Physica A 389 (2010) 3435–3442

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Physica A

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Monte Carlo simulation of the compensation and critical behaviors of aferrimagnetic core/shell nanoparticle Ising modelAhmed Zaim ∗, Mohamed KerouadLaboratoire Physique des Matériaux et Modélisation des Systèmes (LP2MS), Unité Associée au CNRST-URAC, Faculty of Sciences, B.P. 11201,Zitoune, Meknes, Morocco

a r t i c l e i n f o

Article history:Received 18 August 2009Received in revised form 5 March 2010Available online 10 May 2010

Keywords:NanoparticlesCompensation pointIsing modelMonte Carlo simulations

a b s t r a c t

A Monte Carlo simulation has been used to study the magnetic properties and the criticalbehaviors of a single spherical nanoparticle, consisting of a ferromagnetic core of σ =±12 spins surrounded by a ferromagnetic shell of S = ±1, 0 or S = ±

12 , ±

32 spins

with antiferromagnetic interface coupling, located on a simple cubic lattice. A number ofcharacteristic phenomena has been found. In particular, the effects of the shell coupling andthe interface coupling onboth the critical and compensation temperatures are investigated.We have found that, for appropriate values of the system parameters, two compensationtemperatures may occur in the present system.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Magnetic nanoparticles are very important because, with their reduced size and excellent magnetic properties, theycan be used for a variety of nano devices such as a high density magnetic recorders, sensors, molecular imaging devices,etc. [1]. Besides technological applications, the magnetic properties of nanoparticles are scientifically interesting since theirmagnetic properties are quite different from those of the bulk and are greatly affected by the particle size [2]. Much efforthas been devoted to a better understanding of the behavior of magnetic nanoparticles experimentally [3], analytically [4],and in computer simulations [5,6].Ferrimagnetic materials are currently the subject of a great deal of interest due to their potential technological

applications. The occurrence of a compensation point has great technological importance, since at this point only a smalldriving field is required to change the sign of the resultant magnetization. At the compensation temperature, the coercivityof the material increases dramatically facilitating the process of writing and erasing in magneto-optical media [7]. Mucheffort has been devoted to study themagnetic properties of ferrimagnetic systems. Based on the classical Heisenbergmodel,Iglesias et al. studied the spherical maghemite ferrimagnetic nanoparticles by using a Monte Carlo simulation. They foundthat the strong surface anisotropy is responsible for a change in the magnetization reversal mechanism of the particle andmay lead to the spin configuration in the particle forming a hedgehog-like structure [8]. In order to study the magnetizationbehavior of a nanoparticle, a spherical core–shellmodelwith a ferromagnetic core surrounded by a disordered ferrimagneticsurface shell is proposed; it is shown that dynamical effects have a sizeable influence on the exchange bias (EB) properties,provided that a strong shell random anisotropy is assumed [9]. The magnetic properties of the ferrimagnetic nanoparticlehave been investigated theoretically byMonte Carlo simulation [10]. These simulations showed that no compensation pointat finite temperature is found when only the nearest-neighbor interaction between σ and S is taken into account. Leiteet al. [11] have studied the compensation behavior of a ferrimagnetic small particle on an hexagonal substrate, and theyhave shown that the particle exhibits one compensation point. In a recent work [12], we have investigated the critical andcompensation phenomena of a ferrimagnetic core/shell nanocube using a Monte Carlo simulation. We have shown that the

∗ Corresponding author. Tel.: +212 68685982.E-mail addresses: ah_zaim@yahoo.fr (A. Zaim), kerouad@fs-umi.ac.ma (M. Kerouad).

0378-4371/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2010.04.034

3436 A. Zaim, M. Kerouad / Physica A 389 (2010) 3435–3442

−Spin (core)12

− SSpin (shell)

Fig. 1. Schematic drawing of the model of a core/shell spherical nanoparticle with inner spin- 12 core and outer spin-S shell.

compensation temperature exists only below critical values of the shell Jsh/Jc and interface |JInt |/Jc interactions and that itincreases when Jsh/Jc or |JInt |/Jc increases.In recent years, Trohidou et al. [13–15] considered a more realistic model with classical Heisenberg spins. In these

studies, they have considered a completely antiferromagnetic particle, where the core has a uniaxial anisotropy, and thesurface presents both a uniaxial and a radial anisotropy. With this latter model, they have investigated the role playedby the surface anisotropy on the magnetization reversal mechanism. In Refs. [16,17], the authors have demonstrated thatthe remanence and coercivity of dipolar coupled ferromagnetic nanoparticle arrays vary strongly with surface coverage,interparticle distance and the number of deposited monolayers. The modification of the coercive and exchange-bias fieldsin dense nanoparticle arrays with core–shell morphology as a result of the competition between exchange anisotropy andinterparticle dipolar interactions poses a challenging question.Furthermore, the influence of finite-size and surface effects on the magnetic properties of small particles have been

studied by many authors. Bakuzis et al. [18] proposed an Ising-like model to investigate the properties of 2D arrays ofmagnetic nanodots. They obtained that the superferromagnetic ordering temperature increases with the dot size, magneticanisotropy, and dot volume fraction. Within the mean field calculation and Monte Carlo simulation, Leite et al. [19] havestudied the magnetic behavior of an antiferromagnetic small particle of Ising spins on a two dimensional hexagonalstructure, with antiferromagnetic core interactions and disordered magnetic surface. It is shown that at low temperatures,the shifted loop and hysteresis phenomenon appears only for an antiferromagnetic particle with a disordered surface of thespin-glass type. Kaneyoshi [20] has investigated, via the standard mean-field theory, the thermal variations of longitudinaland transverse magnetizations in a ferroelectric nanoparticle described by the transverse Ising model. He shows that themagnetizations of a nanoparticle with size S can be changed with the increase of S and that the magnetizations are ratheraffected by the surface situations. Recently, the phase diagram of Ising nano-particles with cubic structures has beendetermined by Wang and co-workers [21]. The critical temperature of nano-particles with sc, bcc and fcc lattices has beenstudied using the variational cumulant expansion in the Isingmodel [22]. It is found that Tc increaseswith decreasing surfaceto volume ratio.The aim of this paper is to study the effects of the shell coupling, the antiferromagnetic interface coupling, and the crystal

field on the behavior of the critical temperature and the compensation temperature of an Ising ferrimagnetic core/shellnanoparticle, using Monte Carlo simulations. The outline of this paper is as follows: In Section 2, we give the model and theformalism. In Section 3 we present the results and discussions, while Section 4 is devoted to a brief conclusion.

2. Model and Monte Carlo simulation

We consider a Blume Capel ferrimagnetic core/shell nanoparticle model with spherical shape of radius R, located on asimple cubic lattice. Three regions are distinguished inside the particle: a ferromagnetic core with radius Rc , a ferromagneticshell of thickness Rsh = R − Rc and the ferrimagnetic core/shell interface that is formed by the core (shell) spins havingnearest neighbors on the shell (core). The sites of the core are occupied by σ = ± 12 spins, while those of the shell areoccupied by S = ±1, 0 or S = ± 12 , ±

32 spins (see Fig. 1). In the Monte Carlo simulation, based on the heat-bath algorithm

[23,24], we apply free boundary conditions in all directions. The Ising Hamiltonian can be expressed as

H = −Jc∑〈ij〉

σiσj − Jsh∑〈kl〉

SkSl − JInt∑〈ik〉

σiSk − D∑k

S2k . (1)

A. Zaim, M. Kerouad / Physica A 389 (2010) 3435–3442 3437

a

c

b

Fig. 2. The absolute values of the core and shell magnetizations as a function of the temperature T/Jc for different values of JInt/Jc and a fixed value ofJsh/Jc = 0.1.

The first sum runs over pairs neighboring in the core, the second sum runs over pairs in the shell, and the third sumruns over pairs which interact across the interface between the core and the shell of the particle. Jc , Jsh and JInt are theexchange interactions between nearest-neighbors’ magnetic atoms in the core, shell of the particle, and between nearest-neighbors’ spins across the core–shell interface of the nanoparticle (JInt < 0). D represents the crystal field interaction.At each temperature, typically between 2 × 104 and 5 × 104 Monte Carlo steps per spin (MCS) were used for computingthe averages of thermodynamic quantities after 104 initial MCS has been discarded for equilibration. The error bars werecalculated with a Jackknife method [24] by taking all the measurements and grouping them in ten blocks. Althoughnot shown in the figures, the error bars are smaller than the symbol sizes. The implementation of this method is asfollows:

- Calculate average 〈A〉 from the full data set.- Divide data intoM blocks.- For eachm = 1 . . .M , take away blockm and calculate the average 〈Am〉 using the data from all other blocks.- Estimate the error of A by calculating the deviation of 〈Am〉’S from 〈A〉:

δA =

√√√√M − 1M

M∑m=1

(〈Am〉 − 〈A〉)2. (2)

3438 A. Zaim, M. Kerouad / Physica A 389 (2010) 3435–3442

a b

Fig. 3. The phase diagram in the (T/Jc , JInt/Jc ) plane for Jsh/Jc = 0.1 and for different values of D/Jc .

The magnetization per spin in the core and the shell are defined by:

mc =1Nc

Nc∑i=1

σi (3)

and

msh =1Nsh

Nsh∑k=1

Sk, (4)

and the total magnetization per site is,

MT =mc +msh2

. (5)

The total susceptibility χT is defined by

χT = βN(〈M2T 〉 − 〈MT 〉2), (6)

with β = 1kBT, N = Nc + Nsh.

At the compensation point, the totalmagnetizationmust vanish. Then, the compensation temperature can be determinedby the crossing point between the absolute values of the magnetizationsmsh andmc . Therefore, at the compensation point,we must have

|msh(Tcomp)| = |mc(Tcomp)|, (7)

and

sign[msh(Tcomp)] = −sign[mc(Tcomp)]. (8)

The critical temperatures are determined from the maxima of the susceptibility curves.

3. Results and discussions

In this section, we study the effect of the antiferromagnetic interface coupling on the compensation temperature ofa Blume Capel ferrimagnetic spherical nanoparticle model with core/shell morphology using the Monte Carlo simulationtechnique. In Fig. 2, we present the absolute values of the core and shell magnetizations for several values of the interfacecoupling JInt/Jc (−1.2,−1.7 and −2.7) and a fixed value of the shell coupling Jshell/Jc = 0.1 and for a shell with S = ± 12 ,±32 spins. We take a nanoparticle of a core radius Rc = 9 and a shell thickness Rsh = 3, the number of spins in the core

is Nc = 3071 and the number of spins in the shell is Nsh = 4082. In Fig. 2(a) (JInt/Jc = −1.2), the first and second arrowsrepresent the compensation temperature (Tcomp) and critical temperature (Tc), respectively. From the figure, one can seethat below Tcomp, the shell magnetization |msh| and the core magnetization |mc | decrease from their saturation values 1.5and 0.5 when we increase the temperature. These shell and core magnetizations have opposite signs but the cancelation is

A. Zaim, M. Kerouad / Physica A 389 (2010) 3435–3442 3439

a

c

b

Fig. 4. The phase diagram in (T/Jc ,D/Jc ) plane for Jsh/Jc = 0.1 and for two values of JInt/Jc = −1 and JInt/Jc = −1.7.

still incomplete, due to which there is a residual magnetization in the system. As the temperature is increased, the directionof this residual magnetization can switch. Thus, the core becomes more ordered than the shell for temperatures aboveTcomp, |msh| < |mc |. So, there is an intermediate temperature Tcomp such that the cancelation is complete. In Fig. 2(b)(JInt/Jc = −1.7), the saturation values of |msh| and |mc | at the low temperature region take the values of 1.5 and 0.5,respectively. However, as the temperature increases, they cut each other at a temperature Tcomp1 as seen in the figure.Furthermore, as the temperature is increased further, the shell magnetization |msh| is smaller than the core magnetization|mc | and they intersect again at Tcomp2 as seen in the figure. That is, in this case, the system has two compensation points.On the other hand, in Fig. 2(c) (JInt/Jc = −2.7), one can see that the shell and the core magnetizations decrease from theirsaturation values |msh| = 1.5 and |mc | = 0.5 at the low temperature region, with increasing temperature and then vanishat the critical temperature. The shell magnetization is always greater than the core one.To show the influence of the interface coupling |JInt |/Jc on the transition and compensation temperatures, we have shown

in Fig. 3(a), and (b) for S = ± 12 ,±32 and S = ±1, 0 spins respectively, the phase diagrams of the particle in the (T/Jc, |JInt |/Jc)

plane with Jsh/Jc = 0.1, Rc = 9 (Nc = 3071), Rsh = 3 (Nsh = 4082) and for different values of the anisotropy D/Jc . We notethat, there exists a critical value (|JInt |/Jc)c of the ratio |JInt |/Jc below which the critical temperature remains constant, andthen increases with |JInt |/Jc . Concerning the compensation behavior, we can see that the system presents a compensationtemperature for a given range of |JInt |/Jc , which increases when D/Jc is decreased. We can also see that the system canpresent two compensation points for a certain range of |JInt |/Jc ; this range increases when we decrease D/Jc . In Fig. 3(b)

3440 A. Zaim, M. Kerouad / Physica A 389 (2010) 3435–3442

Fig. 5. The phase diagram in (T/Jc , Jsh/Jc ) plane for JInt/Jc = −0.25 and for different values of D/Jc .

(system with S = ±1, 0 spins), we can observe that the system may exhibit only one compensation point and that thecompensation point exists for any value of |JInt |/Jc .To investigate the influence of the crystal field D/Jc on the behavior of the compensation temperature of the particle, we

have presented the phase diagrams in the (T/Jc,D/Jc) plane for Jsh/Jc = 0.1, Rc = 9, Rsh = 3 and JInt/Jc = −1 in Fig. 4(a)and JInt/Jc = −1.7 in Fig. 4(b) for a system with S = ± 12 , ±

32 spins. The filled and open symbols denote the transition

temperature Tc and compensation temperature Tcomp, respectively. It is clear that for JInt/Jc = −1 (Fig. 4(a)), the systemmay present only one compensation temperature. It is also clear that the compensation temperature increases with D/Jcand disappears when D/Jc > 0.32. In Fig. 4(b) (JInt/Jc = −1.7), we can remark that the system exhibits two compensationpoints for−0.225 ≤ D/Jc ≤ 0.025. When D/Jc > 0.025 there is no compensation point, whereas for D/Jc < −0.225, onlyone compensation point exists.We can observe that in both figures, the transition temperature is nearly constant. In Fig. 4(c),we have plotted the phase diagram in the (T/Jc,D/Jc) planewith the same parameters as in Fig. 4(b) and for S = ±1, 0 spins.From this figure, we can clearly see that the system exhibits only one compensation point for any value of D/Jc , and that thecritical temperature is nearly constant.In order to show the effect of the parameter Jsh/Jc on both the compensation and the critical temperatures, we have

presented in Fig. 5 the critical temperature (filled symbols) and compensation temperature (open symbols) versus Jsh/Jcwith JInt/Jc = −0.25, S = ± 12 , ±

32 spins and for different value of D/Jc . From this figure, we find a critical value (Jsh/Jc)c

of the ratio of the shell coupling to the core one Jsh/Jc which depends on the value of D/Jc (for example for D/Jc = −0.3,(Jsh/Jc)c = 0.24).We remark that when Jsh/Jc is less than the critical value (Jsh/Jc)c , Tc is constant and is independent of D/Jc .When Jsh/Jc is greater than (Jsh/Jc)c , Tc increases linearly with Jsh/Jc and depends on D/Jc . On the other hand, the variationsof the compensation temperature are very influenced by the variations of Jsh/Jc and D/Jc . It is seen that the compensationtemperature increases linearly with Jsh/Jc for Jsh/Jc < (Jsh/Jc)c and disappears when Jsh/Jc is greater than (Jsh/Jc)c .To investigate the influence of the size of the core on both the compensation and critical temperatures, we show in

Fig. 6 the compensation and critical temperatures versus the core radius Rc raging from 5 to 16 (Nc = 515–17077) for theparameters Jsh/Jc = 0.06, JInt/Jc = −0.25, D/Jc = 0.0 and S = ± 12 , ±

32 spins and for a fixed shell thickness Rsh = 3

(Nsh = 1594–11594). It is seen that the compensation and critical temperatures increase with increasing core radius andreaches saturation values for high values of Rc . The saturation values of the compensation and the critical temperatures are(Tcomp/Jc)sat = 0.41 and (Tc/Jc)sat = 1.06, respectively.The effect of the shell thickness on both the compensation and critical temperatures is studied. Fig. 7, show the variations

of the compensation and critical temperatures versus the shell thickness Rsh raging from 2 to 8 (Nsh = 2504–17408), forthe parameter Jsh/Jc = 0.06, JInt/Jc = −0.25, D/Jc = 0.0 and S = ± 12 , ±

32 spins and for a fixed core radius Rc = 9

(Nc = 3071). It is seen that the compensation and critical temperatures decrease when we increase the shell thickness andreaches saturation values for high values of Rsh. The saturation values of the compensation and the critical temperatures are(Tcomp/Jc)sat = 0.365 and (Tc/Jc)sat = 1.04, respectively.

4. Conclusion

In summary, we have studied the magnetic properties and the compensation behavior of a spherical ferrimagneticcore/shell nanoparticle on an Ising model, located on a simple cubic lattice, where we have taken a coupling constants

A. Zaim, M. Kerouad / Physica A 389 (2010) 3435–3442 3441

Fig. 6. The compensation and critical temperature versus the core radius Rc ranging from5 to 16, for the parameters Jsh/Jc=0.06, JInt/Jc=−0.25D/Jc = 0.0and for fixed value of shell thickness Rsh = 3.

Fig. 7. The compensation and critical temperature versus the shell thickness Rsh ranging from 2 to 8, for the parameters Jsh/Jc = 0.06, JInt/Jc = −0.25D/Jc = 0.0 and for a fixed value of core radius Rc = 9.

Jc and Jsh for the core and the shell respectively, and an interface coupling constant JInt between nearest-neighbors’ spinacross the core–shell with JInt < 0. Using Monte Carlo techniques, we have discussed the influence of the shell coupling, theinterface coupling, and the crystal field on the critical and compensation temperatures. We have shown that, depending onthe values of the parameters Jsh/Jc , JInt/Jc and D/Jc , the system can exhibit one or even two compensation temperatures.

Acknowledgement

This work has been initiated with the support of PROTARS III D12/08 and URAC 08.

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