Solid state physics, volume 67

The chapter “Collective Effects in Assemblies of Mag-netic Nanoparticles” provides an overview of emergent behavior arisingfrom collections of interacting magnetic particles from the per

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It is our great pleasure to present the 67th edition ofSolid State Physics Thevision statement for this series has not changed since its inception in 1955,andSolid State Physics continues to provide a “mechanism … whereby inves-tigators and students can readily obtain a balanced view of the whole field.”What has changed is the field and its extent As noted in 1955, the knowl-edge in areas associated with solid state physics has grown enormously, and it

is clear that boundaries have gone well beyond what was once, traditionally,understood as solid state Indeed, research on topics in materials physics,applied and basic, now requires expertise across a remarkably wide range

of subjects and specialties It is for this reason that there exists an importantneed for up-to-date, compact reviews of topical areas The intention of thesereviews is to provide a history and context for a topic that has matured suf-ficiently to warrant a guiding overview

The topics reviewed in this volume illustrate the great breadth and sity of modern research into materials and complex systems, while providingthe reader with a context common to most physicists trained or working incondensed matter The chapter “Collective Effects in Assemblies of Mag-netic Nanoparticles” provides an overview of emergent behavior arisingfrom collections of interacting magnetic particles from the perspective ofexperiment, and also in terms of modeling and theory The second chapter,

diver-“Negative Refraction and Imaging from Natural Crystals with HyperbolicDispersion,” describes aspects of material optics with a focus on the fascinat-ing properties of hyperbolic materials whose surprising properties can befound in naturally occurring single-phase materials, as opposed tometamaterials in which these properties are engineered through design.The editors and publishers hope that readers will find the introductionsand overviews useful and of benefit both as summaries for workers in thesefields, and as tutorials and explanations for those just entering

ROBERTE CAMLEY ANDROBERTL STAMPS

ix

Collective Effects in Assemblies

of Magnetic Nanaparticles

D.S Schmool*,1, H Kachkachi†

*Groupe d’Etude de la Matie`re Condens
ee GEMaC, CNRS (UMR 8635) Universit
e de

Versailles/Saint-Quentin, Universit
e Paris-Saclay, Versailles, France

†

PROMES, CNRS-UPR 8521, Universit
e de Perpignan Via Domitia, Perpignan, France

1 Corresponding author: e-mail address: dschmool @fc.up.pt

Solid State Physics, Volume 67 # 2016 Elsevier Inc.

magnetic properties, both in equilibrium and out of equilibrium On the otherhand, assembled nanoparticles into 1D, 2D, or 3D arrays, organized or not,reveal interesting and challenging issues related with their interactions amongthemselves and with their hosting medium, a matrix or a substrate Theensuing collective effects show up through novel features in various measure-ments, such as ferromagnetic resonance (FMR), AC susceptibility and

M€ossbauer spectroscopy, to cite a few Now, for assemblies of small particles(
3–10 nm) one has to deal with the interplay between surface effects andinterparticle interactions whose study requires tremendous efforts In addition,during a few decades one had to struggle with at least two distributions,namely that of the particles size and the anisotropy (effective) easy axes Today,the situation has improved owing to the huge progress in the production ofnearly monodisperse assemblies in well-organized patterns This is one of thereasons for which more theoretical works have appeared recently focusing onsuch newly devised systems

Needless to say that, already at equilibrium, no exact analytical treatment

of any kind is ever possible even in the one-spin approximation (OSP), i.e.,ignoring the internal structure of the particles and thereby surface effects.Only numerical approaches such as the Monte Carlo technique can alleviatethis frustration Indeed, applications of this technique to the case of Isingdipoles can be found in reference [1] The same technique has been used

in reference [2] to study hysteretic properties of monodisperse assemblies

of nanoparticles with the more realistic Heisenberg spin model, withinthe OSP approximation where each particle carries a net magnetic moment

In reference [3], the Landau–Lifshitz thermodynamic perturbation theory[4] is used to tackle the case of weakly dipolar-interacting monodisperseassemblies of magnetic moments with uniformly or randomly distributedanisotropy axes The authors studied the influence of dipolar interactions(DI) on the susceptibility and specific heat of the assembly Today, the lit-erature thrives with theoretical works on the effect of DI on the magneticproperties of assemblies of nanoparticles, most of which make use of numer-ical techniques [2,5–25], because the main interest is for dense assemblies forwhich experimental measurements are relatively easier to perform and theapplications more plausible However, it is important to first build a fairunderstanding of the underlying physics This can only be done upon study-ing model systems that are simple enough for performing analytical devel-opments and still rich enough to capture the main qualitative features of thetargeted systems Analytical expressions come very handy in that they allow

us to figure out what are the main relevant physical parameters and how the

physical observables of interest behave as the former are varied and the ious contributions to the energy compete which other A brief account ofour contribution will be given in the following section.

var-The magnetic properties of magnetic nanoparticles can be rather difficult

to measure, as we saw in the earlier chapter on single particle measurements,where very specialized methods and adaptations are required [26] To over-come some of the problems with the weak experimental signals, many mea-surements are made on assemblies of nanoparticles and elements This meansthat the results obtained are generally an average over the sample and assem-bly and must also be interpreted taking into account the magnetic interac-tions between the particles There have been extensive studies using manytechniques In the following, we aim to give a brief overview of selectedstudies and techniques and will not be an exhaustive review In particular,

we focus on well-known experimental techniques, which have been applied

to the study of nanoparticle systems

Standard techniques, such as magnetometry and AC susceptibility, havebeen applied to the study of magnetic nanoparticle systems Measurementscan be made under the usual conditions since the material quantity is not anissue, as stated previously Where these techniques have shown to be ofimportance is in the study of the superparamagnetic (SPM) behaviorobserved in magnetic nanoparticle assemblies This arises due to the thermalinstability introduced when the magnetic anisotropy, which usually definesthe orientation of the magnetization of the magnetic particle, is insufficient

to maintain its normal orientation In fact the energy barrier is defined as theproduct of the particles magnetic anisotropy constant K and its volume V.Once the thermal energy is of the same order of magnitude as KV, the mag-netization becomes unstable, switching spontaneously between the energyminima of the system As a result, the magnetic measurement, which has acharacteristic measurement time, will sample the magnetic state as being(super)paramagnetic A combination of measurements as a function of tem-perature and applied field allows the system to be defined in terms of itsenergy barrier and the blocking temperature TB, where the magnetization

is stable over the measurement time Indeed, for AC susceptibility ments, a frequency dependence is also important Indeed the averageswitching time between magnetic easy axes is characterized as an attemptfrequency For measurements made with lower characteristic measurementtime, such as M€ossbauer spectroscopy and FMR, corresponding values ofthe blocking temperature will be much higher due to the Arrhenius behav-ior associated with superparamagnetism

measure-Ferromagnetic resonance is a very sensitive method for measuring themagnetic properties of materials via the precessional magnetization dynamicsdefined by the systems magnetic free energy The precessional motion ofthe magnetization is in general strongly influenced by magnetic anisotropiesand exchange effects in solids This is often regarded as the internal effectivemagnetic field experienced by the local magnetic spins of the system Thiscan thus be separated into the various contributions to the local magneticfield, via, magnetocrystalline anisotropy, shape anisotropy, exchange inter-actions, etc In magnetic nanosystems [26–29], this can be adapted to includesurface anisotropy effects as well as magnetic DI between particles This willproduce shifts in the resonance fields and can significantly affect thelinewidth of resonance absorption lines Once again, measurements as afunction of sample temperature can provide further information regardingthe magnetic behavior of nanoparticle assemblies as they move through dif-ferent magnetic regimes.

Nuclear techniques provide another form of probe for the local magneticorder in solids When applied to magnetic nanoparticle systems, information

on the magnetic modifications at a magnetic surface can be established as canthe effects of interparticle interactions One such technique is M€ossbauerspectroscopy, and this has been applied to many Fe-based nanoparticle sys-tems Temperature-dependent measurements provide a sensitive probe ofmagnetic and SPM effects in these low-dimensional systems It has been seen

to be particularly useful for the study of magnetic structures at the surface ofnanoparticles M€ossbauer spectroscopy has also been extensively used toidentify the oxide species which frequently form of metallic Fe and Fe oxidenanoparticles Neutron scattering is another nuclear technique which hasbeen broadly used as a research tool for investigating nanoparticles and mag-netic nanoparticle assemblies This for the most part concerns the scattering

at low angles from the incident neutron beam Such small-angle neutronscattering (SANS) has become a well-established technique in the study

of solids and biological samples Here we consider how it can be applied

to provide information regarding the size and distribution of nanoparticles

in an ensemble Indeed, information regarding the size and shape of samplescan be inferred from scattered intensity distributions Using polarized neu-trons allows magnetic information to be gleaned, which, as in the case of

M€ossbauer spectroscopy, provides information on the surface of the netic particle and with care can be used to establish the spin distribution

mag-or surface anisotropy of magnetic nanoparticles Interparticle interactions

will also affect the magnetic scattering and thus SANS can also provide mation of magnetic interactions between the particles, where studies are fre-quently performed as a function of particle concentration Application of amagnetic field to the sample is also used, where in systems ofmagnetic nanoparticles dispersed in a solvent, or ferrofluid, the interactionbetween the magnetic moments of the particles produces a spatial ordering

infor-of the assembly Core–shell models of magnetic nanoparticles can also beestablished using a combination of SANS and polarized SANS measure-ments, with and without applied magnetic fields

In the following, we focus on some theoretical aspects related to thetreatment of assemblies of magnetic nanoparticles This will discuss theenergy considerations for an ensemble of ferromagnetic nanoparticles,where the individual particle energy is considered as well as the additionalenergy contribution which arises from interparticle (dipolar) interactions.This then allows the equilibrium state of the system to be evaluated andthe magnetization and susceptibility properties to be obtained These con-siderations are followed by a general discussion of dynamic magnetic prop-erties and the AC susceptibility response of an assembly of weakly interactingferromagnetic nanoparticles.Section 3aims to provide a brief overview ofexperimental studies on magnetic nanoparticle assemblies For each of themethods discussed, we will give a short general introduction to the method,where appropriate We will cover both static and dynamic measurementtechniques

2 MAGNETIC NANOPARTICLE ASSEMBLIES:

THEORETICAL ASPECTS

We have recently provided simple expressions for the magnetizationand susceptibility, both in equilibrium and out of equilibrium, which takeaccount of temperature, applied field, intrinsic properties, as well as (weak)

DI [11,12,21,22,30–35] However, this has been done at the price of a fewsimplifying assumptions, either related with the particles themselves or withthe embedding assembly In particular, the study of the effect of DI, which isbased on perturbation theory, applies only to a dilute assembly with an inter-particle separation thrice the mean diameter of the particles In some cases,

we only considered monodisperse assemblies with oriented anisotropy axes.For the calculation of the particle’s relaxation time, we only consider weakfields, small core and surface anisotropies A brief account of these works is

given in the following sections For the study of interplay between dominated intrinsic properties and DI-dominated collective behavior, wemodel a many-spin nanoparticle according to the effective-one-spin problem(EOSP) proposed and studied in Refs [34–39] The EOSP model is a betterapproximation than the OSP model in that it accounts for the intrinsic prop-erties of the nanoparticle, such as the underlying lattice, size, and energyparameters (exchange and anisotropy), via an effective energy potential.

surface-In the simplest case, the latter contains a quadratic and a quartic tions in the components of the particle’s net magnetic moment Thesetwo contributions should not be confused with the core and surface anisot-ropy contributions In fact, the effective model is a result of a competitionbetween several contributions to the energy, namely the spin–spin exchangeinteraction inside the nanoparticle, the on-site anisotropy attributed to thespins in the core and on the surface The outcome of the various competitiveeffects is an effective model for the net magnetic momentm of the nanopar-ticle with a potential energy that contains terms with increasing order in itscomponents mα,α ¼ x, y, z The coefficients of these terms are functions ofthe atomic physical parameters, such as the constant of the on-site anisot-ropies and exchange coupling, together with those pertaining to the under-lying crystal structure

contribu-In the following section, we will give a brief account of these theoreticaldevelopments, related to the intrinsic, as well as collective features of thenanoparticles We will also discuss an excerpt of the main results they lead

to, for the magnetization and susceptibility

2.1 Model

We will illustrate our theoretical developments in the simplest situation of amonodisperse assembly and oriented anisotropy More general situations ofpolydisperse assemblies, with both oriented and random anisotropy, can befound in the cited works, e.g., in Ref [31] We commence with a mono-disperse assembly of N ferromagnetic nanoparticles carrying each a mag-netic moment mi¼ misi, i¼ 1,…,N of magnitude m and direction si,withjsij ¼ 1 Each magnetic moment has a uniaxial easy axis e aligned alongthe same z-direction The energy of a magnetic momentmiinteracting withthe whole assembly, and with a (uniform) magnetic field H ¼ Heh, reads(after multiplying by β 1/kBT)

Ei¼ Eð0Þi +EDI

where the first contributionEð0Þi is the energy of the free (noninteracting)nanocluster at site i, comprising the Zeeman energy and the anisotropy con-tribution, i.e.

Eð0Þi ¼ xisi eh+A sð Þ,i (2)where A sð Þ is a function that depends on the anisotropy model and isi

Cð 0, 0 Þ¼ 4π Dz1

3

(7)

with Dzbeing the demagnetizing factor along the z-axis [3,31] K2and K4

are the constants of the uniaxial and cubic anisotropy, respectively

In the literature, especially for the experimental work, the magneticbehavior of a nanoparticle is often approximated as a macrospin usingOSP or equivalently the Stoner–Wohlfarth model [40] As discussed earlier,

in this approximation the spins of the ferromagnetic particle are considered

to be sufficiently well exchange-coupled that they move together in any

reorientation or reversal of the magnetization In equilibrium the net netic moment is held in a direction according to the uniaxial anisotropy ofthe magnetic nanoparticle, which in the absence of an applied magnetic fieldwill be along the easy axis When a magnetic field is applied, the magneti-zation will reorient in accord with the minimum of the following energy

whereϑ and ψ are the spherical angles between the easy axis and the ticle’s magnetic moment and the applied magnetic field H, respectively.Minimizing this simple energy, one finds two minima and a maximum withtwo energy barriers expressed, for the case where the applied magnetic field

par-is parallel to the easy axpar-is, as follows

to take account of these observations, an effective anisotropy constant Keffisderived on the basis of some arguments borrowed from 2D magnetism.More precisely, Keffis proposed in a form that comprises uniaxial (volume)and surface contributions, namely

Keff ¼ KV+ 6KS

where D is the particle diameter for a spherical particle [44] In fact, it wasshown in Ref [38] that Eq (11) is only valid in elongated nanoparticles and amore general formulation is proposed by the EOSP approach

Then, for an assembly of randomly oriented magnetic noninteractingOSP nanoparticles, the (angular) averaged Stoner–Wohlfarth model is used

to predict the reversal of the collective sample However, many experimentshave shown that surface effects again have to be included and, as such the

SW model should be replaced by a more precise one Accordingly, the effect

of finite size and surface anisotropy on the SW switching mechanisms havebeen extensively studied in Ref [45] In particular, it was shown that forweak surface anisotropy the spins inside the nanoparticle are nearly collinearleading to a coherent switching of the particle’s magnetization However, asthe surface effects become stronger, the spin switching operates via a cluster-wise mechanism

2.2 Equilibrium Properties : Magnetization and Susceptibility

In Ref [31] it was shown that in a dilute assembly the magnetization of ananoparticle at site i (weakly) interacting with the other nanoparticles ofthe assembly is given by (to first order inξ)

szi ’ sz

i 0+XN k¼1

i i0¼ 0, and that this expression is only valid for

a center-to-center interparticle distance larger than thrice the mean diameter

of the nanoparticles[33] This implies that the magnetization of an acting nanoparticle is written in terms of its “free” (with no DI) magnetiza-tion szi 0(and susceptibility@x i szi 0), with of course the contribution of theassembly hosting matrix entering via the lattice sum in Eq (12)

inter-The free-particle magnetizationmð Þi0  hsz

hszi0ðσ 6¼ 0,ξ ¼ 0Þ ¼ C1¼ e

σZð Þ 0 k

There are various asymptotes that can then be derived forhszi0, see Refs.[34, 46] for such developments

Then, in the dilute limit, upon using Eq (12) it is straightforward

to derive an expression for the magnetization of the assembly that takesaccount of the DI Furthermore, as it will be seen later on, an explicit expres-sion for mð Þ 0 allows one to derive an approximate expression for the mag-netization of a (weakly) interacting assembly of EOSP nanoparticles byincluding the cubic anisotropy term with coefficientζ We recall, however,that this applies for relatively weak surface anisotropy and thereby to anequilibrium magnetic state with quasi-collinear spins

Therefore, for monodisperse assemblies we have xi¼ x, σi¼ σ, ξij¼ ξ

In this case, the magnetization of a (weakly) interacting particle, given by Eq.(12), simplifies into the following expression

sz

h i ’ mð Þ 0 1 +ξCð0,0Þ@m@xð Þ0

This indicates that the relevant DI parameter, to this order of approximation,

is in fact the parameter introduced earlier

Note that the lattice sumCð0,0Þis in fact the first of a hierarchy of lattice sums(see Ref [31]).

Next, the longitudinal susceptibility χð Þ 0

k ¼ @mð Þ 0=@x is given by (see,e.g., Ref [46] for the notation)

of the anisotropy increases However, in low fields this is not globally so,

because the competition between Zeeman, thermal and anisotropy butions to the energy results in a crossing between the various magnetizationcurves, as has been observed, e.g., for maghemite particles [48,49] In addi-tion, we see that there is a large deviation from the Langevin law due to sev-eral contributions to the energy, ignored by the Langevin law, especiallyanisotropy Moreover, the results inFig 1B show that the larger the meandiameter of the assembly, the larger the value ofσ, and thereby the larger theexpected deviation from the Langevin curve.

contri-InFig 2we plot the Langevin function (full line) and the Monte Carloresults (symbols) for the magnetization of an interacting assembly of (N ¼

10 10  5) lognormal-distributed moments, with random anisotropy,and for different values of the interparticle separation Here we use the sameassemblies as inFig 1 The intensity of DI, or equivalently the value ofξ, isvaried by varying the lattice parameter a enteringξ [see Eq (6)] More pre-cisely, the parameter a is taken as a real number k times the mean diameter

Dmof the assembly, i.e., a¼ k  Dm Thus, large values of k correspond to anisotropically inflated lattice with large distances between the magnetic

0.2 0.4 0.6 0.8

1 Reduced magnetization (per particle)

3 nm

7 nm Langevin

moments, and thereby weak DI These results, obtained for an oblate ple, confirm the fact that in this case DI suppress the magnetization Thisresult has also been obtained by perturbation theory in Ref [34] whoseresults are shown inFig 3, which are plots of Eq (17) using Eq (14) for mð Þ0.

sam-As discussed in Ref [31] and references therein, DI are anisotropic actions and thus contribute to the effective anisotropy Since the anisotropy

inter-is uniaxial and oriented, i.e., with a common easy axinter-is, its effect leads to amagnetization enhancement In contrast, the DI effect depends on the sign

ofξ
(or more precisely that ofCð 0, 0 Þ), which is related to the sample’s shape.For instance, in the case of oblate samplesCð 0, 0 Þ< 0 leading to a reduction ofthe magnetization, while for prolate samplesCð 0, 0 Þ> 0 and thereby DI con-tribute to enhance the assembly’s magnetization Consequently, for oblatesamples the (oriented) uniaxial anisotropy and DI have opposite effects whilefor prolate samples they play concomitant roles

In the presence of not-too-strong surface anisotropy, one can model thenanoparticle using the EOSP model upon which the free-particle partitionfunction Zkð0Þis replaced by [34]

0.0 0.2 0.4 0.6 0.8 1.0

Fig 2 Reduced magnetization (per particle) of an interacting assembly of N ¼ 10

10  5 lognormal-distributed magnetic moments with mean diameter D m ¼ 7 nm and random anisotropy Monte Carlo in symbols and in lines the analytical expressions (34) of Ref [ 31 ] In the inset, the parameters k is defined in the text while ζ ¼ x/ξ.

Upon performing a double expansion, with respect to x for low field and

to 1/σ for high anisotropy barriers, we obtain the following expression forthe magnetization for the EOSP particle (see Eq (3.39) of Ref [46] for thecaseζ ¼ 0 but arbitrary field)

:(19)Next, writing this in the form

we can easily infer the EOSP corrections to the linear and cubic ities (in the limit of a high anisotropy barrier) due to surface anisotropy ofintensityζ

contribu-ζ < 0 In the former case, the particle’s magnetic moment at equilibriumadopts an intermediate direction between the z-axis and the cube diagonal

So, as ζ increases the magnetic moment gradually rotates away from thez-axis and thereby its statistical average, or the magnetization, decreases

In the case of negative ζ the two anisotropies cooperate to quickly drivethe magnetization toward saturation

Next, using the expression (19) for the free-particle magnetization, as afunction of the applied field x, uniaxial anisotropy (and temperature)σ andsurface anisotropy ζ, in Eq (12) or (15) we can investigate the interplaybetween surface effects and DI, i.e., a competition between the terms in

ζ and ξ
, respectively This was done in Ref [34] The same competitionwas also studied numerically in Ref [21] The outcome of this procedure

is the following approximate expression for the (average) magnetization

of a weakly interacting assembly of EOSP nanoparticles

m x,σ,ζ,ξ

’ χ
1ð Þx +χ
3ð Þx3 (21)where

χ

1 ð Þ’ χð Þ 1 +ξ
12

σ 2 1 

3σ

ζσ

,

(22)

are the linear and cubic susceptibilities (20) augmented by the DI tion of intensity
ξ.

contribu-This asymptotic expression helps understand how surface anisotropycompetes with DI The surface contribution with intensity ζ, which plays

an important role in the magnetization curve, couples to the DI contributionwith intensity
ξvia the term with coefficientξ
ζ Hence, the overall sign ofthe latter determines whether there is a competition between surface and DIeffects or if the changes in magnetization induced by the intrinsic and col-lective contributions have the same tendency Accordingly, plots of themagnetization, which take into account both surface effects and DI, areshown in Fig 4 as a function of the field x, for an oblate sample with

Nx Ny  Nz¼ 20  20  5 and a prolate sample with 10  10  20,with the respective values of Cð0,0Þ’ 4:0856 and 1.7293

As discussed earlier, for oblate samplesξ
< 0, DI tend to suppress netization, whereas for prolate samples
ξ> 0 they enhance it Indeed, we seefrom Eqs (22) that surface anisotropy and DI may have opposite or concom-itant effects depending on their respective signs In Ref [11], it was foundthat the magnetization enhancement in dilute assemblies of maghemitenanoparticles of 3 nm in diameter is suppressed when the concentrationincreases In accordance with the present results, DI tends to smooth outsurface effects, or the other way round, the surface seems to have a screeningeffect on DI

sam-2.3 Dynamic Properties

The dynamics of an assembly of magnetic nanoparticles is a rich environmentfor the study of equilibrium and out-of-equilibrium many-body statisticalphysics Indeed, as discussed earlier, there are physical phenomena whichoccur over wide ranges of spatial and temporal scales The relevant length scalecan range from the Angstr€om, through the nanometer to the millimeter, as we

go from the atoms inside of the nanocrystal, through the nanoparticle, to theassembly thereof On the other hand, the time scale also spans a wide rangethat starts at the femtosecond timescale and ends with a duration of the order

of a few hours, as in relaxation phenomena observed in the isothermal andthermoremanent magnetization Obviously, these time scales are a directconsequence of a competition between short-range and long-range interac-tions operating at different length scales For this reason, among others, it is notpossible to come up with a theory that covers all length and time scales Forshort-time regimes the physics is usually described with the aid of the Landau–Lifshitz equation and its variants, deterministic or stochastic, damped orundamped, local or macroscopic For collective effects, occurring at theassembly scale, the Monte Carlo technique is more appropriate, even thoughthe problem of an efficient algorithm for dynamical processes is not entirelysolved so far, see for instance the works in Refs [51–56] As for analyticalapproaches, there are a very few attempts to tackle the problem, mainlybecause of the tremendous difficulty to calculate the relaxation rate of a many-spin system The main difficulty resides in the fact that it is impossible to ana-lyze the large number of extrema of a multivariate energy potential, in thepresence of several parameters, such as size, shape, applied fields, etc

A way out of this difficulty was proposed in Ref [36] where the EOSP modelwas built for a spherical nanoparticle with N
eel anisotropy on the surface and

no anisotropy in the core, and in Refs [38,39], where it was extended to amore general situation Indeed, the EOSP approach makes it possible to inves-tigate the dynamics of an interacting assembly while taking account of theintrinsic features of the nanoparticles, since this model is a macroscopic modelwhose energy potential depends on the nanoparticle’s parameters This sim-plification allows us to compute the relaxation time taking account of theeffect of surface anisotropy, in addition of course to that of the (effective) uni-axial anisotropy and the applied (static) magnetic field This was done in Ref.[50] Then, in Ref [35] the AC susceptibility of a (weakly) interacting assem-bly of EOSP nanoparticles was computed, after generalizing the calculation ofthe relaxation rate of such particles

The dynamic response of the EOSP assembly is given by the AC tibility which, for an arbitrary angle ψ between the (common) easy axisand the field direction, the effective susceptibility may be written as

suscep-χ ¼ suscep-χkcos2ψ + χ?sin2ψ According to Debye’s model [33,46,57] we have

χ ωð Þ ¼ χkðT,HÞ

1 + iωτk cos2ψ +χ1 + iωτ?ðT,HÞ

? sin2ψ, (23)whereτkandτ?are the longitudinal (inter-well) and transverse (intra-well)relaxation times andχk(T, H) andχ?(T, H) are, respectively, the longitudi-nal and transverse components of the static susceptibility

For an assembly with oriented anisotropy in a longitudinal field (ψ ¼ 0),one assumes that the transverse response is instantaneous, i.e.,τ?’ 0 In thiscase the AC susceptibility is given by Eq (23) or using τk ¼ Γ1 and

withλ being the damping parameter Γðx,σ,ζ,ξ
,λÞ is the relaxation rate of

an EOSP nanocluster weakly interacting within the assembly τD ¼(λγHK)1is the free diffusion time, HK¼ 2K2V/M the (uniaxial) anisotropyfield, andγ ’ 1.76  1011(Ts)1

the gyromagnetic ratio For example, forcobalt particles the anisotropy field is HK
0.3 T, and for λ ¼ 0.1  10, τD

2 1010 2  1012s

Now, if we restrict ourselves to the linear susceptibility,χeqis equal to

χ
1ð Þ given in Eq (22) The second quantity that needs to be calculated inorder to fully evaluate the susceptibility in Eq (24) is the relaxation rate

by [59]

FðαÞ ¼ 1 + 2ð2α2eÞ1 =ð2α 2 Þγð1 +2α12, 1

with γða,zÞ ¼Rz

0dt ta1et, the incomplete gamma function, and where

α ¼ λpffiffiffiσ Asymptotic expressions of F(α) are [59]

FðαÞ ’

ffiffiffiπp

1

3+

ffiffiffiπp

Γ ¼2πj jκ Z

s

Zm

whereZmandZ
s are, respectively, the partition functions in the vicinity ofthe metastable energy minimum and the saddle point, obtained for a qua-dratic expansion of the energy The attempt frequencyκ is computed uponlinearizing the dynamical equation around the saddle point, diagonalizingthe resulting matrix and selecting its negative eigenvalue [60, 61]

In Ref [35] the relaxation rateΓ was calculated in various situations of

an EOSP particle including the effective uniaxial and cubic anisotropy andthe applied magnetic field A detailed analysis of the various energy extrema

is presented in Ref [35], and analytical expressions were given for the ation rate as a function of temperature, effective uniaxial anisotropy (σ),

relax-surface anisotropy (ζ), and applied magnetic field The authors of Ref [35]then investigated the interplay between interparticle DI and intrinsic surfaceanisotropy, in the case ζ > 0 where surface (cubic) anisotropy favors themagnetic alignment along the cube diagonals χ0 and χ00 were computedfor various values of the surface anisotropy coefficient ζ, for both prolateand oblate assemblies Owing to the fact that the effect of increasing ζ is

to draw the particle’s magnetic moment toward the cube diagonals, it cally plays the same role in a prolate sample where the magnetization isenhanced along the z-axis, or in an oblate sample where the magnetization

basi-is enhanced in the xy plane

The results inFig 5 show an example that illustrates the competitionbetween surface anisotropy and DI contribution to the real component ofthe AC susceptibility They were obtained for the finite value ξ
¼ 0:008and an increasing (but small) surface anisotropy parameterζ It can be seenthat the surface anisotropy, in the present case of positiveζ, has the oppositeeffect to that of DI This again shows that there is a screening of DI by surfaceeffects and confirms the results of Ref [34] for equilibrium properties forboth negative and positive ζ, as discussed earlier

Our theoretical calculations of the AC susceptibility of magneticnanoparticles which accounts for the intrinsic properties (e.g., surface

1/s

0 2 4 6 8 10 12

0.01 0.05 0.1

 ωτ D =ð2πÞ ¼ 0:01 h ¼ 0 Source: Reprinted figure with permission from F Vernay,

Z Sabsabi, H Kachkachi, AC susceptibility of an assembly of nanomagnets: combined effects of surface anisotropy and dipolar interactions, Phys Rev B 90 (2014) 094416 Copyright (2009) by the American Physical Society.

effects) as well as the collective effects (due to DI) were then used [35] toprovide a microscopic derivation of the so-called Vogel–Fulcher law [seealso previous works [17, 66–70]]

Our results are in full agreement with previous works [67, 68,70] andfurther extends them in that they take into account: (i) surface anisotropy,(ii) the particles spatial distribution and shape of the assembly, and (iii) thedamping parameter A full discussion can be found in Ref [34] Here weonly report the following expression found there forθVF

to the one derived in Ref [68], includes both the damping parameter and theshape of the assembly, through the expression ofS λð Þ In addition, we notethatξ is proportional to the assembly concentration [34] CVand thereby to

a3, a being the interparticle separation Therefore, we expect that in theabsence of surface anisotropy, θVF would scale asθVF
C2

V
a6 In Ref.[17] experimental estimates of θVF are given for an assembly of Ninanoparticles with varying concentration A comparison of Eq (31) withthe corresponding experimental data is given inFig 6

On the other hand, the first term in Eq (31) accounts for the tion from surface anisotropy In practice it should be possible to adjust theassembly characteristics (assembly shape, particles size and underlying mate-rial) so as to achieve, to some extent, a compensation between surface effectsand the DI contribution This could in principle suppress the dependence of

contribu-θVF on the assembly concentration In addition, the term inζ can also beused to extract from the experimental data an estimate of the (effective) sur-face anisotropy coefficient ζ by reading off the intercept from the plot in

Fig 6 Furthermore, it is worthwhile emphasizing thatθVFis not dent of temperature, as is very often assumed in the literature First, the

indepen-temperature appears in the second term in (31), being related to the DI tribution Even if this term becomes negligible for very diluted assemblies, ifsurface anisotropy is taken into account (ζ6¼0), e.g., for very smallnanoparticles, Eq (31) shows that the phenomenological parameterθVFis

con-in fact a lcon-inear function of temperature via the term con-inζ This can be stood by noting that the surface anisotropy, which is of cubic nature in theEOSP model, drastically modifies the energy potential and thereby affectsthe dynamics of the particle’s magnetization As a consequence, the effect

under-of DI becomes strongly dependent on the thermal fluctuations and the mentary switching processes they induce

ele-Two applications of this formalism have been recently studied by one ofthe authors, namely, on the one hand, the effect of DI on the FMR char-acteristics of a 2D array of nanoparticles and, on the other, the effect of DIand their competition with a DC magnetic field in the behavior of the spe-cific absorption rate (SAR), which is relevant in magnetic hyperthermia.The two corresponding works are in preparation and will be submittedfor publication elsewhere In particular, the analytical expression of the

AC susceptibility obtained with the help of this formalism make it possible

to compute the SAR and study its behavior as a function of various eters pertaining to the assembly Indeed, it is quite easy to show that, in thelinear response, the SAR is proportional to the out-of-phase componentχ00

param-of the AC susceptibility

0 2 4 6 8 10 12

qVF

Data (Masunaga et al.)

Fit qVF = 0.5633 + 0.05405 C v

Fig 6 θ VF against the assembly concentration Experimental data (stars) [ 17 ] and fit of

Eq ( 31 ) (full line) Source: Reprinted figure with permission from F Vernay, Z Sabsabi,

H Kachkachi, AC susceptibility of an assembly of nanomagnets: combined effects of surface anisotropy and dipolar interactions, Phys Rev B 90 (2014) 094416 Copyright (2009) by the American Physical Society.

We recall that in all of these developments involving DI, we have sidered only dilute assemblies and as such we have used perturbation theory

con-to derive (semi-)analytically expressions for the magnetization, ity, and relaxation rate that include the DI contribution In the next section

susceptibil-on the experimental aspects, we will discuss the situatisusceptibil-on of more denseassemblies

As a final discussion it is worth addressing the issue of the relaxation rate(29) and its use in the literature for modeling the dynamics of ensembles ofmagnetic particles In general, as can be seen in the above developments,Γ is

a function of various quantities, such as the anisotropy (core and surface), theapplied magnetic field, the DI, and so on Its calculation has been performed

in various situations and limiting cases However, in the experimental ature the relaxation relate is very often taken in the form of Arrhenius’ law

liter-τDΓ0∝ eσ The main reason evoked is that the behavior ofΓ is dominated

by the exponential Obviously, this is not quite so because this law ignores amajor physical phenomenon that is damping Furthermore, if the (effective)anisotropy is cubic, the switching mechanisms are rather different from those

of a uniaxial anisotropy

This simple Arrhenius’ law and the ensuing simplifications are then used

to interpret, for example, the results for the hysteresis loop and, in particular,for estimating the coercive field More precisely, if the magnetic field isapplied parallel to the easy axis (ψ ¼ 0°), the energy minima become deeper

as the field is increased in this orientation Then, Eq (9) is used to obtain thecoercive field as a function of temperature

HC¼ HK 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lnðτm=τ0Þσ

appears ferromagnetic, while above it, it appears demagnetized and is said to

be SPM Accordingly, within the same approximation, the so-called ing temperature is introduced

charac-Obviously, in order to understand the role of surface anisotropy and itsinterplay with the DI in an ensemble of magnetic nanoparticles, one has toresort to the more general developments discussed earlier, related with thebehavior of the magnetization, the susceptibility, and the relaxation time

As outlined earlier, the proximity and intervening medium betweenmagnetic entities will affect the way they interact Whatever the mechanism

of this interaction, the magnetic properties of the particles will deviate fromthose of their isolated state This adds to the already extrinsic nature of theirbehavior caused by the reduced physical dimensions of the magnetic particles,see the previous section The coupling of magnetic objects will produce a col-lective magnetic behavior of the ensemble of nanoparticles Such a situationcan be considered as being produced by the effective field on an element ofthe assembly due to the interaction fields of all the other elements Indeed,techniques, such as FMR, precisely measures this effective field

Since an assembly of nanoparticles has no specific limit to the number

of particles measured, in addition to the methods discussed earlier, manyother techniques can be used to perform experimental studies Whenconsidering an ensemble of nanoparticles, the principal parameters thatcharacterize the sample will be particle shape, size (average) and size

distribution (mono or polydisperse), as well as the average particle tion The matrix in which the assembly is suspended will also be of impor-tance as interaction mechanisms depend on the intervening media betweenthe magnetic particles For regular arrays of magnetic elements, we can fur-ther consider the element shape and the periodicity and symmetry of thenanostructure.

separa-3.1 Magnetometry

Relaxation effects in assemblies of nanoparticles have been studied by ious methods, such as DC magnetometry, AC susceptibility, as well via thetemperature-induced spontaneous magnetic noise With conventionalmagnetometry, the blocking temperature is frequently determined usingzero-field cooled (ZFC) and field cooled (FC) measurements as a function

var-of sample temperature The ZFC magnetization, MZFC(T), is obtained byheating the sample up to room temperature, then it is cooled in zero appliedfield to low temperature such that the magnetic moments of thenanoparticles are randomly oriented Then a small magnetic field is applied(typically of the order of 200 Oe or less) so that there is a measurable mag-netization as the temperature is then increased MZFCincreases as the ther-mal energy is raised, and there is sufficient energy for the particle to startaligning with the applied field However, as the temperature increases fur-ther thermal fluctuations then effectively reduce the measured magnetiza-tion The MZFC(T) can be used to determine the blocking temperature[71,73] For assemblies of nanoparticles with a size distribution (i.e., poly-dispersion), which is frequently the case, only those particles with TBlessthan the measuring temperature will contribute to the magnetization.The size distribution typically follows a so-called log-normal distribution,given as:

and extract the size distribution from the variation of MZFC(T) [75] The FCmagnetization, MFC(T), is measured by initially applying a small magneticfield to the sample at room temperature The sample is then cooled andthe magnetization increases as the thermal fluctuations reduce In contrastwith the ZFC magnetization, MFC(T) saturates at low temperature Theblocking temperature is then identified as the deviation between MZFC(T)and MFC(T), seeFig 8.

In an isolated magnetic particle or in an assembly of identical interacting particles, the magnetization will decay, due to thermal fluctua-tions, following an exponential decay of the form:

where M0is the magnetization at time t0[77] In a an assembly with a sizedistribution, there will be a subsequent distribution in the energy barrier,which is compounded by variations in particle orientation and particle inter-actions will further compound this distribution, broadening further therange of relaxations times The consideration of this problem led to the con-cept of magnetic viscosity, S The quantitative theory of relaxation was con-sidered by N
eel, and is known as the N
eel model [78] Further analysis byStreet and Woolley [79] and Barbara et al [80] led to a magnetizationexpressed as a function of time after changing the applied magnetic field as:

MðtÞ ¼ M0 1

Zexp
tνeE=k B T

where n(E)dE is the number of particles with an energy between E and E + dE.For a smooth variation of n(E), it has been shown that [80]:

When interactions between nanoparticles in an assembly become verystrong, as would be the case for high concentration, then collective excita-tions and states can be expected This will greatly affect the magnetic

FC

300

Fig 8 Zero-field cooled and field cooled magnetizations as a function of temperature for γ-Fe 2 O 3 nanoparticles of
7 nm diameter It will be noted that the value of the block- ing temperature TBis dependent on the applied magnetic field used in the measurement Source: Reprinted figure with permission from P Dutta, A Manivannan, M.S Seehra, N Shah, G.P Huffman, Magnetic properties of nearly defect-free maghemite nanocrystals, Phys Rev.

B 70 (2004) 174428 Copyright (2004) by the American Physical Society.

properties of the assembly as compared to low concentrations and wheninteractions are negligible The DI, for example in a system of randomlyoriented nanoparticles with a macrospin moment of μ ¼3000μB and acenter-to-center separation of R ¼ 6 nm, yields a dipolar energy of EDDI

¼ (μ0/4π)μ2

/R3[82] With such large particle concentrations, a new netic regime can be identified which is characterized by the crossover fromsingle particle blocking (as described by the SPM state) to collective freezing[72, 83, 84] Two distinct collective states are possible For intermediatestrength interactions, random particle spatial distribution and sufficientlynarrow size distribution, a superspin glass (SSG) state will exist In this casethe superspins collectively freeze into a spin-glass phase below a criticaltemperature, Tg[82,85,86] With more concentrated systems and higherinteraction strengths a superferromagnetic (SFM) state is encountered This

mag-is characterized by ferromagnetic interparticle like correlations [87–89]

A consideration of the DI leads to an expected phase diagram for the ing in a quasi-2D superspin system as obtained from a discontinuousmultilayer (DM) sample This is illustrated inFig 9 For temperatures above

para-T c Source: Reprinted from O Petracic, X Chen, S Bedanta, W Kleemann, S Sahoo, S Cardoso, P.P Freitas, Collective states of interacting ferromagnetic nanoparticles, J Magn Magn Mater 300 (2006) 192–197 Copyright (2006), with permission from Elsevier.

the bulk Curie temperature, Tc, bulk, the system will be paramagnetic Belowthis temperature some form of spontaneous magnetic order will occur insideeach particle Finite size effects can also affect the Curie temperature of a fer-romagnet in an analogous way to that of the melting temperature of a metallicnanoparticle For low particle concentrations (small nominal thickness in the

DM system), the nanoparticle assembly will behave as an SPM The effects ofany interparticle interactions for low concentrations are not significant sinceblocking will disguise any transitions at low temperature We can expect 3Darrays of nanoparticles to behave in a similar manner

As the concentration increases, interparticle or collective ordering can takeplace where the ordering temperature will be greater than the blocking tem-perature For systems with random orientations and size distributions an SSGphase will occur before and longer range ordering with a SFM phase In terms

of magnetometry measurements, the SSG state can be observed in ZFC/FC

M vs T measurements For the DM system of CoFe nanoparticles in an Al2O3

matrix, the experimental curve for the SSG state is shown inFig 10 Here theusual peak in the magnetization is evident; however, a small minimum in

MFC(T) (as marked with an arrow) is observed for the SSG, which arisesfrom small paramagnetic clusters dispersed between the nanoparticles [82]

of SSG systems Source: Reprinted from O Petracic, X Chen, S Bedanta, W Kleemann,

S Sahoo, S Cardoso, P.P Freitas, Collective states of interacting ferromagnetic nanoparticles, J Magn Magn Mater 300 (2006) 192 –197 Copyright (2006), with permis- sion from Elsevier.

In addition to this the sample also exhibit a “memory effect,” where the

MZFC(T) curve shows a dependence on the waiting time at a temperaturebelow the blocking temperature [72,83,85,90]

As the concentration of the nanoparticle assembly further increases, theinteraction strength between the particles will correspondingly grow Thiswill eventually lead to a strong coupling and a fully collective behavior with

an SFM state Such a system will be characterized by domain wall motion as

in a fully ferromagnetic state

3.2 AC Susceptibility

Another measurement frequently employed in the study of magnetic particle assemblies is AC susceptibility Theoretical aspects of this techniquewere outlined in Section 2 One advantage of this method is that a staticmagnetic field is not necessary to perform measurements Ac susceptibilitymeasurements are usually taken in the frequency range below 100 kHz Fornanoparticle systems, a peak in the imaginary component,χ00, of the AC sus-ceptibility, is typically observed at the blocking temperature We note thatthe measurement time, being proportional to the inverse of the frequencyused in the experiment, will be significantly shorter than that used formagnetometry, and hence will provide a larger blocking temperature, see

nano-Eq (33) The frequency dependence will highlight the time scale overwhich the magnetization of the nanoparticle is stable Since low magneticfields are used for the collection of AC susceptibility data, small rotations

of the magnetization can arise as well as thermally assisted reversal For peratures in excess of the blocking temperature,χ00is small andχ0, the realcomponent of the AC susceptibility, will follow a Curie law: χ0∝1=T,indicative of paramagnetic-like behavior The slope of 1/χ0vs temperatureallows the determination of the particle volume Therefore, in polydispersesystems the analysis is more complex, but careful fitting can also yield the sizedistribution [91] In Fig 11A, the real and imaginary data for the AC sus-ceptibility of Co0.1Cu0.9 alloys are shown as a function of temperature atvarious frequencies This nanosystem shows blocking behavior, however,the effect of interactions is seen to be important since the Arrhenius behavioryields unphysical values of the relaxation time, see Fig 11B [92,93].The behavior of the blocking temperature depends on the concentration

tem-of the nanoparticles in the assembly, where a monotonic decrease tem-of theblocking temperature was observed for iron-nitride nanoparticle systemsfor higher concentrations [94] At lower concentration sample showed a

much weaker dependence of TB on the applied field However, otherresearchers have found that the peak temperature of the susceptibility canhave a nonmonotonic dependence on the applied field [30, 95–97] Inthe dilute magnetic assembly of FePt nanoparticles, Zheng et al obtain amaximum in the peak temperature vs applied field which is due to the dis-tribution of energy barrier sizes (and hence particle size) and the slowdecrease of the high-field magnetization above the blocking temperature[98], see also Ref [30] The apparent anomaly is understood in terms ofthe competition between the decrease in the moments of the SPM particlesand the increase in the moments of newly relaxed larger particles and hence

is related to the distribution of particle size [30] The Langevin function isused to describe the SPM behavior of unblocked particles which accountsfor the slower than expected decrease of the magnetization based on theCurie law [30,74]

Masunaga etal have also performed detailed studies of the AC tibility in Ni nanoparticles and in particular for NPs of 4–5 nm in diameter,with mean separations of 14–21 nm [17] The temperature-dependent mea-surements for the real and imaginary components ofχacare shown inFig 12

suscep-A comparison ofFigs 5and12shows a good qualitative agreement betweenexperiment and theory, where we note that 1/σ is proportional to thesample temperature inFig 5 Again SPM behavior is evidenced by the fre-quency independence ofχ0 at high temperature (T≫Tm 0 ) while having afrequency-dependent variation for (T≪Tm 0 ), where the NPs are blocked[72] The frequency dependence can in principle also be attributed to spin

glass-like behavior, arising from strong DI between the randomly orientedparticles The analysis of theχacdata for a range of samples of different con-centrations gives rise to the Arrhenius–N
eel plots shown inFig 13 In thisfigure, we note that the straight line plots are taken from the Arrhenius equa-tion of the form, see Eq (28),

where△E corresponds to the energy barrier between minima, of an tude defined by the anisotropy strength Also shown is the correction to thisrelation, which accounts for interparticle interactions, e.g., via dipolarcoupling, and is considered as a temperature shift T0 This relationship isknown as the Vogel–Fulcher law, see Eq (30), which can also be expressed

sus-As discussed inSection 2.3, Vernay et al [35] interpreted the results based

on a consideration of the surface anisotropy of the particles and the particle interactions, leading to a good agreement with experiments, see

inter-Fig 6

The AC susceptibility has also been experimentally used to distinguishbetween SPM, SSG, and SFM behavior in DM nanoparticle samples usingCole–Cole plots These display somewhat different aspects for the differentinterparticle coupling regimes [82,99]

3.3 Magnetization Dynamics

At much higher frequencies, we enter the regime of FMR, where manystudies have been performed on ferromagnetic nanoparticle assemblies.The basis of FMR theory can be expressed from the dynamical motion

of the magnetization vector at the maximum angle of precession, wherethe equation of motion is expressed in the form of the Landau–Lifshitzequation

composed of the applied external magnetic field as well as contributionswhich take account of internal effective fields due to magnetic anisotropiesand the exchange field This can also include the dipolar field due to mag-netic interactions between the particles of an assembly In this case the dipo-lar field must take into account the spatial distribution of the nanoparticles It

is often useful to define the relation between the effective magnetic field andthe free energy of the system This can be expressed as:

Heff¼  1

where E is the free energy density of the magnetic system and u ¼ M/Mdefines the unit vector in the direction of the magnetization The resonancecondition can be derived from the Landau–Lifshitz equation, which can beexpressed in the form of the Smit–Beljers equation as (a formal derivation ofthis equation can be found in a number of texts, see for example Refs [28,

be a principle component due to static and dynamic (microwave) magneticfield that are required Additional contributions will also be required and aretypically due to magnetostatic (or shape) energy and magneto-crystallineanisotropies

As we have seen, in systems of nanoparticles we generally need to takeinto account the interparticle interactions For a nonmetallic matrix, it isusually sufficient to consider the DI This can then be included in the freeenergy of the system Netzelmann [103] introduced the idea of separatingthe magnetostatic energy into the particle demagnetization term and thesample demagnetization term, where later corrections from Dubowik[104] and Kakazei etal [105] give the magnetostatic energy:

EMS¼1

2ρ 1 ρð ÞM  N$P M +1

where N$p, s represents the demagnetization tensor of the particle ( p) andsample (s), respectively, andρ the volume fraction of magnetic nanoparticles

in the assembly, which can be specified as:

InFig 14the angular variation of the resonance field for a discontinuousCoFe/Al2O3multilayer system illustrates the agreement with theory for dif-ferent effective thicknesses of the nanogranular CoFe layer It will be notedthat the difference in the in-plane to out-of-plane resonance field increases asthe effective thickness increases, meaning that the particles are on averagelarger and closer together This will increase the mean DI strength Weobserved small discrepancies for the smaller particles at angles intermediatebetween the parallel and perpendicular configurations, which may arisefrom deviations of the magnetization from the saturated state due to surfaceanisotropy effects.

The separation of in-plane and out-of-plane interactions was also sidered by Kakazei et al [109, 110] and Majchra´k etal [111,112] In theformer, the authors study the transition from the continuous to the discon-tinuous regime by studying the variation of the effective magnetic layer

(CoFe) thickness, which changes the average particle size and separation.The effective field was obtained using a Kittel analysis of the FMR, asexpressed by:

Heff ¼ 4πMs2KV

Ms

4KS

where KV, Sdenote the surface (N
eel) and volume anisotropies

Temperature-dependent measurements of the FMR were also studied inthe discontinuous magnetic multilayer system [113] The resonance field as afunction of the temperature for these samples is shown inFig 15A Here wesee that the resonance field, Hres, reduces with T, from 300 to 100 K, in amanner expected from classical FMR behavior of ferromagnetic materials

At approximately 90 K there is a significant enhancement of Hres, which isvery marked for lowest effective thicknesses InFig 15B we plot Henhagainst1/t, where Henh¼ Hres Hexpt

res and Hresexptis the resonance field expected from

a normal classical dependence, extrapolating the high temperature trend tolower temperatures Henh(1/t) shows a linear dependence indicating that thisenhancement may have its origin in surface anisotropy, which scales as 1/t It

is clear that the surface contribution should grow with decreasing particlesize, since the surface area to volume ratio increases Therefore, it appearsthat, although the resonance equation used at room temperature is valid,for lower temperatures, <90 K, we need to include an extra termcorresponding to surface anisotropy to account for the effective field From

Fig 15A, it appears that the enhancement vanishes at 15 K

1/t (Å−1)

H enh

Fig 15 (A) Temperature dependence of the resonance field for the [Al2O3 (40 Å)/

Co 80 Fe 20 (t)]10/Al 2 O 3 (30 Å) discontinuous multilayers (B) Maximum resonance field enhancement observed as a function of inverse thickness for the discontinuous multi- layers Source: Reprinted from D.S Schmool, R Rocha, J.B Sousa, J.A.M Santos, G Kakazei, Evidence of surface anisotropy in magnetic nanoparticles, J Magn Magn Mater 300 (2006) e331 Copyright (2006), with permission from Elsevier.