Ferromagnetic order in dipolar systems with anisotropy: application to magnetic nanoparticle supracrystals
Single domain magnetic nanoparticles (MNP) interacting through dipolar interactions (DDI) in addition to the magnetocrystalline energy may present a low temperature ferromagnetic (SFM) or spin glass (SSG) phase according to the underlying structure and the degree of order of the assembly. We study,...
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irk-123456789-1570142019-06-20T01:26:36Z Ferromagnetic order in dipolar systems with anisotropy: application to magnetic nanoparticle supracrystals Russier, V. Ngo, E. Single domain magnetic nanoparticles (MNP) interacting through dipolar interactions (DDI) in addition to the magnetocrystalline energy may present a low temperature ferromagnetic (SFM) or spin glass (SSG) phase according to the underlying structure and the degree of order of the assembly. We study, from Monte Carlo simulations in the framework of the effective one-spin or macrospin models, the case of a monodisperse assembly of single domain MNP fixed on the sites of a perfect lattice with fcc symmetry and randomly distributed easy axes. We limit ourselves to the case of a low anisotropy, namely the onset of the disappearance of the dipolar long-range ferromagnetic (FM) phase obtained in the absence of anisotropy due to the disorder introduced by the latter. Однодоменнi магнiтнi наночастинки (МНЧ) iз дипольною мiжчастинковою взаємодiєю можуть окрiм магнiтокристалiчної фази знаходитися у низькотемпературнiй феромагнiтнiй (ФМ) або спiново-склянiй фазах залежно вiд структури та ступеню впорядкування системи. Методом моделювання Монте-Карло в рамках односпiнової чи макроспiнової моделей авторами дослiджується монодисперсна система однодоменних МНЧ, розташованих на вузлах iдеальної ґратки з симетрiєю типу fcc i з довiльно розподiленими осями. Розглядається випадок низької анiзотропiї, зокрема початку зникнення дипольної далекосяжної ФМ фази. 2017 Article Ferromagnetic order in dipolar systems with anisotropy: application to magnetic nanoparticle supracrystals / V. Russier, E. Ngo // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33703: 1–10. — Бібліогр.: 44 назв. — англ. 1607-324X PACS: 75.50.Tt, 64.60.De DOI:10.5488/CMP.20.33703 arXiv:1710.01532 http://dspace.nbuv.gov.ua/handle/123456789/157014 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
Single domain magnetic nanoparticles (MNP) interacting through dipolar interactions (DDI) in addition to the
magnetocrystalline energy may present a low temperature ferromagnetic (SFM) or spin glass (SSG) phase according to the underlying structure and the degree of order of the assembly. We study, from Monte Carlo simulations in the framework of the effective one-spin or macrospin models, the case of a monodisperse assembly
of single domain MNP fixed on the sites of a perfect lattice with fcc symmetry and randomly distributed easy
axes. We limit ourselves to the case of a low anisotropy, namely the onset of the disappearance of the dipolar
long-range ferromagnetic (FM) phase obtained in the absence of anisotropy due to the disorder introduced by
the latter. |
| format |
Article |
| author |
Russier, V. Ngo, E. |
| spellingShingle |
Russier, V. Ngo, E. Ferromagnetic order in dipolar systems with anisotropy: application to magnetic nanoparticle supracrystals Condensed Matter Physics |
| author_facet |
Russier, V. Ngo, E. |
| author_sort |
Russier, V. |
| title |
Ferromagnetic order in dipolar systems with anisotropy: application to magnetic nanoparticle supracrystals |
| title_short |
Ferromagnetic order in dipolar systems with anisotropy: application to magnetic nanoparticle supracrystals |
| title_full |
Ferromagnetic order in dipolar systems with anisotropy: application to magnetic nanoparticle supracrystals |
| title_fullStr |
Ferromagnetic order in dipolar systems with anisotropy: application to magnetic nanoparticle supracrystals |
| title_full_unstemmed |
Ferromagnetic order in dipolar systems with anisotropy: application to magnetic nanoparticle supracrystals |
| title_sort |
ferromagnetic order in dipolar systems with anisotropy: application to magnetic nanoparticle supracrystals |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157014 |
| citation_txt |
Ferromagnetic order in dipolar systems with anisotropy: application to magnetic nanoparticle supracrystals / V. Russier, E. Ngo // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33703: 1–10. — Бібліогр.: 44 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT russierv ferromagneticorderindipolarsystemswithanisotropyapplicationtomagneticnanoparticlesupracrystals AT ngoe ferromagneticorderindipolarsystemswithanisotropyapplicationtomagneticnanoparticlesupracrystals |
| first_indexed |
2025-07-14T09:21:38Z |
| last_indexed |
2025-07-14T09:21:38Z |
| _version_ |
1837613605170708480 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 3, 33703: 1–10
DOI: 10.5488/CMP.20.33703
http://www.icmp.lviv.ua/journal
Ferromagnetic order in dipolar systems with
anisotropy: application to magnetic nanoparticle
supracrystals
∗
V. Russier†, E. Ngo
ICMPE, UMR 7182 CNRS and UPE 2-8 rue Henri Dunant 94320 Thiais, France
Received June 14, 2017, in final form July 19, 2017, corrected October 9, 2018
Single domain magnetic nanoparticles (MNP) interacting through dipolar interactions (DDI) in addition to the
magnetocrystalline energy may present a low temperature ferromagnetic (SFM) or spin glass (SSG) phase ac-
cording to the underlying structure and the degree of order of the assembly. We study, from Monte Carlo sim-
ulations in the framework of the effective one-spin or macrospin models, the case of a monodisperse assembly
of single domain MNP fixed on the sites of a perfect lattice with fcc symmetry and randomly distributed easy
axes. We limit ourselves to the case of a low anisotropy, namely the onset of the disappearance of the dipolar
long-range ferromagnetic (FM) phase obtained in the absence of anisotropy due to the disorder introduced by
the latter.
Key words:magnetic nanoparticles, Monte Carlo simulation, ferromagnetic order
PACS: 75.50.Tt, 64.60.De
1. Introduction
The physics of nanoscale magnetic materials is still a very active field of research both in view
of the potential applications, especially in nanomedicine [1], and from the fundamental point of view
[2, 3]. The fundamental aspects are all the more important that magnetic nanoparticles (MNP) can be
synthesized in a wide range of size and shapes and their assemblies obtained with different structures:
colloidal suspensions, or ferrofluids embedded in non-magnetic materials where one can tune the in-
terparticle interactions through the concentration, or as powders. Moreover, when the size dispersion
is sufficiently narrow, MNP can self-organize in long-range ordered 3D supracrystals, namely crystals
of nanoparticles [4–6], conversely to the discontinuous metal insulator multilayers (DMIM) where the
underlying structure is disordered [7–9]. The interplay between mutual interactions and the degree of
order of the underlying structure is a key factor in understanding the extrinsic properties of single domain
MNP assemblies [3]. Indeed, due to frustration effects, according to whether the MNP are arranged or
not with a long-range order at high concentration, a ferromagnetic or a spin glass state is expected at a
low temperature [3, 7] (respectively super-ferromagnetic, SFM and super-spin glass, SSG according to
the nanoscale magnetism nomenclature). Hence, one of the purposes of the modelling is to understand
under which conditions the SFM or the SSG state can be reached.
The modelling of the magnetic properties of both nanostructured materials and materials including
nanoscale particles is a multiscale problem since, on the one hand, the local magnetic structure within
MNP at the atomic site scale presents nontrivial features and, on the other hand, the interactions between
MNP play an important role. An important simplification in the modelling occurs for particles of the
radius under a critical size rsd depending on their chemical composition, typically of a few dozens on
∗Paper dedicated to the memory of Dr. J.-P. Badiali.
†E-mail: russier@icmpe.cnrs.fr
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
33703-1
https://doi.org/10.5488/CMP.20.33703
http://www.icmp.lviv.ua/journal
https://doi.org/10.5488/CMP.21.36701
http://creativecommons.org/licenses/by/4.0/
V. Russier, E. Ngo
nanometers, where they reach a single domain regime avoiding both multidomains and vortex structures
(typical values for rsd are 15 nm for Fe, 35 nm for Co, 30 nm for γ-Fe2O3 [2]). In this case, the properties
of the MNP ensemble can be described through an effective one-spin model (EOS) where each MNP is
characterized by its moment and magnetocrystalline anisotropy energy. However, both the moment and
characteristics of the anisotropy term must be considered as effective quantities taking into account some
of the atomic scale features. Hence, the effective saturation magnetization Ms essentially differs from
its bulk value due to crystallinity and/or surface defects, the so-called dead layer, and is expected to be
smaller than the latter. As concerns the characteristics of magnetocrystalline anisotropy energy (MAE),
the situation is more complex since even its symmetry, either uniaxial or cubic, can differ from that of the
bulk material. It is worth mentioning that the non-collinearities of the surface spins can be represented
by an additional cubic term in the MAE in the framework of the EOS [10].
The EOS type of approach appears then as a simplifying level of description but, nevertheless, a nec-
essary step to model the interactingMNP assemblies, especially when dipolar interaction are included. In
the framework of the EOSmodels, the theoretical description ofMNP assembliesmake use of the standard
methods of statistical physics since one deals with a set of particles interacting through a pairwise potential
in a one-body potential including both the MAE and the Zeeman term due to the applied external mag-
netic field. For MNP coated with a non-magnetic layer, the interparticle interaction includes an isotropic
short-range term and the dipole-dipole interaction (DDI), since the exchange term coming from the direct
contact between MNP (referred to as superexchange) vanishes. Hence, one is faced with the models
apparented to the dipolar hard or soft spheres, according to the short ranged potential (DHS and DSS,
respectively) widely developed in a more general framework of dipolar fluids including molecular polar
fluids to which one adds the MAE contribution in the form of a one-body potential. An important distinc-
tion is to bemade concerning the structure of the system under study according towhether it is a liquid-like
suspension, a ferrofluid, or a frozen system which can be an ensemble of MNP in a frozen non-magnetic
embedding medium or a powder sample. In this latter case, the thermally activated degrees of freedom
are the MNP moments and averages over frozen structure realizations that are done in a second step.
In highly concentrated systems, the nature of the state induced at low temperature by the collective
behavior resulting from the DDI strongly depends on the underlying structure and dimensionality due
to the anisotropic character of the DDI. For a well ordered system such as a fully occupied perfect
lattice free of MAE or with a MAE characterized by aligned easy axes, a long-range ferromagnetic (face
centered cubic, fcc, body centered cubic, bcc or body centered tetragonal, bct) or anti-ferromagnetic
(simple cubic, sc) ordered state is reached [11–16]. Notice that the solid dipolar ferromagnetic ground
state presents a bct structure [17], a result in agreement with the solid state phase diagram of the dipolar
hard sphere system [18]. Conversely disordered systems present a spin-glass like (SSG) [7, 19, 20] as
has been experimentally evidenced [21–25]. It is worth mentioning that not only the perfect lattices of
DHS or DSS [15, 26] orient spontaneously at low temperature or high pressure. This is also the case for
the dense liquid state [12, 27–30] where a ferroelectric (or equivalently a FM) nematic state has been
reported in the bulk or in slab geometry.
On a perfect lattice, the structural disorder may be introduced either through the MAE contribution
with a random distribution of easy axes or through dilution. In the limiting case of an infinite magnitude
of the MAE, the DHS on a lattice reduces to the dipolar Ising model, where the exchange coupling
constants Ji j follow the dipolar energy form for moments aligned on the easy axes. In the perfect aligned
dipole (PAD) case, the ordered/disordered transition takes a SG character when reducing the lattice
occupation rate [16], while in the random axes dipoles (RAD), the transition is a SG one whatever the
lattice occupation [31, 32].
Well ordered MNP supracrystals with a fcc structure [4, 5] are presumably good candidates for
experimental realization of a dipolar induced super-ferromagnetic (SFM) phase. The SFM phase was
already looked at through the behavior of the hysteresis in terms of temperature on supracrytals made
of iron oxide nanoparticles [6]. A ferrimagnetic phase was evidenced involving the ferromagnetically
ordered MNP core moments and the spin canting at the MNP surface. In this work we investigate from
Monte Carlo simulations the effect of the magnitude of the MAE on the dipolar SFM state reached by
an ensemble of DHS on a fcc lattice when the MAE easy axes are randomly distributed. We also show
that the partial alignment of the easy axes leads to the recovery of the SFM state. We only focus on the
SPM-SFM transition.
33703-2
Ferromagnetic order in dipolar systems with anisotropy
2. Model and Monte Carlo simulations
We model an assembly of MNP uncoupled exchange, interacting through dipolar interactions (DDI),
characterized by their magnetocrystalline anisotropy (MAE) and self-organized on long-range ordered
supra crystals of fcc symmetry. Although this simple model is not at all specific, we have in mind more
precisely the case of Co or γ-Fe2O3 MNP. The MAE is assumed to be of uniaxial symmetry. This is
justified from a crystallographic point of view in the case of hcp-Co, and from the general experimental
finding for γ-Fe2O3 where the shape or the surface effect leads to an effective uniaxial anisotropy. The
easy axes distribution is random unless otherwise mentioned. In the framework of an effective one-spin
model, we consider a monodisperse ensemble of up to N = 2048 dipolar hard spheres located on the
sites of a fcc lattice with the volume fraction φ. The total energy in a reduced form, after introducing a
reference temperature T0 and β0 = 1/(kBT0) reads
β0E = εd
∑
i< j
m̂im̂j − 3(m̂i r̂i j)(m̂j r̂i j)
(ri j/d)3
− εu
∑
i
(n̂im̂i)
2, (2.1)
where n̂i and m̂i are the easy axes and the unit vectors in the direction of the moments, respectively. The
coupling constants are related to the saturation magnetization Ms, the uniaxial anisotropy constant Ku
and the MNP volume v(d) [the MNP moment is µ = v(d)Ms] by
εd =
β0µ0
4π
(π/6)M2
s v(d), εu = β0Kuv(d), (2.2)
εd is a usual dipolar constant. We also introduce more relevant coupling parameters through a reference
distance, dref = aφ−1/3, and a reduced temperature
λd = εd
(
d
dref
)3
, λu = εu/λd , T∗ = T/(T0λd), (2.3)
where φ is the volume fraction and a convenient choice for the proportionality constant is a = (φ(0)M )
1/3, φ(0)M
being the maximum value of φ for a given structure here taken as the fcc lattice [φ(0)M = φ
(fcc)
M =
√
2π/6].
We perform Monte Carlo simulations in the framework of the parallel tempering scheme [33] with a
distribution of temperatures bracketing the superparamagnetic (SPM)-ordered phase transition. The set
of temperatures {Tn} is chosen either from a geometric distribution, Tn+1/Tn = const or according to the
constant entropy increase scheme [34] which leads to a more homogeneous rate of configuration exchange
in the tempering scheme. Periodic boundary conditions and Ewald summations are usedwith the so-called
conducting external conditions, according to which the system is embedded in a medium characterized
by an infinite permeability, which eliminates the shape dependent depolarizing effects [13, 35]. We
emphasize that this is a necessary condition to get a spontaneous magnetization. At this point, we refer a
reader to the analysis and comments in [27, 36].
We calculate, on the one hand, the mean value of the energy, the spontaneous magnetization, per
particle
〈m〉 =
〈��� 1
N
∑
i
m̂i
���〉, (2.4)
and the nematic order parameter λ [13], revealing the occurrence of an orientational order. The mean
value of the magnetization, defined in equation (2.4) is the order parameter of the SFM phase. When
an anti-ferromagnetic phase is expected, as is the case for the simple cubic (sc) lattice, the staggered
magnetization should be considered in place of 〈m〉 with
〈m〉st =
〈{ ∑
α=1,3
[
1
N
∑
i
m̂iα(−1)piβ+piγ
]2}1/2〉
, (2.5)
where piα denotes the layer in the direction α to which the lattice site i pertains.
33703-3
V. Russier, E. Ngo
On the other hand, we calculate the heat capacity Cv and the susceptibility from the polarization
fluctuation
χm = Nβ〈δm2〉 = Nβ
(
〈m2〉 − 〈m〉2
)
, (2.6)
both showing a characteristic peak at the transition temperature. Finally, the location of transition temper-
ature is determined from finite size scaling analysis through the crossing point of the Binder cumulant [37]
Bm =
1
2
(
5 − 3
〈m4〉
〈m2〉2
)
, (2.7)
and the location of the Cv and 〈δm2〉 versus temperature peaks. Here, as in [38], Bm corresponds to
the normalization such that its limiting values in the (anti)-ferromagnetic and paramagnetic phases are
Bm = 1.0 and Bm = 0.0, respectively.
Since we consider a random distribution of easy axes, by increasing λu we gradually increase the
amount of disorder. Starting from λu = 0 to λu � λd, the model goes from the perfect lattice of dipoles
to the random axes dipoles (RAD) model where the moments are assigned in the direction to the easy
axes m̂i = si n̂i and behave as Ising spins si = ±1. The former presents a well ordered ferromagnetic
phase at low temperature [13], while the latter presents a spin glass state (SSG) [31]. According to this
picture, we expect the system to present a SPM→ SFM and a SPM→ SSG transition for small and large
values of λu , respectively.
3. Results
In what follows, for the sake of simplicity, we shall denote the SPM and SFM as paramagnetic (PM)
and ferromagnetic (FM) phases, respectively. Our purpose is twofold: first, the determination of the
PM-ordered phase transition temperature in the weak λu regime, and, secondly, the characterization of
the ordered phase. In this work we do not focus on the SSG phase which is known to occur at large values
of λu [31].
In the simulations, we use the parameters of cobalt MNP of ca 8 nm in diameter and the volume
fraction corresponding to the experimental supracrystals elaborated in [4] in order to fix the dipolar
coupling constant. Doing this, we get λd ' 2.15. On the other hand, the MAE coupling constant is
seen as a variable parameter. It is clear that the conclusions hold for systems characterized by other
values of the dipolar coupling, since we can adjust the value of the reference temperature T0. Taking into
consideration the fact that the system free of anisotropy presents a FM phase [13–15], we expect the
ordered phase to keep its FM character at small values of λu or at least to present the characteristics of a
quasi-long-range order (QLRO) magnetic phase as it is the case for the random anisotropy model (RAM)
in the weak anisotropy regime [38].
First of all, in the absence of anisotropy, we determine the PM/FM transition temperature in the
framework of the finite size scaling analysis, namely from the crossing point in T∗ of the Binder cumulant
given by equation (2.7), corresponding to different system sizes, see figure 1. We locate the PM/FM
transition at T∗c = 0.618 ± 0.005. This result, leading to ρµ2/(kBTc) = 2.283 where ρ is the number
density, introduced by relating (d/dnn)3 of equation (2.3) to ρ, is in close agreement with the early finding
of Bouchaud and Zerah [14]. Moreover, as shown in figure 1, we find that T∗c is very well bracketed by the
Cv and χm peaks, namelyT∗p(Cv) < T∗c < T∗p(χm)with [T∗p(χm)−T∗p(Cv)] ' 0.02. A precise determination
of Bm at the PM-ordered transition, namely its non-trivial fixed point value, B∗m, is beyond the scope of
the present work. It is worth mentioning that this is a difficult task, which is not unambiguous since it
depends on non-essential parameters of the transition [26, 39–41]. Nevertheless, as expected we get a
value, B∗m = 0.76 ± 0.02 at λu = 0, compatible with that of either the Heisenberg model on the lattice
(0.7916 [42]) or the dipolar hard sphere model (' 0.772 to 0.812 [26]).
Then, we compare the behavior of the simple cubic lattice sc to the behavior of the face centered cubic
fcc lattice. It is well known that the sc lattice is antiferromagnetic [11, 43], and thus one should compare
the staggered spontaneous polarization, given by equation (2.5), of the sc to the usual one, equation (2.4),
33703-4
Ferromagnetic order in dipolar systems with anisotropy
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.5 0.53 0.56 0.59 0.62 0.65
B
m
T
*
λ
u
= 0.0
N = 256
864
2048
0
2
4
6
8
10
12
0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.5
1
1.5
2
2.5
χ
m
C
v
/Nk
B
T
*
N = 256
864
2048
Figure 1. (Color online) Left: the Binder cumulant for the moment, equation (2.7) for λu = 0. Right:
susceptibility, χm, and heat capacity Cv , right for λu = 0.
of the fcc. The result for both the polarization and the nematic order parameter λ, displayed in figure 2
for simulation boxes of similar size shows that the ordering behaviors of the two lattices present very
similar characteristics, at least in the absence of anisotropy. This differs from the situation of the Ising
PAD model corresponding to λu →∞ with aligned easy axes in the z direction where the PM-AFM and
PM-FM transitions are found at T∗c ' 2.5 and 4.0 for the sc and fcc lattices, respectively [43].
Then, we have performed simulations for the fcc lattice and λu in the range 0 6 λu 6 1.28 which
is still in the low anisotropy regime. We still determine the PM-ordered transition temperature from the
finite size behavior of the Binder cumulant and the location of the Cv and χm peaks. An important result
is a very weak dependence of the transition temperature with λu; indeed, we get T∗c = 0.62 ± 0.007 and
0.610±0.007 for λu = 0.465 and 1.28, respectively. The nature of the ordered phase can be deduced from
the low temperature behavior of the magnetization 〈m〉 and the shape of the Cv(T∗) curve. We expect the
magnetization in the FM phase, on the one hand, to continuously increase with a decrease of T∗ down to
vanishing temperatures and, on the other hand, to be size independent below a threshold temperature. In
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
<m> , λ
T
*
Figure 2. (Color online) Comparison of the polarization (fcc) 〈m〉 or staggered polarization (sc) 〈m〉st,
open symbols, and nematic order parameter λ, solid symbols, on the fcc and sc lattices. Triangles: fcc
lattice with N = 1372, squares: sc lattice with N = 1000, λu = 0.
33703-5
V. Russier, E. Ngo
0.1
0.3
0.5
0.7
0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
< m >
T
*
0.85
0.95
0.01 0.04 0.07
Figure 3. Polarization of the fcc lattice in terms of T∗ for λu = 0 and N = 256, squares; 864, circles and
2916, triangles. Inset: low temperature behavior of the nematic order parameter λ, bottom and 〈m〉, top.
particular, the continuous increase of 〈m〉 down to T∗ ' 0 is indicative of the absence of freezing of the
system at any finite temperature. These two criteria are clearly reached in the absence of anisotropy, as
can be seen in figure 3. On the other hand, a lambda-like shape of Cv together with the size dependence
of the Cv peak, as displayed in figure 1 for λu = 0 is indicative of the onset of QLRO state from the PM
one. Indeed, the Cv curve shape of typical SG strongly differs from a lambda-like one and, moreover,
does not follow the characteristic finite size scaling of the PM/FM (or PM/AFM) transition [16, 32].
For λu = 0.465, we get a behavior for both the magnetization 〈m〉 and the nematic order parameter λ
very close to that observed in the absence of anisotropy, λu = 0 and the above two criteria are satisfied.
Moreover, the dependence of 〈m〉 with respect to the number of particles, N is negligible, at least for
N 6 2048, indicating that the mean spontaneous polarization at low temperature remains finite on an
infinite system. Therefore, we can conclude that the ordered phase is FM in this case. This is obviously no
more the case when λu = 1.28, as shown in figure 4 where we compare the corresponding spontaneous
polarization in terms of T∗ for different system sizes. Increasing λu , up to λu = 1.28, the behavior of 〈m〉
strongly differs and the first point is the appearance of a plateau in the 〈m〉(T∗) curve at low temperature.
0.1
0.3
0.5
0.7
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1
0.3
0.5
0.7
0.9
< m >
T
*
λ
u
= 0.465
λ
u
= 1.28
N = 256
500
864
2048
Figure 4. Comparison of the fcc lattice polarization in terms of T∗ for λu = 0.465 and 1.28 and different
values of N as indicated in the figure.
33703-6
Ferromagnetic order in dipolar systems with anisotropy
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7
B
m
T
*
256
500
864
2048
0.8
1
1.2
1.4
1.6
1.8
2
2.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
C
v
/Nk
B
T
*
256
864
1372
2048
Figure 5. (Color online) Left: Binder cumulant for λu = 1.28 and different values of N as indicated in
the figure. Right: heat capacity for λu = 1.28 and different values of N as indicated in the figure.
We also get a different behavior for the Binder cumulant, displayed in figure 5. Indeed, if there is still
a crossing point at the onset of the PM-ordered phase, (which is, however, less pronounced), the low
temperature first Bm no more reaches the limiting value Bm = 1, and its value decreases with an increase
in N which results from the onset of a decrease of Bm with T∗ beyond some value of N . Such a behavior
resembles the one obtained in [44] where it was analyzed as the occurrence of a reentrant SG phase. It is
worth mentioning that the onset of the PM-ordered transition still presents a FM character, as indicated
by the shape of the Cv curve displayed in figure 5 but it is presumably a QLRO one instead of a FM
phase as indicated first by the size dependence of Cv , strongly weakened when compared to the PM-FM
transition case. To confirm this point, we show in figure 6 the behavior of the 〈m〉 in terms of N at a low
temperature. From the fits of 〈m〉(N) at fixed T∗ in the form ' b.N−b and the values of b(λu) obtained
(b = 0.01379 ± 0.0009, 0.04936 ± 0.006 and 0.09904 ± 0.01 for λu = 0.5, 1.0 and 1.28, respectively),
see figure 6, we conclude that the spontaneous magnetization at a low temperature is likely to vanish in
0.6
0.9
100 1000
< m >
N
λu = 1.28; T
*
= 0.14
T
*
= 0.11
λu = 1.0; T
*
= 0.094
λu = 0.5; T
*
= 0.094
Figure 6. Log log plot of the magnetization at low temperature in terms of N . The lines are fits of the
form 〈m〉 = aN−b with b = 0.01379 ± 0.0009 and 0.04936 ± 0.006 at T∗ = 0.094 for λu = 0.5 and
1.0, respectively, and b = 0.09355 ± 0.0.006 and 0.09904 ± 0.01 for λu = 1.28 and T∗ = 0.14 and 0.11,
respectively.
33703-7
V. Russier, E. Ngo
0.1
0.3
0.5
0.7
0.9
0.2 0.6 1 1.4 1.8
< m > , λ
T
*
λu = 1.86; θm = π/6
Figure 7. (Color online) Spontaneous polarization (squares) and nematic order parameter (circles) for a
system with a textured distribution of easy axes centered on the z-axis with a solid angle limited by a
gaussian with variance Θm = π/6; λu = 1.86.
the limit N → ∞ for λu > 0.5. From the low temperature moments configurations obtained with 2048
particles and λu = 1.28, we cannot conclude on the onset of the domain formation at least in this range
of sizes; on the other hand, the 〈µ̂i . µ̂j〉 correlation function still presents a FM character with no sign
inversion but a noticeably reduced range when compared to the λu = 0 case.
Finally, we show that the FM ordered phase is recovered when we consider a preferentially oriented
easy axes distribution, with probability
p(Θ) ∝ sin(Θ)e−Θ
2/2σ2
around the z-axis where one recovers a usual random distribution for σ � 1. In figure 7 we show an
example with σ = π/6 in the case λu = 1.86 a value for which the long-range FM order is lost when the
easy axes is randomly distributed.
To conclude, we have shown that an assembly of MNP located on a perfectly ordered fcc lattice
presents a FM phase at low temperature only in a reduced range of MAE amplitude, λu 6 0.5 for a
random distribution of easy axes. However, there is an onset of a QLRO phase, at a temperature T∗c very
close to the PM-FM transition of the system free of anisotropy. This QLRO phase is characterized by a
spontaneous magnetization vanishing in the limit of an infinite system. The FM phase can be recovered
by aligning, at least partially, the easy axes.
Acknowledgements
We acknowledge important discussions with Dr. J. Richardi, Dr. I. Lisiecki, Dr. T. Ngo, Dr. C. Salze-
mann, Dr. S. Nakamae, Dr. C. Raepsaet, Dr. J.-J. Weis and Pr. J.J. Alonso.
V.Russier, who has been one of theDr. J.-P. Badiali’s PhD students, acknowledges the huge importance
Dr. J.-P. Badiali had on his carrier. The collaboration which lead to about 15 co-authored publications
aimed at a theoretical description of the electrochemical double layer which brought together statistical
physics physicists and electrochemists. It was during this activity that J.P. Badiali urged V. Russier to the
so-called dipolar hard sphere fluid as a model for molecular solvents. The work presented here dealing
with another application of this model is in some sense the continuation of this initial training and
collaboration.
Work performed under grant ANR-CE08-007 from the ANR French Agency. This work was granted
an access to the HPC resources of CINES under the allocation 2016-A0020906180 made by GENCI.
33703-8
Ferromagnetic order in dipolar systems with anisotropy
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https://doi.org/10.1103/PhysRev.70.954
https://doi.org/10.1103/PhysRevLett.68.2043
https://doi.org/10.1103/PhysRevE.48.3728
https://doi.org/10.1103/PhysRevB.47.9095
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V. Russier, E. Ngo
Феромагнiтний порядок у дипольних системах з
анiзотропiєю: застосування до магнiтних наночастинкових
супракристалiв
В. Рус’є, Е. Нго
Iнститут хiмiї та матерiалiв при унiверситетi “Париж-Схiд” (УПС), дослiдницький центр 7182
Нацiонального центру наукових дослiджень та УПС, вул. Анрi Дюнана, 2-8, 94320 Тьє, Францiя
Однодоменнi магнiтнi наночастинки (МНЧ) iз дипольною мiжчастинковою взаємодiєю можуть окрiм ма-
гнiтокристалiчної фази знаходитися у низькотемпературнiй феромагнiтнiй (ФМ) або спiново-склянiй фа-
зах залежно вiд структури та ступеню впорядкування системи. Методом моделювання Монте-Карло в
рамках односпiнової чи макроспiнової моделей авторами дослiджується монодисперсна система одно-
доменнихМНЧ, розташованих на вузлах iдеальної ґратки з симетрiєю типу fcc i з довiльно розподiленими
осями. Розглядається випадок низької анiзотропiї, зокрема початку зникнення дипольної далекосяжної
ФМ фази.
Ключовi слова: магнiтнi наночастинки, моделювання Монте-Карло, феромагнiтне впорядкування
33703-10
Introduction
Model and Monte Carlo simulations
Results
|