Dipolar Interaction Effects in the Magnetic and Magnetotransport Properties of Ordered Nanoparticle Arrays

Transcription

1 Copyrght 2008 Amercan Scentfc Publshers All rghts reserved Prnted n the Unted States of Amerca Journal of Nanoscence and Nanotechnology Vol.8, 1 15, 2008 Dpolar Interacton Effects n the Magnetc and Magnetotransport Propertes of Ordered Nanopartcle Arrays D. Kechrakos and K. N. Trohdou Insttute of Materals Scence NCSR Demokrtos, GR , Athens, Greece Assembles of magnetc nanopartcles exhbt nterestng physcal propertes arsng from the competton of ntrapartcle dynamcs and nterpartcle nteractons. In ordered arrays of magnetc nanopartcles magnetostatc nterpartcle nteractons ntroduce collectve dynamcs actng compettvely to random ansotropy. Basc understandng, characterzaton and control of dpolar nteracton effects n arrays of magnetc nanopartcles s an ssue of central mportance. To ths end, numercal smulaton technques offer an ndspensable tool. We report on Monte Carlo studes of the magnetc hysteress and spn-dependent transport n thn flms formed by ordered arrays of magnetc nanopartcles. Emphass s gven to the modfcatons of the sngle-partcle behavor due to nterpartcle dpolar nteractons as these arse n quanttes of expermental nterest, such as, the magnetzaton, the susceptblty and the magnetoresstance. We nvestgate the role of the structural parameters of an array (nterpartcle separaton, number of stacked monolayers) and the role of the nternal structure of the nanopartcles (sngle phase, core shell). Dpolar nteractons are responsble for ansotropc magnetc behavor between the n-plane and out-of-plane drectons of the sample, whch s reflected on the nvestgated magnetc propertes (magnetzaton, transverse susceptblty and magnetoresstance) and the parameters of the array (remanent magnetzaton, coercve feld, and blockng temperature). Our numercal results are compared to exstng measurements on self-assembled arrays of Fe-based and Co nanopartcles s made. Keywords: Magnetc Nanopartcles, Ordered Arrays, Self-Assembly, Ansotropy, Exchange Bas, Dpolar Interactons, Transverse Susceptblty, Tunnelng Magnetoresstance, Resstor Network, Monte Carlo. 1. INTRODUCTION Laterally confned magnetc nanostructures (dots and nanopartcles) s an mportant class of novel materals wth unque physcal propertes, that emerge because ther sze becomes comparable to varous characterstc physcal lengths (correlaton length, doman wall wdth, etc.). Owng to ther novel physcal propertes they fnd numerous technologcal applcatons n magnetc storage meda, 1 magnetc sensors 2 and magnetc logc devces. 3 Magnetc nanopartcles (NPs) are commonly formed n assembles, wth ether random or ordered structure.in the frst group belong systems such as ferrofluds and granular solds, whle n the second group belong the patterned meda (or magnetc dots) and the self-assembled arrays (SAA) of NPs.The exstence of order n a NP assembly s a decsve property n vew of ther applcaton n magnetc Author to whom correspondence should be addressed. storage meda wth ultrahgh densty ( T b/n 2 ), whch rely on the possblty of treatng the NPs as ndvdually addressable magnetc bts. Magnetc dots are developed by lthographc processes and they are characterzed by lateral dmensons n the range of nm. 4 The major advantage of ths approach to development of ordered nanostructured materals s that t offers great flexblty and good control over the shape, the sze and the arrangement of the dots as well as the choce of the consttuent materal.however, shape mperfectons and formaton of polycrystallne dots are the major factors that determne the devatons from perfect perodcty n the arrays.furthermore, the sze of the magnetc dots, determned by the lmtatons of the lthographc process s n most cases comparable to the exchange length, thus permttng the doman formaton n the ground state or durng magnetzaton reversal, a fact that makes the study of the magnetc behavor of dot arrays qute ntrcate.growth methods and magnetc propertes of J. Nanosc. Nanotechnol. 2008, Vol. 8, No /2008/8/001/015 do: /jnn

2 Magnetc and Magnetotransport Propertes of Ordered Nanopartcle Arrays Kechrakos and Trohdou solated dots and dot arrays have been recently revewed by Martn et al. 4 In magnetc NPs wth dameter D 1 10 nm, the doman wall wdth s well beyond the dameter of the NP, and consequently doman formaton at the ground state or durng magnetzaton reversal s precluded.the magnetzaton of the NPs s practcally at ts saturaton value up to temperatures less than but close to the bulk Cure temperature.they are commonly referred to as sngledoman (SD) partcles, to dstngush them from magnetc dots that exhbt nhomogeneous magnetzaton.hghly monodsperse ( 5%) SD magnetc NPs are produced by soluton chemstry methods and organzed n hexagonal ordered arrays by self-assembly. 5 8 The small sze, the hgh monodspersty and the array perodcty acheved by self-assembly, as well as the low producton cost of SAA, has motvated a great deal of research effort n the feld of synthess and magnetc characterzaton of these systems.the synthetc routes and the structural and magnetc characterzaton methods have been recently revewed by Wllard et al. 9 Assembles of magnetc nanopartcles have been nvestgated n an effort to gan basc understandng of the nterplay between sngle-partcle magnetc ansotropy and nterpartcle magnetostatc nteractons.to ths end, a varety of sample preparaton methods have been adopted by research groups n order to grow and magnetcally characterze nanopartcle assembles, as for example, frozen ferrofluds, dscontnuous metal-nsulator multlayers, co-sputtered metal-nsulator flms, cluster-assembled flms self-organzed partcle arrays on surfaces 22 and chemcally produced self-assembled nanopartcle arrays. Investgatons of the statc and dynamc magnetc propertes of dpolar nteractng nanopartcle assembles brought up fundamental ssues related to the exstence of a ground state whch shares common features wth canoncal spn glasses (slow relaxaton, memory and ageng effects). Dpolar nter partcle nteractons are consdered responsble for the observed complex (spn-glass-lke) behavor of suffcently dense and random nanopartcle assembles.in vew of the technologcal mportance of the hysteress behavor and the thermal stablty of magnetzaton of NP assembles, 28 the mportant ssue of the effects of magnetostatc nteractons on the statc magnetc propertes has also been studed extensvely. Contrary to random assembles, ordered arrays of NPs are deal systems to nvestgate the role of nterpartcle nteractons, for two reasons.frst, the NP arrangement s perodc wth small perturbatons, and the undesred complcatons ntroduced by spatal randomness are substantally suppressed.second, chemcally syntheszed magnetc NPs are often coated by an norganc surfactant layer that prevents agglomeraton durng self-assemblng but also keeps the surfaces of neghborng partcles at a dstance well beyond the range of exchange forces.as a consequence, the prevalng nterpartcle nteractons n a SAA are magnetostatc.fnally, the sphercal, n most cases, shape of the NPs dmnshes the mportance of hgher order multpolar nteractons and the assembly s well descrbed by dpolar nterpartcle forces. Varous experments have demonstrated the presence of magnetostatc nteractons n SAA wth varous degrees of structural dsorder and layered NP assembles.reducton of the remanence at low temperature, 29 ncrease of the blockng temperature, ncrease of the barrer dstrbuton wdth, 32 devatons of the zero-feld cooled magnetzaton curves from the Cure behavor, 8 33 dfference between the n-plane and out-of-plane remanence, 34 and ncrease of the blockng temperature wth frequency of appled feld 35 have been observed and attrbuted to nterpartcle dpole dpole nteractons (DDI).Long-range ferromagnetc order n lnear chans and hexagonal arrays of dpolar coupled sngle-doman mag netc nanopartcles has been demonstrated, supportng the exstence of a dpolar superferromagnetc ground state. Nanopartcle assembles wth random morphology have been studed more as most growth technques developed so far (sputterng, cluster beams, mechancal alloyng) produce random samples.ordered nanopartcle arrays, on the other hand, have been less studed both theoretcally and expermentally due to the dffculty n producng ordered samples.chemcal synthess and self-assembly offer a new and promsng approach to ths drecton. 38 We therefore beleve that basc understandng of the magnetc propertes of dense (nteractng) ordered arrays s currently hghly demanded. In the ongong research effort for development of magnetc nanostructures wth reduced sze and mproved thermal stablty, 28 the explotaton of the exchange bas effect n laterally confned structures (dots and nanopartcles) has attracted a lot of nterest. 39 Atomc scale models of the magnetc structure have been developed n an effort to nterpret expermental observatons of the exchange bas effect n composte NPs wth a ferromagnetc (FM) core and an antferromagnetc (AFM) shell.among the most mportant theoretcal results 40 we menton () the dsappearance of the exchange bas feld (H E )at temperatures above the Néel temperature of the AFM, n agreement wth experments, 41 () the strong dependence of H E on the number unsaturated bonds across the FM-AFM nterface and the dependence of H C on the nterface area, () the ncrease of both H E and H C for a gven NP radus wth ncreasng oxdaton depth, (v) the ncrease of H E and decrease of H C wth ncreasng oxdaton layer thckness and a fxed core radus, (v) the fast stablzaton of H E wth ncreasng core sze, n agreement wth experments, 42 and 2 J. Nanosc. Nanotechnol. 8, 1 15, 2008

3 Kechrakos and Trohdou Magnetc and Magnetotransport Propertes of Ordered Nanopartcle Arrays (v) the reducton of H C and ncrease of H E and ts thermal stablty wth ncreasng exchange constant of the AFM materal and/or at the FM-AFM nterface. Despte the research effort focused on the atomc scale mechansm of magnetzaton reversal n composte nanopartcles, much less attenton has been pad so far to the modfcaton of the magnetc hysteress behavor due to nter-partcle nteractons arsng n assembles.in ths drecton, Fe NPs embedded n ron-oxde matrx 46 were shown to freeze below a temperature owng to the competton between the exchange ansotropy at the core shell nterface and the nterpartcle DDI.Smlarly, ncrease of the exchange bas feld due to magnetostatc nterpartcle couplng was found n strpes of Co/CoO nanopartcles 47 and nter-dot magnetostatc nteractons were shown to produce asymmetrc anomales n the magnetzaton reversal mechansm of Co/CoO dot arrays. 48 The modfcaton of the coercve and exchange-bas felds n dense nanopartcle arrays wth core shell morphology as a result of the competton between exchange ansotropy and nterpartcle dpolar nteractons conssts a challengng ssue. Detecton and quantfcaton of DDI n assembles of magnetc NPs has been addressed so far by a varety of expermental technques ncludng n most cases SQUID magnetometry and AC susceptblty measurements, 10 and more recently small-angle neutron scatterng (SANS) 49 and resonant magnetc X-ray scatterng. 50 The last method s a drect probes of magnetc correlatons at the nterpartcle scale.these studes have provded ample evdence that the nterplay between random ansotropy and DDI determne the magnetc behavor of the NP assembles.more recently dpolar nteracton effects and the resultng collectve dynamcs n SAA of Fe, 51 Fe 2 O 3, 52 and Co NPs was studed by reversble transverse susceptblty (RTS) measurements.the RTS technque s a well establshed and powerful method to obtan nformaton about the ansotropy of magnetc nanopartcles, from consderatons of the peak postons of the felddependent RTS. 55 The mplementaton of RTS to study SAA arrays revealed nformaton regardng the dfferent dynamcal regmes of an nteractng assembly accessed as the temperature ncreases. Electron spn s a degree of freedom whose control and detecton n transport measurements s the bass of the rapdly developng feld of spntroncs. 56 Charge transport measurements n a SAA of Co NPs were performed 57 and revealed a spn-dependent tunnelng mechansm whch s responsble for substantal ( 10%) tunnelng magnetoresstance (TMR) values at low temperature ( 20 K).The tunnelng barrers are provded n an array by the nsulatng surfactant layer surroundng the NPs.TMR measurements probe the nterpartcle correlatons wthn the range of the spn-dffuson length, and are therefore senstve to the magnetc mcrostructure of the assembly.interpartcle nteracton effects are expected to reveal themselves n the TMR sgnal.the feld-dependent magnetzaton and conductvty were dscussed n the experments of Black et al., 57 however a systematc correlaton between the two quanttes remans to be performed. In addton to the expermental work, varous numercal studes that focused on the ground state confguraton and the hysteress behavor of dpolar nteractng nanopartcle arrays have appeared.the nterplay of DDI and perpendcular ansotropy was shown 58 to nduce a reorentaton transton below a crtcal temperature and nteracton-nduced shape ansotropy of a fnte sample controls the magnetzaton reversal mode.dpolar nteractons were found to decrease the coercve feld of magnetc nanopartcle arrays ndependently of the array topology (square or hexagonal) despte the fact that the ground state confguraton s determned by the array topology. 59 The presence of an ncomplete second layer wth hexagonal structure does not destroy the long-range FM orderng of the ground state, 60 whle even slght structural dsorder wthn the array destroys that orderng. 61 On the other hand, hgher order (quadropolar) magnetostatc nteractons were shown to act n synergy wth DDI stablzng the long range order of the ground state n a nanopartcle array. 62 Prevous theoretcal studes of RTS n random assembles of magnetc NPs demonstrated that a wde sze dstrbuton rounds the peaks of RTS, 63 orentatonal texture suppresses the coercvty peak 63 and dpolar nteractons lead to merge of the coercvty and ansotropy peaks More recently the ssue of the structure of the RTS curves of SAA was addressed by Monte Carlo (MC) smulatons 66 that reproduced many of the expermental observatons from RTS measurements n SAA of Fe NPs. 51 Charge transport n nanopartcle arrays has been studed by resstor network models (RN) that nclude n a phenomenologcal way the essental aspects of the thermally-actvated hoppng mechansm, 67 the spn-dependence of the hoppng proccess 68 and the detals of the mcromagnetc confguraton of the sample. 69 In a recent study 69 the sgnature of dpolar nteracton effects n TMR measurements has been nvestgated. In ths artcle we revew our results from MC smulatons of the feld and temperature dependence of the magnetzaton, the RTS and the TMR of hexagonal arrays of dpolar coupled magnetc nanopartcles wth random ansotropy.the consderaton of a hexagonal arrangement of NP s an essental feature of our model as DDI have a well known ansotropc character that relates ther magntude and sgn to the relatve poston of the nteractng dpoles.the man structural parameters we focus on are (a) the nterpartcle separaton, whch s drectly related to the dpolar couplng strength and can be expermentally controlled by varaton of the surfactant layer durng the synthetc process, and (b) the sample thckness, namely the number of stacked monolayers (MLs), whch s a crucal parameter for the collectve response of the array, controlled by the NP concentraton n the collodal dsperson. J. Nanosc. Nanotechnol. 8, 1 15,

4 Magnetc and Magnetotransport Propertes of Ordered Nanopartcle Arrays Kechrakos and Trohdou The am of the present work s to reveal the modfcaton ntroduced to expermentally measured propertes of an ordered NP assembly (magnetzaton, susceptblty, magnetoresstance) due to the presence of DDI.The remanng of the paper s organzed as follows: In Secton 2 we present the structural and magnetc model used n our smulatons.in Secton 3 we dscuss numercal results on the hysteress characterstcs (saturaton remanence, coercvty), the zero feld cooled (ZFC) and feld cooled (FC) curves and and the extracted blockng temperature (T b ). The magnetzaton of the nterestng class of composte nanopartcles wth a core shell morphology s also dscussed.the evoluton of the RTS curves wth temperature and dpolar strength s dscusses next, and fnally magnetoresstance calculatons are presented.whenever expermental results are avalable they are compared wth our smulatons amng to reveal the character of nterpartcle nteractons n dfferent measured samples.fnal conclusons and remarks are gven n the Secton MODEL AND SIMULATION METHOD We proceed wth the defnton of the spn model used to descrbe the magnetc structure of a magnetc NP array formed by ether smple ferromagnetc or composte (FM core/afm shell) nanopartcles.the NPs formng an ordered array are assumed sphercal and monodsperse.the sze dsperson s not expected to ntroduce major modfcatons to the magnetc behavor because partcularly low values are acheved ( 5%) n most samples. The NP dameter s D and they occupy the stes of a trangular lattce n the xy-plane wth lattce constant d D.When more that one MLs are consdered, the partcles are close-packed n an ABC ABC... stackng sequence.ths structure s consstent wth electron mcroscopy studes of Co and Fe 70 NP arrays. Incomplete layer of NPs are formed by random occupancy of the trangular lattce stes.the NPs are sngle doman wth unaxal ansotropy n a random drecton, and they nteract va dpolar forces.the total energy of the system s E = g j S S j 3 S R j S R j k S ê 2 h S H (1) where S s the magnetc moment drecton (spn) of the -th partcle, ê s the easy-axs drecton, and R j s the center-to-center dstance between partcles and j.hats n Eq.(1), and further on, ndcate unt vectors.the energy parameters enterng Eq.(1) are the dpolar energy g = m 2 /d 3, where m = M s V s the partcle moment, the ansotropy energy k = K 1 V, and the Zeeman energy h = mh due to the appled dc feld H.The energy parameters (g k h) enterng Eq.(1), the thermal energy t = k B T, R 3 j and the hstory of the sample determne the mcromagnetc confguraton at a certan temperature and bas feld. Because, our smulaton method reles on mnmzaton of the free energy of the system, multplcaton of all the energy parameters by the same scalng factor does not modfy the results.thus, n all subsequent results we scale the energy parameters enterng Eq.(1) by the sngle partcle ansotropy energy (k = 1).Ths choce makes our numercal results applcable to a class of materals wth the same parameter ratos rather than to a specfc materal.the crucal parameter that determnes the transton from sngle-partcle to collectve behavor s the rato of the dpolar to the ansotropy energy g/k = /6 Ms 2/K 1 D/d 3.The reported values for fcc or hcp Co NPs are g/k = D/d 3, whle for the soft -Co phase, hgher values are expected. 8 ForFeNPs Farrell et al. 25 report g/k = 1 54 D/d 3.Despte the relatve dsperson of the reported values, the mportant ssue s that for most samples of Co and Fe NPs the rato of dpolar to ansotropy strengths s below unty, except for -Co.Thus, n the numercal results presented n the followng secton we consder g/k values less than one. Extensons to the spn model descrbed by Eq.(1) are requred n order to study composte NPs wth a FM core and an AFM shell.in the present work we adopt a model ntroduced by Meklejohn and Bean 71 (further on referred to as the MB model) n ther nterpretaton of shfted loops observed n oxdzed transton metal NPs samples after zero-feld coolng.the MB model provdes a phenomenologcal understandng of the exchange bas effect and the undrectonal ansotropy. 72 Consequently, mportant parameters of the exchange-bas effect such as the nterface structure and nterface magnetzaton are averaged out.despte ts smplcty, the MB model and ts varatons was successfully mplemented n the case of FM/AFM blayers to nterpret the dependence of the exchange bas feld on temperature, 73 on the thckness the AFM layer 74 and on the drecton of the appled feld. 75 The major weaknesses of the MB model beng the overestmaton of the exchange-bas feld values, 72 and the underestmaton of the coercvty values. 76 We adopt the MB model as the smplest possble approach to brng out the essental aspects of the competton between ntra-partcle (unaxal ansotropy) and nterpartcle (dpolar) nteractons.our purpose, s to nvestgate ths nterplay, whch s expected to be mportant n dense samples, rather than revealng the atomc scale mechansm whch s responsble for the exchange bas effect. Accordng to the MB model, coherent rotaton of the atomc spns s assumed n the FM core and the AFM shell, whle the net magnetc moment of the shell s vanshngly small.in addton, the nterface of the AFM s assumed fully uncompensated, namely all spns belong to the same sublattce and s exchanged coupled to the core.consequently, the magnetc state of each NP s descrbed by 4 J. Nanosc. Nanotechnol. 8, 1 15, 2008

5 Kechrakos and Trohdou Magnetc and Magnetotransport Propertes of Ordered Nanopartcle Arrays a par of ansotropc and exchange coupled spns, S FM and S AF M.For a dpolar nteractng assembly the total energy reads S FM S FM j 3 S FM R j S FM j R j E = g j R 3 j k C S FM ê 2 k S S AF M ê 2 J j S FM S AF M h S FM H (2) where S FM and S AF M ndcate the magnetzaton drectons (spns) of the core and the nterface layer of the shell, respectvely. J s the nterface exchange energy and k C, k S are the dstnct values of the core and shell ansotropy, respectvely.notce that owng to the zero net magnetzaton of the AFM shell, only the FM cores couple to the external feld and between them va magnetostatc forces. For smplcty we have assumed n Eq.(2) that the core and shell magnetzatons have a common easy axs, that s therefore labeled by the partcle ndex.due to the vanshng net magnetzaton of the shell there s nether Zeeman nor dpolar contrbutons to the total energy due to the shell.however, the couplng across the FM-AFM nterface makes an exchange contrbuton to the total energy, expressed by the fourth term n Eq.(2). For ether smple or composte NP assembles, the magnetc confguraton s obtaned by a MC smulaton, usng the standard Metropols algorthm. 77 The ntal spn confguraton corresponds to the saturaton state along a chosen axs (x or z).expermentally, to observe a shfted loop a feld-coolng process s performed pror to the hysteress measurement n order to algn the AFM moments parallel to the moments of the FM Furthermore, the value of the observed exchange bas feld.e., the loop shft) ncreases wth the value of the coolng feld (H FC ) The choce of the saturaton state as the ntal state to calculate the hysteress loop of core shell NPs from the fully saturated state, t s equvalent to assumng an nfntely strong coolng feld (H FC J AF M ).Thus, the maxmum value of H E s obtaned.durng relaxaton, the ntal 10 3 MC steps per spn (MCS) are used for relaxaton of the system towards equlbrum and thermal averages are calculated over the subsequent 10 4 MCS, allowng 10 MCS between samplng events to acheve statstcal ndependence.the results are averaged over N c = samples wth dfferent realzatons of the random axes dstrbuton and the thermal fluctuatons.to deal wth the long-range character of the DDI we use perodc boundares n the xy-plane and mplemented the Ewald summaton method adapted to a quas-two-dmensonal system Free boundares along the z-axs are assumed. Transverse susceptblty measurements are performed wth an ac measurng feld (H ac ) perpendcular to the dc bas feld (see, Fg.1).The ac feld s weak ( 10 Oe) and Fg. 1. Sketch of the sample and appled magnetc feld used n our smulatons.(a) In-plane and (b) out-of-plane drectons of the bas feld (H dc ).The ndcated (H ac ) feld s consdered only n the calculaton of the transverse susceptblty.for the magnetoresstance calculatons a dc bas voltage s appled on opposte edges of the sample along the y-axs. ts frequency les n the rf regme (f 10 6 Hz) The weak measurng feld permts us to neglect transverse hysteress effects and calculate the RTS n the zero-feld lmt. Furthermore, snce that the Néel relaxaton tme of the NP magnetzaton s large compared to the nverse frequency of the ac feld the statc approxmaton for the measurng feld s justfed.susceptblty values are obtaned from the fluctuatons of the magnetzaton M y = S y,as 82 T H x z = 1 Nk B T M 2 y M y 2 (3) where T s the n-plane (out-of-plane) RTS and N s the number of NPs n the smulaton cell. In the last part of ths secton we descrbe the resstor network model used to calculate the magnetoresstance of the array.for a gven mcromagnetc confguraton S we ntroduce the spn-dependent conductvty between two nanopartcles and j as, 68 j = P 2 cos j exp R j / E c /k B T (4) where 0 = 2e 2 /h s the conductvty quantum, P s the spn polarzaton of the conducton electrons, cos j = S S j, E c = e 2 /2C s the actvaton energy to charge a neutral NP by addton of a sngle electron, C s the NP capactance relatve to ts surroundng medum, and = / 8m U E F s the electron wave functon decay length n the nsulatng barrer of heght U relatve to the Ferm energy.in all our smulatons we take = d, as a suffcent requrement to allow charge transfer between neghborng nanopartcles and P = 0 34 whch s an approprate for Co NPs Charge conservaton on every node of the network mples j j = 0 (5) j where s the local value of the electrc potental.equaton (5) s solved for the local potentals wth the boundary condtons that set the local potental on opposte sdes of the sample, namely at y = 0 and y = L, (see Fg.1) equal to zero and 0, respectvely.the total conductvty s gven as = j j 2 = 0 (6) 0 j J. Nanosc. Nanotechnol. 8, 1 15,

6 Magnetc and Magnetotransport Propertes of Ordered Nanopartcle Arrays Kechrakos and Trohdou Obvously the values of depend on the spn confguraton S.A thermal average of the conductvty s obtaned by averagng the conductvty values, as obtaned from Eq.(6) over a sequence of equlbrum spn confguratons produced by the MC algorthm.fnally, the tunnelng magnetoresstance of the sample s defned as (a) TMR H = sat H H (7) where sat s the saturaton value of the conductvty.it follows from Eq.(4) that the local conductvty between partcles and j ncreases quadratcally wth spn polarzaton (P) and exponentally wth localzaton length ( ) and partcle capactance (C), the latter dependng on the NP dameter. 57 The TMR values are expected to have a smlar dependence on the parameters and P.However, for a monodsperse sample the TMR values are ndependent of C.Snce we are manly nterested here s the shape of the feld-dependent TMR curves, rather than the actual values of TMR, the dependence of the TMR on and P s not consdered further on. h c (b) 3. NUMERICAL RESULTS AND DISCUSSION 3.1. Isothermal Hysteress and ZFC/FC Magnetzaton of FM Nanopartcles Two mportant characterstcs of the hysteress loop are the saturaton remanence (M r ) and the coercvty (H c ).The former s the magnetzaton of the system after removal of a saturatng feld, and the latter the negatve feld requred to zero the magnetzaton of a system beng n the postve remanent state before applcaton of the feld.in Fgure 2 we show the dependence of M r and H c on the strength of the dpole dpole nteracton.varaton of the dpolar strength n a SAA can be acheved expermentally n two dfferent ways, namely, by varyng the thckness of the surfactant layer, 84 or by synthess of dfferent materal NPs wth the same sze and the same surfactant layer thckness, or combnaton of both.in Fgure 2 we show that dependng on the drecton of the appled feld these quanttes show dfferent varaton wth dpolar strength.ths behavor can be explaned by the ferromagnetc and ansotropc character of the DDI.It s a well establshed fact that DDI on a trangular lattce stablze a FM ground state and create an easy-plane for the array magnetzaton due to ther ansotropc character. Therefore, the n-plane remanence tends towards the saturaton value (M r /M s = 1) as the dpolar strength ncreases, whle the out-of-plane remanence decreases contnuously because the out-of-plane feld s normal to the easy-plane. On the other hand, the coercvty decreases wth ncreasng dpolar strength, ndependently of the appled feld drecton.ths behavor can be understood as due to collectve rotaton of the moments.for an n-plane feld, the Fg. 2. Dependence of (a) saturaton remanence and (b) coercvty on the dpolar strength at low temperature (t/k = 0 01) for a monolayer of FM nanopartcles.closed symbols: n-plane feld.open symbols: out-ofplane feld. effectve ansotropy of a cluster of dpolar coupled NPs s reduced (as a result of an averagng process over many random easy drectons) and the total moment of a cluster s larger than a sngle NP.Due to the synergy of these two factors a weaker reversal feld s requred.for an out-ofplane feld, the development of an easy-plane forces the moments to le n the xy-plane reducng ther projecton along the feld axs, thus a weak feld s requred to fully zero the magnetzaton.ansotropy between the n-plane and out-of-plane remanence (Mr >M r ) has been observed n arrays of Co NPs, 34 where a rato of Mr /M r = 0 30 was found.for these samples the reported 34 dpolar strength s g/k = 0 07 and from the data shown n Fgure 2 we obtan g = 0 07 = 0 3 n good agreement wth the experments.a zero-temperature calculaton 34 on a trangular lattce gave a smlar value for.in the same experments 34 a neglgbly small dependence of the coercvty on the appled feld drecton was found.our smulatons agree wth ths feature showng, that for dpolar strengths n the range g/k = 0 0.1, H C and H C practcally concde (Fg.1).Our MC smulatons and the energy mnmzaton approach 34 justfy the domnant role of DDI n the magnetc propertes of these arrays. An mportant structural parameter n SAA s the number of stacked MLs.Ths parameter can be partally controlled ether by varyng the solvent concentraton 6 J. Nanosc. Nanotechnol. 8, 1 15, 2008

7 Kechrakos and Trohdou Magnetc and Magnetotransport Propertes of Ordered Nanopartcle Arrays h c M r /M s Fg. 3. Dependence of saturaton remanence (upper panel) and coercvty (lower panel) on number of stacked monolayers of FM nanopartcles, at low temperature (t/k = 0 01).Non-nteger values of flm thckness correspond to a randomly occuped uppermost layer.trangles: weak couplng (g/k = 0 25).Crcles: strong couplng (g/k = 10).The appled feld les n-plane.data are taken from Ref.[86]. pror to self-assembly or by formng the arrays usng the Langmur-Blodgett technque. To address the effects of flm thckness on the hysteress propertes of NP arrays we show n Fgure 3 the varaton of the remanence and coercvty wth the number of MLs.For strong dpolar couplng the remanence takes maxmum values when complete monolayers form, whle an ncomplete top layer suppresses the magnetc order due to the competng character of the DDI n a random nanopartcle assembly. 88 Peaks n H c are observed at full coverage, whch ndcate that fully ordered samples are magnetcally harder relatve to samples wth dsordered surfaces.the observed decay of H c wth ncreasng layer thckness marks a transton from a two-dmensonal reversal mode (c = 1 ML), durng whch the moments are forced by the nteracton-nduced easyplane ansotropy to reman n-plane durng reversal, to a three-dmensonal mode (c 2 ML), durng whch the rotaton path of the moments s not restrcted n the plane of the flm.when DDI are weak, t s shown n Fgure 3 that the oscllatory dependence of M r s suppressed and the peaks n H c are washed out.however, the decay of H c wth ncreasng thckness remans, reachng a constant value above c 2 MLs.The ncrease of M r wth coverage, observed for submonolayer coverage n Fgure 3 s n agreement wth experments on dlute samples of Fe NPs 25 wth varable concentratons.also SAA of Fe NPs always showed hgher M r values compared to dlute (dsordered) samples. 25 Ths trend s reproduced by our smulaton results n Fgure 3. As temperature ncreases thermal fluctuatons can assst the magnetc moments to overcome the ansotropy barrer leadng the system to thermal equlbrum.the regme n whch ths s acheved s defned by the blockng temperature of the system.for dpolar nteractng NPs the concept of a sngle-partcle barrer becomes rather vague as the thermal actvaton of a moment s correlated to the moton of all the other moments of the system.however, one can stll refer to the blockng temperature of the system n a phenomenologcal way, namely one can defne t as the temperature above whch the remanence and coercvty vansh.we show n Fgure 4 the temperature dependence of the n-plane and out-of-plane M r and H c for a monolayer of weakly coupled NPs.We frst notce that the effect of nteractons s to ncrease the blockng temperature of the array (t b 0 17) relatve to the non-nteractng case (t b 0 14).Ths s clear n Fgure 4 for the n-plane feld. Most nterestngly, at temperatures above tb 0, M r and H c of the nteractng system are non zero. Ths result defnes an nterestng temperature regme, tb 0 t t b, n whch the thermal energy overwhelms the nomnal ansotropy barrer (E b k), but the hysteress h c Fg. 4. Temperature dependence of (a) saturaton remanence and (b) coercvty of a monolayer of weakly dpolar (g = 0 1) FM nanopartcles.closed symbols: In-plane feld.open symbols: Out-of-plane feld.dashed lne: Non-nteractng nanopartcles.data are taken from Ref.[66]. (a) (b) J. Nanosc. Nanotechnol. 8, 1 15,

8 Magnetc and Magnetotransport Propertes of Ordered Nanopartcle Arrays Kechrakos and Trohdou behavor of the NP assembly perssts due to DDI. Calculatons of the energy barrer dstrbuton n dpolar nteractng NP assembles showed that, DDI broaden the dstrbuton thus producng a range of hgh barrers (E b >k), whch are responsble for the observed enhanced thermal stablty n dense assembles. 89 Notce fnally, that the enhanced thermal stablty s less pronounced, but also exstent n the out-of-plane geometry, despte the fact that n ths geometry dpole nteractons have a demagnetzng effect at low temperature (Fg.2).A smlar enhancement of M r and H c at elevated temperatures was prevously predcted also for NP assembles wth random morphology 88 and ts exstence was verfed expermentally from the measured ncrease of the blockng temperature wth NP densty n frozen ferrofluds. 10 Further nsght nto the collectve behavor at elevated temperatures can be extracted from examnaton of the short-range moment correlaton functon, defned as SRCF = S S j Rj=d S 2 (8) Strctly speakng, the blockng temperature of a system corresponds to the maxmum value of the long range moment correlaton functon, namely the susceptblty. We prefer to examne the short-range correlaton functon because t s drectly accessble by varous spectroscopc experments to be mentoned below.the poston of the SRCF peak of gves a reasonable approxmaton to the blockng temperature of the system.the temperature dependence of the SRCF, shown n Fgure 5, shows that short range FM correlatons exst above the blockng temperature of the non-nteractng array (tb ) and persst up to temperatures above the blockng temperature of the nteractng array (t b 0 17).Recently, Kortrght et al.50 extracted the nterpartcle magnetc correlatons n dense arrays of Co NPs from X-ray scatterng experments.they concluded that for -Co NPs, whch are strongly dpolar, AFM correlatons exst at temperatures above the blockng. SANS studes of Fe NP assembles showed no evdence of AFM correlatons at elevated temperature. 90 Fnally, our smulatons ndcate the exstence of short-range FM nterpartcle correlatons n dpolar coupled arrays.there seems to be dffcult at present to reach a decsve concluson about the nature of nterpartcle correlatons n NP arrays from expermental fndngs.scatterng experments (X-rays, SANS) at lower Q-values, are expected to probe magnetc correlatons at a scale lyng well wthn the nterpartcle separaton regme are most probably requred n order to compare wth our present smulatons. An nterestng feature shown n Fgure 5 s the dfferent temperatures at whch the peaks of SRCF are located.the relatve shft of the peaks ndcates that the array of dpolar nteractng NPs exhbts ansotropc blockng behavor wth T b >T b. The ansotropc blockng can also be demonstrated by examnaton of the ZFC/FC curves shown n Fgure 6.The exstence of ansotropc blockng for dpolar nteractng arrays of Fe NPs has been recently demonstrated expermentally n dense arrays of Fe NPs. 51 In these experments a rato T b /T b 1 15 was obtaned from ZFC/FC measurements.the Fe NP parameters gven 51 are M s = 1360 emu/cc, K = erg/cc, D = 6 8 nmandd 20 nm, whch lead to g/k 0 11.Our smulatons shown n Fgure 6 for g/k = 0 1 gvet b /t b 1 08, whch s n reasonable agreement wth the experments, gven varous factors not consdered n our model, such as the sample thckness, the devatons from perfect stackng of the monolayers and the n-plane structural defects. Changng the structural parameters of a self-assembled array, namely nterpartcle spacng and flm thckness s expected to affect the blockng behavor.the blockng temperature s proportonal to the effectve barrer heght whch s determned by the sngle-partcle ansotropy and the local dpolar feld.dpolar nteractons on the other hand are senstve to nterpartcle spacng (g 1/d 3 ).Decreasng the nterpartcle spacng s equvalent Fg. 5. Temperature dependence of the frst nearest-neghbor correlaton functon at the remanent state for a monolayer of weakly dpolar (g = 0 1) FM nanopartcles.closed symbols: In-plane feld.open symbols: out-of-plane feld.dashed lne: non-nteractng assembly. Fg. 6. ZFC/FC magnetzaton (at h/k = 0 1) for a monolayer of weakly nteractng (g = 0 1) FM nanopartcles.closed symbols: n-plane feld. Open symbols: out-of-plane feld.dashed lnes ndcate the dfferent values of the n-plane and out-of-plane blockng temperatures. 8 J. Nanosc. Nanotechnol. 8, 1 15, 2008

9 Kechrakos and Trohdou Magnetc and Magnetotransport Propertes of Ordered Nanopartcle Arrays to ncreasng the dpolar strength, and consequently modfyng the blockng temperature.smlarly, ncreasng the flm thckness we shown above that t suppresses at lowtemperature the values of M r and H c.in Fgure 7 we show the effect the structural changes have on the blockng temperature of the assembly, the latter beng obtaned from the peak of the correspondng ZFC curve.we see that n the range of dpolar strengths consdered here, T b scales lnearly wth the dpolar strength, or equvalently, t decreases wth the cube of the nterpartcle separaton (T b 1/d 3 ).We can therefore thnk of the effect of weak dpole nteractons (g/k <1) as an ncrease of the snglepartcle ansotropy barrer by an amount proportonal to ther couplng strength. A smlar effect on T b to decreasng the nterpartcle separaton s obtaned by ncreasng the areal coverage and the flm thckness.as shown n Fgure 7 a dramatc ncrease of T b s observed durng formaton of the frst monolayer and a saturaton behavor s reached as soon as the second complete monolayer s formed.reducton of T b upon dluton of chemcally syntheszed assembles has been reported by several groups In partcular, Zhang et al. 31 performed ZFC/FC measurements on self-assembled -Co NP and report 30% ncrease of T b relatve to hghly dlute samples.the NP parameters for these samples 31 correspond to g/k 0 2, thus the predctons of our smulatons (Fg.7) are n agreement wth these experments.smlar dependence of T b on NP concentraton has been also observed T b /T b 0 T b /T b 0 (a) (b) Fg. 7. (a) Dependence of blockng temperature (a) on dpolar strength, for a monolayer of FM nanopartcles, and (b) on flm thckness, for FM nanopartcles wth g/k = 0 25.Measurng n-plane feld h/k = 0 1.The values of T b are normalzed to the value correspondng to non-nteractng nanopartcles.data are taken from Ref.[60]. for NPs dspersed n a sold matrx 10 and reproduced by smulatons for 3D dsordered assembles of magnetc NPs. It s nterestng that despte the demagnetzng character of the dpolar nteracton n the ground state of 3D random assembles, they tend to stablze the FM character of the assembly at elevated temperatures (T < Tb 0).The ncrease of T b wth number of stacked monolayers was observed n dscontnuous Co-Al 2 O 3 multlayers. 18 However, the slow saturaton of T b obtaned after 5 7 monolayers n these experments s probably due to devatons from the deal stackng sequence and the n-plane randomness n NP sze and locaton, nherent to the sample preparaton technque.we also beleve that the lack of perfect stackng s probably the reason that measurements of T b n Langmur-Blodgett flms of Co NPs showed a decrease of T b wth ncreasng thckness Isothermal Hysteress of FM/AFM Nanopartcles We dscuss next the hysteress behavor of a 2D hexagonal array of magnetc NPs wth core shell morphology.to calculate the hysteress loop, the spns are set ntally n the saturaton state along the postve x-axs 93 and the feld s swept from postve to negatve values and back.ths choce produces a negatve shft of the hysteress loop, namely a postve exchange bas feld.the man ssue we address s the dependence of the remanence, the (effectve) coercvty and the exchange bas feld on the strength of the DDI.The total energy of the NP assembly s descrbed by Eq.(2).The coercvty for the shfted loops s defned as H C = 1/2 H C1 H C2, and the exchange bas feld as H E = 1/2 H C1 + H C2, where H C1 H C2 s the upper (lower) branch coercvty correspondng to the backward (forward), wth respect to the exchange bas feld drecton, magnetzaton reversal process. In the MB model the mcroscopc detals (atomc structure, defects, exchange couplng strength) are absorbed nto the value of the exchange constant J.Thus results for dfferent values of J are consdered.the ansotropy of the AFM oxde s assumed much hgher than the core ansotropy, and n the present work we take k S /k C = 5 0. Ths value of k S s hgh enough to ensure blockng of the shell magnetzaton (S AF M ) for appled felds n the range that the core magnetzaton (S FM ) exhbts hysteress behavor.typcal hysteress loops for dpolar nteractng NP arrays are shown n Fgure 8.Smlarly to the case of smple FM NPs dscussed n the prevous secton, M r n core shell NPs ncreases and H C decreases wth ncreasng DDI strength (Fg.9).However the changes of H C due to dpolar couplng are controlled by the value of the nterface exchange.in partcular, larger reducton of H C s observed n systems wth larger nterface exchange J. Nevertheless, the fracton of H C reducton s nearly the same for all values of J. J. Nanosc. Nanotechnol. 8, 1 15,

10 Magnetc and Magnetotransport Propertes of Ordered Nanopartcle Arrays Kechrakos and Trohdou (a) Fg. 9. Dependence of coercvty (h C ) and exchange bas feld (h E )on dpolar strength for a monolayer of core shell NPs at low temperature (t/k C = 0 01).Curves for dfferent values of the nterface exchange are shown.trangles: J/k C = 1 5, crcles: J/k C = 1 0, and squares: J/k C = 0 5.Shell ansotropy k S /k C = 5 0. Fg. 8. Hysteress loops at low temperature (t/k C = 0 01) of a monolayer of core shell NPs wth dfferent dpolar strengths and nterface exchange values.(a) J/k C = 1 5, (b) J/k C = 1 0 and (c) J/k C = 0 5. Squares: g = 0, crcles: g/k C = 0 3, and trangles: g/k C = 1 0.Shell ansotropy k S /k C = 5 0. The behavor of H E wth ncreasng dpolar strength appears more complex (Fg.9).For strong nterface couplng (J/k C = 1 5) H E drops lnearly wth the dpolar strength and for weak nterface couplng (J/k C = 0 5) t s only weakly dependent on the dpolar strength.the most strkng behavor s observed when the nterface exchange s comparable to the core ansotropy (J/k C = 1 0).In ths case, weak DDI enhance H E, whch exhbts a maxmum value around g/k c 0 2.The observed enhancement of H E due to weak DDI s contrary to what one expects on ntutve grounds.namely, DDI that lead to symmetrc backward and forward magnetzaton reversal processes, are shown to enhance the loop asymmetry.insght nto the dependence of the exchange bas feld on dpolar strength can be obtaned from separate examnaton of the two branches of the hysteress loop and the correspondng coercvty values H C1 and H C2.As shown n Fgure 10, H C1 (b) (c) decreases (n absolute value) wth dpolar strength, ndependently of the nterface exchange value.ths behavor s smlar to what s observed n FM NPs (Fg.2) at low temperature.the smlarty stems from the fact that n the case of backward magnetzaton reversal the exchange bas feld ncreases the barrer heght for reversal of an solated moment.the collectve reversal nduced by DDI facltates the reversal process leadng to (absolute) lower coercvty values.on the contrary, the dependence of H C2 on dpolar strength vares sgnfcantly wth the nterface exchange.in the forward reversal process the exchange bas feld reduces the barrer for the reversal of an solated moment.when the value of the exchange s weak (J/k C = 0 5, Fg.10), a reduced but fnte barrer exsts for the forward reversal.as prevously, the collectve reversal, nduced by DDI, reduces further the barrer heght leadng to smaller H C2 values.when the nterface exchange s strong (J/k C = 1 5, Fg.10), the barrer for the forward reversal of an solated moment dsappears and the forward reversal becomes a downhll process n the energy landscape, a fact reflected n the negatve value of H C2.In ths case, DDI ntroduce addtonal barrers due to ther ansotropc character and the collectve moton of the moments obstructs the reversal.the values Fg. 10. Dependence of the upper branch coercvty (h C1 ) and the lower branch coercvty (h C2 ) on dpolar strength for a monolayer of core shell NPs at low temperature (t/k C = 0 01).Curves for dfferent values of the nterface exchange couplng are shown.trangles: J/k C = 1 5, crcles: J/k C = 1 0, and squares: J/k C = 0 5.Shell ansotropy k S /k C = J. Nanosc. Nanotechnol. 8, 1 15, 2008

11 Kechrakos and Trohdou Magnetc and Magnetotransport Propertes of Ordered Nanopartcle Arrays of H C2 approach zero (from negatve values) as the dpolar strength ncreases, reflectng the ncreasng dffculty for forward reversal.for ntermedate nterface exchange values (J/k C = 1 0, Fg.10) the dependence of H C2 on the dpolar strength shows a varyng behavor.for solated moments (g = 0), H C2 assumes small postve values ndcatng a very low barrer for forward reversal.ddi ntally facltate the forward reversal, but wth ncreasng dpolar strength the barrer vanshes and H C2 reaches a negatve value (g/k C 0 2).Further ncrease of the dpolar strength nhbts the forward reversal.the observed dependence of H E on DDI strength shown n Fgure 9 s the net effect from the varaton of the upper and lower branch coercvtes wth dpolar strength.expermentally, enhancement of H E due to DDI has been observed 47 n strpes of Co/CoO NPs where the quas-one-dmensonal morphology of the array enhances the effect of DDI Transverse Susceptblty of FM Nanopartcles Measurements of the feld dependent RTS have been long ago predcted theoretcally 55 to reveal drect nformaton on the magnetc ansotropy of a NP assembly.a typcal RTS curve of a NP assembly wth random ansotropy, obtaned at low temperature by sweepng the bas (dc) feld n one drecton, exhbts three characterstc peaks, one at the coercve feld and two at postons correspondng to the ansotropy feld (±H K ).The latter s related to the sngle-partcle ansotropy as H K = 2K 1 /M s.thus measurement of RTS should, n prncple, be a drect method to measure K 1.However, wde partcle sze dstrbutons, 63 thermal fluctuatons and most mportantly, dpolar nteracton effects modfy the poston and shape of the peaks makng the determnaton of the ansotropy strength uncertan.we dscuss here the evoluton of the feld dependent RTS curves under ncreasng values of the dpolar strength, or equvalently decreasng values of the nterpartcle dstance n SAA of magnetc NPs.We show n Fgure 11 the n-plane T and out-of-plane ( T ) RTS curves at low temperature (t/k = 0 05) for ncreasng dpolar strength.the non-nteractng sample shows clearly the theoretcally predcted three peaks located at the coercve (h c ) and ansotropy (±h K ) felds.downshfted values of h c and h K relatve to the zero-temperature values (h c = 0 98 and h K = 2) are due to thermal fluctuatons effects. The most mportant effects of dpolar nteracton on the RTS curves, shown n Fgures 11 and 12 are: () the suppresson of the h c peak of T, () the locaton of the h K peak of T at hgher felds than the correspondng peaks of T, () the downshft (upshft) of the h K peak of T T wth ncreasng dpolar strength, (v) the slower saturaton wth bas feld of T relatve to T. Fg. 11. Feld dependence of reversble transverse susceptblty of a monolayer of FM nanopartcles at low-temperature (t/k = 0 1).The bas feld s swept from negatve to postve values.closed crcles: n-plane feld.open crcles: out-of-plane feld.the arrows n the out-of-plane data ndcate the poston of the coercve feld.data are taken from Ref.[66]. In recent RTS measurement n Fe NP arrays, 51 observatons smlar to our ponts () and (v) were made.however, the coercvty peak could not be resolved probably due to not suffcent lowerng of the temperature or due to the presence of NP sze dstrbuton.nevertheless, the agreement of our results n ponts (, v) above, consttute suffcent evdence that DDI were responsble for Fg. 12. Dependence of ansotropy feld on dpolar strength at low temperature (t = 0 05) for a monolayer of FM nanopartcles.closed crcles: n-plane feld.open crcles: out-of-plane feld. J. Nanosc. Nanotechnol. 8, 1 15,

12 Magnetc and Magnetotransport Propertes of Ordered Nanopartcle Arrays Kechrakos and Trohdou the observed expermental trends n these measurements. A physcal nterpretaton of the observed opposte trends of the n-plane and out-of-plane h K peaks wth ncreasng dpolar strength (Fg.12) reles on the development of an easy-plane ansotropy nduced by DDI.For an n-plane bas feld, the nteracton-nduced ansotropy reduces the barrer for an rreversble swtchng of the moments leadng to a reducton of the ansotropy feld, whle n the out-of-plane geometry DDI nhbt the rreversble swtchng along the z-axs by developng an easy-plane normal to ths axs, thus ncreasng the ansotropy feld.fnally, the lnear dependence of the ansotropy peaks on the dpolar couplng, or equvalently, on the nverse cube of the nterpartcle spacng (H K 1/d 3 ), shown n Fgure 12 for g/k 0 4, could be used to perform an extrapolaton procedure on measurements taken at dfferent nterpartcle separatons n order to extract the value of sngle-partcle ansotropy (K 1 ) Tunnelng Magnetoresstance n FM Nanopartcle Arrays Tunnelng magnetoresstance refers to a large decrease of a sample s resstvty under applcaton of a bas magnetc feld, observed when charge carrers transmt through two FM regons separated by a non-magnetc nsulatng barrer.the basc mechansm underlyng the effect s spndependent scatterng of the carrers.the frst FM regon acts as a polarzer for the electron spn whle the second regon causes scatterng whose strength s proportonal to the msalgnment of the magnetzaton relatve to the frst regon.in the case consdered here, the magnetc NPs are the relevant FM regons and the surfactant layer separatng them s the nsulatng barrer.thus, n prncple, spn-dependent transport measurements should reflect the underlyng mcromagnetc structure of the NP assembly. In Fgure 13 we plot the feld dependent TMR of a monolayer of dpolar coupled NPs and compare t wth the correspondng branch of the magnetzaton hysteress loop. The sharp peak of TMR occurs very close to the coercve feld, because the spn dsorder n the array s maxmzed at hs feld.the effects of DDI can be observed n the TMR curves.a downshft of the TMR peak poston wth ncreasng dpolar strength s observed followng the reducton of the H c values.the value of TMR at the remanent state decreases wth nteractons, reflectng the ncreasng algnment of the magnetc moments demonstrated also by the ncreased values of the M r (see Fg.2).Fnally, the TMR senstvty, namely the slope of the feld dependent TMR, ncreases wth ncreasng nteracton strength, due to a collectve reversal of the moments durng whch the degree of algnment s hgher for stronger nteractons. The easy-plane ansotropy nduced by DDI n a 2Darray of NPs s expected to produce a strong dependence Fg. 13. Feld-dependence of the magnetzaton (upper panel) and the correspondng tunnelng magnetoresstance (lower panel) of a monolayer of FM nanopartcles, at low temperature (t/k = 0 01).The curves correspond to dfferent dpolar strengths.crcles: g = 0, up-trangles: g/k = 0 1, down-trangles: g/k = 0 2, and damonds: g/k = 1.The bas feld les n-plane and s swept from negatve to postve values.data are taken from Ref.[69]. of the TMR values on the drecton of the bas feld.indeed the TMR curves shown n Fgure 14 vary substantally wth the azmuth angle ( ).The TMR senstvty decreases as the feld approaches the z-axs, reflectng the slow saturaton of the magnetzaton for an out-of-plane bas feld. A strkng feature occurrng for 15 n Fgure 14 s that the peak of TMR does not occur at the coercve feld but at a hgher feld.we state that the peak of TMR occurs at the crtcal feld (H o ), namely the feld for an rreversble swtch of the magnetzaton, 95 rather than at the coercve feld.ths has been verfed by smulatons n purely dpolar arrays 69 or non nteractng arrays wth algned easy axes.in both cases the crtcal feld can be obtaned analytcally and t was found that the peak of TMR occurs exactly at ths feld.when DDI are absent, the random ansotropy leads to H c 0 96H o (Ref.[95]) and the TMR peak occurs very close to the coercve feld. However, as DDI ncrease, they nduce coherent rotaton of the moments and a domnant easy-plane ansotropy. In the case of an easy-plane ansotropy the dfference between the H o and H c s maxmum for 15, 95 whch explans why the maxmum devaton between the feld 12 J. Nanosc. Nanotechnol. 8, 1 15, 2008