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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 À10 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, tÀ1 ¼ 2pf . It is given by thermodynamic phase. No collective inter-particle order the Ne´el–Brown expression [5,6], exists, while the intra-particle spin structure is FM ordered. ÀÁ 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 ÀÁ distance of D ¼ 6 nm yields E À =k ¼ðm =4pk Þm =D ¼ d d B 0 B E ¼KV ðk Á s Þ2 À 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 Á sj À 3ðsi Á qijÞðsi Á 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 60 PM MFC Tc,bulk 40 T T SPM g, c MZFC M (kA/m) 20 Temperature Tb SFM 0 SSG 0 50 100 T (K) Nominal thickness Fig.

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