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: YZhao@ntu.edu.sg.

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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