2) ferrimagnetic nanoparticle

Solid State Communications 158 (2013) 76–81

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Monte Carlo study of the magnetic behavior of a mixed spin (1, 3/2) ferrimagnetic nanoparticle Ahmed Zaim n, Mohamed Kerouad, Mourad Boughrara ´ Associe´e au CNRST-URAC: 08, University Moulay Ismail, Faculty of Sciences, Laboratoire Physique des Mate´riaux et Mode´lisation des Syste mes (LP2MS), Unite B.P. 11201, Zitoune, Meknes, Morocco

a r t i c l e i n f o

a b s t r a c t

Article history: Received 12 July 2012 Received in revised form 8 October 2012 Accepted 11 October 2012 by C. Lacroix Available online 17 October 2012

The Monte Carlo simulation technique has been used to study the magnetic behavior of a ferrimagnetic spherical nanoparticle on a simple cubic lattice. The particle is formed by alternate layers of spins s ¼ 1 and S ¼ 3=2. The interlayer coupling is antiferromagnetic. The influences of the crystal fields, and the exchange couplings, on the critical and compensation behavior of the nanoparticle, are investigated. The results present rich critical behavior, which includes the first- and second-order phase transitions, thus also the tricritical and critical end points. It is also found that the compensation point can appear for appropriate values of the system parameters. & 2012 Elsevier Ltd. All rights reserved.

Keywords: A. Ferrimagnetic C. Nanoparticles D. Critical phenomena E. Monte Carlo techniques

1. Introduction Magnetic nanoparticles are currently a subject of wide basic and applied research due to their technological applications in high-density magnetic storage media [1,2], magnetic resonance imaging [3], non-linear optics [4], catalysis [5], environmental remediation [6], ferrofluids [7], and biomedicine [8]. Magnetic nanoparticles show a wide variety of unusual magnetic properties as compared to the respective bulk materials. Magnetic characteristics of nanoparticles are strongly influenced by the so-called finite-size and surface effects. Their relevance increases with decreasing particle size. Several techniques have been used to study the magnetic properties of these systems, such as mean field approximation [9,10], effective field theory [11–17], variational cumulant expansion [18,19], Green function formalism [20,21], and Monte Carlo simulation [22–24]. Recently, Kaneyoshi [13] has obtained that the existence of re-entrant behavior and compensation point in a decorated Ising nanowire is dependent on the parameter values. Jiang et al. [12] have observed that the positive single-ion anisotropy is a necessary element for the emergence of the accessional magnetization plateaus for the ferrimagnetic mixed-spin nanoparticle.

n

Corresponding author. Tel.: þ212 668 68 59 82; fax: þ 212 535 43 94 82. E-mail addresses: [email protected] (A. Zaim), [email protected] (M. Kerouad). 0038-1098/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2012.10.014

Ferrimagnets are structured in several sublattices with inequivalent moments interacting antiferromagnetically. Under certain condition, the sublattice magnetizations compensate each other, then the resultant moment vanishes at a compensation temperature Tcomp below the critical temperature Tc [25]. The existence of compensation point may have important applications in the area of thermomagnetic recording devices [26]. The critical properties of the mixed spin-3/2 and spin-1 ferrimagneic nanoparticles and the finite-size effect on the compensation point have been studied by Monte Carlo simulation [27]. It was observed that the compensation temperature shows a fluctuating behavior which depends strongly on particle size. Recently, the possibility of two compensation points in a ferrimagnetic core/shell nanoparticle has been investigated using Monte Carlo simulation [28]. Other simulation studied the critical and compensation phenomena of a ferrimagnetic nanocube, consisting of a ferromagnetic core surrounded by a ferromagnetic shell, by introducing an antiferromagnetic core–shell coupling [29]. It is shown that the compensation temperature exists only below critical values of the shell and interface couplings. Kaneyoshi [30] has examined the magnetizations of a cylindrical nanowire where the spins at the surface shell are coupled antiferromagnetically to the spins of the core. He has found that the compensation temperature can appear for appropriate values of the parameters. The aim of this paper is to study the critical and compensation behavior of a ferrimagnetic spherical nanoparticle on a simple cubic lattice, using Monte Carlo simulation. The outline of this paper is as follows: In Section 2, we give the model and Monte

A. Zaim et al. / Solid State Communications 158 (2013) 76–81

77

Carlo simulation. 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 ferrimagnetic nanoparticle model with spherical shape of radius R ¼10, located on a simple cubic lattice. The ferrimagnetic nanoparticle is formed by alternate layers of s and S spins (Fig. 1); hence, the two different types of spins are described by an Ising variable, which can take the values s ¼ 71, 0 and S ¼ 71=2, 73=2. With this particular arrangement of spins, the model presents intersublattice and intrasublattice nearestneighbor interactions of the type ss, sS, and SS. In the Monte Carlo simulation based on the heat-bath algorithm [31], we apply free boundary conditions in all directions. Hamiltonian describing our model can be written as X X X X X H ¼ J1 si sj J2 Sk Sl J3 si Sk DA s2i DB S2k ð1Þ /ijS

/klS

/ikS

i

k

where J1, J2, and J3 are the exchange coupling between nearestneighbor pairs of spins ss, SS, and sS, respectively. The parameter J3 will be taken negative in all the subsequent analyses, that is, the intersublattice coupling is antiferromagnetic. DA and DB are the crystal field interactions acting on the spin sites s and S, respectively. At each temperature, typically between 2  104 and 3  1010 Monte Carlo steps per spin (MCS) were used for computing averages of thermodynamic quantities after 104 initial MSC have been discarded for equilibration. The error bars were calculated with a Jackknife method [31] by taking all the measurement and grouping them into 10 blocks. The sublattice magnetizations per spin are defined by m1 ¼

Fig. 1. Schematic representation of a ferrimagnetic nanoparticle with spherical shape of radius R ¼10. The particle is formed by alternate layers of s (black circles) and S (white circles) spins.

2.5

D A / J 1 = 0.5

N1 1 X s N1 i ¼ 1 i

ð2Þ

2.0

N2 1 X S N2 k ¼ 1 k

ð3Þ

1.5

m2 ¼

T/J1

and

the total magnetization per site is MT ¼

1 ðN1 m1 þN 2 m2 Þ N

ð4Þ

Tcomp

D A / J 1 =-0.6

Tc(SO) Tc(FO)

1.0

J 3 / J 1 = - 0.3

and the total susceptibility is defined as

wT ¼ bNð/M2T S/MT S2 Þ

J 2 / J 1 = 0.2

ð5Þ

with b ¼ 1=kB T and N ¼ N1 þ N2 At the compensation point the total magnetization must vanish. Then, the compensation temperature can be determined by the crossing point between the absolute values of the magnetizations m1 and m2. Therefore, at the compensation point, we must have 9m1 ðT comp Þ9 ¼ 9m2 ðT comp Þ9

ð6Þ

and sign½m1 ðT comp Þ ¼ sign½m2 ðT comp Þ

0.5

0.0 -4

-3

-2

-1

0

1

2

DB/J1 Fig. 2. The phase diagram in ðT=J 1 ,DB =J 1 Þ plane for J 3 =J 1 ¼ 0:3, J 2 =J1 ¼ 0:2 and for two values of DA =J 1 (0.5 and 0.6). Insert shows a zoom of the tricritical and critical end point regions.

ð7Þ

The critical temperatures are determined from the maxima of the susceptibility curves and the first-order phase transitions are obtained by locating the discontinuities of the magnetization curves. 3. Results and discussions In this section, we study the effect of the crystal field interactions (DA =J 1 and DB =J1 ) on the critical and compensation behavior

of a ferrimagnetic spherical nanoparticle. In Fig. 2, we present the phase diagram in the ðT=J 1 ,DB =J 1 Þ plane for J2 =J1 ¼ 0:2, J 3 =J 1 ¼ 0:3 and for two values of the crystal field interactions DA =J 1 ¼ 0:5 and DA =J 1 ¼ 0:6. It is observed that the phase diagram is very rich; it exhibits second-order ðÞ and first-order ðJÞ phase transition temperatures, tricritical point, critical end point, and compensation temperature. We can see that, depending on the value of DA =J1 , we have two kinds of phase diagrams. In the first one ðDA =J 1 ¼ 0:6Þ, we have three critical lines: two of the

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A. Zaim et al. / Solid State Communications 158 (2013) 76–81

second-order and the other of the first-order between the paramagnetic and ferrimagnetic phases linked by two tricritical points, which is denoted by the bold star ð%Þ. The critical temperature increases when we increase DB =J 1 , from a saturation value ðT sat1 ¼ 0:09Þ for large negative value of DB =J 1 to reach another saturation value ðT sat2 ¼ 2:08Þ for large positive value of DB =J 1 . We have also ferri–ferri first-order phase transition at the low temperature region for a certain range of DB =J1 . This firstorder line terminates at a critical end point ð’Þ and separates the ferri ð 7 1=2,0Þ region from the ferri ð 7 3=2, 7 1Þ region. Notice that these results of first-order phase transition temperature, tricritical point, and critical end point have not been observed in Refs. [22,27] where the ferrimagnetic nanoparticle is formed by spin-3/2 and spin-1. The second kind of the phase diagrams ðDA =J 1 ¼ 0:5Þ presented only the second-order phase transition for all values of DB =J 1 . This second-order temperature increases from its saturation value ðT sat1 ¼ 0:81Þ for low value of DB =J1 , to reach another saturation value ðT sat2 ¼ 2:23Þ for higher values of DB =J 1 . It is seen from this two kinds of phase diagrams that the compensation point ðbÞ can appear when DB =J 1 becomes larger than the critical value ðDB =J 1 Þc which is equal to 2.15 in the case of DA =J1 ¼ 0:6. Recently, the effects of the crystal field and transverse field on the phase diagram of the mixed spin (1, 3/2) ferrimagnetic cubic nanowire [32], and the mixed spin (3/2, 5/2) ferrimagnetic nanoparticle [33] are examined. In other work, the magnetic properties of a small particle on a hexagonal lattice are

studied [34]. It is noticed that the first-order phase transition, the tricritical and critical end points are not investigated in these Refs. [32–34]. In order to complete the discussion of the above phase diagram, we study in Fig. 3, the sublattice magnetizations as a function of the crystal field interaction DB =J1 , for selected values of J 2 =J 1 ¼ 0:2, J3 =J1 ¼ 0:3, DA =J1 ¼ 0:6 and for several values of the temperatures (T=J1 ¼ 0:03, 0.55, and 1). In the case of T=J 1 ¼ 1 (Fig. 3a), it is seen that the system presents a continuous passage occurs between the partly ferrimagnetic phase and the paramagnetic phase at DB =J1 ¼ 2:06. In Fig. 3b (T=J1 ¼ 0:55), it is clear that the compensation point appears where the both sublattice magnetizations are equal in magnitude at DB =J1 ¼ 1:82. Since the sublattice magnetizations have opposite signs at this crystal field interaction the compensation point occurs. It is also clear that the system exhibits a discontinuity of the sublattice magnetizations at a first-order point (DB =J1 ¼ 2:23) between the ferrimagnetic phase and the paramagnetic phase. In Fig. 3c (T=J 1 ¼ 0:03), we found a discontinuity of the sublattice magnetizations at a first-order point between two phase ferrimagnetics from ð þ1=2,0Þ to ð þ 3=2,1Þ. In Fig. 4, we show the dependence of the critical (filled symbols) and compensation (open symbols) temperatures on the value of DA =J 1 for J2 =J1 ¼ 0:2 and J 3 =J 1 ¼ 0:3 and for three values of the crystal field interactions (DB =J 1 ¼ 1:5, DB =J 1 ¼ 0:0, and DB =J 1 ¼ 1:5). We can see that in the range of DA =J 1 studied,

Fig. 3. The behavior of the sublattice magnetizations as a function of the crystal field interaction DB =J 1 , for selected values of J 2 =J 1 ¼ 0:2, J3 =J 1 ¼ 0:3, and DA =J1 ¼ 0:6 and for several values of the temperatures: T=J 1 ¼ 1 (a), T=J 1 ¼ 0:55 (b), and T=J 1 ¼ 0:03 (c).

A. Zaim et al. / Solid State Communications 158 (2013) 76–81

the critical temperature increases with DA =J 1 until it seems to reach a constant value that depends on DB =J1 . Concerning the compensation behavior, we can see that for each value of DB =J 1

2.5

2.0

T/J 1

1.5

1.0

0.5

0.0 -1.0

-0.5

0.0

0.5

1.0

1.5

2.0

DA/J1 Fig. 4. The phase diagram in ðT=J 1 ,DA =J 1 Þ plane for J 3 =J 1 ¼ 0:3, J 2 =J1 ¼ 0:2 and for three values of DB =J 1 (1.5, 0.0, and  1.5).

there is a range of values of DA =J 1 for which there are compensation points; this range increases when we increase DB =J 1 . It is also seen from this figure that the compensation temperatures decrease when we decrease the crystal field interaction DB =J 1 and they all combine at the low temperature region at the value of DA =J 1 C0:688. In Fig. 5, we examine the influence of the exchange coupling J2 =J1 on the critical and compensation temperatures of a ferrimagnetic spherical nanoparticle for DA =J 1 ¼ 0:5, DB =J 1 ¼ 1 and for J3 =J1 ¼ 0:5. In Fig. 5a, we plot the total magnetization versus reduced temperature T=J 1 for some selected values of J 2 =J 1 (0.1, 0.3, and 0.6). It is seen that the magnetization curves corresponding to the values J2 =J1 ¼ 0:1 and 0.3 exhibit two successive zeros. Namely, the first zero indicates that the temperature value at which the magnetization of the particle reduces to zero corresponds to the compensation temperature, and the second zero denotes that the temperature value at which the magnetization depresses to zero, corresponding to the critical temperature of the system. It is also seen that the magnetization curve corresponding to the value 0.6 exhibits only one zero at the critical temperature of the system. The style of total magnetization shifts from N-type behavior to Q-type behavior [35] with changing value of J 2 =J 1 ¼ 0:10:6. On the other hand Fig. 5b shows the variation of the two sublattice magnetizations of the system with temperature for some selected values of J 2 =J 1 (0.1, 0.3, and 0.6). We can see clearly that as the temperature increases the magnetization of the two sublattices approaches to zero and vanishes at the critical

0.25 m2

0.20

1.6 1.4 1.2

0.15

1.0

0.10

0.8 0.6

0.05

MT

79

0.4

0.00

0.2 0.0

-0.05

-0.2

-0.10

-0.4 -0.6

-0.15

-0.8

-0.20

m1 -1.0 0

1

2

3

0

4

1

2

T/J1

3

4

T/J1 3.6

3.5

3.4 3.2

3.0

3.0 2.8

2.5

2.6 T/J1

χT

2.4 2.0

2.2 2.0 1.8

1.5

1.6 1.4

1.0

1.2 1.0

0.5

0.8 0.6

0.0 0

1

2 T/J1

3

4

0.0

0.2

0.4

0.6

0.8

1.0

J2/J1

Fig. 5. The influence of the exchange coupling J 2 =J 1 on the total magnetization (a), the sublattice magnetizations (b), and the total susceptibility (c) of the particle for J 3 =J 1 ¼ 0:5, DA =J 1 ¼ 0:5, and DB =J 1 ¼ 1. The phase diagram in ðT=J 1 ,J 2 =J1 Þ plane for the same parameters (d).

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A. Zaim et al. / Solid State Communications 158 (2013) 76–81

temperature. In addition, two sublattice magnetizations take values which are different from zero at the compensation point while, as seen in Fig. 5a, the total magnetization value of the particle is zero at the compensation point for two values of J 2 =J 1 (0.1 and 0.3). In Fig. 5c, we plot the temperature dependence of the total susceptibility wT for some selected values of J 2 =J 1 (0.1, 0.3, and 0.6). It is found that there exists a distinct peak in the susceptibility–temperature curve. Moreover, with increasing value of J 2 =J1 , the peak becomes wider and the position of peak shifts to higher temperature. That is, the critical temperature is enhanced. In order to investigate the influence of the exchange coupling J2 =J1 on the critical and compensation temperatures of a ferrimagnetic spherical nanoparticle, we show in Fig. 5d, the phase diagram in the (T=J 1 , J 2 =J 1 ) plane for DA =J1 ¼ 0:5, DB =J 1 ¼ 1 and for J3 =J1 ¼ 0:5. We have a phase transition line of the second-order from the ferrimagnetic phase at low temperature region to the paramagnetic one at high temperature region. It is clear that when J 2 =J1 increases, both the critical and the compensation temperatures increase but if the strength of J 2 =J 1 is sufficiently large (J 2 =J 1 4 0:46), the compensation temperature disappears, and there exists only a critical temperature for these case. In order to show the effect of the exchange coupling J3 =J1 on the critical and compensation temperatures of a nanoparticle, we present in Fig. 6, the phase diagram in the (T=J1 , 9J3 9=J 1 ) plane for DA =J 1 ¼ 0:5, DB =J 1 ¼ 1 and for J 2 =J 1 ¼ 0:2. We have a phase transition line of the second-order separating the ferrimagnetic phase at low temperature region from the paramagnetic phase at high temperature region. As is evident, both the critical and the compensation temperatures increase with increasing 9J 3 9=J1 , the latter more rapidly, and at a critical value ð9J 3 9=J1 Þc which is this case equal to 1.24, the compensation point disappears. It should be mentioned that the phase diagrams of the mixed spin (1, 3/2) Ising model on honeycomb, square and simple cubic lattices have also discussed in Refs. [36–40]. Some studies show that we can find in these systems a very interesting phenomena. Jiang et al. [36] discussed the phase diagrams and magnetizations of the mixed spin (1, 3/2) transverse Ising model with the presence of the crystal fields on the honeycomb lattice by the use of the effective-field theory. They found that the system can exhibit two tricritical points and existence of two compensation points. Zukovic et al. [40] discussed the critical and compensation properties of the mixed spin (1, 3/2) ferrimagnetic on a square lattice by standard and histogram Monte Carlo simulation. 3.5

They focused on several contradictory results previously obtained by different approximative approaches. In particular, they checked whether the system, for a sufficiently large value of anisotropy strength, can display a re-entrant phenomenon and additional firstorder phase transition within the ordered ferrimagnetic phase, as predicted by the cluster variational theory. Neither of these predictions was confirmed. Instead, they found just one order–disorder phase boundary as a single-valued function of the anisotropy within the entire range of values, in agreement with the mean field theory and effective field theory results. Recently, Deviren et al. [41] studied the layered magnetization curves and phase diagrams for a mixed spin-1 and spin-3/2 Ising system with two alternative layers on a honeycomb lattice by using the framework of the effective field theory. They shown that the system also exhibits a tricritical behavior besides the multicritical point, isolated critical point, and double critical end point, depending on the interaction parameters. Albayrak et al. [42] examined the spin-1 and spin-3/2 model on a bilayer Bethe lattice with crystal field. They observed that the system presents one or two compensation temperatures for appropriate values of the system parameters.

4. Conclusion In summary, we have studied the critical and compensation behavior of a ferrimagnetic spherical nanoparticle on a simple cubic lattice. Using Monte Carlo simulation, we have discussed the effect of the crystal field interactions (DA =J1 and DB =J1 ) and the exchange coupling J2 =J1 on the phase diagrams. The results present rich varieties of phase transitions with first-order and second-order one. These phases are found to be linked by tricritical points. It is observed that a line of first-order transitions separating two distinct ferrimagnetic phases and terminating at a critical end point. It is also observed that the compensation point can appear when DB =J 1 becomes larger than the critical value ðDB =J1 Þc . The compensation points [43] of the ferrimagnetic materials have potential use in magnetic recording materials.

Acknowledgments This work has been initiated with the support of URAC: 08, the Project RS: 02 (CNRST) and the Swedish Research Links Program dnr-348-2011-7264. References

3.0

2.5

T/J1

2.0

1.5

1.0

0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

J3 /J1 Fig. 6. The phase diagram in ðT=J 1 ,9J 3 9=J 1 Þ plane for J 2 =J 1 ¼ 0:2, DA =J 1 ¼ 0:5 and DB =J 1 ¼ 1.

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