Solid state physics, volume 67

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ISSN: 0081-1947
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CONTRIBUTORS
T. Dumelow
Universidade do Estado do Rio Grande do Norte (UERN), Mossoro´, Brazil
H. Kachkachi
PROMES, CNRS-UPR 8521, Universite de Perpignan Via Domitia, Perpignan, France
D.S. Schmool
Groupe d’Etude de la Matie`re Condensee GEMaC, CNRS (UMR 8635) Universite de
Versailles/Saint-Quentin, Universite Paris-Saclay, Versailles, France

vii


PREFACE
It is our great pleasure to present the 67th edition of Solid State Physics. The
vision statement for this series has not changed since its inception in 1955,
and Solid State Physics continues to provide a “mechanism … whereby investigators 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 knowledge 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 important
need for up-to-date, compact reviews of topical areas. The intention of these

reviews is to provide a history and context for a topic that has matured sufficiently to warrant a guiding overview.
The topics reviewed in this volume illustrate the great breadth and diversity of modern research into materials and complex systems, while providing
the reader with a context common to most physicists trained or working in
condensed matter. The chapter “Collective Effects in Assemblies of Magnetic Nanoparticles” provides an overview of emergent behavior arising
from collections of interacting magnetic particles from the perspective of
experiment, and also in terms of modeling and theory. The second chapter,
“Negative Refraction and Imaging from Natural Crystals with Hyperbolic
Dispersion,” describes aspects of material optics with a focus on the fascinating properties of hyperbolic materials whose surprising properties can be
found in naturally occurring single-phase materials, as opposed to
metamaterials in which these properties are engineered through design.
The editors and publishers hope that readers will find the introductions
and overviews useful and of benefit both as summaries for workers in these
fields, and as tutorials and explanations for those just entering.
ROBERT E. CAMLEY AND ROBERT L. STAMPS

ix


CHAPTER ONE

Collective Effects in Assemblies
of Magnetic Nanaparticles
D.S. Schmool*,1, H. Kachkachi†
*Groupe d’Etude de la Matie`re Condensee GEMaC, CNRS (UMR 8635) Universite de
Versailles/Saint-Quentin, Universite Paris-Saclay, Versailles, France
†
PROMES, CNRS-UPR 8521, Universite de Perpignan Via Domitia, Perpignan, France
1
Corresponding author: e-mail address:

Contents
1. Introduction
2. Magnetic Nanoparticle Assemblies: Theoretical Aspects
2.1 Model
2.2 Equilibrium Properties : Magnetization and Susceptibility
2.3 Dynamic Properties
3. Experimental Aspects
3.1 Magnetometry
3.2 AC Susceptibility
3.3 Magnetization Dynamics
€ssbauer Spectroscopy
3.4 Mo
3.5 Neutron Scattering Experiments
4. Summary
References

1
5
6
9
17
24
25
30
33
47
56
85
90

1. INTRODUCTION
Investigating the properties of ensembles of magnetic nanoparticle is a
rich and challenging physics problem, from both the experimental and theoretical points of view. Indeed, one encounters the typical difficult situation
where intraparticle and interparticle effects meet into a formidable manybody problem with both short-range and long-range interactions. The
intraparticle effects are related with the intrinsic properties of the
nanoparticles, such as the underlying material, size, shape, and energy potential. In particular, for small sizes the features of the single-nanoparticle physics
are dominated by finite-size and surface effects that drastically affect their
Solid State Physics, Volume 67
ISSN 0081-1947
/>
#

2016 Elsevier Inc.
All rights reserved.

1


2

D.S. Schmool and H. Kachkachi

magnetic properties, both in equilibrium and out of equilibrium. On the other
hand, assembled nanoparticles into 1D, 2D, or 3D arrays, organized or not,
reveal interesting and challenging issues related with their interactions among
themselves and with their hosting medium, a matrix or a substrate. The
ensuing collective effects show up through novel features in various measurements, 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 and

interparticle 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 of
nearly monodisperse assemblies in well-organized patterns. This is one of the
reasons for which more theoretical works have appeared recently focusing on
such 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 alleviate
this frustration. Indeed, applications of this technique to the case of Ising
dipoles 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, within
the 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 monodisperse
assemblies of magnetic moments with uniformly or randomly distributed
anisotropy axes. The authors studied the influence of dipolar interactions
(DI) on the susceptibility and specific heat of the assembly. Today, the literature thrives with theoretical works on the effect of DI on the magnetic
properties of assemblies of nanoparticles, most of which make use of numerical techniques [2, 5–25], because the main interest is for dense assemblies for
which experimental measurements are relatively easier to perform and the
applications more plausible. However, it is important to first build a fair
understanding of the underlying physics. This can only be done upon studying model systems that are simple enough for performing analytical developments and still rich enough to capture the main qualitative features of the
targeted systems. Analytical expressions come very handy in that they allow
us to figure out what are the main relevant physical parameters and how the


Collective Effects in Assemblies of Magnetic Nanaparticles

3

physical observables of interest behave as the former are varied and the various contributions to the energy compete which other. A brief account of
our contribution will be given in the following section.
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 overcome some of the problems with the weak experimental signals, many measurements are made on assemblies of nanoparticles and elements. This means
that the results obtained are generally an average over the sample and assembly and must also be interpreted taking into account the magnetic interactions between the particles. There have been extensive studies using many
techniques. In the following, we aim to give a brief overview of selected
studies 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, have
been applied to the study of magnetic nanoparticle systems. Measurements
can be made under the usual conditions since the material quantity is not an
issue, as stated previously. Where these techniques have shown to be of
importance is in the study of the superparamagnetic (SPM) behavior
observed in magnetic nanoparticle assemblies. This arises due to the thermal
instability introduced when the magnetic anisotropy, which usually defines
the orientation of the magnetization of the magnetic particle, is insufficient
to maintain its normal orientation. In fact the energy barrier is defined as the
product 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 magnetization becomes unstable, switching spontaneously between the energy
minima of the system. As a result, the magnetic measurement, which has a
characteristic measurement time, will sample the magnetic state as being
(super)paramagnetic. A combination of measurements as a function of temperature and applied field allows the system to be defined in terms of its
energy barrier and the blocking temperature TB, where the magnetization
is stable over the measurement time. Indeed, for AC susceptibility measurements, a frequency dependence is also important. Indeed the average
switching time between magnetic easy axes is characterized as an attempt

frequency. For measurements made with lower characteristic measurement
time, such as M€
ossbauer spectroscopy and FMR, corresponding values of
the blocking temperature will be much higher due to the Arrhenius behavior associated with superparamagnetism.


4

D.S. Schmool and H. Kachkachi

Ferromagnetic resonance is a very sensitive method for measuring the
magnetic properties of materials via the precessional magnetization dynamics
defined by the systems magnetic free energy. The precessional motion of
the magnetization is in general strongly influenced by magnetic anisotropies
and exchange effects in solids. This is often regarded as the internal effective
magnetic field experienced by the local magnetic spins of the system. This
can thus be separated into the various contributions to the local magnetic
field, via, magnetocrystalline anisotropy, shape anisotropy, exchange interactions, etc. In magnetic nanosystems [26–29], this can be adapted to include
surface anisotropy effects as well as magnetic DI between particles. This will
produce shifts in the resonance fields and can significantly affect the
linewidth of resonance absorption lines. Once again, measurements as a
function of sample temperature can provide further information regarding
the magnetic behavior of nanoparticle assemblies as they move through different magnetic regimes.
Nuclear techniques provide another form of probe for the local magnetic
order in solids. When applied to magnetic nanoparticle systems, information
on the magnetic modifications at a magnetic surface can be established as can
the effects of interparticle interactions. One such technique is M€
ossbauer
spectroscopy, and this has been applied to many Fe-based nanoparticle systems. Temperature-dependent measurements provide a sensitive probe of
magnetic 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 of
nanoparticles. M€
ossbauer spectroscopy has also been extensively used to
identify the oxide species which frequently form of metallic Fe and Fe oxide
nanoparticles. Neutron scattering is another nuclear technique which has
been broadly used as a research tool for investigating nanoparticles and magnetic nanoparticle assemblies. This for the most part concerns the scattering
at low angles from the incident neutron beam. Such small-angle neutron
scattering (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 samples
can be inferred from scattered intensity distributions. Using polarized neutrons allows magnetic information to be gleaned, which, as in the case of
M€
ossbauer spectroscopy, provides information on the surface of the magnetic particle and with care can be used to establish the spin distribution
or surface anisotropy of magnetic nanoparticles. Interparticle interactions


Collective Effects in Assemblies of Magnetic Nanaparticles

5

will also affect the magnetic scattering and thus SANS can also provide information of magnetic interactions between the particles, where studies are frequently performed as a function of particle concentration. Application of a
magnetic field to the sample is also used, where in systems of
magnetic nanoparticles dispersed in a solvent, or ferrofluid, the interaction
between the magnetic moments of the particles produces a spatial ordering
of the assembly. Core–shell models of magnetic nanoparticles can also be
established using a combination of SANS and polarized SANS measurements, with and without applied magnetic fields.
In the following, we focus on some theoretical aspects related to the
treatment of assemblies of magnetic nanoparticles. This will discuss the
energy considerations for an ensemble of ferromagnetic nanoparticles,

where the individual particle energy is considered as well as the additional
energy contribution which arises from interparticle (dipolar) interactions.
This then allows the equilibrium state of the system to be evaluated and
the magnetization and susceptibility properties to be obtained. These considerations are followed by a general discussion of dynamic magnetic properties and the AC susceptibility response of an assembly of weakly interacting
ferromagnetic nanoparticles. Section 3 aims to provide a brief overview of
experimental studies on magnetic nanoparticle assemblies. For each of the
methods discussed, we will give a short general introduction to the method,
where appropriate. We will cover both static and dynamic measurement
techniques.

2. MAGNETIC NANOPARTICLE ASSEMBLIES:
THEORETICAL ASPECTS
We have recently provided simple expressions for the magnetization
and susceptibility, both in equilibrium and out of equilibrium, which take
account 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 few
simplifying assumptions, either related with the particles themselves or with
the embedding assembly. In particular, the study of the effect of DI, which is
based on perturbation theory, applies only to a dilute assembly with an interparticle 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 weak
fields, small core and surface anisotropies. A brief account of these works is


6

D.S. Schmool and H. Kachkachi

given in the following sections. For the study of interplay between surfacedominated intrinsic properties and DI-dominated collective behavior, we
model a many-spin nanoparticle according to the effective-one-spin problem

(EOSP) proposed and studied in Refs. [34–39]. The EOSP model is a better
approximation than the OSP model in that it accounts for the intrinsic properties of the nanoparticle, such as the underlying lattice, size, and energy
parameters (exchange and anisotropy), via an effective energy potential.
In the simplest case, the latter contains a quadratic and a quartic contributions in the components of the particle’s net magnetic moment. These
two contributions should not be confused with the core and surface anisotropy contributions. In fact, the effective model is a result of a competition
between several contributions to the energy, namely the spin–spin exchange
interaction inside the nanoparticle, the on-site anisotropy attributed to the
spins in the core and on the surface. The outcome of the various competitive
effects is an effective model for the net magnetic moment m of the nanoparticle with a potential energy that contains terms with increasing order in its
components mα, α ¼ x, y, z. The coefficients of these terms are functions of
the atomic physical parameters, such as the constant of the on-site anisotropies and exchange coupling, together with those pertaining to the underlying crystal structure.
In the following section, we will give a brief account of these theoretical
developments, related to the intrinsic, as well as collective features of the
nanoparticles. 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 a
monodisperse assembly and oriented anisotropy. More general situations of
polydisperse assemblies, with both oriented and random anisotropy, can be
found in the cited works, e.g., in Ref. [31]. We commence with a monodisperse assembly of N ferromagnetic nanoparticles carrying each a magnetic moment mi ¼ mi si , i ¼ 1,…, N of magnitude m and direction si,
with jsij ¼ 1. Each magnetic moment has a uniaxial easy axis e aligned along
the same z-direction. The energy of a magnetic moment mi interacting with
the whole assembly, and with a (uniform) magnetic field H ¼ Heh, reads
(after multiplying by À β À1/kBT)
ð0Þ

E i ¼ E i + E DI
i ,

(1)


Collective Effects in Assemblies of Magnetic Nanaparticles

7

ð0Þ

where the first contribution E i is the energy of the free (noninteracting)
nanocluster at site i, comprising the Zeeman energy and the anisotropy contribution, i.e.
ð0Þ

E i ¼ xi si Á eh + Aðsi Þ,

(2)

where Aðsi Þ is a function that depends on the anisotropy model and is
given by
8
2
>
OSP
< σ i ðsi Á ei Þ ,
!


Aðsi Þ ¼
(3)
ζ

2
>
EOSP:
: σ i ðsi Á ei Þ À s4i, x + s4i, y + s4i, z ,
2
The second term in Eq. (1) represents the DI between nanoclusters,
which can be written as
X
E DI
si Á Dij Á sj
i ¼ξ
(4)
j
where Dij is the usual DI tensor
Dij 

Á
1À
3 3eij eij À 1
rij

(5)

with rij ¼ ri Àrj and eij ¼ rij/rij the unit vector along the i–j bond.
For later convenience, we have introduced the following dimensionless
parameters
 μ m2 =a3 
mH
K2 V

K4
(6)
x
, σ
, ζ , ξ¼ 0
K2
4π
kB T
kB T
kB T
$

together with the DI parameter ξ  ξCð0, 0Þ , where


1
ð0, 0Þ
C
¼ À4π Dz À
3

(7)

with Dz being the demagnetizing factor along the z-axis [3, 31]. K2 and K4
are the constants of the uniaxial and cubic anisotropy, respectively.
In the literature, especially for the experimental work, the magnetic
behavior of a nanoparticle is often approximated as a macrospin using
OSP 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

8

D.S. Schmool and H. Kachkachi

reorientation or reversal of the magnetization. In equilibrium the net magnetic moment is held in a direction according to the uniaxial anisotropy of
the magnetic nanoparticle, which in the absence of an applied magnetic field
will be along the easy axis. When a magnetic field is applied, the magnetization will reorient in accord with the minimum of the following energy
Eðϑ,φÞ ¼ σ cos 2 ϑ + x cosðψ À ϑÞ

(8)

where ϑ and ψ are the spherical angles between the easy axis and the particle’s magnetic moment and the applied magnetic field H, respectively.
Minimizing this simple energy, one finds two minima and a maximum with
two energy barriers expressed, for the case where the applied magnetic field
is parallel to the easy axis, as follows


H 2
(9)
△E ¼ σ 1 Æ
HK
where HK ¼ 2K2/Ms is the anisotropy field and corresponds to the highest
possible value of the switching field for the particle [41]. A full analysis of the
free energy of the system for a general orientation of the magnetic field with
respect to the easy axis yields a switching field which can be expressed as
HSW ¼

HK

3=2

ð sin 2=3 ψ + cos 2=3 ψ Þ

:

(10)

Again, as discussed earlier, the Stoner–Wohlfarthor OSP model is over
simplified since it ignores surface effects that arise from the reduction of
coordination at the boundaries of the magnetic system. In the literature
the anisotropy constant K2 used in HK is in fact an effective constant. Indeed,
experiments show that this model cannot account for the observed surface
effects caused by broken symmetries which significantly alter the local magnetic anisotropy [42, 43]. In an attempt to generalize the SW model in order
to take account of these observations, an effective anisotropy constant Keff is
derived on the basis of some arguments borrowed from 2D magnetism.
More precisely, Keff is proposed in a form that comprises uniaxial (volume)
and surface contributions, namely
Keff ¼ KV + 6

KS
D

(11)

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

9

Collective Effects in Assemblies of Magnetic Nanaparticles

Then, for an assembly of randomly oriented magnetic noninteracting
OSP nanoparticles, the (angular) averaged Stoner–Wohlfarth model is used
to predict the reversal of the collective sample. However, many experiments
have 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 have
been extensively studied in Ref. [45]. In particular, it was shown that for
weak surface anisotropy the spins inside the nanoparticle are nearly collinear
leading to a coherent switching of the particle’s magnetization. However, as
the surface effects become stronger, the spin switching operates via a clusterwise mechanism.

2.2 Equilibrium Properties : Magnetization and Susceptibility
In Ref. [31] it was shown that in a dilute assembly the magnetization of a
nanoparticle at site i (weakly) interacting with the other nanoparticles of
the assembly is given by (to first order in ξ)
 
N
 z  z X
 z
@ szi 0
ξik sk 0 Aki
,
si ’ si 0 +
@xi
k¼1

(12)

where Akl ¼ eh Á Dkl Á eh and hOi is the statistical average of the projection on
the field direction of the quantity O.
Note that Eq. (12) was obtained for an external magnetic field applied in
the z-direction leading to hsix, y i0 ¼ 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 interacting nanoparticle is written in terms of its “free” (with no DI) magnetiza 
 
tion szi 0 (and susceptibility @xi szi 0 ), with of course the contribution of the
assembly hosting matrix entering via the lattice sum in Eq. (12).
The free-particle magnetization mði 0Þ  hszi i0 ¼

Ð Ð Y




ð0Þ
z
s
ds
exp
ÀβE
i
i
i
i

can be either computed numerically or analytically in some limiting cases.

For the particular case of OSP, in a longitudinal magnetic field, i.e., ei
keh kez, the energy reads (dropping the particle’s index i)
E ð0Þ ¼ σs2z + xsz :
Then, one introduces the free-particle probability distribution


10

D.S. Schmool and H. Kachkachi

P 0 ðzÞ ¼

1

E ð0Þ

e
ð0Þ

Zk

,

ð0Þ
Zk ðσ, xÞ ¼

Z

1

À1

E ð0Þ

dsz e

Z


1
À1

dωð0Þ :

(13)

ð0Þ

The free-particle partition function Zk is rewritten in terms of the
Z x
2
Àx2
dt et as [46]
Dawson integral DðxÞ ¼ e
0

pffiffiffiffiffiffi
pffiffiffiffiffiffi
eσ
ð0Þ

Zk ðσ,xÞ ¼ pffiffiffi ½ex Dð σ + Þ + eÀx Dð σ À ފ
σ
with the reduced field h ¼ x/2σ and energy barriers σ Æ  σ ð1 Æ hÞ2 .
Therefore, the free-particle magnetization in the presence of anisotropy
and a longitudinal magnetic field is given by
hsz i0 ðσ 6¼ 0,ξ ¼ 0Þ ¼ C1 ¼

eσ
ð0Þ

σZk

sinhx À h:

(14)

There are various asymptotes that can then be derived for hszi0, 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 takes
account of the DI. Furthermore, as it will be seen later on, an explicit expression for mð0Þ allows one to derive an approximate expression for the magnetization of a (weakly) interacting assembly of EOSP nanoparticles by
including the cubic anisotropy term with coefficient ζ. We recall, however,
that this applies for relatively weak surface anisotropy and thereby to an
equilibrium magnetic state with quasi-collinear spins.
Therefore, for monodisperse assemblies we have xi ¼ x, σ i ¼ σ, ξij ¼ ξ.
In this case, the magnetization of a (weakly) interacting particle, given by Eq.
(12), simplifies into the following expression
!
ð0Þ
ð0, 0Þ @m

z
ð0Þ
(15)
:
hs i ’ m 1 + ξC
@x
This indicates that the relevant DI parameter, to this order of approximation,
is in fact the parameter introduced earlier
$

ξ  ξCð0, 0Þ ¼

N
ξ X
Aij :
N i, j¼1, i6¼j


Collective Effects in Assemblies of Magnetic Nanaparticles

11

Note that the lattice sum Cð0, 0Þ is in fact the first of a hierarchy of lattice sums
(see Ref. [31]).
ð0Þ

Next, the longitudinal susceptibility χ k ¼ @mð0Þ =@x is given by (see,
e.g., Ref. [46] for the notation)
@mð0Þ 1 + 2S2  ð0Þ 2 ð1Þ  ð0Þ 2
¼

À m
¼ a0 À m
@x
3
where
8 l=2
2 ðl À 1Þ!! l=2
>
>
σ + ⋯ , σ ≪1,
<
1 + 2S2
ð1Þ
ð2l + 1Þ!!
:
Sl ðσ Þ ’
, a0 
>
3
>
: 1 À lðl + 1Þ + ⋯ ,
σ ≫1:
4σ

(16)

Consequently, we obtain the approximate expression for the magnetization of a weakly interacting particle within the assembly


 2 !

$
ð1Þ
z
ð0Þ
ð0Þ
:
(17)
hs i ’ m 1 + ξ a0 À m
Obviously, in the absence of any interaction and anisotropy, or at high
temperature (SPM regime), the magnetization is described, as usual, by the
Langevin function


μ0 HMS
z
hs i0 ðσ ¼ 0,ξ ¼ 0Þ ¼ L
,
(18)
kB T
where μ0 is the vacuum permeability introduced so that μ0H is expressed in
tesla. LðxÞ ¼ cothx À 1=x.
Now, we are ready to discuss a few examples of the main results obtained
with the help of these theoretical developments.
In Fig. 1, we present the field behavior of the magnetization for different
values of the anisotropy parameter σ, as rendered by the standard (equilibrium) Monte Carlo calculations [31, 47], for a noninteracting assembly
(ξ ¼ 0). In Fig. 1A, we see that in the high-field regime the higher is σ
the lower is the magnetization. This can easily be understood since randomly
distributed easy axes lead to randomly distributed equilibrium orientations of
the particles magnetic moments and thereby the projection on the field
direction of the assembly net magnetic moment decreases as the strength

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


12

D.S. Schmool and H. Kachkachi

Reduced magnetization (per particle)
1

1

A

B
evin

Lang

vin

0.8

ge

n
La

0.8

3 nm

0.6

0.6

s
0.4

7 nm
7 nm

17
34
68
135

0.4

T = 5K

0.2

0.2

Polydisperse assembly

Polydisperse assembly
Random anisotropy

0

10

20

30

Random anisotropy

40

x

0

10

20

30

40 x

50

Fig. 1 (A) Reduced magnetization (per particle) of an assembly of N ¼ 1024 with
lognormal-distributed magnetic moments with mean diameter Dm ¼ 7 nm and randomly distributed easy axes as obtained from Monte Carlo calculations for different
anisotropy values. xm ¼ nmμBH/kBT, where nm is the mean number of Bohr magnetons
for this assembly. (B) Langevin function together with the Monte Carlo results for Dm ¼ 3

and 7 nm.

because the competition between Zeeman, thermal and anisotropy contributions to the energy results in a crossing between the various magnetization
curves, as has been observed, e.g., for maghemite particles [48, 49]. In addition, we see that there is a large deviation from the Langevin law due to several contributions to the energy, ignored by the Langevin law, especially
anisotropy. Moreover, the results in Fig. 1B show that the larger the mean
diameter of the assembly, the larger the value of σ, and thereby the larger the
expected deviation from the Langevin curve.
In Fig. 2 we plot the Langevin function (full line) and the Monte Carlo
results (symbols) for the magnetization of an interacting assembly of (N ¼
10 Â 10 Â 5) lognormal-distributed moments, with random anisotropy,
and for different values of the interparticle separation. Here we use the same
assemblies as in Fig. 1. The intensity of DI, or equivalently the value of ξ, is
varied by varying the lattice parameter a entering ξ [see Eq. (6)]. More precisely, the parameter a is taken as a real number k times the mean diameter
Dm of the assembly, i.e., a ¼ k  Dm. Thus, large values of k correspond to an
isotropically inflated lattice with large distances between the magnetic


13

Collective Effects in Assemblies of Magnetic Nanaparticles

1.0

Reduced magnetization

vin

ge

n

La

0.8

k=2
k=4
k=5

0.6

1.0

0.8

0.4

T = 5K
0.6
Polydisperse assembly
Random anisotropy 0.4

Symbols: MC

0.2

Low field
SDA (k = 2)

0.2

0.0
0.0

0.0
0.0

5.0

0.2

10.0

0.5

15.0

ζ

<x>

0.8

20.0

Fig. 2 Reduced magnetization (per particle) of an interacting assembly of N ¼ 10Â
10 Â 5 lognormal-distributed magnetic moments with mean diameter Dm 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.

moments, and thereby weak DI. These results, obtained for an oblate sample, confirm the fact that in this case DI suppress the magnetization. This

result has also been obtained by perturbation theory in Ref. [34] whose
results are shown in Fig. 3, which are plots of Eq. (17) using Eq. (14) for mð0Þ .
As discussed in Ref. [31] and references therein, DI are anisotropic interactions and thus contribute to the effective anisotropy. Since the anisotropy
is uniaxial and oriented, i.e., with a common easy axis, its effect leads to a
magnetization enhancement. In contrast, the DI effect depends on the sign
$

of ξ (or more precisely that of Cð0, 0Þ ), which is related to the sample’s shape.
For instance, in the case of oblate samples Cð0, 0Þ < 0 leading to a reduction of
the magnetization, while for prolate samples Cð0, 0Þ > 0 and thereby DI contribute to enhance the assembly’s magnetization. Consequently, for oblate
samples the (oriented) uniaxial anisotropy and DI have opposite effects while
for prolate samples they play concomitant roles.
In the presence of not-too-strong surface anisotropy, one can model the
nanoparticle using the EOSP model upon which the free-particle partition
ð0Þ

function Zk is replaced by [34]


14

D.S. Schmool and H. Kachkachi

1

m

0.8
Without DDI
With DDI Oblate

With DDI prolate

0.6

0.4
D = 3 nm
T=5K

0.2

0
0

1

2

3

4

x

5

Fig. 3 Reduced magnetization of two assemblies of equivalent sizes but one is prolate
and the other oblate.

Z
Z¼

σζ

dφdωð0Þ eÀ 2 Σα¼x, y, z sα :
4

Then we assume that the cubic anisotropy remains small and proceed
with a perturbative calculation of Z. Indeed, the condition of validity for
the EOSP model [36–39] (obtained for a nanoparticle with an SC or
FCC crystal lattice) is roughly ζ ¼ K4 =K2 ≲1=4. As such, the spin noncollinearities induced by surface anisotropy are not too strong and thereby
the anisotropy energy minima are mainly defined by the uniaxial contribution, whereas the cubic contribution only introduces saddle points. This
leads to larger relaxation rates [50] but does not affect the equilibrium
properties.
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 for
the magnetization for the EOSP particle (see Eq. (3.39) of Ref. [46] for the
case ζ ¼ 0 but arbitrary field)





 
 !
1
2 x3 ζ
2
5 x3
ð0Þ
m ðx,σ,ζ Þ ’ 1 À x À 1 À

+ À 1À x+ 2À
:
σ
σ 3 σ
σ
σ 3
(19)
Next, writing this in the form
mð0Þ ’ χ ð1Þ x + χ ð3Þ x3


Collective Effects in Assemblies of Magnetic Nanaparticles

15

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




1
ζ
2
ð1Þ
χ ’ 1À
+
À1 +
,
σ

σ
σ



!
1
2
ζ
5
ð3Þ
À1 +
+
2À
:
(20)
χ ’
3
σ
σ
σ
The competition between the uniaxial and cubic anisotropy contributions is easy to understand. As has been discussed earlier (see also Ref.
[21]), for ζ > 0 the energy minima of the cubic contribution are along
the cube diagonals ½Æ1, Æ 1, Æ 1Š while for ζ < 0 they are along the cube
edges ½1,0, 0Š, ½0,1,0Š, ½0,0,1Š. Hence, the uniaxial anisotropy with an easy
axis along the z-direction, i.e., ½0, 0,1Š, competes with the cubic anisotropy
when ζ > 0, whereas the two anisotropies have concomitant effects when
ζ < 0. In the former case, the particle’s magnetic moment at equilibrium
adopts an intermediate direction between the z-axis and the cube diagonal.
So, as ζ increases the magnetic moment gradually rotates away from the

z-axis and thereby its statistical average, or the magnetization, decreases.
In the case of negative ζ the two anisotropies cooperate to quickly drive
the magnetization toward saturation.
Next, using the expression (19) for the free-particle magnetization, as a
function of the applied field x, uniaxial anisotropy (and temperature) σ and
surface anisotropy ζ, in Eq. (12) or (15) we can investigate the interplay
between surface effects and DI, i.e., a competition between the terms in
$

ζ and ξ , respectively. This was done in Ref. [34]. The same competition
was 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 ð1Þ x + χ ð3Þ x3
(21)
where

 !
2
3 ζ
χ ’χ + ξ 1À À2 1À
,
σ
σ σ



!
4$
3
3ζ
$ð3Þ
ð3Þ
À
,
χ ’χ À ξ 1À
3
σ
σ
$ð1Þ

ð1Þ

$

(22)


16

D.S. Schmool and H. Kachkachi

are the linear and cubic susceptibilities (20) augmented by the DI contribu$

tion of intensity ξ .
This asymptotic expression helps understand how surface anisotropy
competes with DI. The surface contribution with intensity ζ, which plays

an important role in the magnetization curve, couples to the DI contribution
$

$

with intensity ξ via the term with coefficient ξ ζ. Hence, the overall sign of
the latter determines whether there is a competition between surface and DI
effects or if the changes in magnetization induced by the intrinsic and collective contributions have the same tendency. Accordingly, plots of the
magnetization, which take into account both surface effects and DI, are
shown in Fig. 4 as a function of the field x, for an oblate sample with
Nx  Ny  Nz ¼ 20  20  5 and a prolate sample with 10  10  20,
with the respective values of Cð0, 0Þ ’ À4:0856 and 1.7293.
$

As discussed earlier, for oblate samples ξ < 0, DI tend to suppress mag$

netization, whereas for prolate samples ξ > 0 they enhance it. Indeed, we see
from Eqs. (22) that surface anisotropy and DI may have opposite or concomitant effects depending on their respective signs. In Ref. [11], it was found
that the magnetization enhancement in dilute assemblies of maghemite
nanoparticles of 3 nm in diameter is suppressed when the concentration
increases. In accordance with the present results, DI tends to smooth out
surface effects, or the other way round, the surface seems to have a screening
effect on DI.

m

m

0.8

0.8
Without DDI, z = 0
Without DDI, z = 0.25
With DDI, z = 0.25

0.6

0.4

0.4
20 × 20 × 5, D = 3 nm

0.2

0

Without DDI, z =0
Without DDI, z = 0.25
With DDI, z = 0.25

0.6

0

1

2

3

4

x

10 × 10 × 20, D = 3 nm

0.2

5

0

0

1

2

3

4

x

5

Fig. 4 Left: magnetization as a function of the (dimensionless) field x for an oblate sample (20 Â 20 Â 5). Right: magnetization as a function of the reduced field x for a prolate
sample (10 Â 10 Â 5). Here ξ ’ 0.18.

Collective Effects in Assemblies of Magnetic Nanaparticles

17

2.3 Dynamic Properties
The dynamics of an assembly of magnetic nanoparticles is a rich environment
for the study of equilibrium and out-of-equilibrium many-body statistical
physics. Indeed, as discussed earlier, there are physical phenomena which
occur over wide ranges of spatial and temporal scales. The relevant length scale
can 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 the
assembly thereof. On the other hand, the time scale also spans a wide range
that 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 and
thermoremanent magnetization. Obviously, these time scales are a direct
consequence of a competition between short-range and long-range interactions operating at different length scales. For this reason, among others, it is not
possible to come up with a theory that covers all length and time scales. For
short-time regimes the physics is usually described with the aid of the Landau–
Lifshitz equation and its variants, deterministic or stochastic, damped or
undamped, local or macroscopic. For collective effects, occurring at the
assembly scale, the Monte Carlo technique is more appropriate, even though
the problem of an efficient algorithm for dynamical processes is not entirely
solved so far, see for instance the works in Refs. [51–56]. As for analytical
approaches, there are a very few attempts to tackle the problem, mainly
because of the tremendous difficulty to calculate the relaxation rate of a manyspin system. The main difficulty resides in the fact that it is impossible to analyze the large number of extrema of a multivariate energy potential, in the
presence of several parameters, such as size, shape, applied fields, etc.
A way out of this difficulty was proposed in Ref. [36] where the EOSP model
was built for a spherical nanoparticle with Neel anisotropy on the surface and
no anisotropy in the core, and in Refs. [38, 39], where it was extended to a

more general situation. Indeed, the EOSP approach makes it possible to investigate the dynamics of an interacting assembly while taking account of the
intrinsic features of the nanoparticles, since this model is a macroscopic model
whose energy potential depends on the nanoparticle’s parameters. This simplification allows us to compute the relaxation time taking account of the
effect of surface anisotropy, in addition of course to that of the (effective) uniaxial anisotropy and the applied (static) magnetic field. This was done in Ref.
[50]. Then, in Ref. [35] the AC susceptibility of a (weakly) interacting assembly of EOSP nanoparticles was computed, after generalizing the calculation of
the relaxation rate of such particles.


18

D.S. Schmool and H. Kachkachi

The dynamic response of the EOSP assembly is given by the AC susceptibility which, for an arbitrary angle ψ between the (common) easy axis
and the field direction, the effective susceptibility may be written as
χ ¼ χ k cos 2 ψ + χ ? sin 2 ψ. According to Debye’s model [33, 46, 57] we have
χ ðωÞ ¼

χ k ðT ,HÞ
χ ðT, HÞ 2
cos 2 ψ + ?
sin ψ,
1 + iωτk
1 + iωτ?

(23)

where τk and τ? are the longitudinal (inter-well) and transverse (intra-well)
relaxation times and χ k(T, H) and χ ?(T, H) are, respectively, the longitudinal 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 this

case the AC susceptibility is given by Eq. (23) or using τk ¼ ΓÀ1 and
eq

$ eq

χ k ¼ χ eq ¼ χ free + ξ χ int ,

$ 
χ x,σ, ζ, ξ ,η ¼

χ eq
:
1 + iωΓÀ1

Next, we introduce the reduced frequency

$ 
η x,σ,ζ, ξ ,λ  ωτk ¼ ðωτD ÞðτD ΓÞÀ1 ,

(24)

(25)

$

with λ 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)À1 is the free diffusion time, HK ¼ 2K2V/M the (uniaxial) anisotropy
field, and γ ’ 1.76 Â 1011 (TÁs)À1 the gyromagnetic ratio. For example, for
cobalt particles the anisotropy field is HK $ 0.3 T, and for λ ¼ 0.1 À 10, τD $

2 Â 10À10 À 2 Â 10À12 s.
Now, if we restrict ourselves to the linear susceptibility, χ eq is equal to
$ð1Þ
χ given in Eq. (22). The second quantity that needs to be calculated in
order to fully evaluate the susceptibility in Eq. (24) is the relaxation rate

$ 
Γ x, σ, ζ, ξ ,λ .
Accordingly, in Ref. [58], J€
onsson and Garcia-Palacios derived the following approximate expression for Γ for a weakly interacting assembly
!
 2
1 D 2E 1
Γ ’ Γ0 1 + Ξk + FðαÞ Ξ? 0 :
(26)
0
2
4


19

Collective Effects in Assemblies of Magnetic Nanaparticles

D E
 
Ξ2k and Ξ2? 0 are the spin averages of the longitudinal and transverse
0
P
components of the dipolar field Ξi ¼ ξ j6¼i Dij Á sj . The subscript 0 is a

reminder of the fact that the averages are computed with the Gibbs distribution of the noninteracting assembly [33]. The function F(α) is given
by [59]
FðαÞ ¼ 1 + 2ð2α2 eÞ1=ð2α Þ γð1 +
2

1 1
,
Þ,
2α2 2α2

(27)

Rz
with γða,zÞ ¼ 0 dt t aÀ1 eÀt , the incomplete gamma function, and where
pffiffiffi
α ¼ λ σ . Asymptotic expressions of F(α) are [59]
8 pffiffiffi
pffiffiffi
π 1
π
>
>
À +
α, α ≪ 1,
<
α 3
6
FðαÞ ’
>
>

:1 + 1 À 1 ,
α ≫ 1:
α 4α2
The free-particle relaxation rate Γ0 that was used in Ref. [58] is given by
2
τD Γ0 ¼ pffiffiffi σ 3=2 eÀσ :
π

(28)

Then, the relaxation rate (28) was generalized in Ref. [35] in order to take
into account the magnetic field as well as the core and surface anisotropies.
For intermediate-to-high damping Langer’s approach allows us to compute the relaxation rate Γ of a system with many degrees of freedom related
with its transition from a metastable state through a saddle point [60–65]
$

jκj Z s
Γ¼
,
2π Z m
$

(29)

where Z m and Z s are, respectively, the partition functions in the vicinity of
the metastable energy minimum and the saddle point, obtained for a quadratic expansion of the energy. The attempt frequency κ is computed upon
linearizing the dynamical equation around the saddle point, diagonalizing
the resulting matrix and selecting its negative eigenvalue [60, 61].
In Ref. [35] the relaxation rate Γ was calculated in various situations of
an EOSP particle including the effective uniaxial and cubic anisotropy and

the applied magnetic field. A detailed analysis of the various energy extrema
is presented in Ref. [35], and analytical expressions were given for the relaxation rate as a function of temperature, effective uniaxial anisotropy (σ),


20

D.S. Schmool and H. Kachkachi

surface anisotropy (ζ), and applied magnetic field. The authors of Ref. [35]
then investigated the interplay between interparticle DI and intrinsic surface
anisotropy, in the case ζ > 0 where surface (cubic) anisotropy favors the
magnetic alignment along the cube diagonals. χ 0 and χ 00 were computed
for various values of the surface anisotropy coefficient ζ, for both prolate
and 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 basically plays the same role in a prolate sample where the magnetization is
enhanced along the z-axis, or in an oblate sample where the magnetization
is enhanced in the xy plane.
The results in Fig. 5 show an example that illustrates the competition
between surface anisotropy and DI contribution to the real component of
$

the AC susceptibility. They were obtained for the finite value ξ ¼ 0:008
and an increasing (but small) surface anisotropy parameter ζ. It can be seen
that the surface anisotropy, in the present case of positive ζ, has the opposite
effect to that of DI. This again shows that there is a screening of DI by surface
effects and confirms the results of Ref. [34] for equilibrium properties for
both negative and positive ζ, as discussed earlier.
Our theoretical calculations of the AC susceptibility of magnetic
nanoparticles which accounts for the intrinsic properties (e.g., surface
∼

x = 0.008

12
10

c′

8
z=
0.01
0.05
0.1

6
4
2
0

0.1

0.15
1/s

0.2

0.25

Fig. 5 χ 0 for an interacting prolate ð10 Â 10 Â 20Þ assembly with a fixed DI strength
$

ξ ¼ 0:008 and varying surface anisotropy coefficient ζ, for the frequency

$

f  ωτD =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.


Collective Effects in Assemblies of Magnetic Nanaparticles

21

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

kB ðT ÀθVF Þ
Γ ¼ τÀ1
0 e

(30)

9
12
where ν0 ¼ τÀ1
0 ’ 10 À 10 Hz and θ VF represents an effective temperature
supposed to account for the DI correction; ΔE is the energy barrier, which

reads ΔE ¼ KV in the case of uniaxial anisotropy and zero field.
Our results are in full agreement with previous works [67, 68, 70] and
further extends them in that they take into account: (i) surface anisotropy,
(ii) the particles spatial distribution and shape of the assembly, and (iii) the
damping parameter. A full discussion can be found in Ref. [34]. Here we
only report the following expression found there for θVF

θVF ζ 1 À 2 Á
¼ +
ξS
T
4 6σ

(31)

where S is a function of the lattice and damping through the function F(α).
Expression (31) provides a somewhat microscopic description of the phenomenological parameter θVF in terms of the interparticle interactions, the
surface anisotropy and damping. Indeed, the last term in (31), which is similar
to the one derived in Ref. [68], includes both the damping parameter and the
shape of the assembly, through the expression of S ðλÞ. In addition, we note
that ξ is proportional to the assembly concentration [34] CV and thereby to
aÀ3, a being the interparticle separation. Therefore, we expect that in the
absence of surface anisotropy, θVF would scale as θVF $ CV2 $ aÀ6 . In Ref.
[17] experimental estimates of θVF are given for an assembly of Ni
nanoparticles with varying concentration. A comparison of Eq. (31) with
the corresponding experimental data is given in Fig. 6.
On the other hand, the first term in Eq. (31) accounts for the contribution from surface anisotropy. In practice it should be possible to adjust the
assembly characteristics (assembly shape, particles size and underlying material) so as to achieve, to some extent, a compensation between surface effects
and the DI contribution. This could in principle suppress the dependence of
θVF on the assembly concentration. In addition, the term in ζ can also be

used to extract from the experimental data an estimate of the (effective) surface anisotropy coefficient ζ by reading off the intercept from the plot in
Fig. 6. Furthermore, it is worthwhile emphasizing that θVF is not independent of temperature, as is very often assumed in the literature. First, the


22

D.S. Schmool and H. Kachkachi

12
10

Data (Masunaga et al.)
Fit q VF = 0.5633 + 0.05405 Cv2

q VF(K)

8
6
4
2
0
0

2

4

6

8

10

12

14

Concentration Cv (%)

Fig. 6 θVF 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.

temperature appears in the second term in (31), being related to the DI contribution. Even if this term becomes negligible for very diluted assemblies, if
surface anisotropy is taken into account (ζ6¼0), e.g., for very small
nanoparticles, Eq. (31) shows that the phenomenological parameter θVF is
in fact a linear function of temperature via the term in ζ. This can be understood by noting that the surface anisotropy, which is of cubic nature in the
EOSP model, drastically modifies the energy potential and thereby affects
the dynamics of the particle’s magnetization. As a consequence, the effect
of DI becomes strongly dependent on the thermal fluctuations and the elementary switching processes they induce.
Two applications of this formalism have been recently studied by one of
the authors, namely, on the one hand, the effect of DI on the FMR characteristics of a 2D array of nanoparticles and, on the other, the effect of DI
and their competition with a DC magnetic field in the behavior of the specific absorption rate (SAR), which is relevant in magnetic hyperthermia.
The two corresponding works are in preparation and will be submitted
for 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 parameters pertaining to the assembly. Indeed, it is quite easy to show that, in the
linear response, the SAR is proportional to the out-of-phase component χ 00

of the AC susceptibility.