq. 共6兲by inse ing in o i he de ini ion o ⌬as he hal di e ence be ween he b anches o he loop: m±,i=L 冉 ␮ iH kT 冊 ±⌬i.共11兲 The hys e esis loop o he whole ensemble o pa icles can be exp essed as m±=兺 i piL 冉 ␮ iH kT 冊 ±pi⌬i=m±⌬.共12兲 When conside ing ha he way in which each pa icle e- sponds o he mean in e ac ion ield 共con olled by he cu o unc ion Fas well as by he mean in e ac ion ield兲is a cha ac e is ic o he sys em and i s mean magne ic momen , we a i e a Eq. 共10兲 o he exp ession o ⌬, which has been shown no o be able o ep oduce accu a ely he expe imen- al da a. Howe e , as he cu o unc ion de e mining he e- sponse o he pa icle depends on he magne ic momen , we should conside ha i is di e en o di e en pa icles. A na u al conclusion o his easoning is ha he ⌬共m兲cu e should be i ed by he sum o di e en con ibu ions weigh ed acco ding o he g ain size dis ibu ion: ⌬=兺 i pi⌬i,共13兲 wi h ⌬i= ␮ iH0 kT F共mi兲u⬘共mi兲.共14兲 No e ha he magne ic momen o each pa icle is used o each con ibu ion, in con as o he o iginal model, whe e he a e age magne ic momen was used. To check hese assump ions agains he expe imen al da a, we will ep esen he g ain size dis ibu ion by only wo componen s, in o de o minimize numbe o he i ing pa- ame e s and ye allow a easonable physical in e p e a ion o he esul s. Figu e 8共b兲p esen s he i o he expe imen al ⌬共m兲cu e o Eq. 共13兲using wo con ibu ions. The mag- ne ic momen s o bo h componen s we e ex ac ed om he i , and he consis ency o he p ocedu e has been checked by using hose alues o i he anhys e e ic magne iza ion cu e 共Fig. 9兲. I has o be no ed ha he linea con ibu ion used in he i ings o ep esen he signal no coming om he nano- pa icles does no in luence he shape o he ⌬共m兲cu e. The e o e, his i in ol es one ee pa ame e less han he Lange in i o he anhys e e ic cu e. The b anches o he hys e esis loop can be calcula ed om M±=N1 ␮ 1m±,1 +N2 ␮ 2m±,2,共15兲 wi h FIG. 8. Fi o ⌬共m兲o he Fe76C 4Mo4Cu1B15 sample p ehea ed up o 715 K and measu ed a 475 K, using one 共a兲o wo 共b兲 magne ic momen s. FRANCO e al. PHYSICAL REVIEW B 72, 174424 共2005兲 174424-6 m±,i=L 冉 ␮ iH kT ± ␮ iH0 kT F共mi兲 冊 .共16兲 Figu e 9 shows he expe imen al poin s oge he wi h he i o he loop, e idencing he much be e i o he hys e esis loops by he modi ied model wi h he use o a g ain size dis ibu ion 共compa ison should be made wi h he co e- sponding dashed loop o Fig. 6, as ha is he one ob ained om he p edic ed Fo a single a e age magne ic momen 兲. The emaining di e ences be ween he i ed and expe imen- al loops migh o igina e om he simpli ied ep esen a ion o he g ain size dis ibu ion, wi h only wo componen s. Howe e , magne ic elaxa ion can also ha e an in luence on hese di e ences,24,25 al hough he loops ha e been measu ed a a educed speed 共abou 150 min pe hys e esis loop兲and ield inc emen s om poin o poin a e ela i ely small, o y o minimize ha e ec . V. TEMPERATURE DEPENDENCE OF THE MAGNETIC MOMENT OF THE NANOPARTICLES To be able o check how plausible hese i ed magne ic momen s a e, we should i s conside ha he dipola in e - ac ions p o oke no only hys e esis bu also a dis o ion o he empe a u e e olu ion o he magne ic momen o he pa icles. In ac , as he mic os uc u e o he samples does no e ol e in he conside ed empe a u e ange, he obse ed inc ease o he magne ic momen as measu ing empe a u e inc eases canno be co ela ed wi h g ain g ow h 共Fig. 10兲. The e o e, we should conside ha he momen s ob ained om he Lange in i ings a e no he ac ual momen s bu he appa en ones. I has been shown ha his e ec can be ep esen ed by an e ec i e empe a u e T*which should be ela ed o he dipo- la in e ac ion be ween he pa icles.8Ini ially p oposed as an independen model wi h espec o ha o he hys e e ic be- ha io o he nanopa icles men ioned abo e, he e was a i s a emp o co ela e hese physically in e dependen beha io s,7which p oduced plausible esul s. In he ollow- ing, we will p esen a modi ica ion o his uni ied model, which sol es some o he p oblems o he p e ious e sion. Based on he ac ha he calcula ed alues o he sa u a- ion magne iza ion and he ini ial suscep ibili y o he sample should be independen o he ep esen a ion ha we make o he sys em 共using he appa en Nand ␮ oge he wi h he ac ual empe a u e, o he eal Nand ␮ wi h he help o an in e ac ion empe a u e兲, he ela ionship be ween he appa - en and eal pa ame e s is ␮ a= ␮ 1+共T*/T兲,共17兲 Na=N 冉 1+T* T 冊 .共18兲 The ela ionship be ween he wo in e ac ion models was based on he co ela ion be ween he he mal ene gy associ- a ed wi h he in e ac ion empe a u e and he magne ic en- e gy associa ed wi h he in e ac ion ield. Keeping his in mind, we should conside ha his la e ene gy has o be ela ed o he eal magne ic momen o he pa icles in he p esence o he in e ac ion ield, so we can se ␰ ␮ H0=kT*.共19兲 No e ha in he i s a emp a co ela ing bo h models, he appa en momen was used. Al hough his di e ence is minu e in he empe a u e egion close o he pu e supe pa a- magne ic egime, as was he case in he C -con aining Fineme alloy s udied in Re . 7, he disc epancies ge bigge FIG. 9. Lange in i ing 共uppe 兲o he anhys e e ic magne iza- ion cu e o he Fe76C 4Mo4Cu1B15 sample p ehea ed up o 715 K, using he same wo magne ic momen s ob ained om he i ing o Fig. 8共b兲. Lowe panel p esen s he co esponding educed hys e - esis loop. C osses, expe imen al da a; solid lines, i . FIG. 10. Tempe a u e dependence o he appa en 共open, igh axis兲and eal 共solid, le axis兲magne ic momen s o he Fe76C 4Mo4Cu1B15 samples p ehea ed up o 715 and 730 K. Lines a e a guide o he eyes. E o ba s o he appa en magne ic mo- men s a e compa able o he size o he symbols. RELATIONSHIP BETWEEN COERCIVITY AND …PHYSICAL REVIEW B 72, 174424 共2005兲 174424-7 o a b oade empe a u e ange, as in his case 共in he o igi- nal pape , howe e , a de ia ion om he expec ed linea be- ha io was al eady no ed in he egion close o he coe ci i y maximum兲. Taking in o accoun ha he momen o Eq. 共5兲is he appa en one 共as i is he one ex ac ed om he Lange in i 兲 and using Eq. 共17兲, he in e ac ion ield can be exp essed in e ms o he in e ac ion empe a u e and he eal magne ic momen : H0=3k共T+T*兲m ␮ .共20兲 Combining Eqs. 共18兲–共20兲, he ela ionship be ween he ap- pa en and eal densi y o pa icles is 1 Na =1 N共1−3 ␰ m 兲.共21兲 The e o e, he plo o he ecip ocal appa en densi y o pa - icles s he educed emanence o he loops should be a s aigh line om which he ac ual densi y o pa icles and he p opo ionali y coe icien ␰ can be ex ac ed 共Fig. 11兲.In he case o he 715 K sample, he pa ame e s co esponding o each o he magne ic con ibu ions ha e been plo ed, o- ge he wi h he a e age alue o he densi y. The pa ame e s ex ac ed om he linea i s a e shown in Table II. I is wo h men ioning ha all he linea i s p oduce he same alue o he p opo ionali y coe icien ␰ close o 1. This sugges s he accu acy o his app oach, as he meaning o Eq. 共19兲is ha bo h e ec s 共hys e esis and il o he anhys e e ic cu e兲a e a ep esen a ion o he dipola in e ac ion ene gy. Howe e , as he magne ic ene gy implies he a e age o he ac ual dis ibu ion o pa icles, i is easonable ha ␰ sligh ly depa s om uni y. The eal alues o he magne ic momen s 共 he a e age one in he case o he 715 K sample兲a e plo ed in Fig. 10. As expec ed, he magne ic momen 共g ain size兲inc eases wi h annealing empe a u e, while he magne ic momen de- c eases as measu ing empe a u e inc eases. Assuming ha he nanoc ys als a e o pu e Fe, he a e age magne ic mo- men s would co espond o mean g ain sizes o 14 and 16 nm, o he samples p ehea ed a 715 and 730 K, espec- i ely. In he i s case, he wo con ibu ions used o he i ings co espond o 13 and 17 nm. These esul s a e in ag eemen wi h a sa u a ion o he g ain size, which p oduces a na owing o he g ain size dis ibu ion o he highe an- nealing empe a u e. TEM obse a ions 共Fig. 12兲show ha in he sample p ehea ed up o 715 K g ains wi h signi ican ly di e en sizes coexis , whe eas o he sample p ehea ed up o 730 K he g ains a e mo e uni o m in size and he smalle ones a e no de ec ed. The maximum g ain size ound o he la e sample is o he o de o 30 nm. Howe e , he g ain FIG. 11. Tempe a u e dependence o he appa en densi y o pa icles o he Fe76C 4Mo4Cu1B15 samples p ehea ed up o 715 and 730 K. Lines a e linea i s o he da a. Fo he 715 K sample, he a e age densi y is plo ed oge he wi h hose o bo h con ibu ions o he magne ic momen . TABLE II. Real densi y o pa icles and p opo ionali y ac o in Eq. 共21兲 o he Fe76C 4Mo4Cu1B15 samples p ehea ed up o 715 and 730 K. In he i s case, he alues co esponding o he a e age magne ic momen as well as o each con ibu ion a e p esen ed. Ta=715 K Ta=730 K N共1/g兲共5.0±0.2兲⫻1014 共4.45±0.03兲⫻1015 N1共1/g兲共8.8±0.8兲⫻1013 N2共1/g兲共4.5±0.3兲⫻1014 ␰ 1.11±0.09 1.11±0.02 ␰ 11.1±0.2 ␰ 21.1±0.2 FIG. 12. TEM images o he samples p ehea ed up o 715 共a兲 and 735 共b兲K. FRANCO e al. PHYSICAL REVIEW B 72, 174424 共2005兲 174424-8 sizes ob ained by TEM a e bigge han hose calcula ed om he i ings o he magne iza ion cu es, which is usual in his kind o s udy.8,18 I is wo h men ioning ha al hough he 30 nm g ain size is conside ed as a limi diame e o single- domain Fe pa icles,26 he ac ual alue s ongly depends on he geome y.27 Mo eo e , he ele an size o magne ic measu emen s is he magne ic size, which is below ha limi . I pa icles we e mul idomain, he model would no be ap- plicable, bu also, coe ci i y should dec ease wi h inc easing g ain size, which is he opposi e beha io o he one p e- sen ed in Fig. 3, suppo ing he applicabili y o he model. As an in e nal check, once he in e ac ion pa ame e s a e ex ac ed om he i s and he eal magne ic momen s a e calcula ed, Eq. 共17兲can be used o p edic he appa en mag- ne ic momen s co esponding o hese samples. Figu e 13 shows he ema kable co ela ion be ween he appa en mo- men s ob ained om he Lange in i o he expe imen al da a and hose p edic ed by he model. VI. CONCLUSIONS The magne ic in e ac ions be ween magne ic nanopa - icles ha e been s udied in Nanope m- ype alloys measu ed a high empe a u es. As empe a u e is inc eased, hese al- loys show a ansi ion om a so magne ic beha io o pa - icles coupled by exchange o ha o supe pa amagne ic pa - icles wi h dipola in e ac ion, he ansi ion being ma ked by a maximum in coe ci i y. The composi ion o he s udied alloys was chosen in o de o shi he coe ci i y maximum o lowe empe a u es, hus ex ending he empe a u e ange whe e he in e ac ing supe pa amagne ic pa icles can be s udied. The mean- ield and e ec i e empe a u e models o Allia e al.5,8 could be adap ed o i he obse ed beha io . I has been con i med ha a empe a u es abo e he coe ci i y maximum, dipola in e ac ions be ween he supe pa amag- ne ic nanopa icles a e esponsible o bo h he hys e esis and he di e ences be ween he eal magne ic momen o he nanopa icles and hose appa en ones ob ained om he Lange in i o he magne iza ion cu es. While in he case o C -con aining Fineme - ype alloys he hys e esis loops could be i ed by using an a e age mag- ne ic momen , in he case o Nanope m- ype alloys a ea ly s ages o nanoc ys alliza ion, a dis ibu ion o pa icle sizes has o be aken in o accoun . I has been shown ha he mean in e ac ion ield is ela ed o he mean magne ic momen bu he esponse o he sys em canno be subs i u ed by ha o an a e age sys em: he esponse o he pa icles o he addi ional in e ac ion ield depends also on he magne ic cha ac e is ics o he pa icles. The ela ionship be ween magne ic in e ac ion ene gy and e ec i e he mal ene gy has been e ined o ake in o accoun he beha io o he sys em a empe a u es mo e dis an om he ully supe pa amagne ic egime. The consis ency o he nume ical p ocedu e, shown by he co ela ion be ween he appa en magne ic momen s ob ained om he Lange in i s o he anhys e e ic magne iza ion cu es and he appa en magne ic momen s p edic ed by he model, is inc eased wi h espec o he p e ious app oaches o he p oblem. ACKNOWLEDGMENTS This wo k was suppo ed by he Spanish Go e nmen and EU-FEDER 共P ojec No. MAT 2004-04618兲, he PAI o Jun a de Andalucía, he Hunga ian Resea ch Fund 共OTKA G an s No. T-038383 and No. T-046795兲and he Hispano- Hunga ian Bila e al Coope a ion P og am 共G an s No. HH2004-0015 and No. E-21/04兲. The au ho s a e indeb ed o D . T. Kemény o ui ul discussion. 1P. Ta aj, M. P. Mo ales, T. González-Ca eño, S. Vein emillas- Ve dague , and C. J. Se na, J. Magn. Magn. Ma e . 290-291,28 共2005兲. 2J. Do mann, D. Fio ani, and E. T onc, Ad . Chem. Phys. 98, 283 共1997兲. 3J. Bansmann, S. H. Bake , C. Binns, J. A. Blackman, J.-P. Buche , J. Do an es-Dá ila, V. Dupuis, L. Fa e, D. Kech akos, A. Kleibe , K.-H. Meiwes-B oe , G. M. Pas o , A. Pe ez, O. Toulemonde, K. N. T ohidou, J. Tuaillon, and Y. Xie, Su . Sci. Rep. 56, 189 共2005兲. 4M. Vopsa oiu and P. R. Bissell, J. Phys. D 35, 1296 共2002兲. 5P. Allia, M. Coisson, M. Knobel, P. Tibe o, and F. Vinai, Phys. Re . B 60, 12207 共1999兲. 6M. Knobel, W. C. Nunes, A. L. B andl, J. M. Va gas, L. M. Socolo sky, and D. Zanche , Physica B 354,80共2004兲. 7V. F anco, L. F. Kiss, T. Kemény, I. Vincze, C. F. Conde, and A. Conde, Phys. Re . B 66, 224418 共2002兲. 8P. Allia, M. Coisson, P. Tibe o, F. Vinai, M. Knobel, M. A. No- ak, and W. C. Nunes, Phys. Re . B 64, 144420 共2001兲. 9M. Bahiana, J. P. Pe ei a Nunes, D. Al bi , P. Va gas, and M. FIG. 13. Co ela ion be ween he appa en magne ic momen ob ained om he Lange in i s and he appa en magne ic momen p edic ed by he model. Line co esponds o slope 1. RELATIONSHIP BETWEEN COERCIVITY AND …PHYSICAL REVIEW B 72, 174424 共2005兲 174424-9