Collective States of Interacting Ferromagnetic Nanoparticles

ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 300 (2006) 192–197 www.elsevier.com/locate/jmmm

Collective states of interacting ferromagnetic nanoparticles

O. Petracica, X. Chena, S. Bedantaa, W. Kleemanna,Ã, S. Sahoob, S. Cardosoc, P.P. Freitasc

aAngewandte Physik, Universita¨t Duisburg-Essen, D-47048 Duisburg, Germany bDepartment of Physics and Astronomy, University of Nebraska, Lincoln, Nebraska 68588, USA cINESC, Rua Alves Redol 9-1, 1000 Lisbon, Portugal

Available online 15 November 2005

Abstract

Discontinuous magnetic multilayers [CoFe/Al2O3] are studied by use of magnetometry, susceptometry and numeric simulations. Soft ferromagnetic Co80Fe20 nanoparticles are embedded in a diamagnetic insulating a-Al2O3 matrix and can be considered as homogeneously magnetized superspins exhibiting randomness of size (viz. moment), position and anisotropy. Lacking intra-particle core-surface ordering, generic freezing processes into collective states rather than individual particle blocking are encountered. With increasing particle density one observes ﬁrst superspin glass and then superferromagnetic domain state behavior. The phase diagram resembles that of a dilute disordered ferromagnet. Criteria for the identiﬁcation of the individual phases are given. r 2005 Elsevier B.V. All rights reserved.

PACS: 75.10.Nr; 75.50.Lk; 75.40.Gb; 75.60.Ej

Keywords: Magnetic nanoparticles; Dipolar interaction; Superspin glass; Superferromagnetism

10 1. Introduction where t010 s is the inverse attempt frequency, K an effective anisotropy constant and V the volume of the The physics of nanoscale magnetic materials is a vivid nanoparticle. The energy barrier is here approximated by subject in current magnetism research. This is partially due EB ¼ KV. to the promised potential in modern data storage applica- The magnetic behavior of the particle is characterized by tions [1] but mainly due to the wide spectrum of novel the so-called ‘‘blocking’’ temperature, Tb, below which the effects found [2–4]. One particularly interesting topic is the particle moments appear blocked on the time scale of the study of assemblies of magnetic nanoparticles, where each measurement, tm. This is the case, when tmEt. Using particle is in a magnetic single-domain state. In the simplest Eq. (1) one obtains case of ferromagnetic (FM) particles showing coherent T KV=k lnðt =t Þ. (2) reversal of the moments, one can assign to each granule a b B m 0 single moment or ‘‘superspin’’ being usually in the order of An ensemble of nanoparticles is denoted as SPM, when the 1000 mB, where mB is Bohr’s magneton. A FM nanoparticle magnetic interactions between the particles are sufficiently is defined as superparamagnetic (SPM), when the energy small [7]. Then the magnetic behavior of the ensemble is barrier, EB, for a magnetization reversal is comparable to essentially given by the configurational average over a set the thermal energy, kBT, during the measurement. The of independent particles. More generally, one can denote direction of the superspin then fluctuates with a frequency f the magnetic behavior as SPM in the sense of a or a characteristic relaxation time, t1 ¼ 2pf . It is given by thermodynamic phase. No collective inter-particle order the Neél–Brown expression [5,6], exists, while the intra-particle spin structure is FM ordered. In the case of small concentrations of particles, only SPM t ¼ t0 exp KV=kBT , (1) behavior is observed. However, for increasing concentrations the role of ÃCorresponding author. Tel.: +49 203 379 2809; fax: +49 203 379 1965. magnetic interactions becomes non-negligible. The mean E-mail address: [email protected] (W. Kleemann). (point) dipolar energy of two interacting nanoparticles, e.g.

0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.10.061 ARTICLE IN PRESS O. Petracic et al. / Journal of Magnetism and Magnetic Materials 300 (2006) 192–197 193 each with a moment of m ¼ 3000 mB and center-to-center is given by 2 3 distance of D ¼ 6 nm yields E =k ¼ðm =4pk Þm =D ¼ d d B 0 B E ¼KV ðk s Þ2 m M V Hsz þ m M2 4p 26 K. Considering all neighbors, it is obvious that the effect i X i i i 0 s i i 0 s 3 of dipolar interactions can be observed even at tempera- V iV j½si sj 3ðsi qijÞðsi qijÞ=rij, ð3Þ tures in the order of 100 K. In the case of imperfectly fjg spherical particles one also needs to take into account where K is the effective anisotropy constant, V the volume higher-order multipole terms [6,8]. Consequently, with i and Ms the saturation magnetization of a particle. ki, si and increasing particle density one finds a crossover from q are the unit vectors of the anisotropy axis, the superspin single-particle blocking to collective freezing [2–4,7]. One ij direction and the distance vector to particle j, qij ¼ rij=jrijj, can distinguish two kinds of collective states. For an respectively. The summation in the third term includes all intermediate strength of dipolar interactions, randomness particles {j} surrounding i within a cut-off radius of 25 nm. of particle positions and sufficiently narrow size distribu- In this study, we focused on the spatial superspin tion one can observe a superspin glass (SSG) state. Here configuration of a 2D array of nanoparticles after proper the superspins freeze collectively into a spin glass phase equilibration of the ensemble, i.e. 20,000 Monte Carlo below a critical temperature, Tg [2–4,10–13]. For higher steps. The positions of approximately 100 particles were densities of particles and hence stronger interactions, one directly transferred from transmission electron microscopy can observe a superferromagnetic (SFM) state. It is (TEM) images of a SSG DMIM sample with tn ¼ 0:9nm characterized by a FM inter-particle correlation [12,14–18]. [25]. The volumes Vi were calculated from the diameters as There exist various experimental realizations of magnetic found from the TEM images assuming spherical particles. nanoparticle assemblies, e.g. frozen ferrofluids [2–4], Values for the effective anisotropy were K ¼ 4 105 J=m3 discontinuous metal-insulator multilayers (DMIMs) [25] and for the saturation magnetization Ms ¼ [12,19], co-sputtered metal-insulator films [20], self-orga- 1:44 MA=m (bulk Co). nized particle arrays on surfaces [21], focused ion-beam In order to simulate a dense SFM system (tn ¼ 1:4 nm), structured thin films [22] and mechanically alloyed the particle volumes were manually scaled by a factor materials [23]. 1.53E(1.4 nm/0.9 nm)3 under the assumption of Vollmer– In this article, we discuss experimental studies on a series Weber particle growth with constant areal density of of DMIMs, [Co80Fe20(tn)/Al2O3(3 nm)]10, where the nom- growth nuclei [12,27]. Moreover, an easy in-plane axis was inal thickness is varied in the range 0.5ptnp1.8 nm. A introduced by choosing random anisotropy axis vectors crossover from SSG to SFM behavior is observed and a from a 3D unit sphere, where the probability for the x-, y-, phase diagram, i.e. transition temperature vs. nominal and z-component corresponds to a Gaussian function with thickness, is constructed. Moreover, results from numeric width sy ¼ 0:6 for the y and sz ¼ 0:3 for the z-component. simulations on a SFM system are presented.

3. Phase diagram of interacting nanoparticles 2. Details of experiment and simulation A cartoon of the expected phase diagram, transition temperature vs. nominal thickness, is depicted in Fig. 1.At We performed experimental studies on DMIMs, [Co80- high temperatures, T4Tc,bulk (i.e. bulk Curie temperature), Fe20(tn)/Al2O3(3 nm)]10, with nominal thickness in the the system is paramagnetic (PM). Below Tc,bulk sponta- range 0.5ptnp1.8 nm. The samples were prepared by neous FM order builds up in each particle. Here one sequential Xe-ion beam sputtering from Co80Fe20 and should note that a possible finite size effect of the Curie Al2O3 targets on glass substrates [24]. Due to the non- temperature might be included. At very small nominal wetting properties of the soft FM Co80Fe20, it grows as an ensemble of nearly spherical particles being eventually thicknesses and, hence, small concentrations, the system behaves SPM. Any possible effect of inter-particle interac- embedded in the insulating Al2O3 matrix. The average diameter of CoFe particles can be tuned by the nominal tions is not apparent, since the single-particle blocking at Tb ¼ T bðtn; tmÞ disguises any transition at temperatures thickness, tn. E.g. for tn ¼ 0:9 nm the mean diameter of CoFe particles is hDi¼2:8nm[25]. During growth a weak ToTb. magnetic field (m0H10 mT) was applied in order to induce an in-plane easy-axis. Magnetization and AC susceptibility 3.1. Superspin glass (SSG) ordering measurements were performed by use of a commercial superconducting quantum interference device (SQUID) For higher nominal thicknesses collective inter-particle magnetometer (Quantum Design, MPMS-5S). ordering occurs, where the ordering temperature exceeds Ensembles of dipolarly interacting nanoparticles were the blocking temperature. For particle arrangements studied by Monte-Carlo simulations with the Metropolis exhibiting randomness of size (viz. moment), position and algorithm [26]. The simulation essentially minimizes the anisotropy first a SSG phase is encountered. The transition total energy of the entire system at a given field, H, parallel line, T g ¼ T gðtnÞ, is a phase transition line separating the to the z-axis and temperature, T. The energy of a particle i SPM and SSG phase. ARTICLE IN PRESS 194 O. Petracic et al. / Journal of Magnetism and Magnetic Materials 300 (2006) 192–197

PM MFC Tc,bulk

T T SPM g, c MZFC M (kA/m) 20 Temperature

SFM 0 SSG 0 50 100 T (K) Nominal thickness Fig. 2. ZFC magnetization, MZFC, and FC magnetization, MFC, vs. T of [Co Fe (0.9 nm)/Al O (3 nm)] measured in a ﬁeld of 0.4 mT. The Fig. 1. Schematic phase diagram, transition temperature vs. nominal 80 20 2 3 10 arrow marks a dip in M being typical of SSG systems. thickness, with paramagnetic (PM), superparamagnetic (SPM), superspin FC glass (SSG) and superferromagnetic (SFM) phase. Relevant lines are the blocking temperature of the individual particles, Tb,andthecollective 6 transition line, i.e. the glass transition, Tg, or SFM transition temperature, Tc.

0.1 M

5 ∆ Evidence for SSG ordering in nanoparticle assemblies is found in several systems [2–4,10–13] using a wide spectrum 0.0 20 30 40 50 of various methods, i.e. 4

(1) A first simple test to identify SSG ordering is achieved from zero-field-cooled/field-cooled (ZFC/FC) curves of 3 the magnetization, M vs. T. Fig. 2 shows MZFC and ref MZFC

MFC vs. T obtained on the SSG system [Co80Fe20(tn)/ M (arb.units) Al2O3(3 nm)]10 with tn ¼ 0:9 nm. While a typical peak 2 in the ZFC curve is found in both SPM and SSG MZFC samples, the observation of a decrease in MFC(T) upon cooling (see arrow) can only be observed in SSG systems [28]. The appearance of a dip in this example is 1 due to an additional PM signal from small clusters or atoms dispersed between the particles [27]. (2) SSG and more generally any spin glass system exhibits 0 the so-called aging effect in the ZFC magnetization 20 30 40 50 60 [29]. Here the sample is rapidly cooled down in zero T (K)

field to a temperature below Tg and a certain waiting Fig. 3. Memory effect observed in MZFC(T) in a field of 0.01 mT on the time, tw, spent at this temperature. Finally, a small field SSG sample [Co80Fe20(0.9 nm)/Al2O3(3 nm)]10. A stop of duration is applied and the magnetization recorded as function 2 104 s was applied at 32 K. The inset shows the difference curve, ref DM ¼ MZFC MZFC. of time, t. Then the derivative SðtÞ¼ð1=m0HÞðqMðtÞ= qln tÞ shows a peak at tEtw [2–4,13,28]. Hence, the system ‘‘remembers’’ the time spent in zero field before shows a dip exactly at Ts. This is best seen in the the field is applied. difference between the curves without and with inter- ref (3) A similar experiment reveals the so-called memory mittent stop, DM ¼ MZFC MZFC [2–4,13,28]. Fig. 3 effect [29]. The sample is rapidly cooled in zero field to shows results on the SSG system with tn ¼ 0:9 nm, 4 a certain stop temperature, T soT g. After waiting a where a stop of 2 10 satT s ¼ 32 K was applied. certain time at Ts the cooling procedure is resumed to (4) Spin glasses exhibit a divergence of the non-linear some lower temperature. Then a small field is applied susceptibility, w3, at the critical temperature, Tg, i.e. [29] and the ZFC magnetization curve, MZFC(T), is g recorded upon heating up. The obtained ZFC curve w3 / T Tg 1 , (4) ARTICLE IN PRESS O. Petracic et al. / Journal of Magnetism and Magnetic Materials 300 (2006) 192–197 195

where g is the corresponding critical exponent. w3 is (a) 50K tn = 0.9nm obtained from isothermal magnetization curves, which 1.0 55K can be expanded as odd power series, M ¼ w1H 3 5

w3H +O(H ). From ﬁts of the w3(T) values to Eq. (4) ) 60K 2 one can then extract the parameters Tg and g. SSG

''(10 0.5 systems typically exhibit glass temperatures in the range χ f Tg50 K [12,30,31]. (5) Another test for SSG ordering is based on measure- 0 00 ments of the complex AC susceptibility, w ¼ w iw ,in 0.0 zero ﬁeld. The real part, w0ðTÞ shows a peak at the 0.0 0.5 1.0 1.5 2.0 2.5 ‘‘freezing’’ temperature Tf. With decreasing AC fre- t = 1.3nm (b) quency, f, the peak position, Tf, shifts to lower 320K n temperatures. In a SSG there will be a limit value, 40 350K - - f Tf Tg for f 0. This is expressed as a critical power ) 2 380K law [29,32],

''(10 n zv χ 20 t ¼ t Tf T g 1 , (5) where t is the characteristic relaxation time of the 1 * system being probed by the AC frequency, 2pf ¼ t ,t 0 is the relaxation time of the individual particle moment, 0 20406080100 n the critical exponent of the correlation length, x ¼ (T/ χ'(102) T 1)n, and z relates t and x as t / xz. E.g. in the g Fig. 4. Cole–Cole plots, w00(f) vs. w0(f), for a SSG DMIM sample with DMIM with tn ¼ 0:9 nm we found reasonable values, t ¼ 0:9 nm (a) at T ¼ 50, 55 and 60 K and for a SFM DMIM with t ¼ 81 n n Tg ¼ 61 2K, zn ¼ 10:2 and t ¼ 10 s [25]. 1:3 nm (b) at T ¼ 320, 350 and 380 K. The arrows indicate the order of AC (6) One can achieve data collapse of the AC susceptibility frequencies corresponding to each data point. Note that both plots are on one master curve in the dynamical scaling plot, deliberately in a 1:1 scale for the w00 and w0 axes. b 00 n zn ðT=T g 1Þ w =weq vs. ot ðT=Tg 1Þ , and thus obtain values for T , zn, and b. Here w (T) is the g eq t ¼ 1:3 nm is shown in Fig. 4(b). For low frequencies hypothetical equilibrium curve, which can be approxi- n (right-hand side) one finds a quarter circle, whereas at mated by a Curie–Weiss hyperbola at high tempera- high f an increasing part with positive curvature tures, b the critical exponent of the order parameter appears. These features can be successfully modeled and o ¼ 2pf [30,31]. in the framework of the motion of a pinned DW in a (7) In the case of a SSG system, a so-called Cole–Cole plot, random FM exhibiting relaxation, creep, slide and w00 vs.w0, shows a strongly flattened (viz. down-shifted) switching [17,34,35,37]. semi-circle [29]. An example is presented in Fig. 4(a) (2) Another strong indication of SFM behavior can be showing results on a DMIM with t ¼ 0:9nm [33].In n found from analyzing relaxation curves of the thermo- contrast, a SPM system would exhibit an almost remanent magnetic moment, m vs. t. Ulrich et al. [38] complete (not shifted) semi-circle [34]. introduce a universal relaxation behavior for any nanoparticle system according to a decay law of the form 3.2. Superferromagnetic (SFM) ordering d=dt mðtÞ¼WðtÞmðtÞ. (6) If the nominal thickness or the particle concentration is Here the time-dependent logarithmic decay rate, increased, higher-order multipole terms of the dipolar WðtÞ¼ðd=dtÞ ln mðtÞ, obeys a power law, interaction become relevant [8,9]. This can lead to SFM n ordering [14–17,35] (see Fig. 1), where regions of FM WðtÞ¼At with tXt0. (7) correlated particle moments (SFM domains) are found An SFM system can be distinguished from a SSG by [18,36]. The magnetic behavior is then essentially given by the exponent n. A is an arbitrary constant. While SSG the nucleation and motion of domain walls (DWs). We samples exhibit no1, SFM systems appear to have assume such a SFM domain state to occur in our DMIMs values n41 [39]. In the latter case m(t) obeys a power for tn41:1 nm. Analogously to SSG systems, there exist law, pertinent methods to identify SFM behavior in nanopar- 1n ticle systems. mðtÞ¼m1 þ m1t , (8) where mN and m1 are proper fit constants. We find (1) The Cole–Cole plot, w00 vs. w0, appears to be completely excellent agreement with extracted values for n dissimilar to that of a SPM and SSG system [34].An from experimental relaxation curves in our DMIM experimental Cole–Cole plot of a SFM DMIM with samples [39]. ARTICLE IN PRESS 196 O. Petracic et al. / Journal of Magnetism and Magnetic Materials 300 (2006) 192–197

(3) Relaxation curves, m(t), of a SFM sometimes show a peculiar behavior after rapidly FC the sample. One can observe an intermediate increase of m(t) after switching off the ﬁeld [40]. In this case the relaxation is characterized by two processes: ﬁrst, a decaying part according to a power law being due to DW motion; second, an increasing contribution following a stretched exponential due to the post-alignment of the particle moments inside the SFM domains, i.e. 1n b mðtÞ¼m0 þ m1t þ m2½1 exp ððt=tÞ Þ, (9)

where m0, m1 and m2 are ﬁt constants. t and b are the effective relaxation time and exponent, respectively, of the post-alignment process.

Similar to regular FM, the transition temperature, Tc, Fig. 6. Spin structure at zero field of a 2D array of dipolar interacting can be obtained from analyzing hysteresis loops, i.e. nanoparticles after ZFC from 300 to 5 K. The particle moments are recording the remanent and coercive fields vs. T. Finally, represented as cones. The color code (red, white) corresponds to the x- the results on the DMIMs can be summarized in an component of the moment. experimental phase diagram as presented in Fig. 5 resembling that of a dilute disordered ferromagnet [41]. One finds SSG behavior for a nominal thickness study neglects higher-order multipole terms of the dipolar tno1:1 nm, while a SFM state is encountered at interaction [8,9]. Probably a more advanced simulation tn41:1 nm. Moreover, a crossover to a random field including those contributions would drastically enhance domain state (RFDS) is found at lower temperatures [39] the SFM ordering tendency. in the SFM state. The borders of this phase diagram have recently been specified to be 0:5otno0:7 nm for 4. Conclusion SPM-to-SSG [27] and 1:6otno1:8 nm for SFM-to-perco- lated FM [42]. Ensembles of FM nanoparticles can exhibit collective In addition, Monte-Carlo simulations on a 2D array of behavior. When the dipolar interaction becomes non- dipolar interacting nanoparticles were performed in order negligible, superparamagnetism is overcome below a to study SFM ordering. Fig. 6 shows the spin structure of certain threshold inter-particle clearance (at constant approximately 100 particles in zero field after ZFC from density). In our granular DMIMs, [Co80Fe20(tn)/ 2 300 to 5 K. The simulation area is 50 50 nm . One Al2O3(3 nm)]10 we can observe SSG and SFM behavior observes FM correlated regions including several particles at tn40:5 nm and 41.1 nm, respectively. Both collective [18,36], which can be interpreted as SFM domains. This states are clearly identified by various kinds of experimental methods. 400 Acknowledgment

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