Coercivity control in finite arrays of magnetic particles
Bo Yang1,2 and Yang Zhao1,a)
1School of Materials Science and Engineering, Nanyang Technological University, Singapore 6397982Centre for Optical and Electromagnetic Research, State Key Laboratory of Modern Optical Instrumentation,Zhejiang University, Hangzhou 310058, China
(Received 7 August 2011; accepted 12 October 2011; published online 23 November 2011)
Micromagnetic simulation has been performed for two-dimensional arrays of single-domain
magnetic particles using the Landau-Lifshitz equation of motion and an energy minimization
method. Effects of array anisotropy and spin positional disorder on the hysteresis loop and coercivity
of the particle systems are investigated. Simulation results show that the hysteresis loop can be
largely modified by breaking geometric symmetry of square arrays, and coercivity in general is
found to increase with array disorder. Magnetic hysteresis is strongly affected by disorder when the
array contains only a few particles. VC 2011 American Institute of Physics. [doi:10.1063/1.3662950]
I. INTRODUCTION
In the past years technologies to fabricate magnetic par-
ticles on the submicrometer and nanometer scales had been
emerging rapidly thanks to their potent magnetic properties
and potential applications in the high density recording indus-
try and related areas.1–3 Among various magnetic nanopar-
ticles systems, thin film or monolayer magnetic media are
widely studied both experimentally and theoretically.4–7 In
these systems, particles are well coated with no direct con-
tacts. For small-sized and high-density particle arrays, single-
domain states become favorable8,9 and static magnetic fields
assume more significance as inter particle spacings are
reduced, and no exchange or weak exchange coupling exists
between the grains.
A number of two-dimensional (2D) micromagnetic
models have been developed for single-domain grains with
highly ordered arrangements, in which magnetic properties
are ultimately determined by the structure of the film and
the strength of dipole-dipole interactions between discrete
grains.5,10–13 The system energy of a 2D magnetic particle
system generally consists of the anisotropy energy, the
demagnetizing energy from dipole-dipole interactions, and
the Zeeman energy. Competition of these energy contribu-
tions leads to hysteresis loops and coercive forces that
hinge on a number of factors, such as the shape and size of
particles, the lattice structures, and the direction of magnet-
ization. This is particularly true in a finite 2D array of a few
magnetic nanoparticles, where magnetic properties exhibit
a strong dependence on the strength of the dipole
fields.14–18 Previously, despite quite a number of unsettled
issues, it was found that interaction-induced frustrations
give rise to even-odd oscillations in the magnetic properties
of square magnetic dots arrays as the system size is varied
due to the boundary effects.15–17 In addition, the array size,
the orientation of the applied field,15–17 the magnetocrystal-
line anisotropy,17 and the symmetry of the array15,17 also
have important effects on the magnetization process. To
investigate static and dynamic properties of the arrays, a
commonly used method is to numerically integrate the
Landau-Lifshitz equations of motion for magnetic dots
arrays.14–18
In this work, two approaches, namely, a method of
energy minimization and the Landau-Lifshitz equation of
motion, are adopted and compared to study first ordered mag-
netic particle arrays on a square lattice. Effects of geometric
asymmetry on array coercivity and hysteresis behavior in 2D
particles system are also investigated. In particular, two struc-
tures breaking array symmetry are examined, i.e., rectangular
arrays with different lattice constants along perpendicular
directions, and square arrays with defects. Another way to
control the coercivity of finite 2D magnetic arrays is to intro-
duce structural disorder (from size and shape dispersions),
random anisotropy axes, and positional disorder. Positional
disorder alters dipolar interactions between two magnetic par-
ticles, and its influence on three-dimensional (3D) magnetic
particle systems19 were studied previously. In order to esti-
mate the influence of positional disorder on finite 2D mag-
netic particle arrays, in-plane particles are randomly displaced
from square-lattice points, and their coercivity and hysteresis
behavior is studied here.
This paper is organized as follows: Section II introdu-
ces the methodology to calculate hysteretic behavior of
magnetic particles systems. Section III displays numerical
results for both ordered and disordered arrays of magnetic
point dipoles, for which discussion of magnetization proc-
esses and hysteresis are also presented. A brief summary is
given in Section IV.
II. METHODOLOGY
In our model, particles are assumed to be of single
domains, circularly shaped, and possess no intrinsic anisot-
ropy, while neighboring spins are not exchange coupled. We
consider only the case of zero temperature. Particles are ei-
ther placed on, or randomly displaced from, lattice positions.
The particles are characterized by a bulk saturation magnet-
ization Ms, and a volume v. The dots are assumed to be iden-
tical and sufficiently small that the average magnetization of
a dot can be reasonably approximated by a single effectivea)Electronic mail: [email protected].
0021-8979/2011/110(10)/103908/7/$30.00 VC 2011 American Institute of Physics110, 103908-1
JOURNAL OF APPLIED PHYSICS 110, 103908 (2011)
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magnetic moment M. To give a measurement of particle posi-
tion randomness, we define Ar as the maximum distance a
certain spin can deviate from its lattice position, which can
be called the random positional perturbation amplitude. By
varying Ar, we can go from an ordered system to a disordered
one.
The system energy is comprised of the magnetostatic
energy and the Zeeman energy in this model, while the
exchange-coupling is neglected,
Etot ¼ Emag þ Eext; (1)
Emag ¼ �1
2tX
i 6¼j
3ðrij �MiÞðrij �MjÞ �Mi �Mj
r3ij
; (2)
Eext ¼ �Hext �X
i
Mi: (3)
If the magnetization for each dipole is given by Msmi with
mi the unit vector of the magnetic moment, the system
energy can be rewritten in a reduced form,
Etot
pM2s
¼� 1
2
X
i 6¼j
3ðrij � miÞðrij � mjÞ � mi � mj
r3ij
l3
� Ms
pMsHext �
X
i
mi; (4)
where l3¼V/N with V the sample volume and N the number
of spins. The volume packing faction p is defined as Nv/V.The effective field on each dipole is the sum of the
applied field, Hext, and the total dipolar field
HeffðiÞ ¼ �@Etot
@Msmi
¼ pMs
X
j 6¼i
3rijðrij � mjÞ � mj
ðrij=lÞ3þHext: (5)
Two approaches are employed in this work to study the hys-
teresis loops and detailed configurations of the spin
moments for a given external magnetic field. One is based
on the interior-point method to minimize the total energy
using Mathematica,20,21 and the other solves the gyromag-
netic equation of motion (EOM) with the Landau-Lifshitz
damping,
dMi
dt¼ cMi �Heff �
kMi
Mi � ðMi �HeffÞ: (6)
Here c is the gyromagnetic ratio, and k is the damping con-
stant which controls the rate of dissipation. The first term
which represents the gyromagnetic rotation was neglected in
the interest of fast convergence. In our calculations, mag-
netic fields are measured in units of Ms and the energy is
measured in units of Ms2.
III. RESULTS AND DISCUSSIONS
A. Symmetric ordered system
Static and dynamic magnetic properties of n� n ordered
arrays of magnetic particles have been examined previously
by solving the Landau-Lifshitz equation.15–17 However, con-
tention still surrounds a number of issues. For example, both
the shape of the calculated hysteresis loops and the area
enclosed by the loops are in dispute between Refs. 16 and 17.
In our work, the aforementioned two simulation methods are
performed for n� n (n¼ 2, 3,…, 11) square arrays of mag-
netic particles arranged well on a 2D lattices. An external field
is applied in the plane of the arrays. Without loss of general-
ity, the external field is applied in the y-direction throughout
this paper (as shown in Fig. 1). All spins are aligned along the
direction of a sufficiently large external field in the initial
state. In this section, the packing factor p is chosen to be 0.5.
In the energy minimization approach, the unit vector of
magnetic moment mi is specified by its orientation hi, as
defined in Fig. 1. The total system energy, rewritten as a func-
tion of hi, is minimized by using the FindMinimum function
in Mathematica which is based on the interior-point
FIG. 1. Geometry of a square array of small magnetic particles lattice with
lattice constant l. The angle h specifies the unit vector of magnetic moment m.
FIG. 2. Hysteresis loops obtained by energy minimization method with
DHext¼ 0.002Ms (solid curves) and DHext¼ 0.005Ms (dashed curves) for
5� 5 in-plane magnetic spins array. The left-top and right-down insets show
the configurations of the magnetic moments at Hext¼�0.330Ms for
DHext¼ 0.002Ms and DHext¼ 0.005Ms, respectively.
103908-2 B. Yang and Y. Zhao J. Appl. Phys. 110, 103908 (2011)
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method.20,21 Initially, a strong external field (2Ms) is applied
to initialize the array to magnetic saturation. The external
field Hext is then reduced to �2Ms stepwise with a step size
DHext. For each value of the external field, converged results
from a previous step are used to initialize hi. It is found that
the value of DHext in our simulation has considerable influ-
ence over the shape of the calculated hysteresis loops. As
shown in Fig. 2, the hysteresis loop obtained by using field-
step of DHext¼ 0.002Ms (solid curves) has one more abrupt
jump than the one obtained by using DHext¼ 0.005Ms
(dashed curves). The loops are similar to the hysteresis loops
in Fig. 1(h) of Ref. 16 and Fig. 7(d) of Ref. 17, respectively.
The one additional abrupt jump in the hysteresis loop
obtained by using DHext¼ 0.002Ms is related to a spin config-
uration at Hext¼�0.330Ms, characterized by the formation
of two vortices of magnetic moments as shown in the left-top
inset of Fig. 2. The spins arrangement at Hext¼�0.330Ms
obtained by using a field step size DHext¼ 0.005Ms, as shown
in the right-bottom inset in Fig. 2, has a larger energy and is
therefore a metastable state. Similar results can be found in
other magnetic particles arrays with different array sizes.
Therefore, the field-step must be sufficiently small in order to
obtain reliable hysteresis loops. We adopt the field-step
DHext¼ 0.002Ms for n� n (n¼ 2, 3,…, 11) square arrays for
the remainder of the paper. Calculated magnetization loops
using the energy minimization scheme are shown in Fig. 3 by
solid red lines (plotted in form of M/Ms as a function of
Hext/Ms) for square arrays of n� n spins (n¼ 2, 3,…, 11).
In the Landau-Lifshitz damping model, the integration
over time is carried out for a damping constant k¼ 0.005
and a fixed time step Dt¼ 0.005/pMs. Iterations are consid-
ered converged when the change in magnetic moment orien-
tation for each spin becomes negligibly small. The external
field is reduced stepwise from the positive maximum to the
negative maximum, and then back to the positive maximum
with a step size DHext. Typically, convergent solutions can
be found with DHext on the order of 0.005pMs. Magnetiza-
tion loops of arrays with various sizes are shown with the
black solid curves in Fig. 3.
From Fig. 3, one can see that the results of hysteresis
loops obtained with the two methods coincide. Moreover,
our simulation results for hysteresis loops are in agreement
with those given in Ref. 16. Initially, all spins are aligned
along the direction of the external field for a range of large
values of the applied field Hext. As the applied field is
reduced, spins start to rotate away from the field direction
and relax to some equilibrium states. These equilibrium
FIG. 3. (Color online) Magnetization loops obtained by the energy minimi-
zation method (red solid) and the dynamic method (black solid) for n� n(n¼ 2, 3,…, 11) in-plane magnetic spins arrays. Computed loops from the
two approaches agree so well with each other that those from energy mini-
mization are only displayed in (i) and (j), and as the lower line in (c) and the
upper line in (h). The rest are the loops calculated using the dynamic
method. The right insets of (a)-(j) show the spin configurations when
Hext¼ 0; The left insets of (d)-(j) show the spin configurations at the jump in
hysteresis loops around Hext¼�Ms.
FIG. 4. Hysteresis loops of n� n rectangular magnetic particles array with
lx/ly¼ 0.8 (dashed), lx/ly¼ 1 (solid), lx/ly¼ 1.25 (dotted-dashed) for (a)
n¼ 5; (b) n¼ 10. The external field is applied in the y-direction as men-
tioned above.
103908-3 B. Yang and Y. Zhao J. Appl. Phys. 110, 103908 (2011)
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states exist in a range around the zero external field and
result in abrupt jumps in the hysteresis loops. For Hext¼ 0,
the formation of vortex states, as shown in the right insets of
Figs. 3(a) and 3(c), results in zero magnetization for n¼ 2, 4.
Systems with odd n show nonzero remanence Mr due to
unpaired spins, such as those found in a “barrel” state for
n¼ 3 as shown in the inset of Fig. 3(b), and in “head-to-tail”
state for n¼ 5 as shown in the right inset of Fig. 3(d). For
n¼ 6,…, 11, the magnetic behavior of the arrays with odd-nand even-n becomes similar. There are nonzero values of Mr
for both odd-n and even-n values due to the internal frustra-
tion imposed by the array boundary, as shown in the right
insets of Figs. 3(e)–3(j). We can also find that the arrange-
ment of magnetic moments at the center of the arrays is per-
pendicular to the direction of the external field for the even
numbers of n, whereas it is along the direction of the external
field for odd numbers of n. As the external field is further
reduced to around �Ms, the chain reversals at the array
boundary cause the last giant jumps in the magnetization
loops [cf. the left insets of Figs. 3(d)–3(j)]. With increasing
the array size, the boundary effects would be less important.
Both the external field needed for the chain reversals and
the last jump in hysteresis loops it causes a decrease with
increasing n.
B. Asymmetric ordered system
The previous section is dedicated to hysteresis and coer-
civity behaviors of orderly 2D systems of magnetic particles
due to the boundary effect. As shown previously,19 orderly
spin arrangements yield no hysteresis or coercivity for 3D
magnetic particle systems, while positional disorder introdu-
ces a hysteresis loop thanks to clustering of magnetic
dipoles. Since vertical particle pairs experience twice as
much magnetostatic interaction as the horizontal pairs in a
saturated state,19 it is therefore interesting to remove the geo-
metric symmetry of 2D magnetic spin arrays, which may
lead to unexpected patterns of magnetization.
Magnetization curves were previously calculated for
asymmetric arrays consisting of N� (Nþ 1) and N� (Nþ 2)
nanomagnets arrays on a square lattice.17 In Ref. 17, it is
found that hysteresis is absent along the short side of the
asymmetric arrays when the array size is small (2� 3, 3� 4,
and 2� 4). In this work, we consider the n� n rectangular
arrays with varying lattice constants lx and ly. In Fig. 4, the
magnetization curves are shown for 5� 5 and 10� 10 arrays
with lx/ly¼ 0.8, 1, and 1.25. When lx/ly¼ 0.8, which means
the external field are imposed along the long sides of the
arrays, the dipolar interactions between the particles in the
FIG. 5. Hysteresis loops of 5� 5 magnetic particles array with defect located at the i row and the j column of the array (i, j¼ 1, 2, 3). The right insets show
the spin configurations when Hext¼ 0.
103908-4 B. Yang and Y. Zhao J. Appl. Phys. 110, 103908 (2011)
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direction perpendicular to the external field are stronger than
between those in the field direction. Spin clusters in rows
perpendicular to the applied field have overall negative inter-
actions with each other, and nucleation of these clusters
occurs before the external field is reduced to zero.10 These
clusters do not contribute to the coercivity. Consequently,
the in-plane spins possess no hysteresis for both 5� 5 and
10� 10 arrays, as shown by the dashed curves in Figs. 4(a)
and 4(b), respectively. When lx/ly¼ 1.25, spin clusters in col-
umns parallel to the applied field have an overall positive
interaction. Reversal of these spin clusters results in abrupt
jumps in the hysteresis loop for 5� 5 and 10� 10 arrays (as
shown by the dotted-dashed curves in Fig. 4). The magnet-
ization curves display much stronger hysteretic behavior
than those of a square lattice (lx/ly¼ 1).
In addition to lattice anisotropy, defects can also occur
in magnetic spin arrays on a square lattice. For a 5� 5 array
with only one defect, for example, Fig. 5 displays changes in
the hysteresis behavior by varying the position of the defect,
labeled by (i, j) with i, j¼ 1, 2, 3 (i.e., on the ith row and the
jth column). A comparison of Figs. 3(d) and 5 demonstrates
that a single defect at the boundary of a square array can alter
the hysteresis loop substantially, while defects in the interior
of the array have less influence on the hysteretic behavior.
The magnetization loops of arrays with defects on an array
edge that is perpendicular to the external field exhibit less
FIG. 6. Reduced coercive force Hc/pMs (solid curves) and remanence Mr/Ms
(dashed curves) vary with normalized random perturbation amplitude Ar/l for
3� 3 (circles) and 10� 10 (squares) arrays.
FIG. 7. (Color online) Various of hys-
teresis loops of different 3� 3 magnetic
particles arrays when Ar¼ 0.4. The
insets show spin configurations of these
disordered arrays when Hext¼ 0.
103908-5 B. Yang and Y. Zhao J. Appl. Phys. 110, 103908 (2011)
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number of abrupt jumps [cf. Figs. 5(b) and 5(c)]. A small
remanence can be found in the magnetization loops of arrays
with defects on an array edge that is parallel to the external
field [cf. Figs. 5(d) and 5(g)]. The absence of a magnetic par-
ticle on the corner of an array has the most drastic effect,
shrinking the loop to a relatively small area near the origin.
This is similar to the influence of a small field misalignment
demonstrated in Ref. 16.
C. Disordered system
As described in previously,19 coercivity increases with
positional randomness of dipolar particle, while orderly spin
arrangements yield no hysteresis or coercivity for 3D mag-
netic particle systems. In this subsection, micro-magnetic
properties of in-plane particles randomly displaced from lat-
tice point will be examined. We consider magnetic spin
arrays with spins randomly displaced from square-lattice
points (Ar/l¼ 0.0, 0.1, 0.15, 0.2, 0.3, 0.4). One hundred real-
izations of random particle arrays are selected for a given
random perturbation amplitude Ar, and hysteretic behavior is
calculated for each Ar. Average coercivity Hc (in unit of
pMs) and remanence Mr (in unit of Ms) are obtained for each
Ar (in unit of l) for 3� 3 and 10� 10 arrays, as shown in
Fig. 6. From Fig. 6 it is found that the coercive field and the
remanence increase with increasing disorder (for both disor-
dered 3� 3 and 10� 10 arrays). However, an ordered 3� 3
array still has larger remanence than disordered arrays with
Ar/l¼ 0.1, 0.15, 0.2.
Coercivity which increases with positional disorder may
be attributed to the clustering of magnetic dipoles. As the
array is randomly perturbed, some particles get close to each
other and form a relatively isolated subset from the rest of
the array. The in-plane hysteresis loops are dominated by the
strong dipolar interaction between these particles, especially
for small arrays. As shown in Fig. 7, for the 3� 3 array,
Ar¼ 0.4l both the shape of hysteresis loops and the coerciv-
ity are modified considerably when the spins are randomly
perturbed. The insets of Figs. 7(a), 7(b), and 7(e) show spin
configurations of the disordered arrays with large remanence
due to formation of anti-ferromagnetic chains for this case.
In Fig. 7(a), clusters with positive effective interactions
along the external field10 formed at the upper-left and the
right of the arrays result in a large coercivity and two abrupt
jumps in the hysteresis loop. In Figs. 7(c) and 7(d), no
obvious clusters along the external field can be found in the
spin configurations, attributing to a small coercivity and rem-
anence. In Fig. 7(f), clusters are formed perpendicular to the
applied field, and nucleation of these clusters occur before
the external field is reduced to zero, yielding a negative rem-
anence and coercivity. For the 10� 10 array, Ar¼ 0.4l,increased cluster sizes bring about a larger coercivity (cf.
Fig. 8). Moreover, reversals of these clusters before the sys-
tem reaches saturation result in many small jumps near the
tails of the hysteresis loop.
IV. CONCLUSION
We have studied the hysteretic behavior for n� n arrays
of magnetic particles interacting via the dipole-dipole inter-
actions using two approaches, the energy minimization
method and the Landau-Lifshitz equation of motion. Mag-
netic propertis of finite 2D magnetic-particle arrays are
mainly determined by dipolar interactions and array geome-
try, and in particular, by the array boundaries. Isotropic par-
ticle arrays with a size equal to or greater than 5� 5 show
hysteresis behavior and coercivity due to the boundary
effect. For anisotropic arrays, no hysteresis is found when
the external fields are imposed along the long sides of the
arrays. Conversely, strong hysteresis behavior can be found
when the external fields are applied along the short sides of
the anisotropic arrays. Defects at the boundary of a square
array can also influence the coercivity and the shape of hys-
teresis loop dramatically. We have also shown that coerciv-
ity increases with array randomness due to clustering of
magnetic dipoles in disordered systems. Magnetic hysteresis
is strongly affected by disorder when the array has only a
few particles.
ACKNOWLEDGMENTS
Support from the Singapore National Research Founda-
tion through the Competitive Research Programme under
Project No. NRF-CRP5-2009-04 is gratefully acknowledged.
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103908-7 B. Yang and Y. Zhao J. Appl. Phys. 110, 103908 (2011)
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