# Dipolar Interaction and Sample Shape Effects on the Hysteresis Properties of
2$d$ Array of Magnetic Nanoparticles

Manish Anand<EMAIL_ADDRESS>Department of Physics, Bihar National
College, Patna University, Patna-800004, India.

###### Abstract

We study the ground state and magnetic hysteresis properties of 2$d$ arrays
($L_{x}\times L_{y}$) of dipolar interacting magnetic nanoparticles (MNPs) by
performing micromagnetic simulations. Our primary interest is to understand
the effect of sample shape, $\Theta$\- the ratio of the dipolar strength to
the anisotropy strength, and the direction of the applied field
$\vec{H}=H_{o}\hat{e}_{H}$ on the ground state and the magnetic hysteresis in
an array of MNPs. To study the effect of shape of the sample, we have varied
the aspect ratio $A_{r}=L_{y}/L_{x}$ which in turn, is found to induce shape
anisotropy in the system. Our main observations are: (a) When the dipolar
interaction is strong $(\Theta>1)$, the ground state morphology has in-plane
ordering of magnetic moments. (b) The ground state morphology has randomly
oriented magnetic moments which is robust with respect to system sizes and
$A_{r}$ for weakly interacting MNPs ($\Theta<1$). (c) Micromagnetic
simulations suggests that the dipolar interaction decreases the coercive field
$H_{c}$. (d) The remanence magnetization $M_{r}$ is found to be strongly
dependent not only on the strength of dipolar interaction but also on the
shape of the sample. (e) Due to anisotropic nature of dipolar interaction, a
strong effect of shape anisotropy is observed when the field is applied along
longer axis of the sample. The dipolar interaction in such a case induces an
effective ferromagnetic coupling when the aspect ratio is very large. These
results are of vital importance in high-density recording systems, magneto-
impedance sensors, etc.

##  I Introduction

Magnetic nanoparticles (MNPs) arrays are of profound importance not only
because of interesting physics but also due to their numerous technological
applications wu2016 ; ulbrich ; hervault ; bustamante ; schutz ; dejardin ;
capiod ; puntes ; jordanovic ; zhong ; cowburn ; manish2020 . For instance, a
two-dimensional array of MNPs is a suitable system for high-density digital
storage and perpendicular recording media pardavi ; mohtasebzadeh ; xue . The
magnetic properties in such a case depend strongly on the shape, size,
geometry of the systems and the magnetic interactions. The primary magnetic
interaction in such systems is dipole-dipole interactions. The dipolar
interaction is long-ranged and favours ferromagnetic as well as
antiferromagnetic coupling. As a result, unconventional morphologies are
observed when the size of the system is comparable to the range of dipolar
interactions. Due to anisotropic nature of dipolar interactions, there have
also been observations of ferromagnetic, striped, checkerboard patterns,
vortices, etc. depending on the geometry of the lattice Low ; Kbell ; Edlund ;
anand2019 ; alivisatos ; collier . The magnetic hysteresis properties in such
systems are found to be greatly influenced by the dipolar interactions
kechrakos2005 ; usov2017 ; branquinho2013 ; anand2020

There are several studies in the literature which make an implicit and
explicit reference to the ubiquitous dipolar interactions and anisotropy in
ordered arrays of MNPs. We summarize below few of them which is relevant to
our present work: (a) Li et al. studied the ground state, magnetic specific
and magnetic hysteresis properties for three types of closely spaced
nanomagnet arrays using Monte Carlo simulation li . They observed vortex state
due to the dipolar interactions in these arrays. For face-centred cubic
nanomagnet arrays, a slight jump occurs in the hysteresis curve. (b) Using the
micromagnetic simulation, Yang et al. studied the magnetic properties in two-
dimensional arrays of MNPs yang . They found that coercivity increases with
arrays disorder. (c) Using the Landau-Lifshitz-Gilbert approach, Morales-Meza
et al. studied the magnetization reversal in a two-dimensional array of MNPs
morales . They showed that coercivity is reduced even if the particle position
in the array is random. (d) Chinni et al. performed experiments and
micromagnetic simulation using Object Oriented MicroMagnetic Framework (OOMMF)
to study magnetic properties in nanocomposite films chinni . They observed an
unexpected hysteretic behaviour characterized by in-plane anisotropy and
crossed branches in the hysteresis curves measured along the hard direction.
(e) Faure et al. studied the magnetic properties of ordered arrays of MNPs
using experiment and Monte Carlo simulation faure . The dipolar interaction is
found to induce a ferromagnetic coupling that increases in strength with
decreasing thickness of the array. (f) Using theoretical calculations, Xue et
al. studied the magnetic hysteresis properties of the two-dimensional
hexagonal array of MNPs xue . The hysteresis curves are found to vary their
shapes from a rectangle to a non-hysteresis straight line through a set of
complicated loops, in accordance with the magnetization reversal process.

Although the above studies imply that dipolar interaction and geometrical
arrangement of MNPs affect the magnetic properties of an ordered array of
MNPs, a systematic study as a function of dipolar interaction strength, shape
anisotropy of the system and direction of the applied field is still missing.
In this work, we attempt to understand the effect of dipolar interaction
manipulated by changing the interparticle separation, the direction of applied
magnetic field and shape anisotropy induced by varying the aspect ratio on the
magnetic properties of two dimensional ($2d$) arrays of MNPs. The main
questions which we have attempted to address are : (1) What are the
consequences of dipolar interaction on ground state organizations of MNPs? (2)
How are the magnetic hysteresis properties modified due to the dipolar
interactions, shape anisotropy of the sample and direction of the applied
magnetic field? To answer these questions, we consider 2$d$ ($L_{x}\times
L_{y}$) arrays of cubical shaped Fe3O4 MNPs. It is well known fact that
uniformly magnetized cubical MNPs do not possess shape anisotropy moskowitz .
But we have been able to induce it in the system by just varying the sample
shape. It will be shown that the latter has a drastic effect on the hysteresis
behaviour. These ordered arrays of MNPs have many interesting properties due
to structural order, well defined interparticle interactions and geometry
confinement Held . In such a case, MNPs are coated with an inorganic
surfactant to prevent agglomeration during self-assembling. Due to this MNPs
are at a well distance beyond the range of exchange interaction. As a
consequence, the predominant interaction in such arrays is dipolar interaction
Dkech . The energy of such an assembly is, therefore, given by the sum of the
anisotropy energy and the dipolar interaction energy.

To vary the relative dipolar interaction strength with respect to anisotropy
strength, we have defined a ratio $\Theta=D/KV$ where $D$ is the strength of
dipolar interaction, $K$ is the anisotropy constant, and $V$ is the volume of
the nanoparticle. The value of $\Theta$ for various interparticle separation
$a$ for Fe3O4 has been given in Table 1. We refer to $\Theta>1$ as the strong
dipolar interaction regime, $\Theta<1$ as the weak dipolar interaction regime.
We have performed micromagnetic simulation using OOMMF code from NIST donahue
. In OOMMF code, the finite difference method is employed, which requires
discretization of a chosen geometry over a grid of identical prism-cells, and
the magnetization is supposed to be uniform in each cell.  In the
implementation of OOMMF, the continuous magnetic material is divided in
discrete cubes (i.e. grid cells), which are in geometrical contact. This
discretization scheme suits to a continuous magnetic film rather than an
assembly of non-touching nanoparticles. Due to this reason, this type of
modelling does not discriminate between the magnetic behavior of 2$d$
continuous films and 2$d$ nanoparticle arrays. To overcome this, we have put
exchange interaction to zero to mimic the seperated MNPs arrays magnetically.
The absence of exhange interaction is also needed for the MNPs where they are
coated with a surfactant to avoid agglomeration. We have sucessfully
implemented it in our earlier studies anand2016 ; anand2018 . We have used
Landau-Lifshitz-Gilbert (LLG) equation which is used to describe the
precessional motion of a moment in magnetic field at $T=0$ K. We have solved
the coupled equation of motion for the given lattice to obtain the minimum
energy configurations. To study the hysteresis, we have applied a dc magnetic
field $\mu_{o}\vec{H}=H_{o}\hat{e}_{H}$, where $\hat{e}_{H}=\hat{x}$,
$\hat{y}$ and $\hat{z}$ is the direction of the applied field along $x$, $y$
and $z$-axis respectively. We also study the role of aspect ratio
$A_{r}=L_{y}/L_{x}$ to induce shape anisotropy in the system and the direction
of the field $\hat{e}_{H}$.

The interplay of anisotropy, dipolar energy and aspect ratio creates unusual
morphologies which have profound implications on the magnetic properties. Our
work, therefore, throws light on this subject from a microscopic picture and
provides a basis for results obtained in experimental and theoretical studies.
This paper is organized as follows. In Section II, we introduce the model for
an array of MNPs, the LLG equation, which provides the prototypical ground
state (GS) morphologies and equilibrium morphologies. In Sec. III, we present
our numerical results and discuss the dependence of the magnetic hysteresis on
$\Theta$, aspect ratio $A_{r}=L_{y}/L_{x}$ and the direction of the applied
field $\hat{e}_{H}$. A conclusion of our results is provided in Section IV.

##  II Model and Methodology

###  II.1 Model for Dipolar Interacting Arrays of MNPs

We consider a self-assembled 2$d$ array ($L_{x}\times L_{y}$) of cubical
shaped magnetic nanoparticles in the $xy$-plane. The total number of the MNPs
in the assmebly are $N=L_{x}/a\times L_{y}/a$, where $l$ is the length of an
edge of the particle and $a$ is the lattice spacing. Each MNP has magnetic
moment $\vec{\mu_{i}}=\mu\hat{e_{i}}$, $i=1,2,\cdot\cdot\cdot,N$. The
magnitude of the magnetic moment $\mu=M_{s}V$ where $M_{s}$ is the saturation
magnetization and $V=l^{3}$ is the volume of the magnetic nanoparticle. The
MNPs are assumed to have uniaxial anisotropy $\vec{K}=K\hat{k_{i}}$ where $K$
is the anisotropy constant, and $\hat{k_{i}}$ is the direction of the
anisotropy. Generally, MNPs are coated with a surfactant to prevent
agglomeration, which suppresses the exchange interactions. The energy of such
a system, therefore, includes constribution from anisotroy energy $E_{a}$ and
dipolar interaction $E_{d}$ Bedanta ; Haase :

$E=E_{a}+E_{d}=-KV\sum_{i}(\hat{k_{i}}\cdot{\hat{e}_{i}})^{2}-D\sum_{i,j}\frac{3(\hat{e_{i}}\cdot{{\hat{r}}_{ij}})(\hat{e_{j}}\cdot{\hat{r}_{ij}})-(\hat{e_{i}}\cdot\hat{e_{j}})}{\left(r_{ij}/a\right)^{3}},$
(1)

where $\mu_{o}$ is the permeability of free space, $\hat{e}_{i}$ is the unit
vector along the magnetic moment, $r_{ij}$ is the distance between particles
$i$ and $j$, and $\hat{r}_{ij}$ is the unit vector along with it. The
$1/r_{ij}^{3}$ dependance means that the dipolar interaction is long ranged in
nature. The analytical calculation of dipolar interaction between cubical
shaped MNPs can be found in the reference schabes . It is well known fact that
self assembled MNPs usually form $2d$ close-packed hexagonal lattices but
magnetic properties are found to be nearly independent of the structure of the
lattice russier .

Here we define the dipolar interaction strength $D=\mu_{o}{\mu}^{2}/4\pi
a^{3}$. As magnetic properties of such an assembly are governed by the
relative strength of the anisotropy and dipolar energy, we have defined a
ratio $\Theta=D/KV$. Although dipolar interactions and anisotropy energy
depend on various system parameters, the behaviour of the assembly will be
dictated by $\Theta$ rather than the precise values of parameters such as $a$,
$V,$ $\mu$ and $K$. When $D$ is greater than $KV$, the dipolar interaction is
stronger than anisotropy i.e $\Theta>1$. Similarly, $\Theta<1$ can be termed
as the weak dipolar regime.

In the presence of an external magnetic field $\mu_{o}\vec{H}$, there is an
additional contribution to energy $E$ given by Haase

$E_{H}=-H_{o}\sum_{i=1}\vec{\mu_{i}}\cdot\hat{e}_{H},$ (2)

here $H_{o}$ is the magnitude of field and $\hat{e}_{H}$ is the unit vector in
the direction of applied field.

###  II.2 Landau-Lifshitz-Gilbert Equation

The precessional motion of the magnetic moment $\vec{\mu}_{i}$ in a magnetic
field can be described by the LLG equation Dantas :

$\frac{\mbox{d}\vec{\mu_{i}}}{\mbox{dt}}=-\gamma\vec{\mu_{i}}\times\vec{H}_{i}^{e}-\lambda\vec{\mu_{i}}\times\left(\vec{\mu_{i}}\times\vec{H}_{i}^{e}\right),\quad
i=1,2,\cdot\cdot\cdot,N,$ (3)

where $\gamma$ is the electron gyromagnetic ratio,
$\lambda=\gamma\alpha/M_{s}$ is a phenomenological dimensionless damping
factor, and $\vec{H}_{i}^{e}=-\partial E_{T}/\partial\vec{\mu}_{i}$ is the
effective field experienced by magnetic moment, where $E_{T}=E+E_{H}$. While
studying ground state properties, $E_{H}$ is taken to be zero. The first term
in Eq. (3) takes care of the precession of $\vec{\mu}_{i}$ around
$\vec{H}_{i}^{e}$. The second term is due to a phenomenological dissipative
motion: the magnetic moment $\vec{\mu}_{i}$ precesses around
$\vec{H}_{i}^{e}$. The solution of these coupled differential equations yields
the GS configuration $\\{\vec{\mu_{i}}\\}$, of the assembly.

To perform micromagnetic simulation using OOMMF code, the entire system is
discretized into cells where each has lateral dimension $l$ say. Each cell in
such case represents a magnetic moment $\mu=M_{s}V$. The centre-to-centre
separation between moments is, therefore, $l\equiv a_{0}$. By our formulation
in Eq. (1), the strength of the dipolar interaction can be manipulated by
varying the centre-to-centre separation $a$ of the MNPs. However, with the
protocol implemented in OOMMF, a change in the centre-to-centre separation
from $a_{0}$ to $a_{\beta}$ say, changes the cell volume from $V\equiv V_{0}\
(=a_{0}^{3})$ to $V_{\beta}\ (=a_{\beta}^{3})$. As a consequence, the magnetic
moment gets altered to $\mu_{\beta}=M_{s}V_{\beta}$, which ultimately modifies
the magnetic properties of the particles under study. This undesirable
artefact in the simulation needs to be overcome. To overcome this artefact, we
formulated a rescaling method for saturation magnetization:

$M_{s}^{\beta}=M_{s}^{0}\frac{V_{0}}{V_{\beta}},$ (4)

where $M_{s}^{0}$ is the saturation magnetization for nano particle of volume
$V_{0}=a_{0}^{3}$. It is easy to see that now
$\mu=M_{s}^{0}V_{0}=M_{s}^{\beta}V_{\beta}$ as desired. Corresponding changes
need to be incorporated in other related variables of interest such as the
coercive field $H_{c}$ and the anisotropy field $H_{K}=2K/M_{s}$ carrey  for
non-interacting or weakly interacting MNPs which play an important role in
hysteresis:

$\displaystyle H_{c}^{\beta}$ $\displaystyle=$ $\displaystyle
H_{c}^{0}\frac{V_{\beta}}{V_{0}},$ (5) $\displaystyle H_{K}^{\beta}$
$\displaystyle=$ $\displaystyle H_{K}^{0}\frac{V_{\beta}}{V_{0}}.$ (6)

We have successfully implemented this scaling procedure to study the heat
dissipation and spin transport properties in the assembly of dipolar
interacting MNPs in our earlier work anand2016 ; anand2018 .

##  III Numerical Results

We consider cubical shaped MNPs of Fe3O4 arranged on a $2d$ ($L_{x}\times
L_{y}$) lattice. The particle size is chosen to be $l=10$ nm. We have used
anisotropy constant $K=13\times 10^{3}$ J$\mathrm{m}^{-3}$ and saturation
magnetization $M_{s}=4.77\times 10^{5}$ A/m for numerical evaluations. The six
values of interparticle separation $a$ are considered for magnetic hysteresis
study: $a_{0}=10$ nm, $a_{1}$ = 12 nm, $a_{2}$ = 16 nm, $a_{3}$ = 20 nm,
$a_{4}$ = 30 nm and $a_{5}$ = 40 nm. Table 1 provides the values of $\Theta$
for these interparticle separations. The initial condition that we choose for
the assembly of MNPs is random orientations of magnetic moments and anisotropy
axes. All the data obtained using the simulations are averaged over 50 sets of
initial conditions. The ground state morphologies are obtained by solving LLG
equation using OOMMF code in the absence of external magnetic field. To study
the magnetic hysteresis properties, we apply a dc magnetic field of strength
-150 mT to 150 mT for $a=10$ nm. For other interparticle separations, the
scaled saturation magnetization $M_{s}^{\beta}$ were obtained using Eq. (4).
The corresponding $H_{c}^{\beta}$ and $H_{K}^{\beta}$ were calculated using
Eqs. (5) and (6), respectively. We have chosen the magnetic field to be large
enough as comapred to $H_{c}$ to achieve saturation and the ramping up or
slowing down has been appropriately fine-tuned to capture the magnetic
properties near $H_{c}$. We study the dependence of magnetic hysteresis on the
$\Theta$, aspect ratio $A_{r}=L_{y}/L_{x}$ and the direction of the applied
field $\hat{e}_{H}$. In our simulations, $L_{x}=120$ nm; $A_{r}=1,2,4,8,16$
and 32; $\Theta=1.75$, 1.01, 0.42, 0.22, 0.06 and 0.03; $\hat{e}_{H}=\hat{x}$,
$\hat{y}$ and $\hat{z}$.

###  III.1 Ground State (GS) Morphologies

We first study the effect of $\Theta$ and $A_{r}$ on ground state spin
morphologies. For this purpose, we look square samples i.e $A_{r}=1.0$ such
that $L_{x}\times L_{y}\equiv L_{x}\times L_{x}$. Fig. (1) depicts GS
morphologies for $L_{x}$ = 120 nm. The interparticle separation $a$ is chosen
to be 10 nm [see Fig. 1(a)] and 20 nm [see Fig. 1(b)]. These correspond to
$\Theta=1.75$ and 0.22, respectively. So the number of spins in Fig. 1(a) is
$12\times 12$ and $6\times 6$ in Fig. 1(b). we depict the spins with noz-zero
$z$-component by the green cone. Those lying in the $xy$ plane with a positive
$x$-component have been indicated by the red coloured cone, while those with
negative $x$-component by the blue coloured cone. It is quite evident that for
strongly interacting MNPs ($\Theta=1.75$), all the spins are in the $xy$-plane
[see Fig. 1(a)]. They exhibit locally ordered regions. On the other hand, when
magnetic interactions are weak ($\Theta=0.22$), the moments are randomly
oriened, signifying lack of magnetic order [see Fig. 1(b)]. Magnetic moments
also tend to align normal to the plane of the sample. It means that dipolar
interaction favours in-plane ordering. Then, we study the effect of aspect
ratio $A_{r}$ on the spin morphologies. For $\Theta<1$, the features are
unchanged as $A_{r}$ is increased to 2, 3, 4, etc. and are prototypically
represented by Fig. 1(b). We do not show them to avoid repetition. In Fig.
(2), we depict morphologies corresponding to the strongly dipolar interacting
MNPs ($a=10$ nm, $\Theta=1.75$) for $L_{x}$ = 120 nm and two aspect ratio (a)
$A_{r}$ = 2 [see Fig. 2(a)] and (b) $A_{r}$ = 4 [see Fig. 2(b)]. In both the
cases, the MNPs exhibit local order and perfer to lie in the $xy$-plane. The
morphology of the magnetic moments near the sample edges is very distinct from
that in the bulk. The moments tend to align along the edges as ferromagnetic
chains.

###  III.2 Magnetic Hysteresis Study

Next, we study the effect of dipolar strength $\Theta$, shape anisotropy of
the sample and direction of applied field $\hat{e}_{H}$ on the magnetic
hysteresis in a systematic manner. To vary the dipolar interaction strength,
we have varied the interparticle separation. To induce shape anisotropy in the
system, we have changed the aspect ratio $A_{r}$ of the sample. In Fig. (3),
we have plotted magnetic hysteresis curves for square sample, i.e.,
$L_{x}=L_{y}=1.0$ as a function of $\Theta$ and $\hat{e}_{H}$. We have
considered six values of dipolar strength $\Theta=1.75$, 1.01, 0.42, 0.22,
0.06 and 0.03 in each case. The magnetic field axis ($x$-axis) has been scaled
by $H_{K}$, the anisotropy field (for non-interacting MNPs with randomly
oriented anisotropy axes). The direction of the external field is (a)
$\mu_{o}\vec{H}=H_{o}\hat{x}$ [see Fig. 3(a)], (b)
$\mu_{o}\vec{H}=H_{o}\hat{y}$ [see Fig. 3(b)] and (c)
$\mu_{o}\vec{H}=H_{o}\hat{z}$ [see Fig. 3(c)]. It is quite evident that when
the field is applied in the plane of the sample, i.e., either along $x$ or
$y$-direction, the coercive field $H_{c}$ decreases with an increase in the
strength of dipolar interaction. As a consequence, the area under the
hysteresis curve diminishes. When the field is applied normal to the plane of
the sample ($z$-direction), magnetic moment ceases to follow the applied field
for $\Theta>1$. As a result, non-hysteresis is observed, so $H_{c}$ and
$M_{r}$ tend to zero for strongly interacting MNPs. It is clearly seen that
the coercive field $H_{c}\approx 0.48H_{K}$ and $M_{r}\approx 0.5$ for weak
interaction or non-interacting MNPs ($\Theta<1$) irrespective of the direction
of the applied field. These values of $H_{c}$ and $M_{r}$ correspond to single
particle hysteresis with randomly oriented anisotropy axis carrey ; stoner1948
. It means that magnetic hysteresis follows the Stoner-Wohlfarth model when
the interaction among the MNPs is weak as expected carrey ; stoner1948 .

To study the dependence of shape anisotropy on the magnetic properties, we
study the magnetic hysteresis as a function of aspect ratio $A_{r}$ for
various values of $\Theta$ and three directions of the applied field, i.e.,
along ${x}$, ${y}$ and ${z}$-axis respectively. We have considered six values
of aspect ratio $A_{r}=1.0$, 2.0, 4.0, 8.0, 16.0 and 32.0 in each case. In
Fig. (4), the direction of the applied field is in the plane of the sample
[$\mu_{o}\vec{H}=H_{o}\hat{x}$ and $\mu_{o}\vec{H}=H_{o}\hat{y}$]. The values
of $\Theta$ are (a) $\Theta=1.75$ [see Fig. 4(a) and 4(e)], (b) $\Theta=1.01$
[see Fig. 4(b) and 4(f)], (a) $\Theta=0.42$ [see Fig. 4(c) and 4(g)] and (d)
$\Theta=0.22$ [see Fig. 4(d) and 4(h)]. The direction of the applied field is
normal to the plane of the sample (along the $z$-axis) in Fig. (5). The other
parameters, $A_{r}$ and $\Theta$ in Fig. (5) remain the same as that of Fig.
(4). When the applied field is along the shorter axis (along the $x$-axis),
dipolar interaction decreases the value of $H_{c}$ and $M_{r}$ [see Fig.
4(a)-(d)]. There is a weak dependence of $H_{c}$ on $A_{r}$. For weakly
interacting MNPs, hysteresis curves follow the Stoner-Wohlfarth model
stoner1948 . A strong effect of shape anisotropy is observed when the field is
applied along the increasing length of the sample, i.e., along the $y$-axis.
The remanent magnetization $M_{r}$ increases with increase in $A_{r}$, and it
reaches to 0.9 for strongly interacting MNPs [see Fig. 4(e)]. In this case,
also, the larger is the strength of dipolar interaction, the smaller is the
value of $H_{c}$ [see Fig. 4 (e)-(h)]. The magnetic moments cease to follow
the applied magnetic field when the field is applied along the $z$-axis for
strongly interacting MNPs. As a consequence, almost no hysteresis is observed
for strongly interacting MNPs with applied field normal to the plane of the
sample [see Fig. 5(a)-(b)]. Like the other two cases stated above, weakly
interacting MNPs follow the Stoner-Wohlfarth model irrespective of $A_{r}$.

###  III.3 Characterization of Magnetic Hysteresis Curves

Finally, we study the variation of the coercive field $H_{c}$ and $M_{r}$ as a
function of $\Theta$, $A_{r}$ and $\hat{e}_{H}$ by extracting their values
from the simulated magnetic hysteresis curves. In Fig. (6), we have plotted
variation of $H_{c}$ (scaled by $H_{K}$) and $M_{r}$ as function of dipolar
strength $\Theta$ for six values of aspect ratio $A_{r}=1.0$, 2.0, 4.0, 8.0,
16.0 and 32.0. The direction of the applied magnetic field is (a)
$\mu_{o}\vec{H}=H_{o}\hat{x}$ [see Fig. 6(a) and 6(d)], (b)
$\mu_{o}\vec{H}=H_{o}\hat{y}$ [see Fig. 6(b) and 6(e)] and (c)
$\mu_{o}\vec{H}=H_{o}\hat{z}$ [see Fig. 6(c) and 6(f)]. With the field applied
along the $x$-axis, $H_{c}$ decreases with an increase in the strength of
dipolar interaction $\Theta$ and its value drops to $0.1H_{K}$ for
$A_{r}=32.0$ and $\Theta=1.75$ [see Fig. 6(a)]. The variation of $H_{c}$ is
similar as with field along $x$-axis when the field is applied longer axis of
the sample (along the $y$-axis), but the value of $H_{c}$ is slightly higher
in this case for $A_{r}=32.0$ and $\Theta=1.75$ [see Fig. 6(b)]. When the
field is applied normal to the plane of the sample, $H_{c}$ decreases very
fast with an increase in $\Theta$ and $A_{r}$. Its value drops down to zero
for strongly interacting MNPs, and $A_{r}=32.0$ [see Fig. 6(c)]. There is a
weak dependence of $M_{r}$ on $A_{r}$ when the field is applied along the
$x$-axis [see Fig. 6(d)]. There is a strong effect of shape anisotropy when
the field is applied along $y$-axis [see Fig. 6(e)]. For strongly interacting
MNPs, $M_{r}$ increases as the aspect ratio $A_{r}$ is increased and it
reaches to 0.9 for $\Theta=1.75$, and $A_{r}=32.0$ [see Fig. 6(e)]. $M_{r}$
decreases very sharply with $\Theta$, and its value drops to zero for
$\Theta=1.75$ when the magnetic field is applied normal to the plane of the
sample [see Fig. 6(f)]. For weakly interacting MNPs, magnetic hysteresis
curves follow the Stoner-Wohlfarth model irrespective of aspect ratio and
direction of the applied field, which is reflected in $H_{c}$ ($\approx
0.48H_{K}$) and $M_{r}$ ($\approx 0.5$).

These results can be explained by probing the effect of dipolar interaction
and shape anisotropy of the sample induced by varying aspect ratio. It is
evident from the ground state morphologies that the dipolar interactions
favour in-plane ordering of magnetic moments. When $A_{r}$ is increased, the
dipolar interaction favours ferromagnetic coupling between the magnetic
moments. It is because of the fact the demagnetization field decreases as
shape anisotropy is increased ($A_{r}$ increases in our case), which is also
reported by Wysin wysin2012 . As a consequence, magnetic moments tend to align
along the longer axis of the sample as it costs less energy as compared to
other configurations. It has also been reported by jordanovic et al.
jordanovic . This ferromagnetic coupling for strongly interacting MNPs can
also be explained by that fact that the exchange interaction between the MNPs
is assumed to be zero in the present work. Due to the absence of exchange
interaction, the energetics of domain walls are entirely controlled by the
dipolar interaction. It makes energetically favourable to form domain walls
along the increasing length of the sample. When an external field is applied
to these systems system, the response of this system not only depends on the
dipolar strength but also on the shape anisotropy of the sample which has been
induced by increasing $A_{r}$ in our work. When the field is applied along the
shorter axis of the sample, i.e., along the $x$-axis, the natural tendency to
get aligned along the longer axis of the sample is hindered. As a result,
magnetic moment ceases to follow the external field, which is reflected in a
decrease in the value of $M_{r}$ ($\approx 0.1$) and $H_{c}$ to 0.1$H_{K}$ for
strongly interacting MNPs with $A_{r}=32.0$. Magnetic moments are found to
follow the field when the field is applied along the longer axis of the
sample, but due to the increase in the strength of dipolar interaction as
well, $H_{c}$ decreases, but its value remains slightly higher as that of the
previous situation. Due to the same reason, $M_{r}$ increases with $A_{r}$ for
$\Theta>1$. This increase of $M_{r}$ and decrease of $H_{c}$ when the field is
applied along the longer axis of the sample is in qualitative agreement with
the work done by García-Arribas et al. garcia2013 . They have studied the
shape anisotropy effect in a thin film permalloy microstrips.

When we force the magnetic moments to get aligned normal to the plane of the
sample by applying the field along the $z$-axis, they oppose strongly for a
large value of $\Theta$ and $A_{r}$. This happens due to the fact that the
natural tendency of the magnetic moments is to get organized in the plane of
the sample ($xy$-plane in our case). As a result, non-hysteresis is observed
in this case, which is reflected in zero value of $H_{c}$ and $M_{r}$. This
non-hysteresis behaviour has also been reported by Xue et al. xue . They have
studied the hysteresis properties in a two-dimensional hexagonal array with
aligned uniaxial anisotropy. From all the cases, it is quite evident that
dipolar interaction always decreases the value of coercive field $H_{c}$. It
is because of that dipolar interactions cause a collective reversal of the
magnetic moments under an applied magnetic field and as a consequence, the
coercive field decreases with an increase in dipolar interaction strength.

##  IV Conclusion

To conclude, we have studied the effect of dipolar interactions and shape of
the sample on the magnetic properties of 2$d$ ($L_{x}\times L_{y}$) array of
cubical shaped magnetic nanoparticles (MNPs) by performing micromagnetic
simulation using OOMMF code donahue . It is well known that uniformly
magnetized cubical shaped particles do not possesses shape anisotropy
moskowitz  but we have been able to induce the latter in the system by varying
aspect ratio $A_{r}$. Our primary aim was to study ground state (GS)
morphologies, and understand the magnetic hysteresis properties as a function
of $\Theta$, shape anisotropy induced by varying the aspect ratio
$A_{r}(=L_{y}/L_{x})$ and the direction of the applied field
$\mu_{o}\vec{H}=H_{o}\hat{e}_{H}$. Our observations are as follows: (a) For
weakly interacting MNPs ($\Theta<1$), the magnetic moments are randomly
oriented, and the morphology is unaffected by $A_{r}$. MNPs also tend to align
normal to the plane of the sample. (b) For strong dipolar strengths
($\Theta>1$), magnetic moments prefer to orient in the plane of the sample.
The morphology, in this case, comprises regions of correlated moments. (c)
Magnetic moment favours ferromagnetic coupling along the longer axis of the
sample when $A_{r}$ is increased provided $\Theta>1$. (d) When the dipolar
interaction is weak; magnetic hysteresis curves follow the Stoner-Wohlfarth
model ($H_{c}\approx 0.48H_{K}$ and $M_{r}\approx 0.5$) irrespective of the
direction of applied field $\hat{e}_{H}$ and aspect ratio $A_{r}$. (e) A
strong effect of shape anisotropy is observed when the field is applied along
the increasing length of the sample, which is reflected in the very large
value of remanent magnetization $M_{r}\approx 0.9$. (f) The dipolar
interaction always decreases the coercive field $H_{c}$. (g) For strongly
interacting MNPs, $H_{c}$ and $M_{r}$ tend to zero when the field is applied
normal to the plane of the sample.

We have succeeded in obtaining well guidelines regarding how magnetic
hysteresis properties may change in the dipolar interacting MNPs arrays. The
obtained results contribute to expanding the fundamental comprehension of two-
dimensional dipolar interacting system and offer exciting implications for the
creation of self-assembled arrays of magnetic nanoparticles of desired
magnetic response. The micromagnetic simulation showed that the dipolar
interaction induces an effective ferromagnetic coupling in thin arrays (when
the aspect ratio is very large). It means that one can modulate the magnetic
properties of the ordered arrays of MNPs by varying the strength of shape
anisotropy of the system. The latter can be changed by varying the width and
length of the system. Our results are also relevant for the experimental
samples, which can be divided into various categories depending on how they
behave when an external magnetic field is applied to them.

## ACKNOWLEDGMENTS

Most of the numerical simulations presented in this work have been carried out
in the Department of Physics, Indian Institute of Technology (IIT) Delhi. I am
grateful to Prof. Varsha Banerjee for providing the computational facility at
IIT Delhi.

## References

  * (1) L. Wu, A. Mendoza-Garcia, Q. Li, and S. Sun,  Chemical reviews 116, 10473 (2016).
  * (2) K. Ulbrich, K. Hola, V. Subr, A. Bakandritsos, J. Tucek, and R. Zboril,  Chemical reviews 116, 5338 (2016).
  * (3) A. Hervault and N. T. K. Thanh,  Nanoscale 6, 11553 (2014).
  * (4) R. Bustamante, A. Millán, R. Piñol, F. Palacio, J. Carrey, M. Respaud, R. Fernandez-Pacheco, and N. J. O. Silva,  Physical Review B 88, 184406 (2013).
  * (5) C. A. Schütz, L. Juillerat-Jeanneret, H. Mueller, I. Lynch, and M. Riediker,  Nanomedicine 8, 449 (2013).
  * (6) J.-L. Déjardin, A. Franco, F. Vernay, and H. Kachkachi,  Physical Review B 97, 224407 (2018).
  * (7) P. Capiod, L. Bardotti, A. Tamion, O. Boisron, C. Albin, V. Dupuis, G. Renaud, P. Ohresser, and F. Tournus,  Physical Review Letters 122, 106802 (2019).
  * (8) V. F. Puntes and K. M. Krishnan,  IEEE transactions on magnetics 37, 2210 (2001).
  * (9) J. Jordanovic, M. Beleggia, J. Schiøtz, and C. Frandsen,  Journal of Applied Physics 118, 043901 (2015).
  * (10) Z. Zhong, B. Gates, Y. Xia, and D. Qin,  Langmuir 16, 10369 (2000).
  * (11) R. Cowburn, A. Adeyeye, and M. Welland,  New Journal of Physics 1, 16 (1999).
  * (12) M. Anand,  Journal of Magnetism and Magnetic Materials 522, 167538 (2021).
  * (13) M. Pardavi-Horvath,  Journal of magnetism and magnetic materials 177, 213 (1998).
  * (14) A. Mohtasebzadeh, L. Ye, and T. Crawford,  International journal of molecular sciences 16, 19769 (2015).
  * (15) D. Xue and Z. Yan,  Journal of applied physics 100, 103906 (2006).
  * (16) U. Löw, V. Emery, K. Fabricius, and S. Kivelson,  Physical Review Letters 72, 1918 (1994).
  * (17) K. De’Bell, A. MacIsaac, I. Booth, and J. Whitehead,  Physical Review B 55, 15108 (1997).
  * (18) E. Edlund and M. N. Jacobi,  Physical review letters 105, 137203 (2010).
  * (19) M. Anand, V. Banerjee, and J. Carrey,  Physical Review B 99, 024402 (2019).
  * (20) A. P. Alivisatos,  science 271, 933 (1996).
  * (21) C. Collier, R. Saykally, J. Shiang, S. Henrichs, and J. Heath,  Science 277, 1978 (1997).
  * (22) D. Kechrakos and K. Trohidou,  Physical Review B 71, 054416 (2005).
  * (23) N. Usov, O. Serebryakova, and V. Tarasov,  Nanoscale research letters 12, 489 (2017).
  * (24) L. C. Branquinho, M. S. Carrião, A. S. Costa, N. Zufelato, M. H. Sousa, R. Miotto, R. Ivkov, and A. F. Bakuzis,  Scientific reports 3, 2887 (2013).
  * (25) M. Anand,  Journal of Applied Physics 128, 023903 (2020).
  * (26) Y. Li, T. Wang, H. Liu, f. Dai, X. Yu, and G. Liu,  Journal of Nanomaterials 16, 331 (2015).
  * (27) B. Yang and Y. Zhao,  Journal of Applied Physics 110, 103908 (2011).
  * (28) M. Morales-Meza, P. P. Horley, A. Sukhov, and J. Berakdar,  The European Physical Journal B 87, 186 (2014).
  * (29) F. Chinni, F. Spizzo, F. Montoncello, V. Mattarello, C. Maurizio, G. Mattei, and L. D. Bianco,  Materials 10, 717 (2017).
  * (30) B. Faure et al.,  Nanoscale 5, 953 (2013).
  * (31) R. Moskowitz and E. Della Torre,  IEEE Transactions on Magnetics 2, 739 (1966).
  * (32) G. Held, G. Grinstein, H. Doyle, S. Sun, and C. Murray,  Physical Review B 64, 012408 (2001).
  * (33) D. Kechrakos and K. Trohidou,  Journal of nanoscience and nanotechnology 8, 2929 (2008).
  * (34) M. J. Donahue and D. G. Porter,  National Institute of Standards and Technology, Gaithersburg, MD (1999).
  * (35) M. Anand, J. Carrey, and V. Banerjee,  Physical Review B 94, 094425 (2016).
  * (36) M. Anand, J. Carrey, and V. Banerjee,  Journal of Magnetism and Magnetic Materials 454, 23 (2018).
  * (37) S. Bedanta and W. Kleemann,  Journal of Physics D: Applied Physics 42, 013001 (2008).
  * (38) C. Haase and U. Nowak,  Physical Review B 85, 045435 (2012).
  * (39) M. Schabes and A. Aharoni,  IEEE Transactions on Magnetics 23, 3882 (1987).
  * (40) V. Russier,  Journal of Applied Physics 89, 1287 (2001).
  * (41) C. C. Dantas and L. A. de Andrade,  Physical Review B 78, 024441 (2008).
  * (42) J. Carrey, B. Mehdaoui, and M. Respaud,  Journal of Applied Physics 109, 083921 (2011).
  * (43) E. C. Stoner and E. Wohlfarth,  Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 240, 599 (1948).
  * (44) G. Wysin,  available on https://www. phys. ksu. edu/personal/wysin/notes/demag. pdf (2012).
  * (45) A. García-Arribas, E. Fernández, A. V. Svalov, G. V. Kurlyandskaya, A. Barrainkua, D. Navas, and J. M. Barandiaran,  The European Physical Journal B 86, 136 (2013).

$a$ (nm) | $\Theta=D/KV$
---|---
$10$ | $1.75$
$12$ | $1.01$
$16$ | $0.42$
$20$ | $0.22$
$30$ | $0.06$
$40$ | $0.03$
Table 1: Evaluation of the ratio $\Theta=D/KV$ for Fe3O4 as a function of
interparticle separation $a$ for cubical shaped particle. The lateral
dimension of the particle is $l=10$ nm.

Figure 1: Typical GS morphologies for: (a) $L_{x}=L_{y}=120$ nm, $a=10$ nm and
$\Theta=1.75$; (b) $L_{x}=L_{y}=120$ nm, $a=20$ nm and $\Theta=0.22$. The
magnetic moments are coloured with red colour cone, which has a positive $x$
component, and the magnetic moments with negative $x$ component are coloured
blue colour cone. Those with a non-zero $z$ component, and pointing normal to
the $xy$-plane, are indicated with shown with green colour cone. For
$\Theta\simeq 0$, moments are randomly oriented, and they also tend to orient
normal to the plane of the sample. For $\Theta>1$, the moments predominantly
lie in the $xy$-plane and exhibit local order.

Figure 2: Typical GS morphologies for strongly interacting MNPs ($a=10$ nm,
$\Theta=1.75$) as a function of aspect ratio $A_{r}$. (a) $L_{x}=120$ nm and
$A_{r}=2$ (b) $L_{x}=120$ nm and $A_{r}=4$. Due to large dipolar strength,
there is an order in the morphology, and they also tend to orient in the plane
of the sample. As $A_{r}$ is increased, magnetic moments also favour chain
arrangement along the longer the axis of the sample. The colour coding is the
same as described in Fig. (1).

Figure 3: Magnetic hysteresis behaviour as a function of dipolar interaction
strength and direction of the applied field for square sample, i.e.,
$A_{r}=1.0$ ($L_{x}=L_{y}=120$ nm). (a) The external field $\mu_{o}\vec{H}$ is
applied along the $x$-direction, (b) $\mu_{o}\vec{H}=H_{o}\hat{y}$ and (c)
$\mu_{o}\vec{H}=H_{o}\hat{z}$. The strength of dipolar interaction has been
varied from very large value $1.75$ to $0.03$. When the field is applied in
the plane of the sample, coercive field $H_{c}$ and remanent magnetization
decrease with an increase in the strength of the dipolar interaction. When the
field is applied is normal to the plane of the sample, $H_{c}$ and $M_{r}$
decrease to zero for large dipolar strength. The coercive field $H_{c}\approx
0.48H_{K}$ and $M_{r}\approx 0.5$ are observed for weakly interacting MNPs
irrespective of the direction of the applied field as expected (signature of
the Stoner-Wohlfarth model).

Figure 4: Dependence of magnetic hysteresis on aspect ratio $A_{r}$ and
dipolar interaction strength $\Theta$ when the field is applied in the plane
of the sample the sample, i.e., along $x$-direction [Left Panel: Figure
4(a)-(d)] or $y$-direction [Right Panel: Figure 4(e)-(h)]. Four values of
$\Theta=$ 1.75, 1.01, 0.42 and 0.22 have been considered. The aspect ratio
$A_{r}$ is varied from 1.0 to 32.0 in each case. The dipolar interaction
decreases the coercive field $H_{c}$ and $M_{r}$. There is a strong dependence
of $M_{r}$ on the shape anisotropy introduced by increasing $A_{r}$, $M_{r}$
reaches to $0.9$ when the field is applied along the longer axis of the sample
in the presence of strong dipolar interaction. For weak dipolar interaction,
i.e., $\Theta=0.22$, magnetic hysteresis follow the Stoner-Wohlfarth model
($H_{c}\approx 0.48H_{K}$ and $M_{r}\approx 0.5$) irrespective of $A_{r}$.

Figure 5:  Magnetic hysteresis as a function of dipolar interaction strength
$\Theta$ and aspect ratio $A_{r}$ when the field is applied normal to the
plane of the sample $\mu_{o}\vec{H}=H_{o}\hat{z}$. Four values of $\Theta$=
1.75, 1.01, 0.42 and 0.22 have been considered. The aspect ratio $A_{r}$ is
varied from 1.0 to 32.0 in each case. For large dipolar interaction strength,
the magnetic moments cease to align normal to the sample as a result, non-
hysteresis is observed. The coercive field $H_{c}$ and $M_{r}$ drop down to
zero in this case. For weak dipolar interaction, i.e., $\Theta=0.22$, magnetic
hysteresis follow the Stoner-Wohlfarth model ($H_{c}\approx 0.48H_{K}$ and
$M_{r}\approx 0.5$) irrespective of the aspect ratio $A_{r}$.

Figure 6: Coercive field $H_{c}$ and remanent magnetization $M_{r}$ variation
as a function of dipolar interaction strength $\Theta$, aspect ratio $A_{r}$
and direction of the applied field. $H_{c}$ has been scaled by single particle
anisotropy field $H_{K}$. $H_{c}$ decreases with an increase in dipolar
strength $\Theta$ and $A_{r}$ when the field is applied in the plane of the
sample (either along $\hat{x}$ or $\hat{y}$-axis). $H_{c}$ drops down to zero
when the field is applied normal to the plane of the sample for strongly
interacting MNPs. When the field is applied along the $x$-direction, $M_{r}$
is decreased with an increase in aspect ratio and dipolar interaction strength
$\Theta$. There is a strong shape anisotropy effect is observed when the field
is applied along the longer axis of the sample. $M_{r}$ increases with
increase in $\Theta$ and $A_{r}$, and it reaches to 0.9 with $A_{r}=32.0$ and
$\Theta=1.75$. When the external field is applied normal to the plane of the
sample $\mu_{o}\vec{H}=H_{o}\hat{z}$, $M_{r}$ decreases with increase in
$A_{r}$ and $\Theta$, and it drops down to zero for $\Theta=1.75$ and
$A_{r}=32.0$.