Pramana – J. Phys. (2021) 95:181 © Indian Academy of Scienceshttps://doi.org/10.1007/s12043-021-02222-w

Dipolar interaction and sample shape effects on the hysteresisproperties of 2d array of magnetic nanoparticles

MANISH ANAND∗

Department of Physics, Bihar National College, Patna University, Patna 800 004, India∗Corresponding author. E-mail: itsanand121@gmail.com

MS received 4 May 2021; revised 1 July 2021; accepted 5 July 2021

Abstract. We study the ground-state and magnetic hysteresis properties of 2d arrays (Lx × Ly) of dipolarinteracting magnetic nanoparticles (MNPs) by performing micromagnetic simulations. Our primary interest isto understand the effect of sample shape, �, the ratio of the dipolar strength to the anisotropy strength and thedirection of the applied field �H = H0eH on the ground state and the magnetic hysteresis in an array of MNPs.To study the effect of the shape of the sample, we have varied the aspect ratio Ar = Ly/Lx , which in turn, isfound to induce shape anisotropy in the system. Our main observations are: (a) When the dipolar interaction isstrong (� > 1), the ground-state morphology has an in-plane ordering of magnetic moments, (b) the ground-statemorphology has randomly oriented magnetic moments that are robust regarding system sizes and Ar for weaklyinteracting MNPs (� < 1), (c) micromagnetic simulations suggest that the dipolar interaction decreases the coercivefield Hc, (d) the remanence magnetisation Mr is found to be strongly dependent not only on the strength of dipolarinteraction but also on the shape of the sample and (e) due to the anisotropic nature of dipolar interaction, a strongshape anisotropy effect is observed when the field is applied along the long axis of the sample. In such a case, thedipolar interaction induces an effective ferromagnetic coupling when the aspect ratio is enormous. These resultsare of vital importance in high-density recording systems, magneto-impedance sensors, etc.

Keywords. Self-assembled arrays; dipolar interaction; hysteresis; micromagnetic simulations; ferromagnetism.

PACS Nos 78.67.Pt; 71.36.+c; 44.90.+c; 61.46.+w

1. Introduction

Magnetic nanoparticle (MNP) arrays are of profoundimportance because of their exciting physics and theirnumerous technological applications [1–12]. Forinstance, a two-dimensional array of MNPs is suit-able for high-density digital storage and perpendicularrecording media [13–15]. The magnetic properties insuch a case depend strongly on the shape, size, geom-etry of the systems and the magnetic interactions.The primary magnetic interactions in such systems aredipole–dipole interactions. The dipolar interaction arelong-ranged and favours ferromagnetic as well as anti-ferromagnetic couplings. As a result, unconventionalmorphologies are observed when the size of the systemis comparable to the range of dipolar interactions. Dueto the anisotropic nature of dipolar interactions, therehave also been observations of ferromagnetic, striped,checkerboard patterns, vortices, etc., depending on thegeometry of the lattice [16–21]. The magnetic hysteresis

properties in such systems are found to be significantlyinfluenced by the dipolar interactions [22–25]

Several studies in the literature make implicit andexplicit references to the ubiquitous dipolar interactionsand anisotropy in ordered arrays of MNPs. We sum-marise a few of them, relevant to our present work, asfollows: (a) Li et al studied the ground-state, magnetic-specific and magnetic hysteresis properties for threetypes of closely spaced nanomagnet arrays using MonteCarlo simulation [26]. They observed the vortex statedue to the dipolar interactions in these arrays. Forface-centred cubic nanomagnet arrays, a slight jumpoccurs in the hysteresis curve. (b) Yang et al studiedthe magnetic properties in two-dimensional arrays ofMNPs using the micromagnetic simulation [27]. Theyfound that coercivity increases with arrays disorder. (c)Using the Landau–Lifshitz–Gilbert approach, Morales-Meza et al studied the magnetisation reversal in atwo-dimensional array of MNPs [28]. They showed thatcoercivity is reduced even if the particle position in the

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array is random. (d) Chinni et al performed experimentsand micromagnetic simulation using object-orientedmicromagnetic framework (OOMMF) to study mag-netic properties in nanocomposite films [29]. Theyobserved an unexpected hysteretic behaviour charac-terised by in-plane anisotropy and crossed branches inthe hysteresis curves measured along the hard direc-tion. (e) Faure et al studied the magnetic properties ofordered arrays of MNPs using experiment and MonteCarlo simulation [30]. The dipolar interaction inducesa ferromagnetic coupling that increases in strength withdecreasing thickness of the array. (f) Xue et al studied themagnetic hysteresis properties of the two-dimensionalhexagonal array of MNPs using theoretical calculations[15]. The hysteresis curves are found to vary their shapesfrom a rectangle to a non-hysteresis straight line througha set of complicated loops, in accordance with the mag-netisation reversal process.

Although the aforementioned studies imply that dipo-lar interaction and geometrical arrangement of MNPsaffect the magnetic properties of an ordered array ofMNPs, a systematic study as a function of dipolar inter-action strength, shape anisotropy of the system anddirection of the applied field is still missing. In thiswork, we attempt to understand the effect of dipolarinteraction manipulated by changing the interparticleseparation, the direction of applied magnetic field andshape anisotropy induced by varying the aspect ratio onthe magnetic properties of two-dimensional (2d) arraysof MNPs. The main questions which we have attemptedto address are: (1) What are the consequences of dipo-lar interaction on ground-state organisations of MNPs?(2) How are the magnetic hysteresis properties modi-fied due to the dipolar interactions, shape anisotropy ofthe sample and direction of the applied magnetic field?To answer these questions, we consider 2d (Lx × Ly)arrays of cubical shaped Fe3O4 MNPs. It is a well-known fact that uniformly magnetised cubical MNPsdo not possess shape anisotropy [31]. But we have beenable to induce it in the system by just varying the sam-ple shape. It will be shown that the latter has a drasticeffect on hysteresis behaviour. These ordered arrays ofMNPs have many interesting properties due to struc-tural order, well-defined interparticle interactions andgeometry confinement [32]. In such a case, MNPs arecoated with an inorganic surfactant to prevent agglomer-ation during self-assembling. Due to this, MNPs are at areasonable distance beyond the range of exchange inter-action. As a consequence, the predominant interactionin such arrays is dipolar [33]. Therefore, the energy ofsuch an assembly is given by the sum of the anisotropyenergy and the dipolar interaction energy.

To vary the relative dipolar interaction strength withrespect to anisotropy strength, we have defined a ratio� = D/KV where D is the dipolar interaction strength,K is the anisotropy constant and V is the volume of thenanoparticle. The value of � for various interparticleseparation a for Fe3O4 is given in table 1. We referto � > 1 as the strong dipolar interaction regime and� < 1 as the weak dipolar interaction regime. We haveperformed micromagnetic simulation using OOMMFcode from NIST [34]. In the OOMMF code, the finitedifference method is employed, which requires discreti-sation of a chosen geometry over a grid of identicalprism cells. The magnetisation is supposed to be uni-form in each cell. In the implementation of OOMMF,the continuous magnetic material is divided into dis-crete cubes (i.e. grid cells), which are in geometricalcontact. This discretisation scheme suits a continuousmagnetic film rather than an assembly of non-touchingnanoparticles. Due to this reason, this type of modellingdoes not discriminate between the magnetic behaviourof 2d continuous films and 2d nanoparticle arrays. Toovercome this, we have put exchange interaction tozero to mimic the separated MNP arrays magnetically.The absence of exchange interaction is also neededfor the MNPs where they are coated with a surfactantto avoid agglomeration. We have successfully imple-mented it in our earlier studies [35,36]. We have usedthe Landau–Lifshitz–Gilbert (LLG) equation, which isused to describe the precessional motion of a moment inthe magnetic field at T = 0 K. We have solved the cou-pled equation of motion for the given lattice to obtainthe minimum energy configurations. To study hystere-sis, we have applied a DC magnetic field μ0

�H = H0eH ,where eH = x , y and z is the direction of the appliedfield along the x-, y- and z-axes respectively. We alsostudy the role of aspect ratio Ar = Ly/Lx to induceshape anisotropy in the system and the direction of thefield eH .

The interplay of anisotropy, dipolar energy and aspectratio creates unusual morphologies that profoundlyaffect magnetic properties. Our work, therefore, throwslight on this subject from a microscopic picture and pro-vides a basis for results obtained in experimental andtheoretical studies. This paper is organised as follows:First, §2 introduces the model for an array of MNPs, theLLG equation, which provides the prototypical groundstate (GS) morphologies and equilibrium morphologies.Then, in §3, we present our numerical results and dis-cuss the dependence of the magnetic hysteresis on �,aspect ratio Ar = Ly/Lx and the direction of the appliedfield eH . Finally, a conclusion of our results is providedin §4.

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Table 1. Evaluation of the ratio � = D/KV forFe3O4 as a function of interparticle separation a forcubical shaped particle. The lateral dimension of theparticle is l = 10 nm.

a (nm) � = D/KV

10 1.7512 1.0116 0.4220 0.2230 0.0640 0.03

2. Model and methodology

2.1 Model for dipolar interacting arrays of MNPs

We consider a self-assembled 2d array (Lx × Ly) ofcubical shaped MNPs in the xy-plane. The total numberof the MNPs in the assembly are N = Lx/a × Ly/a,where l is the length of an edge of the particle and ais the lattice spacing. The magnetic moment of eachMNP is �μi = μei , i = 1, 2, . . . , N . The magnitudeof the magnetic moment μ = MsV where Ms is thesaturation magnetisation and V = l3 is the volumeof the MNP. The MNPs are assumed to have uniaxialanisotropy �K = K ki , where K is the anisotropy con-stant and ki is the direction of anisotropy. Generally,MNPs are coated with a surfactant to prevent agglomer-ation, which suppresses the exchange interactions. Theenergy of such a system, therefore, includes contribu-tions from anisotropy energy Ea and dipolar interactionenergy Ed [37,38]:

E = Ea + Ed

= −KV∑

i

(ki · ei )2

− D∑

i, j

3(ei · ri j )(e j · ri j ) − (ei · e j )(ri j/a)3 , (1)

where μ0 is the permeability of the free space, ei is theunit vector along the magnetic moment, ri j is the dis-tance between particles i and j and ri j is the unit vectoralong with it. The 1/r3

i j dependence means that the dipo-lar interaction is long-ranged in nature. The analyticalcalculation of dipolar interaction between the cubicalshaped MNPs can be found in ref. [39]. It is well-knownthat self-assembled MNPs usually form 2d close-packedhexagonal lattices, but magnetic properties are nearlyindependent of the lattice structure [40].Here we define the dipolar interaction strength D =μ0μ

2/4πa3. As magnetic properties of such an assem-bly are governed by the relative strength of the anisotropy

and dipolar energy, we have defined a ratio � = D/KV .Although dipolar interactions and anisotropy energydepend on various system parameters, the behaviour ofthe assembly will be dictated by � rather than by theprecise values of parameters such as a, V, μ and K .When D is more significant than KV , the dipolar inter-action is stronger than anisotropy, i.e., � > 1. Similarly,� < 1 can be termed as the weak dipolar regime.

In the presence of an external magnetic field μ0�H ,

there is an additional contribution to energy E given by[38]

EH = −H0

∑

i=1

�μi · eH , (2)

where H0 is the magnitude of the field and eH is the unitvector in the direction of the applied field.

2.2 Landau–Lifshitz–Gilbert equation

The precessional motion of the magnetic moment �μi ina magnetic field can be described by the LLG equation[41]:

d �μi

dt= −γ �μi × �He

i − λ �μi × ( �μi × �Hei ),

i = 1, 2, . . . , N , (3)

where γ is the electron gyromagnetic ratio, λ = γα/Msis a phenomenological dimensionless damping factorand �He

i = −∂ET /∂ �μi is the effective field experi-enced by the magnetic moment, where ET = E + EH .While studying ground-state properties, EH is taken tobe zero. The first term in eq. (3) takes care of the pre-cession of �μi around �He

i . The second term is due toa phenomenological dissipative motion: the magneticmoment �μi precesses around �He

i . The solution of thesecoupled differential equations yields the GS configura-tion { �μi }, of the assembly.

The entire system is discretised into cells to performmicromagnetic simulation using OOMMF code, wherethe lateral dimension of each cell is l. Each cell in sucha case represents a magnetic moment μ = MsV . Thecentre-to-centre separation between moments is, there-fore, l ≡ a0. By eq. (1), the strength of the dipolarinteraction can be manipulated by varying the centre-to-centre separation a of the MNPs. However, withthe protocol implemented in OOMMF, a change in thecentre-to-centre separation from a0 to aβ , changes thecell volume from V ≡ V0 (=a3

0) to Vβ (=a3β). Con-

sequently, the magnetic moment gets altered to μβ =MsVβ , which ultimately modifies the magnetic proper-ties of the particles under study. This undesirable artefactin the simulation needs to be overcome. To overcome

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this artefact, we formulated a rescaling method for sat-uration magnetisation:

Mβs = M0

sV0

Vβ

, (4)

where M0s is the saturation magnetisation for nanopar-

ticle of volume V0 = a30. It is easy to see that now μ =

M0s V0 = Mβ

s Vβ as desired. Corresponding changesneed to be incorporated in other related variables ofinterest such as the coercive field Hc and the anisotropyfield HK = 2K/Ms [42] for non-interacting or weaklyinteracting MNPs, which plays an important role in hys-teresis:

Hβc = H0

cVβ

V0, (5)

HβK = H0

KVβ

V0. (6)

We have successfully implemented this scaling pro-cedure to study the heat dissipation and spin transportproperties in the assembly of dipolar interacting MNPsin our earlier work [35,36].

3. Numerical results

We consider cubical shaped MNPs of Fe3O4 arrangedon a 2d (Lx × Ly) lattice. The particle size is chosento be l = 10 nm. We have used anisotropy constantK = 13 × 103 J m−3 and saturation magnetisationMs = 4.77 × 105 A/m for numerical evaluations. Sixvalues of interparticle separation a are considered formagnetic hysteresis study: a0 = 10 nm, a1 = 12 nm,a2 = 16 nm, a3 = 20 nm, a4 = 30 nm and a5 = 40nm. Table 1 provides the values of � for these interpar-ticle separations. The initial condition that we choosefor the assembly of MNPs is random orientations ofmagnetic moments and anisotropy axes. All the dataobtained using the simulations are averaged over 50sets of initial conditions. The ground-state morpholo-gies are obtained by solving the LLG equation usingthe OOMMF code without an external magnetic field.To study the magnetic hysteresis properties, we applya DC magnetic field of strength −150 mT to 150 mTfor a = 10 nm. For other interparticle separations, thescaled saturation magnetisation Mβ

s was obtained usingeq. (4). corresponding Hβ

c and HβK were calculated using

eqs (5) and (6), respectively. We have chosen the mag-netic field to be large enough compared to Hc to achievesaturation. The ramping up or slowing down has beenappropriately fine-tuned to capture the magnetic prop-erties near Hc. We study the dependence of magnetichysteresis on �, aspect ratio Ar = Ly/Lx and the

direction of the applied field eH . In our simulations,Lx = 120 nm; Ar = 1, 2, 4, 8, 16 and 32; � = 1.75,1.01, 0.42, 0.22, 0.06 and 0.03; eH = x , y and z.

3.1 Ground-state (GS) morphologies

We first study the effect of � and Ar on ground-statespin morphologies. For this purpose, we took squaresamples, i.e. Ar = 1.0, such that Lx × Ly ≡ Lx × Lx .Figure 1 depicts GS morphologies for Lx =120 nm. Theinterparticle separation a is chosen to be 10 nm (see fig-ure 1a) and 20 nm (see figure 1b). These correspond to� = 1.75 and 0.22, respectively. So the number of spinsin figure 1a is 12 × 12 and 6 × 6 in figure 1b. We depictthe spins with non-zero z-component by the green cone.Those lying in the xy plane with a positive x-componenthave been indicated by the red coloured cone, whilethose with negative x-component by the blue colouredcone. It is quite evident that for strongly interactingMNPs (� = 1.75), all the spins are in the xy-plane(see figure 1a). They exhibit locally ordered regions.On the other hand, when magnetic interactions are weak(� = 0.22), the moments are randomly oriented, signi-fying a lack of magnetic order (see figure 1b). Magneticmoments also tend to align normal to the plane of thesample. It means that dipolar interaction favours in-plane ordering. Then, we study the effect of aspect ratioAr on the spin morphologies. For � < 1, the features areunchanged as Ar is increased to 2, 3, 4, etc. and are proto-typically represented by figure 1b. We do not show themto avoid repetition. In figure 2, we depict morphologiescorresponding to the strongly dipolar interacting MNPs(a = 10 nm, � = 1.75) for Lx = 120 nm and twoaspect ratios (a) Ar = 2 (see figure 2a) and (b) Ar = 4(see figure 2b). In both cases, the MNPs exhibit localorder and prefer to lie in the xy-plane. The morphologyof the magnetic moments near the sample edges is verydistinct from that in bulk. The moments tend to alignalong the edges as ferromagnetic chains. These obser-vations can be explained as follows. As the anisotropyaxes are assumed to have random orientations in three-dimensional space in the present work, the magneticmoments tend to follow the anisotropy directions forsmall dipolar interaction strength, i.e. � < 1. Conse-quently, there is no preferential direction of magneticmoments for weakly interacting MNPs, as evident infigure 1b. It is also well-known that the dipolar interac-tion favours in-plane ordering of magnetic moments asthe energy cost is very high for the orientations normalto the sample plane [43]. As a result, there is an in-plane ordering of magnetic moments for large � (seefigure 1a). Interesting physics emerges in the highlyanisotropic system (huge aspect ratio Ar ). In such a case,the dipolar interaction of enough strength induces shape

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Figure 1. Typical GS morphologies for (a) Lx = Ly = 120nm, a = 10 nm and � = 1.75 and (b) Lx = Ly = 120 nm,a = 20 nm and � = 0.22. The red cones represent magneticmoments with a positive x component and the blue conesrepresent magnetic moments with a negative x component.Those with a non-zero z components, and pointing normalto the xy-plane, are represented by green colour cones. For� � 0, moments are randomly oriented, and they also tendto orient normal to the plane of the sample. For � > 1, themoments predominantly lie in the xy-plane and exhibit localorder.

anisotropy in the system [44]. In other words, the dipo-lar interaction promotes ferromagnetic coupling amongthe magnetic moments in such a case [44]. As a con-sequence, magnetic moments tend to form chain-likeconfigurations for large �, as depicted in figure 2.

3.2 Magnetic hysteresis study

Next, we study the effect of dipolar strength �, shapeanisotropy of the sample and direction of the appliedfield eH on the magnetic hysteresis in a systematicmanner. To vary the dipolar interaction strength, wehave varied the interparticle separation. To induce shapeanisotropy in the system, we have changed the aspectratio Ar of the sample. In figure 3, we have plottedmagnetic hysteresis curves for the square sample, i.e.,Lx = Ly = 1.0 as a function of � and eH . We have con-sidered six values of dipolar strength �(= 1.75, 1.01,0.42, 0.22, 0.06 and 0.03) in each case. The magneticfield axis (x-axis) has been scaled by HK , the anisotropyfield (for non-interacting MNPs with randomly orientedanisotropy axes). The direction of the external field is(a) μ0 �H = H0 x (see figure 3a), (b) μ0 �H = H0 y (seefigure 3b) and (c) μ0 �H = H0 z (see figure 3c). It is quiteevident that when the field is applied in the plane of thesample, i.e., either along the x- or y-direction, the coer-cive field Hc decreases with an increase in the strength ofdipolar interaction. As a consequence, the area under thehysteresis curve diminishes. When the field is appliednormal to the sample (z-direction) plane, the magneticmoment ceases to follow the applied field for � > 1.

Figure 2. Typical GS morphologies for strongly interactingMNPs (a = 10 nm, � = 1.75) as a function of aspect ratioAr . (a) Lx = 120 nm and Ar = 2 and (b) Lx = 120 nm andAr = 4. Due to large dipolar strength, there is an order in themorphology, and they also tend to orient in the plane of thesample. As Ar is increased, magnetic moments also favourchain arrangement along the longer axis of the sample. Thecolour coding is the same as described in figure 1.

As a result, non-hysteresis is observed, and so Hc andMr tend to zero for strongly interacting MNPs. It isclearly seen that the coercive field Hc ≈ 0.48HK andMr ≈ 0.5 for weak interaction or non-interacting MNPs(� < 1) irrespective of the direction of the applied field.These values of Hc and Mr correspond to single-particlehysteresis with the randomly oriented anisotropy axes[42,45]. It means that the magnetic hysteresis follows theStoner–Wohlfarth model when the interaction amongthe MNPs is weak as expected [42,45].

To study the dependence of shape anisotropy on themagnetic properties, we study the magnetic hysteresisas a function of aspect ratio Ar for various values of �

and three directions of the applied field, i.e., along x-, y-and z-axes, respectively. We have considered six valuesof aspect ratio Ar (= 1.0, 2.0, 4.0, 8.0, 16.0 and 32.0)in each case. In figure 4, the direction of the appliedfield is in the plane of the sample [μ0 �H = H0 x andμ0 �H = H0 y]. The values of � are (a) � = 1.75 (seefigures 4a and 4e), (b) � = 1.01 (see figures 4b and 4f),(c) � = 0.42 (see figures 4c and 4g) and (d) � = 0.22(see figures 4d and 4h). The direction of the applied

181 Page 6 of 9 Pramana – J. Phys. (2021) 95:181

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-0.5

0

0.5

1

1.751.010.420.220.060.03

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-0.5

0

0.5

1

1.751.010.420.220.060.03

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1.751.010.420.220.060.03

(a)

(b)

(c)

Figure 3. Magnetic hysteresis behaviour as a function ofdipolar interaction strength and direction of the applied fieldfor square sample, i.e., Ar = 1.0 (Lx = Ly = 120 nm).

(a) The external field μ0�H is applied along the x-direction,

(b) μ0�H = H0 y and (c) μ0

�H = H0 z. The strength of dipo-lar interaction has been varied from a very large value of1.75–0.03. When the field is applied in the sample plane,coercive field Hc and remanent magnetisation decrease withan increase in the strength of the dipolar interaction. Whenthe field applied is normal to the sample plane, Hc and Mrdecrease to zero for large dipolar strength. The coercive fieldHc ≈ 0.48HK and Mr ≈ 0.5 are observed for weakly inter-acting MNPs irrespective of the direction of the applied fieldas expected (signature of the Stoner–Wohlfarth model).

field is normal to the plane of the sample (along the z-axis) in figures 5. The other parameters, Ar and � infigures 5, remain the same as that of figures 4. Whenthe applied field is along the shorter axis (along the x-axis), dipolar interaction decreases the value of Hc andMr (see figures 4a–4d). There is a weak dependenceof Hc on Ar . For weakly interacting MNPs, hystere-sis curves follow the Stoner–Wohlfarth model [45]. Astrong effect of shape anisotropy is observed when thefield is applied along the increasing length of the sample,i.e., along the y-axis. The remanent magnetisation Mrincreases with an increase in Ar , and it reaches 0.9 forstrongly interacting MNPs (see figure 4e). In this casealso, the larger is the strength of dipolar interaction, the

Figure 4. Dependence of magnetic hysteresis on aspect ratioAr and dipolar interaction strength � when the field is appliedin the plane of the sample, i.e., along x-direction (left panel:figures 4a–4d) or y-direction (right panel: figures 4e–4h).Four values of �(= 1.75, 1.01, 0.42 and 0.22) have beenconsidered. The aspect ratio Ar is varied from 1.0 to 32.0in each case. The dipolar interaction decreases the coercivefield Hc and Mr . There is a strong dependence of Mr on theshape anisotropy introduced by increasing Ar , Mr reaches0.9 when the field is applied along the long axis of the samplein the presence of strong dipolar interaction. For weak dipo-lar interaction, i.e., � = 0.22, magnetic hysteresis followsthe Stoner–Wohlfarth model (Hc ≈ 0.48HK and Mr ≈ 0.5)irrespective of Ar .

smaller is the value of Hc (see figures 4e–4h). The mag-netic moments cease to follow the applied magnetic fieldwhen the field is applied along the z-axis for stronglyinteracting MNPs. As a consequence, almost no hys-teresis is observed for strongly interacting MNPs withapplied field normal to the plane of the sample (see fig-ures 5a and 5b). Like the other two cases stated already,weakly interacting MNPs follow the Stoner–Wohlfarthmodel irrespective of Ar .

3.3 Characterisation of magnetic hysteresis curves

Finally, we study the variation of the coercive field Hcand Mr as a function of �, Ar and eH by extracting theirvalues from the simulated magnetic hysteresis curves.Figure 6 shows the variation of Hc (scaled by HK ) andMr as a function of dipolar strength � for six values of

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(a) (b)

(c) (d)

Figure 5. Magnetic hysteresis as a function of dipolar inter-action strength � and aspect ratio Ar when the field isapplied normal to the plane of the sample μ0 �H = H0 z.Four values of � (= 1.75, 1.01, 0.42 and 0.22) have beenconsidered. The aspect ratio Ar is varied from 1.0 to 32.0 ineach case. For large dipolar interaction strength, the magneticmoments cease to align normal to the sample and as a result,non-hysteresis is observed. The coercive field Hc and Mr dropdown to zero in this case. For weak dipolar interaction, i.e.,� = 0.22, magnetic hysteresis follows the Stoner–Wohlfarthmodel (Hc ≈ 0.48HK and Mr ≈ 0.5) irrespective of theaspect ratio Ar .

aspect ratio Ar (= 1.0, 2.0, 4.0, 8.0, 16.0 and 32.0). Thedirection of the applied magnetic field is (a) μ0 �H =Hox (see figures 6a and 6d), (b) μ0 �H = H0 y (see fig-ures 6b and 6e) and (c) μ0 �H = Hoz (see figures 6c and6f). With the field applied along the x-axis, Hc decreaseswith an increase in the strength of dipolar interaction� and its value drops to 0.1HK for Ar = 32.0 and� = 1.75 (see figure 6a). The variation of Hc is indepen-dent of the external field’s direction as along it is appliedin the sample plane (x- or y-axis). Notably, Hc is slightlylarger for Ar = 32.0 and � = 1.75, provided the mag-netic field is applied along the long axis of the sample (y-axis) (see figure 6b). When the field is applied normal tothe sample plane, Hc decreases very fast with an increasein � and Ar . Its value drops down to zero for stronglyinteracting MNPs, and Ar = 32.0 (see figure 6c). Thereis a weak dependence of Mr on Ar when the field isapplied along the x-axis (see figure 6d). There is a strongeffect of shape anisotropy when the field is appliedalong the y-axis (see figure 6e). For strongly interactingMNPs, Mr increases as the aspect ratio Ar is increased,and it reaches 0.9 for � = 1.75 and Ar = 32.0 (see fig-ure 6e). Mr decreases very sharply with �, and its valuedrops to zero for � = 1.75 when the magnetic field isapplied normal to the sample plane (see figure 6f). Forweakly interacting MNPs, magnetic hysteresis curvesfollow the Stoner–Wohlfarth model irrespective of theaspect ratio and direction of the applied field, reflectedin Hc (≈ 0.48HK ) and Mr (≈ 0.5).

These results can be explained by probing the effectof dipolar interaction and shape anisotropy of the sam-ple induced by varying the aspect ratios. It is evidentfrom the ground-state morphologies that the dipolarinteractions favour the in-plane ordering of magneticmoments. When Ar is increased, the dipolar interactionfavours ferromagnetic coupling between the magneticmoments. The demagnetisation field decreases as shapeanisotropy is increased (Ar increases in our case), whichis also reported by Wysin [46]. Consequently, mag-netic moments tend to align along the long axis of thesample as it costs less energy compared to other con-figurations. It has also been reported by Jordanovic etal [9]. This ferromagnetic coupling for strongly inter-acting MNPs can also be explained by the fact thatthe exchange interaction between the MNPs is assumedto be zero in the present work. Due to the absence ofexchange interaction, the energetics of domain walls areentirely controlled by the dipolar interaction. It makesit energetically favourable to form domain walls alongthe increasing length of the sample. When an externalfield is applied to this system, the response of this sys-tem not only depends on the dipolar strength but alsoon the shape anisotropy of the sample, which has beeninduced by increasing Ar in our work. When the field isapplied along the shorter axis of the sample, i.e., alongthe x-axis, the natural tendency to get aligned along thelong axis of the sample is hindered. As a result, mag-netic moment ceases to follow the external field, whichis reflected in a decrease in the value of Mr (≈ 0.1)and Hc to 0.1HK for strongly interacting MNPs withAr = 32.0. Magnetic moments are found to follow thefield when the field is applied along the long axis of thesample, but due to the increase in the strength of dipolarinteraction as well, Hc decreases, but its value remainsslightly higher than that of the previous situation. Dueto the same reason, Mr increases with Ar for � > 1.This increase of Mr and decrease of Hc when the fieldis applied along the long axis of the sample is in quali-tative agreement with the work done by García-Arribaset al [47]. They have studied the shape anisotropy effectin thin-film permalloy microstrips.

When we force the magnetic moments to get alignednormal to the plane of the sample by applying the fieldalong the z-axis, they oppose strongly for a large valueof � and Ar . This happens due to the fact that thenatural tendency of the magnetic moments is to getorganised in the plane of the sample (xy-plane in ourcase). As a result, non-hysteresis is observed in thiscase, which is reflected in the zero value of Hc and Mr .This non-hysteresis behaviour has also been reportedby Xue et al [15]. They have studied the hysteresisproperties in a two-dimensional hexagonal array withaligned uniaxial anisotropy. From all the cases, it is quite

181 Page 8 of 9 Pramana – J. Phys. (2021) 95:181

Figure 6. Variation of coercive field Hc and remanent mag-netisation Mr as a function of dipolar interaction strength�, aspect ratio Ar and direction of the applied field. Hc hasbeen scaled by the single-particle anisotropy field HK . Hcdecreases with an increase in dipolar strength � and Ar whenthe field is applied in the plane of the sample (either along x-or y-axis). Hc drops down to zero when the field is appliednormal to the plane of the sample for strongly interactingMNPs. When the field is applied along the x-direction, Mr isdecreased with an increase in aspect ratio and dipolar inter-action strength �. There is a strong shape anisotropy effectwhen the field is applied along the long axis of the sample.Mr increases with an increase in � and Ar , and it reaches 0.9with Ar = 32.0 and � = 1.75. When the external field isapplied normal to the plane of the sample μ0 �H = H0 z, Mrdecreases with increase in Ar and �, and it drops down tozero for � = 1.75 and Ar = 32.0.

evident that dipolar interaction always decreases thecoercive field Hc because the dipolar interaction causesa collective reversal of the magnetic moments underan applied magnetic field. Consequently, the coercivefield decreases with an increase in dipolar interactionstrength.

4. Conclusion

To conclude, we have studied the effect of dipolarinteractions and shape of the sample on the magneticproperties of the 2d (Lx × Ly) array of cubical shapedmagnetic nanoparticles (MNPs) by performing micro-magnetic simulation using OOMMF code [34]. It iswell-known that uniformly magnetised cubical shapedparticles do not possess shape anisotropy [31]. Still, wehave been able to induce the latter in the system byvarying the aspect ratio Ar . Our primary aim was tostudy ground-state (GS) morphologies and understand

the magnetic hysteresis properties as a function of �,shape anisotropy induced by varying the aspect ratioAr (= Ly/Lx ) and the direction of the applied field

μ0�H = H0eH . Our observations are as follows: (a)

For weakly interacting MNPs (� < 1), the magneticmoments are randomly oriented, and the morphologyis unaffected by Ar . MNPs also tend to align nor-mal to the plane of the sample. (b) For strong dipolarstrengths (� > 1), magnetic moments prefer to ori-ent in the sample plane. The morphology, in this case,comprises regions of correlated moments. (c) Mag-netic moment favours ferromagnetic coupling along thelonger axis of the sample when Ar is increased pro-vided � > 1. (d) When the dipolar interaction is weak,magnetic hysteresis curves follow the Stoner–Wohlfarthmodel (Hc ≈ 0.48HK and Mr ≈ 0.5) irrespectiveof the direction of the applied field eH and aspectratio Ar . (e) A strong effect of shape anisotropy isobserved when the field is applied along the increasinglength of the sample, reflected in the very large valueof remanent magnetisation Mr ≈ 0.9. (f) The dipolarinteraction always decreases the coercive field Hc. (g)Thus, for strongly interacting MNPs, Hc and Mr tendto zero when the field is applied normal to the sampleplane.

We have succeeded in obtaining guidelines regardinghow magnetic hysteresis properties may change in thedipolar interacting MNPs arrays. The obtained resultscontribute to expanding the fundamental comprehen-sion of two-dimensional dipolar interacting systems.Furthermore, they offer exciting implications for cre-ating self-assembled arrays of magnetic nanoparticlesof desired magnetic response. The micromagnetic sim-ulation showed that the dipolar interaction induces aneffective ferromagnetic coupling in thin arrays (whenthe aspect ratio is enormous). It means that one canmodulate the magnetic properties of the ordered arraysof MNPs by varying the strength of shape anisotropyof the system. The latter can be changed by varyingthe width and length of the system. Our results arealso relevant for the experimental samples, which canbe divided into various categories depending on howthey behave when an external magnetic field is appliedto them.

Acknowledgements

Most of the numerical simulations presented in this workhave been carried out in the Department of Physics,Indian Institute of Technology (IIT) Delhi. The authoris grateful to Prof Varsha Banerjee for providing thecomputational facility at IIT Delhi.

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References

[1] L Wu, A Mendoza-Garcia, Q Li and S Sun, Chem.Rev. 116(18), 10473 (2016)

[2] K Ulbrich, K Hola, V Subr, A Bakandritsos, J Tucek andR Zboril, Chem. Rev. 116(9), 5338 (2016)

[3] A Hervault and N T K Thanh, Nanoscale 6(20), 11553(2014)

[4] R Bustamante, A Millán, R Piñol, F Palacio, J Carrey,M Respaud, R Fernandez-Pacheco and N J O Silva,Phys.Rev. B 88(18), 184406 (2013)

[5] C A Schütz, L Juillerat-Jeanneret, H Mueller, I Lynchand M Riediker, Nanomedicine 8(3), 449 (2013)

[6] J L Déjardin, A Franco, F Vernay and H Kachkachi,Phys. Rev. B 97(22), 224407 (2018)

[7] P Capiod, L Bardotti, A Tamion, O Boisron, C Albin,V Dupuis, G Renaud, P Ohresser and F Tournus, Phys.Rev. Lett. 122(10), 106802 (2019)

[8] V F Puntes and K M Krishnan, IEEE Trans. Mag. 37(4),2210 (2001)

[9] J Jordanovic, M Beleggia, J Schiøtz and C Frandsen, J.Appl. Phys. 118(4), 043901 (2015)

[10] Z Zhong, B Gates, Y Xia and D Qin, Langmuir 16(26),10369 (2000)

[11] R Cowburn, A Adeyeye and M Welland, New J. Phys.1(1), 16 (1999)

[12] M Anand, J. Magn. Magn. Mater. 522, 167538 (2021)[13] M Pardavi-Horvath, J. Magn. Magn. Mater. 177, 213

(1998)[14] A Mohtasebzadeh, L Ye and T Crawford, Int. J. Mol.

Sci. 16(8), 19769 (2015)[15] D Xue and Z Yan, J. Appl. Phys. 100(10), 103906 (2006)[16] U Löw, V Emery, K Fabricius and S Kivelson, Phys.

Rev. Lett. 72(12), 1918 (1994)[17] K De’Bell, A MacIsaac, I Booth and J Whitehead, Phys.

Rev. B 55(22), 15108 (1997)[18] E Edlund and M N Jacobi, Phys. Rev. Lett. 105(13),

137203 (2010)[19] M Anand, V Banerjee and J Carrey, Phys. Rev. B 99(2),

024402 (2019)[20] A P Alivisatos, Science 271(5251), 933 (1996)[21] C Collier, R Saykally, J Shiang and S Henrichs, J. Health

Sci. 277(5334), 1978 (1997)[22] D Kechrakos and K Trohidou, Phys. Rev. B 71(5),

054416 (2005)[23] N Usov, O Serebryakova and V Tarasov, Nanoscale Res.

Lett. 12(1), 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, Sci.Rep. 3, 2887 (2013)

[25] M Anand, J. Appl. Phys. 128(2), 023903 (2020)[26] Y Li, T Wang, H Liu, f Dai, X Yu and G Liu, J. Nano-

mater. 16(1), 331 (2015)[27] B Yang and Y Zhao, J. Appl. Phys. 110(10), 103908

(2011)[28] M Morales-Meza, P P Horley, A Sukhov and J Berakdar,

The Eur. Phys. J. B 87(8), 186 (2014)[29] F Chinni, F Spizzo, F Montoncello, V Mattarello,

C Maurizio, G Mattei and L D Bianco, Materials 10(7),717 (2017)

[30] B Faure et al, Nanoscale 5(3), 953 (2013)[31] R Moskowitz and E Della Torre, IEEE Trans.

Magn. 2(4), 739 (1966)[32] G Held, G Grinstein, H Doyle, S Sun and C Murray,

Phys. Rev. B 64(1), 012408 (2001)[33] D Kechrakos and K Trohidou, J. Nanosci. Nanotechnol.

8(6), 2929 (2008)[34] M J Donahue and D G Porter, Technical Report (National

Institute of Standards and Technology, Gaithersburg,MD, 1999) vol. 29, p. 53

[35] M Anand, J Carrey and V Banerjee, Phys. Rev. B 94(9),094425 (2016)

[36] M Anand, J Carrey and V Banerjee, J. Magn. Magn.Mater. 454, 23 (2018)

[37] S Bedanta and W Kleemann, J. Phys. D 42(1), 013001(2008)

[38] C Haase and U Nowak, Phys. Rev. B 85(4), 045435(2012)

[39] M Schabes and A Aharoni, IEEE Trans. Magn. 23(6),3882 (1987)

[40] V Russier, J. Appl. Phys. 89(2), 1287 (2001)[41] C C Dantas and L A de Andrade, Phys. Rev. B 78(2),

024441 (2008)[42] J Carrey, B Mehdaoui and M Respaud, J. Appl.

Phys. 109(8), 083921 (2011)[43] A Bailly-Reyre and H T Diep, J. Magn. Magn.

Mater. 528, 167813 (2021)[44] M Anand, 2104.02961 (2021)[45] E C Stoner and E Wohlfarth, Philos. Trans. R. Soc. Lon-

don A Math. Phys. Sci. 240(826), 599 (1948)[46] G Wysin, Demagnetisation fields, available on https://

www.phys.ksu.edu/personal/wysin/notes/demag.pdf(2012)

[47] A García-Arribas, E Fernández, A V Svalov, G VKurlyandskaya, A Barrainkua, D Navas and J M Baran-diaran, Eur. Phys. J. B 86(4), 136 (2013)