Magnetic ordering in arrays of one-dimensional nanoparticle chains

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2009 J. Phys. D: Appl. Phys. 42 215003

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J. Phys. D: Appl. Phys. 42 (2009) 215003 (9pp) doi:10.1088/0022-3727/42/21/215003

Magnetic ordering in arrays ofone-dimensional nanoparticle chainsD Serantes1, D Baldomir1, M Pereiro1, B Hernando2, V M Prida2,J L Sanchez Llamazares2, A Zhukov3, M Ilyn3 and J Gonzalez3

1 Departamento de Fısica Aplicada, Facultade de Fısica, Universidade de Santiago de Compostela,Campus Sur s/n, 15782 Santiago de Compostela, Spain2 Departamento de Fısica, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, 33007 Oviedo,Spain3 Departamento Fısica de Materiales, Facultad de Quımica, UPV, 1072, 20080 San Sebastian, Spain

Received 29 July 2009, in final form 1 September 2009Published 13 October 2009Online at stacks.iop.org/JPhysD/42/215003

AbstractThe magnetic order in parallel-aligned one-dimensional (1D) chains of magnetic nanoparticlesis studied using a Monte Carlo technique. If the easy anisotropy axes are collinear along thechains a macroscopic mean-field approach indicates antiferromagnetic (AFM) order evenwhen no interparticle interactions are taken into account, which evidences that a mean-fieldtreatment is inadequate for the study of the magnetic order in these highly anisotropic systems.From the direct microscopic analysis of the evolution of the magnetic moments, we observespontaneous intra-chain ferromagnetic (FM)-type and inter-chain AFM-type ordering at lowtemperatures (although not completely regular) for the easy-axes collinear case, whereas arandom distribution of the anisotropy axes leads to a sort of intra-chain AFM arrangement withno inter-chain regular order. When the magnetic anisotropy is neglected a perfectly regularintra-chain FM-like order is attained. Therefore it is shown that the magnetic anisotropy, andparticularly the spatial distribution of the easy axes, is a key parameter governing the magneticordering type of 1D-nanoparticle chains.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Nanometre-sized magnetic systems give rise to a huge varietyof physical phenomena and magnetic behaviour in comparisonwith the bulk counterparts [1]. The magnetic properties ofnanostructured and nanoscaled materials are strongly modifiedby slight variations both of the material characteristics (size,shape, anisotropy, chemical composition) and external factors(applied field, spatial assembly), providing a rich scenario withnovel properties of potential applications in many different andinterdisciplinary research fields, ranging from biomedicine [2]to data storage [3]. Recently, much investigation has beenfocused on the synthesis and magnetic characterization of1D arrangements of magnetic nanoparticles [4–6]. The latteris motivated by their promising technological potential [7],particularly to increase the storage density data [8] and fordigital processing [9]. In such 1D-nanostructured systems, theinterplay between the magnetic anisotropy and the interparticle

dipole–dipole interaction is the main key point determining themagnetic properties of the system. The relationship betweenboth energy terms constitutes a complex problem that turns outto be non-trivial. Although ferromagnetic (FM) order alongthe nanoparticle chain can be easily predicted from minimumenergy arguments [10], however, the opposite behaviour, thatis antiferromagnetic (AFM) order along the chain, has beenobserved experimentally [5]. Also, it has been theoreticallycalculated that in non-interacting particle systems, uniaxialmagnetic anisotropy can be misinterpreted as FM or AFMcoupling depending on the relative angle between the magneticfield and the anisotropy easy axes [11]. Therefore, the interplaybetween the magnetic anisotropy and the dipolar interactionenergies not only constitutes a necessary problem to solve fromthe applied point of view due to its relevance for technologicalapplications, but also an interesting scientific problem by itself.

On this basis, we report in this work a detailed analysis ofthe magnetic properties of a system of dipolarly interacting

0022-3727/09/215003+09$30.00 1 © 2009 IOP Publishing Ltd Printed in the UK

J. Phys. D: Appl. Phys. 42 (2009) 215003 D Serantes et al

0.6 0.5 0.4 0.3 0.2 0.1 0.0

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0.00.20.4

0.60.0 0.2 0.4 0.6 0.8

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z

x y H

Figure 1. Schematic representation of the simulated system. Left panel stands for the particle chains in the YZ-plane square distributionthat extend along the X-direction. Right panel exemplifies a set of five chains in the XZ-plane view, showing a snapshot of the magneticmoments of the nanoparticles.

magnetic nanoparticles collinearly arranged in 1D chains.By means of a Monte Carlo (MC) technique we study thenature of its magnetic ordering resulting from the interplaybetween dipolar and anisotropy energies. The computationaltechnique provides us with two different data sources tostudy the system: a macroscopic approach (averaging themacroscopic parameters, magnetization versus temperature)and a microscopic one (direct observation of the temperatureevolution of the magnetic moment of each nanoparticle). Theinformation provided by both approaches will be discussedand compared with other results previously reported in theliterature. The article is organized as follows. Section 2describes the computational method employed. In section 3,we consider the simple case of easy anisotropy axes collinearalong the chains and characterize its magnetic propertiesboth from the macroscopic (section 3.1) and microscopic(section 3.2) points of view. In section 4, a non-collinear(random) easy-axes distribution is studied and compared withthe collinear case. Finally, in section 5 we briefly summarizethe results obtained.

2. Computational details

To study the magnetic ordering in 1D chains of magneticnanoparticles we have used the same model reported in[12]. We consider an ensemble of N = 250 nanoparticlesdistributed in a net of relative positions (�x, �y, �z) =(0.067, 0.200, 0.200). Each single chain consists of 10particles aligned along the x-direction. So, as shown infigure 1, the system consists of 5 × 5 parallel chains of 10particles in length in a square distribution on the YZ-plane.We assume all the nanoparticle ensembles to be spherical andequal in size, with the magnetic moment for each i particleproportional to its volume, V , being defined as | �µi | = MSV ,with MS the saturation magnetization. Each single particle hasuniaxial anisotropy (characterized by the uniaxial anisotropyconstant K), and we first consider the simple case of theeasy anisotropy axes aligned with the x-direction, i.e. along

the chain. For simplicity we assume that MS and K aretemperature-independent.

The thermomagnetic evolution of the system has beensimulated considering periodic boundary conditions, inwhich the energy of each i particle has three differentenergy contributions: Zeeman, magnetic anisotropy andmagnetostatic dipolar interactions. The coupling of themagnetic moment of the ith particle with an externalfield �H is given as usual by the expression: E

(i)Z =

−�µi�H , whilst the uniaxial anisotropy contribution is E

(i)A =

−KV (( �µini)/| �µi |)2, with ni ≡ x since the easy axes of allthe nanoparticles are pointing along the x-axis. The dipolarinteraction energy between two particles i and j is computedas E

(i,j)

D = ( �µi �µj)/r3ij − 3( �µi�rij )( �µj �rij )/r

5ij , where �rij is

the vector connecting both particles. We use the Metropolisalgorithm to drive the individual movement of the magneticmoment of each particle: we select a particle i at randomand generate a new orientation of its magnetic moment. Thenew orientation is accepted as min[1, exp(−�E/kBT )], with�E the energy difference between the old and the attemptedorientation and kB the Boltzmann constant, i.e. if �E � 0 theparticle moment reaches the new orientation, and if �E > 0the probability of the moment to shift to the new orientationis given by exp(−�E/kBT ). The repetition of this process N

times (where N = 250) defines one computational unit time,referred to as a MC step hereafter.

In order to treat the problem in a general manner wehave chosen normalized units for the different magnitudesemployed. Hence, the magnetization of the sample is replacedby its reduced magnitude m = M/MS, M being the averagedprojection of the individual magnetic moments of the particlesalong the external magnetic field �H direction. The magnitudeof the applied field is related to the anisotropy field of theparticles, HA, by means of the reduced parameter h = H/HA,with HA = 2K/MS. The temperature (T ) is treated ast = kBT /2KV . Normalizing the above energies to 2 KV,

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J. Phys. D: Appl. Phys. 42 (2009) 215003 D Serantes et al

the total energy of the system in reduced units is given by

eT =N∑

i=1

{− cos2 ϕi

2− h cos θi

+c

2Nc0

N∑j �=i

[eµi

eµj

a3ij

− 3(eµi

�aij )(eµj�aij )

a5ij

]}. (1)

The three terms of equation (1) correspond to the reducedanisotropy, Zeeman and dipolar energy, respectively. Theangle ϕi gives the orientation between the magnetic momentof the ith particle and its respective easy anisotropy axis, andθi is the angle between the magnetic moment (unit vector eµi

)

and the applied magnetic field, whereas �aij = �rij /L stands forthe vector connecting particles i and j normalized by the sideof the unit cubic cell, L. The c value is the volume sampleconcentration and c0 = 2K/M2

S is a dimensionless parameterthat accounts for the intrinsic magnetic characteristics of theparticles [13]. For all the results presented in this work,the values of magnetization are averaged over 500 differentindependent configurations, and the temperature variation ratiois 0.026 97 every 1000 MC steps. It is important to recall herethat the results reported in this work do not intend to describequantitatively a particular system, but instead to show a generalapproach to the magnetic ordering properties of magneticnanoparticles with the characteristics and spatial distributionassumed above. Because of this, the main parameter values,such as the temperature variation ratio, applied magneticfield or dipolar interaction energy, do not have particularvalues, since they will acquire different values depending onthe specific characteristics of the nanoparticles fulfilling thefeatures of the studied system.

3. Collinear easy axes

With the purpose of understanding the physical natureof the magnetic interaction in the 1D-nanoparticle chainarrangement we have analysed its temperature evolutionfrom two different approaches: a macroscopic mean-fieldtreatment on the averaged magnetization of the system anda microscopic single-particle magnetic moment analysis. Forthe macroscopic study, we have simulated zero field cooling(ZFC) and field cooling (FC) magnetization curves for differentvalues of the applied field and analysed the evolution of themacroscopic averaged magnetization value with temperature.For the microscopic one, we have recorded the evolution alongthe temperature of the magnetic moment of each single particlein the same ZFC/FC processes, to directly study the magneticevolution of the system versus temperature from its individualcomponents. In the following subsections we present eachapproximation employed in each scenario and discuss theresults obtained.

3.1. Macroscopic approach

We simulated the magnetic response of the system underZFC and FC conditions and analysed its field dependence.Such processes are commonly used to study the magneticproperties of magnetic nanoparticle systems due to the useful

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ZFC FC h 1.50 1.00 0.75 0.50 0.40 0.30 0.20 0.10

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h=H/HA

-

tmax

m=

M/M

S

t=kBT/2KV

tmax

Figure 2. Magnetization versus temperature curves in reduced unitsm(t) for different values of the applied magnetic field. Inset showsthe field dependence of the maxima of the curves (vertical arrows inthe main panel) and the θ values obtained from a Curie–Weiss fitting(solid curves in the main panel).

information they provide about the magnetic properties of thesystem. They are usually characterized by (i) a paramagnetic-like dependence at high temperatures, where both ZFC andFC curves coincide, (ii) a splitting between both curves witha decrease in the temperature, with the ZFC one slightlydiminishing its growing rate until it reaches a maximumand (iii) below this maximum the ZFC curve decreases withdecreasing T , while the FC one keeps growing althoughat a lower rate. The maximum in the ZFC curve roughlycorresponds to the so-called blocking temperature, TB, afundamental magnitude characterizing systems of magneticnanoparticles [1]. It is related to the magnetic anisotropybarrier of the particles and is proportional to the reducedmagnetic field as TB ∝ KV (1 − H/HA)1.5 [14]. In the idealnon-interacting case TB is expected to disappear at the appliedfields H > HA (h > 1.0 in reduced units), although otherfactors such as the interparticle interactions may extend itsexistence to larger fields [15]. Above TB the particles are said tobe in the superparamagnetic state, while they remain ‘blocked’below TB.

In a previous work, we reported that these characteristicsin the ZFC and FC curves are not observed in a system ofdipole–dipole interacting magnetic nanoparticles distributedinto parallel 1D chains with collinear anisotropy axisunder a perpendicular magnetic field [12]. Althoughthe paramagnetic-like dependence is observed at hightemperatures and also the maximum in the ZFC curve, thereare, however, eye-catching features different from the above-described characteristics, such as the coincidence of bothZFC and FC curves at low temperatures and the existenceof the maximum even for fields h > 1.0. In figure 2 weshow the ZFC and FC m(t) curves for different field valuesapplied perpendicular to the easy axes of the particles (i.e.perpendicular to the nanoparticle chains).

The curves plotted in figure 2 show two distinctivefeatures. First, both ZFC and FC processes coincide for thewhole temperature range, indicating reversibility. Second, the

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J. Phys. D: Appl. Phys. 42 (2009) 215003 D Serantes et al

curves exhibit a maximum, tmax, above which the ZFC–FCcurves follow a paramagnetic-like behaviour that shifts towardssmaller t values with increasing h. The coincidence of the ZFCand FC curves in the whole temperature range means that tmax

stands for a feature different from the usual one obtained fora superparamagnetic system below its blocking temperature,TB: it does not indicate the transition between the blockedand the superparamagnetic states. As the magnetic field isperpendicularly applied to the anisotropy axes of the particlesnone of the anisotropy directions are favoured, and thereforethe peak of the curves has a different physical meaning from thetemperature at which the thermal energy equals the anisotropybarrier and therefore allows the jumping of the magneticmoment between the two anisotropy wells. The general shapeof the curves strongly recalls that one observed for AFMmaterials, which exhibits an initial magnetization increaseuntil the ordering temperature (θ ) is reached, followed bya paramagnetic-like dependence. To analyse the possibilityof the peak accounting for an ordering temperature (�), wehave fitted the thermal dependence of magnetization in therange of temperatures well above tmax to the Curie–Weisslaw m = c/(t − θ). The results are shown in the inset offigure 2 for the different values of applied field considered,together with the values of tmax. A negative θ value is obtainedsuggesting an AFM-type magnetic ordering. The coincidencebetween tmax and θ in the low-field regime reinforces the ideaof magnetic ordering temperature. Simulations carried outat different computational times indicate that tmax and θ aretime-independent, a fact that also supports an AFM-orderingtemperature scenario. The error in the determination of tmax

is given by the temperature variation step, whereas the errorof θ is smaller than the symbol size. Moreover, it mustbe emphasized that the c-fitting parameter grows linearlywith increasing field, as expected from the Curie–Weissdependence.

From the results discussed above, the AFM characterof the system seems to be clear, with the magnetic orderingtemperature found as

limh→0[θ ] ≡ limh→0[tmax] ≡ �. (2)

However, at this stage it is very important to make someobservations about the adequacy of using a mean-field theorywhen dealing with assemblies of magnetic nanoparticles, andabout the reliability of the conclusions drawn. It has beentheoretically demonstrated that a Curie–Weiss approach leadsto AFM-type order in a non-interacting particle system whenthe direction of the applied magnetic field is perpendicular tothe easy anisotropy axes of the particles [11]. With the aimto understand the role played by the magnetic anisotropy, thedipolar interaction and its combined influence on the shapeof the ZFC and FC m(t) curves, we show in figure 3 bothprocesses for the same magnetic field value of h = 0.1considering the three following possibilities: interplay betweenanisotropy and dipolar energies (circles), dipolar energy only(triangles) and anisotropy energy only (squares).

The thermomagnetic curves shown in figure 3 illustratethe peculiar features associated with the different energeticcontributions. Both the ZFC and FC processes practically

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10

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t=kBT/2KV

eA

eA+e

D

eD

1/m

(a.

u.)

ZFC FC e

A

eA+e

D

eD

m=

M/M

S

t=kBT/2KV

Figure 3. ZFC (full symbols) and FC (empty symbols)magnetization as a function of temperature curves in reduced unitsm(t) considering different energetic contributions: magneticanisotropy and dipolar energies both together (circles); dipolarenergy only (triangles) and anisotropy energy only (squares). Insetshows the inverse of the ZFC magnetization curves versustemperature.

coincide for the three cases considered, and so from now on weshall refer to the different curves making no difference betweenthem, just among the different considered energies. Whensolely the magnetic anisotropy is present in the system thecurve exhibits a maximum, and the inverse of the susceptibilitypoints to an AFM-like alignment in the Curie–Weiss mean-field treatment (as the inset states). This result coincides withthat described by Vargas et al [11] that uniaxial magneticanisotropy acts similarly to AFM coupling under a magneticfield applied perpendicular to the uniaxial easy anisotropyaxes. When the magnetic anisotropy is not taken into accountand the dipole–dipole interacting energy is the only internalcontribution governing the magnetic behaviour of the system,the shape of the curve is quite different. At high temperatures(roughly above the maximum of the only-anisotropy curve)the dipolar interacting curve follows a paramagnetic-likedependence. However, at temperatures below the maximumthe curve rapidly reduces its growth rate, showing somethingsimilar to a plateau. The inverse-susceptibility curve forthis case (see the inset) shows the same tendency of theanisotropy contribution, indicating an AFM-type couplingwhen extrapolating the curve from high-temperature values.Interestingly, the value of θ obtained is quite comparable to thatof the only-anisotropy curve. When the combined influenceof magnetic anisotropy and dipolar interaction is taken intoaccount for the magnetic behaviour of the system, the curveexhibits the characteristics displayed and discussed in figure 2.

In summary, our MC simulations agree with the previousresults reported by Vargas et al [11], establishing that theCurie–Weiss approximation is not an adequate scenario tostudy the magnetic ordering of magnetic nanoparticles sinceit leads to wrong interpretations when the magnetic field isapplied perpendicular to the easy anisotropy axes. In view ofthis, we shall study the magnetic ordering of the system in adifferent manner, directly observing the thermal evolution of

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J. Phys. D: Appl. Phys. 42 (2009) 215003 D Serantes et al

z

x

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t ~~~~

1t0.

max

t 0

.5t m

axt ≡

t max

t = 2

t max

t = 5

t max

h=1.0h=0.1

h=0.0

0.0

0.2

0.4

0.6

0.8

0.6 0.4 0.2 0.0

0.6 0.4 0.2 0.0

0.6 0.4 0.2 0.0

0.0

0.2

0.4

0.6

0.8

Figure 4. XZ-snapshot of the magnetic moments for the particularset of chains located at y = 0.8, at some selected temperatureschosen above and below the maximum of the ZFC/FC curve andthe applied magnetic field values, h = 0.0, 0.1 and 1.0.

the individual magnetic moments. The advantage of using aMC technique is that we can analyse the system not only fromthe macroscopic but also from a microscopic point of view,going directly into the details of the evolution of the magneticcomponents as will be presented and discussed in the nextsection.

3.2. Microscopic approach

We have recorded the temperature evolution of themagnetization of each particle while varying the temperature,and analysed the snapshots of the magnetization of the wholesystem at different temperatures. We have focused this analysison the magnetization behaviour when cooling with zero fieldin order to observe any spontaneous ordering in the magneticconfiguration of the nanoparticles ensemble. In figure 4, wedisplay the snapshots of the magnetic configuration of someselected nanoparticle chains (y = 0.8) on the XZ-plane.

In figure 4, we show the snapshots of the magneticconfiguration for the y = 0.8 chains projected in the XZ-planeat some selected temperatures with respect to the maxima (tmax)

of the h = 0.1 ZFC curve. Different external conditions for

the applied field are shown. In the left column we showthe h = 0.0 case, the one which we are more interestedin, since it may show some spontaneous magnetic orderingeffect. The centre and right columns show the h = 0.1and 1.0 cases, respectively, to give an idea of the effect ofthe external magnetic field on the orientation of the singlemagnetic moments of individual particles. Focusing on theh = 0.0 case, we clearly observe the transition from anoverall disordered state at high temperatures to a FM-like intra-chain ordering at low temperatures, where AFM-type inter-chain ordering is also observed. Similar types of behaviourare observed for the other chains of particles, although weonly plot this case for the sake of simplicity on its qualitativeunderstanding. Interestingly we note that some chains are‘broken’ into two different FM-like groups having oppositeorientations. The origin of the ‘broken’ intra-chain orderingmight be found in the influence of the magnetic anisotropy,which hinders the intra-chain magnetization reversal processleading to the minimal-energy FM-like parallel array of theparticles. We shall come back to this point later when analysingthe behaviour of the system without magnetic anisotropy. Letus also mention here the absence of a regular AFM inter-chain ordering. The magnetic arrangement of the chainsexhibits neither a perfectly nor a regular nearest-neighbourAFM-like alignment in the YZ-plane. This effect can beexplained by the large difference between the intra-chaindipolar field favouring the FM-type alignment and the inter-chain dipolar field that favours the inter-chain AFM-likearrangement. The latter is about 26 times smaller than thefirst one, and consequently it is unable to generate a completechain reversal magnetization process leading to a perfectlyregular inter-chain AFM-like arrangement. The intra- andinter-chain reversal mechanisms are time-dependent (largertimes imply more chances to reverse the magnetization) andtherefore we have simulated the evolution of the magnetizationfor much larger number of MC steps (until 140 000 MCsteps) in order to check the time dependence of the inter-chain arrangement. No noticeable effects on the magneticconfiguration were observed, suggesting that larger time-scalesare required in order to reach the magnetization reversal of theentire chains [17].

The effect of the applied magnetic field (h = 0.1, h = 1.0)on the magnetic configuration is included to illustrate theinfluence of the external field on the magnetic state of thesample. It points out the practically negligible deviation fromthe easy axes at the small field value h = 0.1, which supportsthe use of this value in a low-field inverse-susceptibility mean-field approximation. For the h = 1.0 case the magnetostaticdipolar interaction opposes the alignment of the magneticmoments along the field direction. In the case when themagnetic moments are allowed to reach an angle larger than45◦ from the easy anisotropy axes, the dipolar-field componentin the external field would try to reverse the neighbouringparticles’ moments, and reciprocally.

From figure 4 the existence of intra-chain FM-like andinter-chain AFM-like ordering is clear. With the purpose ofstudying both aspects and going further into the magneticbehaviour study of the system, we now select each chain as a

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J. Phys. D: Appl. Phys. 42 (2009) 215003 D Serantes et al

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10

⟨ m

⊥ chai

n⟩⟨

m

chai

n⟩

t=kBT/2KV

(c)

(b)

(a)

m=

M/M

S

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2

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t=kBT/2KV

m⊥ ch

ain

m ch

ain

t=kBT/2KV

Figure 5. Magnetization projection versus temperature for h = 0.0,perpendicular (b) and along (c) the chains. (a) Shows the ZFC curvefor h = 0.1. The vertical solid line is a guide to indicate the positionof tmax. Insets of (b) and (c) show the averaged absolute values ofthe magnetization perpendicular and along the chains, respectively.

proper magnetic entity and analyse its temperature evolution.We show in figure 5 the magnetization versus temperature dataalong and perpendicular to the axis of the 1D chains at zeroapplied field.

The total magnetization of every YZ chain is defined as

�m(yz) =10∑i=1

�mi,

therefore as a result the maximum possible value ofmagnetization for each 1D chain is 10. The y, z-variationstands for every different particle chain that extends alongthe x-direction. In figure 5, the temperature evolution of themagnetization of the nanoparticle chains is recorded along theaxis of the chains (m‖

chain) and perpendicular to them (m⊥chain),

the respective averaged magnetization values, 〈|m⊥chain|〉 and

〈|m||chain|〉, being introduced as the corresponding macroscopic

order parameters. Figure 5(a) shows the temperaturedependence of the magnetization along the field direction forh = 0.1, while cooling the system at zero field (i.e. h = 0.0,and so the average magnetization is zero) and when the sampleis heated under the applied field. This process is shown tofacilitate the comparison of the magnetization processes insideevery chain with respect to the maxima observed in figure 2.The vertical solid line signals the maximum of the ZFC curvethat roughly corresponds to −θ , as shown in the inset of figure 2and discussed in the text (see section 3.1). Figure 5(b) shows

the situation with perpendicular magnetization for the wholearray of 1D chains. The horizontal dotted lines indicate themaximum magnetization possible for every chain (i.e. 10),which would stand for a parallel orientation of all the particles’moments inside one chain. The overall non-zero momentat high temperatures due to the thermal disorder must benoted, and the disorder reduces when the temperature tendsto zero. This is more visible in the inset of figure 5(b), where〈|m⊥

chain|〉 is plotted as a function of temperature. Interestingly,this value approximately decreases with temperature for t <

tmax (tmax indicated by the vertical solid line). Figure 5(c)shows the magnetization of the chains along their longitudinaldirection, two main features being observed differing fromthe perpendicular case. First, the disorder observed at hightemperatures is greater than that observed in the perpendiculardirection. Second, at low temperatures the magnetizationof the chains starts to grow, reflecting an intra-chain FMorder at low temperatures. This FM-type internal orderingsplits the chains into two subsystems of opposite signs, whichindicates the occurrence of AFM-type ordering between them.Both components tend to saturate to the maximum possiblevalue (i.e. 10), although some chains do not reach saturationand keep low magnetization values (around 0, 4) roughlyconstant with temperature. The inset of figure 5(c) shows theaveraged absolute values of these tendencies, 〈|m||

chain|〉. Athigh temperatures a larger average magnetization value thanfor the perpendicular case is observed as well as a tendency toincrease at low temperatures, although the maximum valueattained does not reach the maximum possible value. Noclear correlation was observed with the ordering temperature� as discussed from figure 2 (indicated by the vertical solidline).

The simulations shown in figure 5 demonstrate that inthe absence of an external field the magnetization of thesystem spontaneously orders at low temperatures, formingintra-chain FM-like entities with inter-chain AFM order. Theexistence of ‘broken’ chains discussed after figure 4 is reflectedin the non-saturating magnetization value of some chains.At this point some open questions still arise. What mightbe the origin of the broken magnetization inside some ofthe chains? Why is the AFM-like disposition between thechains irregular? It is clear that it must be the magnetostaticdipolar interaction, the only interparticle interaction presentin the system, the mechanism responsible for the observedspontaneous orientation of the magnetic moments inside thechains. Also noticeable is the important role that the magneticanisotropy plays in the magnetization reversal process of themagnetic moments along the chain (as discussed in figure 3).For a better understanding of the role played by magneticanisotropy in the resulting magnetic ordering of the system andtrying to answer the above-mentioned questions, the followingtwo steps were considered. Firstly, we have analysed the sameproperties as in figure 5, but considering only the magnetostaticdipolar interaction between the particles versus temperatureto eliminate the influence of the magnetic anisotropy as apreferential orientation for the magnetic moments inside thechains. Secondly, we also analysed the magnetic behaviourof the system for a random distribution of the anisotropy easy

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J. Phys. D: Appl. Phys. 42 (2009) 215003 D Serantes et al

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m=

M/M

S

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.0

0.1

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0.3

0.4

0.5

-θ

h=H/HA

t

eA=0.0

m⊥ ch

ain

t=kBT/2KV

(c)

(b)

(a)

m ch

ain

⟨ m

⊥ chai

n⟩⟨

m

chai

n⟩

Figure 6. Magnetization projection versus temperature for h = 0.0,perpendicular (b) and along (c) the chains axes, when the dipolarinteraction is the only internal energy driving the magnetic behaviourof the system. (a) shows the ZFC h = 0.1 m(t) curve. The verticalsolid line serves as a guide to illustrate limh→0[θ ]. Insets of (b) and(c) show the averaged absolute values of the magnetizationperpendicular and along the chains axes, respectively..

axes. Figure 6 displays a scenario similar to that shown infigure 5, but considering the dipolar interaction as the onlyenergy driving the magnetic behaviour of the system as afunction of the temperature.

Figure 6(a) shows the ZFC m(t) curve, where nomaximum should appear as discussed in figure 3, and itsinset shows the ordering temperature as a function of theapplied magnetic field. Figure 6(b) shows the perpendicularcomponent of the magnetization, which exhibits similarbehaviour to the case with magnetic anisotropy. But it is infigure 6(c) where the difference between both cases is pointedout: all the magnetic chains reach the maximum possiblemagnetization value at low temperatures (see also the inset offigure 6(c)). Therefore, it is demonstrated that in the absence ofmagnetic anisotropy complete intra-chain FM-type order canbe found. However, the spontaneous FM-type order observeddoes not arise at a well-defined value, but extends over a widetemperature range as in the case with anisotropy (figure 5).The lack of a well-defined change in the macroscopic orderparameters, 〈|m⊥

chain|〉 and 〈|m||chain|〉, reflects the complexity of

the system, where the magnetic order generated by the dipolarinteractions is distorted by the dynamical effects arising fromthe anisotropy barrier, or, in the case analysed in figure 6, by

the anisotropy induced by the dipolar interaction itself 4. Withregard to the YZ-plane, we observe that the AFM arrangementis still not perfectly regular. In this inter-chain scenario wereproduce results similar to those with magnetic anisotropy,which indicates that larger time-scales are also required toreverse an entire chain even when no magnetic anisotropy ispresent [17].

4. Non-collinear easy axes

Until now, we have evidenced that magnetic nanoparticlechains exhibit intra-chain FM order together with AFM inter-chain one at low temperatures. The 1D chains exhibita ‘broken’ FM order when the nanoparticles arrange theiruniaxial magnetic anisotropy axes aligned parallel to thechains, while the AFM order is not fully regular regardlessof the existence or not of magnetic anisotropy. However,our results indicate a different magnetic ordering from theone recently reported by Bliznyuk et al [5]. These authorsexperimentally found intra-chain AFM order in 1D chains of Ninanoparticles. We believe that this different magnetic orderingis originated by a different orientation of the easy anisotropyaxes, since the splitting between the FC and field heatingmagnetization curves indicates non-perpendicular orientationof the magnetic field with respect to the anisotropy axes. Infact, AFM coupling between nanomagnets arranged in linearchains remaining confined to a plane can be observed if themagnetic anisotropy axes lie transverse to the longitudinalchain direction [18]. In order to investigate the influence ofnon-collinear anisotropy axis distribution we treat the problemin the most general case, when considering randomly orientedanisotropy easy axes and analysing the magnetic propertiesof the system in the same way as for the parallel-alignedcase. With this aim, we have simulated similar scenarios tothose considered in figure 2, but in this case for random easy-axes distribution. In this particular situation, the ZFC and FCmagnetization curves resemble the usual superparamagneticfeatures, with the characteristic splitting between the ZFC andFC curves at low temperatures. In figure 7, we show the mostrepresentative features of the magnetic behaviour of the systemfor comparison with the collinear-anisotropy case.

The ZFC and FC curves for the h = 0.1, 0.2 cases areshown in the top panel of figure 7 to illustrate the differencefrom the parallel-aligned anisotropy case. The fitting of theinverse susceptibility to a Curie–Weiss-like dependence inthe high-temperature regime (i.e. well above tmax) indicatesAFM-like behaviour with negative θ -ordering parameter, thecoincidence between tmax and (−θ) in the h → 0 limit(as shown in the inset) also being observed. The bottompanel of figure 7 shows the XZ-snapshots of the magneticconfiguration for three different representative temperaturevalues for the magnetic properties of the system: ∼0.2tmax, tmax

and ∼5tmax. The snapshots clearly illustrate that the systemdoes not attain any type of intra-chain FM-like order at anytemperature, but instead of that shows some kind of irregular

4 For more information about the interplay between the dynamical effects(associated with the blocking temperature) and phase transitions (associatedwith dipolar interactions) in nanoparticle systems see for example [16].

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J. Phys. D: Appl. Phys. 42 (2009) 215003 D Serantes et al

Figure 7. Top panel: ZFC and FC magnetization curves m(t) forfield values h = 0.1, 0.2. Inset shows the field dependence of θ andtmax. Bottom panel: snapshots of the magnetic configuration at somerepresentative temperatures: (a) ∼tmax/5, (b) ∼tmax and (c) ∼5tmax.

AFM-like intra-chain order. Therefore, this demonstrates thatthe spatial distribution of the easy anisotropy axes is the keyparameter responsible for the magnetic ordering in these kindsof 1D-nanoparticle system assemblies: a collinear distributionleads to a FM-like intra-chain order, whereas the random easy-axes distribution leads to an AFM-like order. However, itstill remains an open unsolved question with respect to thetype of magnetic ordering in these kinds of 1D structures,which is to determine the frontier between the FM-like andthe AFM-like magnetic ordering. At this point of researchwe recognize that the next issue we must face to characterizethe resulting magnetic ordering in 1D assemblies of magneticnanoparticle systems is the effect of the relative orientation ofthe easy anisotropy axes with respect to the nanoparticle-chainlongitudinal direction, together with the relative strength of thedipolar interaction related to the anisotropy energy.

5. Conclusions

In this work we study the magnetic ordering in a systemof dipolarly interacting magnetic nanoparticles distributedalong 1D chains, and particularly the interplay betweenthe nanoparticles magnetostatic dipolar interaction and theirmagnetic anisotropy. With such a purpose, we have useda computational Monte Carlo technique that provides uswith accurate control of the system characteristics. Thistechnique allowed us to study the magnetic evolution of thesystem using both a macroscopic mean-field approach anda microscopic direct analysis of the temperature evolutionof each magnetic moment of the particles. We startedby choosing the simple case of collinear uniaxial magneticanisotropy easy axes, and we have analysed its magnetic

response from a macroscopic point of view, interpreting themagnetic behaviour of the system throughout its macroscopicaveraged parameters. By fitting the magnetization to aCurie–Weiss mean-field approximation, we obtained thateven for the non-interacting case the Curie–Weiss θ -orderingtemperature stands for the AFM order. Therefore, weconcluded that a mean-field treatment is not adequate tostudy magnetic nanoparticle systems, in agreement with [11],since it may lead to misinterpretations regarding the magneticbehaviour of the system. Next we analysed the system bya microscopic approach, recording the temperature evolutionof the magnetic moments of each single particle. Withthe decrease in temperature, we obtained the occurrenceof spontaneous intra-chain FM ordering combined with aninter-chain AFM one. Both magnetic dispositions are notcompletely regular. In the intra-chain case some chains aredivided into two sections having opposite FM alignments,while in the inter-chain side some neighbour chains are alignedparallel in a FM-type form. In order to better understandthe role played by magnetic anisotropy in the magneticproperties of the system and particularly in this case withincompletely regular magnetic ordering features, we simulatedits magnetic evolution considering the magnetic anisotropy ofthe nanoparticles to be negligible. In this case a complete intra-chain FM-like order, although no noticeable changes in theirregular inter-chain AFM order, was appreciated. We believethat a complete AFM-like arrangement of the system would beachieved for a much longer time scenario, for which a completeor collective single-chain magnetization reversal mechanismcould take place. Finally, with a view to understandingthe intra-chain AFM ordering experimentally observed byBliznyuk et al [5], we also performed calculations similar tothose carried out for the study of the collinear-axes and thenegligible-anisotropy cases, but taking a random distributionof the anisotropy easy axes. In this case we obtained an intra-chain AFM-like ordering, both from a macroscopic Curie–Weiss type fitting and from microscopic analysis, which pointsto the primary importance of the anisotropy easy axes on theresulting magnetic properties of the system.

Acknowledgments

The authors thank the Xunta de Galicia (XdeG), ProjectINCITE 08PXIB236052PR, and the Spanish Ministry ofEducation and Science, projects No MAT2006-10027,MAT2006-13925-C02-01, NAN2004-09203-C04-03, NAN2004-09203-C04-04. They thank the Centro de Super-computacion de Galicia (CESGA) for computational facilities.D Serantes and M Pereiro also acknowledge XdeG for financialsupport (Maria Barbeito and Isabel Barreto Programmes,respectively). FICYT is acknowledged by J L SanchezLlamazares (contract No COF07-013).

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