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2 core and spin-1 shell structure
Journal of Magnetism and Magnetic Materials 323 (2011) 3168–3175
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Thermal and magnetic properties of a ferrimagnetic nanoparticle with spin-3/2 core and spin-1 shell structure a,c ¨ Yusuf Yuksel , Ekrem Aydıner b, Hamza Polat a, Department of Physics, Dokuz Eyl¨ ul University, Tr-35160 _Izmir, Turkey Department of Physics, Istanbul University, 34134 Vezneciler Istanbul, Turkey c Dokuz Eyl¨ ul University, Graduate School of Natural and Applied Sciences, Turkey a
b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 June 2011 Received in revised form 7 July 2011 Available online 21 July 2011
Intensive Monte Carlo simulations based on standard Metropolis algorithm have been applied to investigate the phase diagrams of a ferrimagnetic cubic nanoparticle (nanocube) with a spin-3/2 core surrounded by a spin-1 shell layer with antiferromagnetic interface coupling. It has been shown that occupation of sites of particle core by spin-3/2 plays an important role on the shape of the phase diagrams. In addition, effects of the crystal field interaction as well as exchange interactions on the thermal and magnetic properties of the system have also been discussed in detail, and some interesting features have been observed for the temperature dependence of total magnetization curves of particle. & 2011 Elsevier B.V. All rights reserved.
Keywords: Compensation temperature Monte Carlo simulation Nanoparticle
1. Introduction Recently, magnetic nanoparticles attracted great theoretical and experimental interest, due to their technological [1,2] and biomedical [3–5] applications such as information storage devices and drug delivery in cancer thermotherapy. These fine particles serve as a bridge between bulk materials and atomic or molecular structures. Physical properties of a bulk material are independent from its size; however, nanoparticles often exhibit size-dependent properties. In addition, due to their good coercivity, they are also utilized for potential application in high density magnetic recording instead of conventional limited capacity thin film materials. From the theoretical point of view, these systems have been studied by a variety of techniques such as mean field theory (MFT) [6], effective field theory EFT [7–9], Green functions (GF) formalism [10], variational cumulant expansion (VCE) [11,12] and Monte Carlo (MC) simulations [13–27]. Monte Carlo simulation [28] is regarded as a powerful numerical approach for simulating the behavior of many complex systems, including magnetic nanoparticle systems. For instance, by utilizing this tool, Refs. [13–24] investigated the exchange bias effect in magnetic core/shell nanoparticles where the hysteresis loop exhibits a shift below the Nee´l temperature of the antiferromagnetic shell due to the exchange coupling at the interface region of ferromagnetic core and antiferromagnetic shell. In recent papers, the authors paid attention to the investigation of dependence on the core, shell and interface couplings, as
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E-mail address: [email protected] (H. Polat). 0304-8853/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2011.07.011
well as particle size of critical and compensation temperatures of core/shell nanoparticles and many interesting results have been obtained. For example, Zaim et al. [25] studied the magnetic properties and hysteresis loops of a single nanocube with a ferromagnetic spin-1/2 core surrounded by a ferromagnetic spin-1 shell with antiferromagnetic interface coupling. They have applied MC simulations and examined the effects of shell coupling and interface coupling on both the compensation and magnetization profiles but they did not include any crystal field interaction term on the model Hamiltonian. Particularly, they have observed that as the shell thickness increases then both the critical and compensation temperatures of the system increase and reach a saturation value for high values of thickness. Similarly, Zaim and Kerouad [26] have simulated a spherical particle consisting of a ferromagnetic spin-1/2 core and a ferromagnetic spin-1 or 3/2 shell with antiferromagnetic interface coupling. They have focused on the effects of the shell and interface couplings on both the critical and compensation temperatures. They have found that there may be two compensation temperatures when the sites of shell sublattice are occupied by S¼3/2 spins and they observed that when the shell thickness increases, both the critical and compensation temperatures of the system decrease and reach their saturation values for high values of shell thickness. Similarly, Jiang et al. [27] have investigated the magnetic properties of the ferrimagnetic nanoparticles on a bodycentered cubic lattice based on a classical Heisenberg spin Hamiltonian in which S ¼3/2 and s ¼ 1 spins are distributed in the two interpenetrating square sublattices. They have made a detailed discussion on the conditions for the occurrence of a compensation temperature in the system. They have also
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observed that compensation temperature of system fluctuates as the particle size increases, and it becomes relatively independent of particle size when the particle size is increased further. More recently, Kaneyoshi [9] has studied magnetic properties of a cylindrical nanowire or nanotube by using EFT with correlations and investigated the effects of dilution at the surface, surface exchange interaction, and the shell coupling on the magnetization profiles, and found a number of characteristic phenomena. As shown in many of the previously published works mentioned above, critical and compensation temperatures of a ferrimagnetic nanoparticle exhibit a size-dependent behavior which is a characteristic property of these systems. On the other hand, a ferrimagnetic nanoparticle may also exhibit different phase diagrams depending on the occupation of core layer by s 41 spins. Therefore, the purpose of the present work is to investigate the phase diagrams of a ferrimagnetic nanoparticle with a spin-3/2 core surrounded by a spin-1 shell layer. For this aim, we organized the paper as follows: in Section 2, we briefly present our model. The results and discussions are presented in Section 3, and finally Section 4 contains our conclusions.
2. Model and simulation technique We consider a cubic ferrimagnetic nanoparticle composed of a spin-3/2 ferromagnetic core which is surrounded by a spin-1 ferromagnetic shell layer. At the interface, we define an antiferromagnetic interaction between core and shell spins (Fig. 1). Hamiltonian describing our model can be written as X X X X X H ¼ Jint si Sk Jc si sj Jsh Sk Sl Dc ðsi Þ2 Dsh ðSk Þ2 , ð1Þ /ikS
/ijS
/klS
i
k
where s ¼ 7 3=2, 7 1=2 and S¼ 71,0. Jint, Jc and Jsh define antiferromagnetic interface and ferromagnetic core and shell exchange interactions, respectively. Dc and Dsh represent single ion anisotropy terms of core and shell sublattices, respectively. / S denotes the nearest neighbor interactions on the lattice and we fixed the value of Jsh throughout the simulations. For simplicity, we also assume Dc ¼Dsh ¼D as a limited case. In order to simulate the system, we employ Metropolis Monte Carlo simulation algorithm [29] to Eq. (1) on an L L L cubic lattice with free boundary conditions. Configurations were generated by selecting the sites in sequence through the lattice and making single-spin-flip attempts, which were accepted or rejected according to the Metropolis algorithm. Data were generated over 50–100 realizations by using 25 000 Monte Carlo
Fig. 1. Schematic representation of core/shell nanoparticle on a simple cubic lattice with a spin-3/2 core and spin-1 shell.
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steps per site after discarding the first 2500 steps. Error bars were calculated by using the jackknife method [30]. Our program calculates the sublattice magnetizations Mc and Msh, the total magnetization M, magnetic susceptibility w and the specific heat C. These quantities are defined as Mc ¼
M¼
C¼
Nc 1 X s, Nc i ¼ 1 i
Mc þ Msh , 2 1 kT
2
Msh ¼
w¼
Nsh 1 X S , Nsh k ¼ 1 k
L3 ð/M 2 S/MS2 Þ, kT
ð/E2 S/ES2 Þ,
ð2Þ
ð3Þ
ð4Þ
where k is the Boltzmann constant (here k¼1) and Nc and Nsh denote the number of spins in core and shell layers, respectively. To determine the compensation temperature Tcomp from the computed magnetization data, the intersection point of the absolute values of the sublattice magnetizations was found using jMc ðTcomp Þj ¼ jMsh ðTcomp Þj,
ð5Þ
signðMc ðTcomp ÞÞ ¼ signðMsh ðTcomp ÞÞ,
ð6Þ
with Tcomp o Tc , where Tc is the critical temperature i.e. Ne´el temperature. Eqs. (5) and (6) indicate that the sign of the sublattice magnetizations is different, however, absolute values of them are equal to each other at the compensation point.
3. Results and discussion In this section, we examine the phase diagrams and temperature dependencies of thermal and magnetic properties of the system for some selected values of Hamiltonian parameters. However, in order to make a comparison with previously published works, we select the number of core and shell spins as Nc ¼ 113 and Nsh ¼ 153 113 (the same values as in Ref. [25]) throughout Figs. 2–7. Fig. 2 represents the effect of antiferromagnetic interface coupling between core and shell spins on the thermal and magnetic properties of the particle for some selected values of Jint/Jsh with Jc/Jsh ¼0.25 and D/Jsh ¼0.0. In Fig. 2(a) we plot the total magnetization versus reduced temperature T/Jsh. As seen in this figure, there are two zeros of magnetization curves for different Jint/Jsh values. 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 the temperature value at which magnetization depresses to zero, corresponding to critical temperature of the system. Besides, both the compensation point and critical temperature of the system increase as the absolute value of Jint/Jsh increases and the total magnetization curves exhibit an apparent minimum for weak Jint/Jsh values. On the other hand, Fig. 2(b) shows the variations of the core and shell magnetizations of the system with temperature for selected values of Jint/Jsh with Jc/Jsh ¼0.25 and D/Jsh ¼0.0. We can clearly see that as the temperature increases then the magnetizations of the particle core and shell approach to zero and vanish at the critical temperature. In addition, core and shell magnetization curves take values which are different from zero at the compensation point while, as seen in Fig. 2(a), the total magnetization value of the particle is zero at the compensation point for all values of Jint/Jsh. On the other hand, in Fig. 2(c), we plot the temperature dependence of the specific heat of the particle for some selected values of Jint/Jsh with Jc/Jsh ¼0.25 and D/Jsh ¼ 0.0. As seen in Fig. 2(c), as the temperature increases starting from zero, then the specific heat curves exhibit a hump at low temperatures and a sharp peak
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Fig. 2. Effect of the antiferromagnetic interface coupling Jint =Jsh on the temperature dependencies of (a) total magnetization M, (b) core and shell magnetizations Mc, Msh and (c) specific heat C of the system for Jc =Jsh ¼ 0:25 and D=Jsh ¼ 0:0 with Nc ¼ 113 and Nsh ¼ 153 113 spins. (d) Phase diagram of the system in ðTc ,Tcomp Jint =Jsh Þ plane for the same parameters.
Fig. 3. Temperature dependence of specific heat of the particle for Jc =Jsh ¼ 0:25 and D=Jsh ¼ 0:0 with some selected values of Jint =Jsh . The curves are calculated for Nc ¼ 113 and Nsh ¼ 153 113 spins.
occurs at a second-order phase transition temperature. Particularly, the shoulder which is observed in the specific heat of the system probably originates from the thermal fluctuations of the internal energy and magnetization of the particle core due to antiferromagnetic interface interaction Jint/Jsh, and according to our numerical calculations these phenomena are closely related to the transition temperature of particle core. Namely, even if Jint =Jsh ¼ 0:1 (i.e. positive), the shoulder is still obtained in the specific heat curves, since specific heat must be independent of the sign of Jint =Jsh . Besides, if we take avalue of Jint =Jsh ¼ 0:001, the height of the shoulder is increased and it takes much sharper form, like divergence. In addition, if we take other values larger than jJint =Jsh j 40:1, such as jJint =Jsh j ¼ 1:0, such a shoulder tends to disappear. Hence, whether the shoulder may be obtained or not in the specific heat is essentially dependent on the value of jJint =Jsh j. These observations are depicted in Fig. 3. Moreover, maxima of the specific heat curves slide to higher temperature values which confirms that the critical temperature increases as the absolute value of Jint =Jsh increases. In order to investigate the influence of Jint =Jsh on both critical and compensation temperatures of the particle, we plot the phase diagram of the system in a ðTc ,Tcomp Jint =Jsh Þ plane in Fig. 2(d).
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Fig. 4. Effect of the ferromagnetic core spin coupling Jc/Jsh on the temperature dependencies of (a) total magnetization M, (b) core and shell magnetizations Mc, Msh and (c) specific heat C of the system for Jint =Jsh ¼ 0:1 and D=Jsh ¼ 0:0 with Nc ¼ 113 and Nsh ¼ 153 113 spins. (d) Phase diagram of the system in ðTc ,Tcomp Jc =Jsh Þ plane for the same parameters.
Fig. 5. Temperature dependence of total magnetization M, core and shell magnetizations Mc, Msh, and magnetic susceptibility w of the particle for Jint =Jsh ¼ 0:1 and D=Jsh ¼ 0:0 with Jc =Jsh ¼ 0:3 (left panel) and Jc =Jsh ¼ 0:55 (right panel). The curves are calculated for Nc ¼ 113 and Nsh ¼ 153 113 spins.
As seen in Fig. 2(d), compensation and critical temperatures of the particle increase gradually as the absolute value of Jint =Jsh increases. Contribution to the transition temperature of the particle comes
Fig. 6. Effect of the ferromagnetic core spin coupling Jc/Jsh on the temperature dependency of total magnetization M (left panel) of the system for Jint =Jsh ¼ 1:0 and D=Jsh ¼ 0:0 with Nc ¼ 113 and Nsh ¼ 153 113 spins. Right panel shows phase diagram in ðTc ,Tcomp Jc =Jsh Þ plane for the same parameters.
from core and shell sublattices of the particle, since the two sublattices have the same transition temperature values. Furthermore, we see that the system exhibits a compensation temperature
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Fig. 7. Effect of the crystal field D/Jsh on the temperature dependencies of (a) total magnetization M, (b) core and shell magnetizations Mc, Msh and (c) specific heat C of the system for Jint =Jsh ¼ 0:1 and Jc =Jsh ¼ 0:25 with Nc ¼ 113 and Nsh ¼ 153 113 spins. (d) Phase diagram of the system in ðTc ,Tcomp D=Jsh Þ plane for the same parameters.
for all values of Jint =Jsh . This type of behavior of compensation temperature has not been observed in Ref. [25] where the core layer of the particle is occupied by s ¼ 1=2 spins. Next, in Fig. 4, we represent the influence of the ferromagnetic exchange interaction Jc/Jsh on the thermal and magnetic properties of particle for some selected values of Jc/Jsh with Jint =Jsh ¼ 0:1 and D=Jsh ¼ 0:0. In Fig. 4(a), the total magnetization versus T=Jsh curves are plotted. As seen in this figure, for Jc =Jsh ¼ 0:3,0:35 and 0.40, the magnetization curves exhibit two successive zeros. Namely, the first one of them which emerges at lower temperatures corresponds to compensation point, and the second one occurs at the critical temperature of the system. On the other hand, for Jc =Jsh ¼ 0:45,0:50 and 0.55, total magnetization of the particle exhibits a cusp, instead of a minima and the compensation temperature disappears. In order to make a reasonable explanation to this behavior, we should investigate the variations of the core and shell magnetizations of the system with temperature which are depicted in Fig. 4(b) for selected values of Jc/Jsh with Jint =Jsh ¼ 0:1 and D=Jsh ¼ 0:0. In this figure, we see that an increasing trend in the value of Jc/Jsh parameter strengthens the ferromagnetic interaction between core spins, while up to a threshold value of Jc/Jsh, shell magnetizations are not affected from this circumstance. For the values of Jc =Jsh ¼ 0:30,0:35 and 0.40, magnetization of core sublattice reaches to a paramagnetic state before shell sublattice and in this case the system exhibits a compensation
temperature and also a constant critical temperature. At some threshold value, the two sublattice magnetizations have the same temperature dependence and absolute values of sublattice magnetizations do not cancel each other. Hence, the compensation temperature turns into a critical temperature. On the other hand, for Jc =Jsh ¼ 0:45,0:50 and 0.55, shell sublattice undergoes a phase transition from a ferromagnetic to a paramagnetic phase before core sublattice does. Therefore, critical temperature of the system increases for Jc =Jsh Z 0:45. The temperature dependence of the specific heat of the particle is plotted in Fig. 4(c) for some selected values of Jc/Jsh with Jint =Jsh ¼ 0:1 and D=Jsh ¼ 0:0. In Fig. 4(c), we observe that specific heat exhibits a hump at lower temperatures and a sharp peak at the transition temperature. Moreover, locations of these sharp peaks do not change up to a threshold value of Jc/Jsh, and then slides to the increasing temperature direction by turning into rounded peaks for the values of Jc/Jsh greater than the threshold value, which means that critical temperature of the system is increasing. We can see the effect of Jc/Jsh on the critical and compensation temperatures in Fig. 4(d) where the phase diagram of the system is plotted in ðTc ,Tcomp Jc =Jsh Þ plane. From Fig. 4(d) we can clearly see that as Jc/Jsh increases, critical temperature of the system remains constant ðTc ¼ 2:36Þ and compensation temperature increases linearly up to a threshold value of Jc/Jsh. If Jc/Jsh exceeds this threshold value then compensation temperature disappears and critical
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temperature of the system increases linearly. We also note that, when the antiferromagnetic interface coupling constant Jint =Jsh is weak (i.e. Jint =Jsh ¼ 0:1), critical temperature curve shown in Fig. 4(d) exhibits similar features with critical temperature of a semi-infinite ferromagnet [31]. Namely, horizontal line ðTc ¼ 2:36Þ in Fig. 4(d) is the transition temperature of shell sublattice, since the contribution of the particle core to the transition temperature is very weak. On the other hand, when Jc =Jsh 40:45, the effect of ferromagnetic core interactions becomes dominant, and hence, the linearly increasing part of Tc line originates from the transition temperature of the particle core (see Fig. 5). Furthermore, we can clearly see in Fig. 6 that, for large values of interface coupling such as Jint =Jsh ¼ 1:0, the transition temperature Tc of the particle does not exhibit any horizontal line, since both the core and shell sublattices of the particle have the same transition temperature values. Consequently, the results mentioned above are qualitatively in a good agreement with those of Ref. [9], where the same phenomenon has been obtained for the phase diagrams of a cylindrical Ising nanowire (or nanotube). In Fig. 7, we examine the effect of crystal field interaction D/Jsh on the thermal and magnetic properties of particle for some selected values of D/Jsh with Jint =Jsh ¼ 0:1 and Jc =Jsh ¼ 0:25. In Fig. 7(a), we show the temperature dependence of the total magnetization of the particle. We see that the magnetization curves exhibit two zeros indicating the presence of a compensation temperature on the system and as D/Jsh increases starting from 0.5, zeros of magnetization curves shift to high temperatures which means that both compensation points and critical temperatures of the particle increase. In Fig. 7(b), we see the effect of D/Jsh on the temperature dependence of core and shell magnetizations of the particle for several values of D/Jsh with Jint =Jsh ¼ 0:1 and Jc =Jsh ¼ 0:25. As seen in this figure, increasing values of D/Jsh cause the critical temperature of core and shell sublattices to increase. Furthermore, for all values of D/Jsh, as the temperature of the system increases, core magnetization of the particle decreases and accordingly, core sublattice of the particle reaches to a paramagnetic phase at a critical temperature, at which shell magnetization remains at a ferromagnetic state. At this point, we note that for sufficiently negative crystal fields ðD=Jsh r 0:72Þ, s spins which occupy the ferromagnetic core sublattice of the particle behaves like spin-1/2. Accordingly, there is no temperature value at which the two sublattice magnetizations cancel each other. Therefore, the system cannot exhibit a compensation temperature. Fig. 7(c) shows the influence of the crystal field interactions on the specific heat curves of the system with Jint =Jsh ¼ 0:1 and Jc =Jsh ¼ 0:25. As seen in Fig. 7(c), the two peaks observed in the specific heat curves tend to slide to higher temperatures with increasing D/Jsh values. The phase diagram of the system in ðTc ,Tcomp D=Jsh Þ plane (see Fig. 7(d)) shows the variation of critical and compensation temperatures of the particle with D/Jsh. As seen in this figure, values of compensation point and critical temperature regularly increases with increasing D/Jsh values. For all D/Jsh values, transition temperature of the particle comes from the transition temperature of shell layer, since core sublattice has a weak ferromagnetic interaction, in comparison to shell layer. It is also worth to note that for D=Jsh r0:72, no compensation point emerges in the system. In Fig. 8, we examine the size-dependent properties of the particle for Jint =Jsh ¼ 0:1, Jc =Jsh ¼ 0:25 and D=Jsh ¼ 0:0 with Nc ¼ 113 spins in the core layer and we select different numbers of shell layers (N). In Fig. 8(a), we depict the total magnetization of the particle as a function of the temperature with different shell thickness values. In Fig. 8(a), total magnetization curves decline to zero at two temperature values; one at a compensation temperature and the other at a critical temperature at which the system undergoes a phase transition from a ferrimagnetic phase to a paramagnetic phase. As seen in Fig. 8(a), it is hard to make a comment about dependence of
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compensation point of the particle on the shell thickness. However, it might be predicted that critical temperature of the system increases as N increases. As we represent in Fig. 8(b), this behavior of total magnetization curves originates from the temperature dependence of shell magnetization of the particle. In other words, increasing thickness of shell sublattice almost does not have any significant effect on the core magnetization, whereas increasing thickness of shell layers has an immediate influence on the temperature dependence of shell magnetization of the particle. However, it is not yet possible to make a comment about the reaction of compensation temperature of the particle to increasing N values. In Fig. 8(c), we plot the temperature dependence of the specific heat of the particle with some selected values of shell thickness. As seen in this figure, the first peak which has a rounded shape emerges at the transition temperature of the particle core and tends to disappear as the number of shell layers increases, whereas the second peak which has a relatively sharp shape slides to high temperatures as N increases which confirms that the critical temperature increases with increasing shell thickness values. In order to investigate the effect of shell thickness N on behavior of critical and compensation temperatures, we plot the phase diagram of the system in Fig. 8(d) in ðTc ,Tcomp NÞ plane for Jint =Jsh ¼ 0:1, Jc =Jsh ¼ 0:25 and D=Jsh ¼ 0:0 with a fixed number of Nc ¼ 113 spins in the particle core. We locate the compensation point by mapping the core magnetization curve to the absolute value of shell magnetization. In Ref. [25], the authors observed that as the shell thickness increases then both the critical and compensation temperatures of the system increase and reach a saturation value for high values of thickness for a particle which is composed of a spin-1/2 core surrounded by a spin-1 shell layer, whereas in Ref. [26], it has been reported that when the sites of shell sublattice are occupied by S¼3/2 spins, increasing the shell thickness causes a decrement in the values of both the critical and compensation temperatures for a spherical particle. However, our numerical results indicate that compensation temperature of the particle decreases rapidly as shell thickness increases and remains more or less unchanged at a saturation value for large N values. On the other hand, critical temperature which originates from the transition temperature of shell layer of the particle increases with increasing number of shell layers N, and reaches a saturation value for large N values. Hence, we can conclude that behavior of critical and compensation temperatures of a ferrimagnetic nanoparticle particle may exhibit different characteristics when the core layer is occupied by s 4 1 spins. At this point, we note that saturation values of critical and compensation temperatures are found as Tc ¼3.11 and Tcomp ¼1.289, respectively. As a final investigation, let us examine some ferrimagnetic properties of the total magnetization curves depicted throughout Figs. 2–8. In the bulk ferrimagnetism of Ne´el [32], it is possible to classify the thermal variation of the total magnetization curves in five categories, namely the type Q, the type P, the type N, the type L and the type M. As is specified in a recent paper by Strecˇka [33], conventional Q-type dependency shows a monotonic decrease of the magnetization with increasing temperature with a steep decrease of the magnetization just in the vicinity of critical temperature. P-type dependence shows the temperature-induced maximum as the temperature raises, whereas the N-type curve is being characterized by one compensation point at which resultant magnetization disappears due to the complete cancelation of the sublattice magnetization. The L-type curve is very analogous to the P-type dependence, however, the resultant magnetization starts from zero in this particular case. Similarly, the M-type dependence also begins from zero but it shows two separate maxima before reaching the critical temperature. According to this classification scheme, magnetization curves shown in Figs. 2 and 8(a) correspond to the type N in the bulk ferrimagnetism. On the other hand, as seen in Fig. 4(a), magnetization curves
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Fig. 8. Effect of the number of shell layers N on the temperature dependencies of (a) total magnetization M, (b) core and shell magnetizations Mc, Msh and (c) specific heat C of the system for Jint =Jsh ¼ 0:1, Jc =Jsh ¼ 0:25 and D=Jsh ¼ 0:0 with Nc ¼ 113 spins. (d) Phase diagram of the system in ðTc ,Tcomp NÞ plane for the same parameters.
exhibit a type N behavior for weak Jc/Jsh values, and as Jc/Jsh value increases then the magnetization curves turn into P-type curves. Similarly, magnetization curves depicted in Fig. 6 exhibit the N-type, the Q-type, and the P-type behaviors, respectively, with increasing Jc/Jsh values. The curves in Fig. 7(a) exhibit the N-type behavior for D=Jsh 4 0:72. More interestingly, if D=Jsh r 0:72, s spins which occupy the ferromagnetic core sublattice of the particle behaves like spin-1/2, and hence, no compensation point emerges. As a result of this behavior of core sublattice, total magnetization curve for D=Jsh ¼ 0:75 in Fig. 7(a) could not be classified by the five types of behaviors. The same novel phenomenon has been obtained in Ref. [9] for the magnetization curves of a cylindrical Ising nanowire (or nanotube).
4. Conclusions In this work, by employing Monte Carlo simulation method to the system, we have investigated the phase diagrams and thermal and magnetic properties of a ferrimagnetic nanoparticle which is defined on a simple cubic lattice with a ferromagnetic spin-3/2 core which is interacting antiferromagnetically with a ferromagnetic spin-1 shell layer. In particular, we have examined the temperature dependence of magnetization curves and specific
heat of the particle and we have focused our attention on the effect of exchange interactions and crystal field interaction, as well as thickness of the shell layer of the particle, on the behavior of critical and compensation temperatures. As shown in total magnetization curves throughout Figs. 2–8, some characteristic features, observed in bulk ferrimagnetism, namely the type Q, the type P and the type N behavior of magnetization curves obtained for the temperature dependence of magnetization curves. In addition, as shown in Fig. 7(a), a new type of thermal variation, namely not classified by the five types in bulk [32] has been found in the present system. We have also found some interesting properties which are also observed in semi-infinite ferromagnets [31]. For example, specific heat curves may exhibit a hump which is obviously related to the phase transition properties of the core sublattice of the particle. Moreover, our numerical results indicate that in contrast to the results obtained by previously published works, compensation temperature of the particle decreases rapidly as shell thickness increases and remains more or less unchanged at a saturation value for large N values. On the other hand, critical temperature increases with increasing number of shell layers N and reaches a saturation value for large N values. As a result, behavior of critical and compensation temperatures of a ferrimagnetic nanoparticle may exhibit different characteristics when the core layer is occupied by s 41 spins.
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Acknowledgments One of the authors (Y.Y.) would like to thank the Scientific and ¨ _ITAK) for partial Technological Research Council of Turkey (TUB ¨ financial support. This work has been completed at Dokuz Eylul University, Graduate School of Natural and Applied Sciences. The partial financial supports from SRF (Scientific Research Fund) of ¨ University (Grant no. 2009.KB.FEN.077) (H.P.), and Dokuz Eylul ¨ _ITAK (Grant no. 109T681) (E.A.) are also acknowledged. TUB References [1] T.Y. Kim, Y. Yamazaki, T. Hirano, Physica Status Solidi B 241 (2004) 1601. [2] R.H. Kodama, Journal of Magnetism and Magnetic Materials 200 (1999) 359. [3] Q.A. Pankhurst, J. Connolly, S.K. Jones, J. Dobson, Journal of Physics D: Applied Physics 36 (2003) R167. [4] A.H. Habib, C.L. Ondeck, P. Chaudhary, M.R. Bockstaller, M.E. McHenry, Journal of Applied Physics 103 (2008) 07A307. [5] N. Sounderya, Y. Zhang, Recent Patents on Biomedical Engineering 1 (2008) 34. [6] T. Kaneyoshi, Journal of Magnetism and Magnetic Materials 321 (2009) 3430. [7] T. Kaneyoshi, Physica Status Solidi B 242 (2005) 2938. [8] T. Kaneyoshi, Journal of Magnetism and Magnetic Materials 323 (2011) 2483. [9] T. Kaneyoshi, Physica Status Solidi B 248 (2011) 250. [10] D.A. Garanin, H. Kachkachi, Physical Review Letters 90 (2003) 65504. [11] H. Wang, Y. Zhou, E. Wang, D.L. Lin, Chinese Journal of Physics 39 (2001) 85. [12] H. Wang, Y. Zhou, D.L. Lin, C. Wang, Physica Status Solidi B 232 (2002) 254. [13] E. Eftaxias, K.N. Trohidou, Physical Review B 71 (2005) 134406.
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Iglesias, X. Batlle, A. Labarta, Physical Review B 72 (2005) 212401. [14] O. [15] D. Kechrakos, K.N. Trohidou, M. Vasilakaki, Journal of Magnetism and Magnetic Materials 316 (2007) e291. [16] Y. Hu, A. Du, Journal of Applied Physics 102 (2007) 113911. [17] K.N. Trohidou, M. Vasilakaki, L. Del Bianco, D. Fiorani, A.M. Testa, Journal of Magnetism and Magnetic Materials 316 (2007) e82. [18] M.H. Wu, Q.C. Li, J.M. Liu, Journal of Physics Condensed Matter 19 (2007) 186202. Iglesias, X. Battle, A. Labarta, Journal of Physics Condensed Matter [19] O. 19 (2007) 406232. Iglesias, X. Battle, A. Labarta, Journal of Nanoscience and Nanotechnology [20] O. 8 (2008) 2761. [21] M. Vasilakaki, E. Eftaxias, K.N. Trohidou, Physica Status Solidi A 205 (2008) 1865. [22] M. Vasilakaki, K.N. Trohidou, Physical Review B 79 (2009) 144402. [23] Y. Hu, A. Du, Physica Status Solidi B 246 (2009) 2384. [24] Y. Hu, L. Liu, A. Du, Physica Status Solidi B 247 (2010) 972. [25] A. Zaim, M. Kerouad, Y. El Amraoui, Journal of Magnetism and Magnetic Materials 321 (2009) 1083. [26] A. Zaim, M. Kerouad, Physica A 389 (2010) 3435. [27] L. Jiang, J. Zhang, Z. Chen, Q. Feng, Z. Huang, Physica B 405 (2010) 420. [28] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, Journal of Chemical Physics 21 (1953) 1087. [29] K. Binder, in: K. Binder (Ed.), Monte Carlo Methods in Statistical Physics, Springer, Berlin, 1979. [30] M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Statistical Physics, Clarendon Press, Oxford, 2001. [31] T. Kaneyoshi, Introduction to Surface Magnetism, CRC Press, Boca Raton, FL, 1991. [32] L. Ne´el, Annales de Physique 3 (1948) 137. [33] J. Strecˇka, Physica A 360 (2006) 379 and references therein.
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Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Thermal and magnetic properties of a ferrimagnetic nanoparticle with spin-3/2 core and spin-1 shell structure a,c ¨ Yusuf Yuksel , Ekrem Aydıner b, Hamza Polat a, Department of Physics, Dokuz Eyl¨ ul University, Tr-35160 _Izmir, Turkey Department of Physics, Istanbul University, 34134 Vezneciler Istanbul, Turkey c Dokuz Eyl¨ ul University, Graduate School of Natural and Applied Sciences, Turkey a
b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 June 2011 Received in revised form 7 July 2011 Available online 21 July 2011
Intensive Monte Carlo simulations based on standard Metropolis algorithm have been applied to investigate the phase diagrams of a ferrimagnetic cubic nanoparticle (nanocube) with a spin-3/2 core surrounded by a spin-1 shell layer with antiferromagnetic interface coupling. It has been shown that occupation of sites of particle core by spin-3/2 plays an important role on the shape of the phase diagrams. In addition, effects of the crystal field interaction as well as exchange interactions on the thermal and magnetic properties of the system have also been discussed in detail, and some interesting features have been observed for the temperature dependence of total magnetization curves of particle. & 2011 Elsevier B.V. All rights reserved.
Keywords: Compensation temperature Monte Carlo simulation Nanoparticle
1. Introduction Recently, magnetic nanoparticles attracted great theoretical and experimental interest, due to their technological [1,2] and biomedical [3–5] applications such as information storage devices and drug delivery in cancer thermotherapy. These fine particles serve as a bridge between bulk materials and atomic or molecular structures. Physical properties of a bulk material are independent from its size; however, nanoparticles often exhibit size-dependent properties. In addition, due to their good coercivity, they are also utilized for potential application in high density magnetic recording instead of conventional limited capacity thin film materials. From the theoretical point of view, these systems have been studied by a variety of techniques such as mean field theory (MFT) [6], effective field theory EFT [7–9], Green functions (GF) formalism [10], variational cumulant expansion (VCE) [11,12] and Monte Carlo (MC) simulations [13–27]. Monte Carlo simulation [28] is regarded as a powerful numerical approach for simulating the behavior of many complex systems, including magnetic nanoparticle systems. For instance, by utilizing this tool, Refs. [13–24] investigated the exchange bias effect in magnetic core/shell nanoparticles where the hysteresis loop exhibits a shift below the Nee´l temperature of the antiferromagnetic shell due to the exchange coupling at the interface region of ferromagnetic core and antiferromagnetic shell. In recent papers, the authors paid attention to the investigation of dependence on the core, shell and interface couplings, as
Corresponding author. Tel.: þ 90 2324128672; fax: þ 90 2324534188.
E-mail address: [email protected] (H. Polat). 0304-8853/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2011.07.011
well as particle size of critical and compensation temperatures of core/shell nanoparticles and many interesting results have been obtained. For example, Zaim et al. [25] studied the magnetic properties and hysteresis loops of a single nanocube with a ferromagnetic spin-1/2 core surrounded by a ferromagnetic spin-1 shell with antiferromagnetic interface coupling. They have applied MC simulations and examined the effects of shell coupling and interface coupling on both the compensation and magnetization profiles but they did not include any crystal field interaction term on the model Hamiltonian. Particularly, they have observed that as the shell thickness increases then both the critical and compensation temperatures of the system increase and reach a saturation value for high values of thickness. Similarly, Zaim and Kerouad [26] have simulated a spherical particle consisting of a ferromagnetic spin-1/2 core and a ferromagnetic spin-1 or 3/2 shell with antiferromagnetic interface coupling. They have focused on the effects of the shell and interface couplings on both the critical and compensation temperatures. They have found that there may be two compensation temperatures when the sites of shell sublattice are occupied by S¼3/2 spins and they observed that when the shell thickness increases, both the critical and compensation temperatures of the system decrease and reach their saturation values for high values of shell thickness. Similarly, Jiang et al. [27] have investigated the magnetic properties of the ferrimagnetic nanoparticles on a bodycentered cubic lattice based on a classical Heisenberg spin Hamiltonian in which S ¼3/2 and s ¼ 1 spins are distributed in the two interpenetrating square sublattices. They have made a detailed discussion on the conditions for the occurrence of a compensation temperature in the system. They have also
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observed that compensation temperature of system fluctuates as the particle size increases, and it becomes relatively independent of particle size when the particle size is increased further. More recently, Kaneyoshi [9] has studied magnetic properties of a cylindrical nanowire or nanotube by using EFT with correlations and investigated the effects of dilution at the surface, surface exchange interaction, and the shell coupling on the magnetization profiles, and found a number of characteristic phenomena. As shown in many of the previously published works mentioned above, critical and compensation temperatures of a ferrimagnetic nanoparticle exhibit a size-dependent behavior which is a characteristic property of these systems. On the other hand, a ferrimagnetic nanoparticle may also exhibit different phase diagrams depending on the occupation of core layer by s 41 spins. Therefore, the purpose of the present work is to investigate the phase diagrams of a ferrimagnetic nanoparticle with a spin-3/2 core surrounded by a spin-1 shell layer. For this aim, we organized the paper as follows: in Section 2, we briefly present our model. The results and discussions are presented in Section 3, and finally Section 4 contains our conclusions.
2. Model and simulation technique We consider a cubic ferrimagnetic nanoparticle composed of a spin-3/2 ferromagnetic core which is surrounded by a spin-1 ferromagnetic shell layer. At the interface, we define an antiferromagnetic interaction between core and shell spins (Fig. 1). Hamiltonian describing our model can be written as X X X X X H ¼ Jint si Sk Jc si sj Jsh Sk Sl Dc ðsi Þ2 Dsh ðSk Þ2 , ð1Þ /ikS
/ijS
/klS
i
k
where s ¼ 7 3=2, 7 1=2 and S¼ 71,0. Jint, Jc and Jsh define antiferromagnetic interface and ferromagnetic core and shell exchange interactions, respectively. Dc and Dsh represent single ion anisotropy terms of core and shell sublattices, respectively. / S denotes the nearest neighbor interactions on the lattice and we fixed the value of Jsh throughout the simulations. For simplicity, we also assume Dc ¼Dsh ¼D as a limited case. In order to simulate the system, we employ Metropolis Monte Carlo simulation algorithm [29] to Eq. (1) on an L L L cubic lattice with free boundary conditions. Configurations were generated by selecting the sites in sequence through the lattice and making single-spin-flip attempts, which were accepted or rejected according to the Metropolis algorithm. Data were generated over 50–100 realizations by using 25 000 Monte Carlo
Fig. 1. Schematic representation of core/shell nanoparticle on a simple cubic lattice with a spin-3/2 core and spin-1 shell.
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steps per site after discarding the first 2500 steps. Error bars were calculated by using the jackknife method [30]. Our program calculates the sublattice magnetizations Mc and Msh, the total magnetization M, magnetic susceptibility w and the specific heat C. These quantities are defined as Mc ¼
M¼
C¼
Nc 1 X s, Nc i ¼ 1 i
Mc þ Msh , 2 1 kT
2
Msh ¼
w¼
Nsh 1 X S , Nsh k ¼ 1 k
L3 ð/M 2 S/MS2 Þ, kT
ð/E2 S/ES2 Þ,
ð2Þ
ð3Þ
ð4Þ
where k is the Boltzmann constant (here k¼1) and Nc and Nsh denote the number of spins in core and shell layers, respectively. To determine the compensation temperature Tcomp from the computed magnetization data, the intersection point of the absolute values of the sublattice magnetizations was found using jMc ðTcomp Þj ¼ jMsh ðTcomp Þj,
ð5Þ
signðMc ðTcomp ÞÞ ¼ signðMsh ðTcomp ÞÞ,
ð6Þ
with Tcomp o Tc , where Tc is the critical temperature i.e. Ne´el temperature. Eqs. (5) and (6) indicate that the sign of the sublattice magnetizations is different, however, absolute values of them are equal to each other at the compensation point.
3. Results and discussion In this section, we examine the phase diagrams and temperature dependencies of thermal and magnetic properties of the system for some selected values of Hamiltonian parameters. However, in order to make a comparison with previously published works, we select the number of core and shell spins as Nc ¼ 113 and Nsh ¼ 153 113 (the same values as in Ref. [25]) throughout Figs. 2–7. Fig. 2 represents the effect of antiferromagnetic interface coupling between core and shell spins on the thermal and magnetic properties of the particle for some selected values of Jint/Jsh with Jc/Jsh ¼0.25 and D/Jsh ¼0.0. In Fig. 2(a) we plot the total magnetization versus reduced temperature T/Jsh. As seen in this figure, there are two zeros of magnetization curves for different Jint/Jsh values. 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 the temperature value at which magnetization depresses to zero, corresponding to critical temperature of the system. Besides, both the compensation point and critical temperature of the system increase as the absolute value of Jint/Jsh increases and the total magnetization curves exhibit an apparent minimum for weak Jint/Jsh values. On the other hand, Fig. 2(b) shows the variations of the core and shell magnetizations of the system with temperature for selected values of Jint/Jsh with Jc/Jsh ¼0.25 and D/Jsh ¼0.0. We can clearly see that as the temperature increases then the magnetizations of the particle core and shell approach to zero and vanish at the critical temperature. In addition, core and shell magnetization curves take values which are different from zero at the compensation point while, as seen in Fig. 2(a), the total magnetization value of the particle is zero at the compensation point for all values of Jint/Jsh. On the other hand, in Fig. 2(c), we plot the temperature dependence of the specific heat of the particle for some selected values of Jint/Jsh with Jc/Jsh ¼0.25 and D/Jsh ¼ 0.0. As seen in Fig. 2(c), as the temperature increases starting from zero, then the specific heat curves exhibit a hump at low temperatures and a sharp peak
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Fig. 2. Effect of the antiferromagnetic interface coupling Jint =Jsh on the temperature dependencies of (a) total magnetization M, (b) core and shell magnetizations Mc, Msh and (c) specific heat C of the system for Jc =Jsh ¼ 0:25 and D=Jsh ¼ 0:0 with Nc ¼ 113 and Nsh ¼ 153 113 spins. (d) Phase diagram of the system in ðTc ,Tcomp Jint =Jsh Þ plane for the same parameters.
Fig. 3. Temperature dependence of specific heat of the particle for Jc =Jsh ¼ 0:25 and D=Jsh ¼ 0:0 with some selected values of Jint =Jsh . The curves are calculated for Nc ¼ 113 and Nsh ¼ 153 113 spins.
occurs at a second-order phase transition temperature. Particularly, the shoulder which is observed in the specific heat of the system probably originates from the thermal fluctuations of the internal energy and magnetization of the particle core due to antiferromagnetic interface interaction Jint/Jsh, and according to our numerical calculations these phenomena are closely related to the transition temperature of particle core. Namely, even if Jint =Jsh ¼ 0:1 (i.e. positive), the shoulder is still obtained in the specific heat curves, since specific heat must be independent of the sign of Jint =Jsh . Besides, if we take avalue of Jint =Jsh ¼ 0:001, the height of the shoulder is increased and it takes much sharper form, like divergence. In addition, if we take other values larger than jJint =Jsh j 40:1, such as jJint =Jsh j ¼ 1:0, such a shoulder tends to disappear. Hence, whether the shoulder may be obtained or not in the specific heat is essentially dependent on the value of jJint =Jsh j. These observations are depicted in Fig. 3. Moreover, maxima of the specific heat curves slide to higher temperature values which confirms that the critical temperature increases as the absolute value of Jint =Jsh increases. In order to investigate the influence of Jint =Jsh on both critical and compensation temperatures of the particle, we plot the phase diagram of the system in a ðTc ,Tcomp Jint =Jsh Þ plane in Fig. 2(d).
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Fig. 4. Effect of the ferromagnetic core spin coupling Jc/Jsh on the temperature dependencies of (a) total magnetization M, (b) core and shell magnetizations Mc, Msh and (c) specific heat C of the system for Jint =Jsh ¼ 0:1 and D=Jsh ¼ 0:0 with Nc ¼ 113 and Nsh ¼ 153 113 spins. (d) Phase diagram of the system in ðTc ,Tcomp Jc =Jsh Þ plane for the same parameters.
Fig. 5. Temperature dependence of total magnetization M, core and shell magnetizations Mc, Msh, and magnetic susceptibility w of the particle for Jint =Jsh ¼ 0:1 and D=Jsh ¼ 0:0 with Jc =Jsh ¼ 0:3 (left panel) and Jc =Jsh ¼ 0:55 (right panel). The curves are calculated for Nc ¼ 113 and Nsh ¼ 153 113 spins.
As seen in Fig. 2(d), compensation and critical temperatures of the particle increase gradually as the absolute value of Jint =Jsh increases. Contribution to the transition temperature of the particle comes
Fig. 6. Effect of the ferromagnetic core spin coupling Jc/Jsh on the temperature dependency of total magnetization M (left panel) of the system for Jint =Jsh ¼ 1:0 and D=Jsh ¼ 0:0 with Nc ¼ 113 and Nsh ¼ 153 113 spins. Right panel shows phase diagram in ðTc ,Tcomp Jc =Jsh Þ plane for the same parameters.
from core and shell sublattices of the particle, since the two sublattices have the same transition temperature values. Furthermore, we see that the system exhibits a compensation temperature
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Fig. 7. Effect of the crystal field D/Jsh on the temperature dependencies of (a) total magnetization M, (b) core and shell magnetizations Mc, Msh and (c) specific heat C of the system for Jint =Jsh ¼ 0:1 and Jc =Jsh ¼ 0:25 with Nc ¼ 113 and Nsh ¼ 153 113 spins. (d) Phase diagram of the system in ðTc ,Tcomp D=Jsh Þ plane for the same parameters.
for all values of Jint =Jsh . This type of behavior of compensation temperature has not been observed in Ref. [25] where the core layer of the particle is occupied by s ¼ 1=2 spins. Next, in Fig. 4, we represent the influence of the ferromagnetic exchange interaction Jc/Jsh on the thermal and magnetic properties of particle for some selected values of Jc/Jsh with Jint =Jsh ¼ 0:1 and D=Jsh ¼ 0:0. In Fig. 4(a), the total magnetization versus T=Jsh curves are plotted. As seen in this figure, for Jc =Jsh ¼ 0:3,0:35 and 0.40, the magnetization curves exhibit two successive zeros. Namely, the first one of them which emerges at lower temperatures corresponds to compensation point, and the second one occurs at the critical temperature of the system. On the other hand, for Jc =Jsh ¼ 0:45,0:50 and 0.55, total magnetization of the particle exhibits a cusp, instead of a minima and the compensation temperature disappears. In order to make a reasonable explanation to this behavior, we should investigate the variations of the core and shell magnetizations of the system with temperature which are depicted in Fig. 4(b) for selected values of Jc/Jsh with Jint =Jsh ¼ 0:1 and D=Jsh ¼ 0:0. In this figure, we see that an increasing trend in the value of Jc/Jsh parameter strengthens the ferromagnetic interaction between core spins, while up to a threshold value of Jc/Jsh, shell magnetizations are not affected from this circumstance. For the values of Jc =Jsh ¼ 0:30,0:35 and 0.40, magnetization of core sublattice reaches to a paramagnetic state before shell sublattice and in this case the system exhibits a compensation
temperature and also a constant critical temperature. At some threshold value, the two sublattice magnetizations have the same temperature dependence and absolute values of sublattice magnetizations do not cancel each other. Hence, the compensation temperature turns into a critical temperature. On the other hand, for Jc =Jsh ¼ 0:45,0:50 and 0.55, shell sublattice undergoes a phase transition from a ferromagnetic to a paramagnetic phase before core sublattice does. Therefore, critical temperature of the system increases for Jc =Jsh Z 0:45. The temperature dependence of the specific heat of the particle is plotted in Fig. 4(c) for some selected values of Jc/Jsh with Jint =Jsh ¼ 0:1 and D=Jsh ¼ 0:0. In Fig. 4(c), we observe that specific heat exhibits a hump at lower temperatures and a sharp peak at the transition temperature. Moreover, locations of these sharp peaks do not change up to a threshold value of Jc/Jsh, and then slides to the increasing temperature direction by turning into rounded peaks for the values of Jc/Jsh greater than the threshold value, which means that critical temperature of the system is increasing. We can see the effect of Jc/Jsh on the critical and compensation temperatures in Fig. 4(d) where the phase diagram of the system is plotted in ðTc ,Tcomp Jc =Jsh Þ plane. From Fig. 4(d) we can clearly see that as Jc/Jsh increases, critical temperature of the system remains constant ðTc ¼ 2:36Þ and compensation temperature increases linearly up to a threshold value of Jc/Jsh. If Jc/Jsh exceeds this threshold value then compensation temperature disappears and critical
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temperature of the system increases linearly. We also note that, when the antiferromagnetic interface coupling constant Jint =Jsh is weak (i.e. Jint =Jsh ¼ 0:1), critical temperature curve shown in Fig. 4(d) exhibits similar features with critical temperature of a semi-infinite ferromagnet [31]. Namely, horizontal line ðTc ¼ 2:36Þ in Fig. 4(d) is the transition temperature of shell sublattice, since the contribution of the particle core to the transition temperature is very weak. On the other hand, when Jc =Jsh 40:45, the effect of ferromagnetic core interactions becomes dominant, and hence, the linearly increasing part of Tc line originates from the transition temperature of the particle core (see Fig. 5). Furthermore, we can clearly see in Fig. 6 that, for large values of interface coupling such as Jint =Jsh ¼ 1:0, the transition temperature Tc of the particle does not exhibit any horizontal line, since both the core and shell sublattices of the particle have the same transition temperature values. Consequently, the results mentioned above are qualitatively in a good agreement with those of Ref. [9], where the same phenomenon has been obtained for the phase diagrams of a cylindrical Ising nanowire (or nanotube). In Fig. 7, we examine the effect of crystal field interaction D/Jsh on the thermal and magnetic properties of particle for some selected values of D/Jsh with Jint =Jsh ¼ 0:1 and Jc =Jsh ¼ 0:25. In Fig. 7(a), we show the temperature dependence of the total magnetization of the particle. We see that the magnetization curves exhibit two zeros indicating the presence of a compensation temperature on the system and as D/Jsh increases starting from 0.5, zeros of magnetization curves shift to high temperatures which means that both compensation points and critical temperatures of the particle increase. In Fig. 7(b), we see the effect of D/Jsh on the temperature dependence of core and shell magnetizations of the particle for several values of D/Jsh with Jint =Jsh ¼ 0:1 and Jc =Jsh ¼ 0:25. As seen in this figure, increasing values of D/Jsh cause the critical temperature of core and shell sublattices to increase. Furthermore, for all values of D/Jsh, as the temperature of the system increases, core magnetization of the particle decreases and accordingly, core sublattice of the particle reaches to a paramagnetic phase at a critical temperature, at which shell magnetization remains at a ferromagnetic state. At this point, we note that for sufficiently negative crystal fields ðD=Jsh r 0:72Þ, s spins which occupy the ferromagnetic core sublattice of the particle behaves like spin-1/2. Accordingly, there is no temperature value at which the two sublattice magnetizations cancel each other. Therefore, the system cannot exhibit a compensation temperature. Fig. 7(c) shows the influence of the crystal field interactions on the specific heat curves of the system with Jint =Jsh ¼ 0:1 and Jc =Jsh ¼ 0:25. As seen in Fig. 7(c), the two peaks observed in the specific heat curves tend to slide to higher temperatures with increasing D/Jsh values. The phase diagram of the system in ðTc ,Tcomp D=Jsh Þ plane (see Fig. 7(d)) shows the variation of critical and compensation temperatures of the particle with D/Jsh. As seen in this figure, values of compensation point and critical temperature regularly increases with increasing D/Jsh values. For all D/Jsh values, transition temperature of the particle comes from the transition temperature of shell layer, since core sublattice has a weak ferromagnetic interaction, in comparison to shell layer. It is also worth to note that for D=Jsh r0:72, no compensation point emerges in the system. In Fig. 8, we examine the size-dependent properties of the particle for Jint =Jsh ¼ 0:1, Jc =Jsh ¼ 0:25 and D=Jsh ¼ 0:0 with Nc ¼ 113 spins in the core layer and we select different numbers of shell layers (N). In Fig. 8(a), we depict the total magnetization of the particle as a function of the temperature with different shell thickness values. In Fig. 8(a), total magnetization curves decline to zero at two temperature values; one at a compensation temperature and the other at a critical temperature at which the system undergoes a phase transition from a ferrimagnetic phase to a paramagnetic phase. As seen in Fig. 8(a), it is hard to make a comment about dependence of
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compensation point of the particle on the shell thickness. However, it might be predicted that critical temperature of the system increases as N increases. As we represent in Fig. 8(b), this behavior of total magnetization curves originates from the temperature dependence of shell magnetization of the particle. In other words, increasing thickness of shell sublattice almost does not have any significant effect on the core magnetization, whereas increasing thickness of shell layers has an immediate influence on the temperature dependence of shell magnetization of the particle. However, it is not yet possible to make a comment about the reaction of compensation temperature of the particle to increasing N values. In Fig. 8(c), we plot the temperature dependence of the specific heat of the particle with some selected values of shell thickness. As seen in this figure, the first peak which has a rounded shape emerges at the transition temperature of the particle core and tends to disappear as the number of shell layers increases, whereas the second peak which has a relatively sharp shape slides to high temperatures as N increases which confirms that the critical temperature increases with increasing shell thickness values. In order to investigate the effect of shell thickness N on behavior of critical and compensation temperatures, we plot the phase diagram of the system in Fig. 8(d) in ðTc ,Tcomp NÞ plane for Jint =Jsh ¼ 0:1, Jc =Jsh ¼ 0:25 and D=Jsh ¼ 0:0 with a fixed number of Nc ¼ 113 spins in the particle core. We locate the compensation point by mapping the core magnetization curve to the absolute value of shell magnetization. In Ref. [25], the authors observed that as the shell thickness increases then both the critical and compensation temperatures of the system increase and reach a saturation value for high values of thickness for a particle which is composed of a spin-1/2 core surrounded by a spin-1 shell layer, whereas in Ref. [26], it has been reported that when the sites of shell sublattice are occupied by S¼3/2 spins, increasing the shell thickness causes a decrement in the values of both the critical and compensation temperatures for a spherical particle. However, our numerical results indicate that compensation temperature of the particle decreases rapidly as shell thickness increases and remains more or less unchanged at a saturation value for large N values. On the other hand, critical temperature which originates from the transition temperature of shell layer of the particle increases with increasing number of shell layers N, and reaches a saturation value for large N values. Hence, we can conclude that behavior of critical and compensation temperatures of a ferrimagnetic nanoparticle particle may exhibit different characteristics when the core layer is occupied by s 4 1 spins. At this point, we note that saturation values of critical and compensation temperatures are found as Tc ¼3.11 and Tcomp ¼1.289, respectively. As a final investigation, let us examine some ferrimagnetic properties of the total magnetization curves depicted throughout Figs. 2–8. In the bulk ferrimagnetism of Ne´el [32], it is possible to classify the thermal variation of the total magnetization curves in five categories, namely the type Q, the type P, the type N, the type L and the type M. As is specified in a recent paper by Strecˇka [33], conventional Q-type dependency shows a monotonic decrease of the magnetization with increasing temperature with a steep decrease of the magnetization just in the vicinity of critical temperature. P-type dependence shows the temperature-induced maximum as the temperature raises, whereas the N-type curve is being characterized by one compensation point at which resultant magnetization disappears due to the complete cancelation of the sublattice magnetization. The L-type curve is very analogous to the P-type dependence, however, the resultant magnetization starts from zero in this particular case. Similarly, the M-type dependence also begins from zero but it shows two separate maxima before reaching the critical temperature. According to this classification scheme, magnetization curves shown in Figs. 2 and 8(a) correspond to the type N in the bulk ferrimagnetism. On the other hand, as seen in Fig. 4(a), magnetization curves
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Fig. 8. Effect of the number of shell layers N on the temperature dependencies of (a) total magnetization M, (b) core and shell magnetizations Mc, Msh and (c) specific heat C of the system for Jint =Jsh ¼ 0:1, Jc =Jsh ¼ 0:25 and D=Jsh ¼ 0:0 with Nc ¼ 113 spins. (d) Phase diagram of the system in ðTc ,Tcomp NÞ plane for the same parameters.
exhibit a type N behavior for weak Jc/Jsh values, and as Jc/Jsh value increases then the magnetization curves turn into P-type curves. Similarly, magnetization curves depicted in Fig. 6 exhibit the N-type, the Q-type, and the P-type behaviors, respectively, with increasing Jc/Jsh values. The curves in Fig. 7(a) exhibit the N-type behavior for D=Jsh 4 0:72. More interestingly, if D=Jsh r 0:72, s spins which occupy the ferromagnetic core sublattice of the particle behaves like spin-1/2, and hence, no compensation point emerges. As a result of this behavior of core sublattice, total magnetization curve for D=Jsh ¼ 0:75 in Fig. 7(a) could not be classified by the five types of behaviors. The same novel phenomenon has been obtained in Ref. [9] for the magnetization curves of a cylindrical Ising nanowire (or nanotube).
4. Conclusions In this work, by employing Monte Carlo simulation method to the system, we have investigated the phase diagrams and thermal and magnetic properties of a ferrimagnetic nanoparticle which is defined on a simple cubic lattice with a ferromagnetic spin-3/2 core which is interacting antiferromagnetically with a ferromagnetic spin-1 shell layer. In particular, we have examined the temperature dependence of magnetization curves and specific
heat of the particle and we have focused our attention on the effect of exchange interactions and crystal field interaction, as well as thickness of the shell layer of the particle, on the behavior of critical and compensation temperatures. As shown in total magnetization curves throughout Figs. 2–8, some characteristic features, observed in bulk ferrimagnetism, namely the type Q, the type P and the type N behavior of magnetization curves obtained for the temperature dependence of magnetization curves. In addition, as shown in Fig. 7(a), a new type of thermal variation, namely not classified by the five types in bulk [32] has been found in the present system. We have also found some interesting properties which are also observed in semi-infinite ferromagnets [31]. For example, specific heat curves may exhibit a hump which is obviously related to the phase transition properties of the core sublattice of the particle. Moreover, our numerical results indicate that in contrast to the results obtained by previously published works, compensation temperature of the particle decreases rapidly as shell thickness increases and remains more or less unchanged at a saturation value for large N values. On the other hand, critical temperature increases with increasing number of shell layers N and reaches a saturation value for large N values. As a result, behavior of critical and compensation temperatures of a ferrimagnetic nanoparticle may exhibit different characteristics when the core layer is occupied by s 41 spins.
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Acknowledgments One of the authors (Y.Y.) would like to thank the Scientific and ¨ _ITAK) for partial Technological Research Council of Turkey (TUB ¨ financial support. This work has been completed at Dokuz Eylul University, Graduate School of Natural and Applied Sciences. The partial financial supports from SRF (Scientific Research Fund) of ¨ University (Grant no. 2009.KB.FEN.077) (H.P.), and Dokuz Eylul ¨ _ITAK (Grant no. 109T681) (E.A.) are also acknowledged. TUB References [1] T.Y. Kim, Y. Yamazaki, T. Hirano, Physica Status Solidi B 241 (2004) 1601. [2] R.H. Kodama, Journal of Magnetism and Magnetic Materials 200 (1999) 359. [3] Q.A. Pankhurst, J. Connolly, S.K. Jones, J. Dobson, Journal of Physics D: Applied Physics 36 (2003) R167. [4] A.H. Habib, C.L. Ondeck, P. Chaudhary, M.R. Bockstaller, M.E. McHenry, Journal of Applied Physics 103 (2008) 07A307. [5] N. Sounderya, Y. Zhang, Recent Patents on Biomedical Engineering 1 (2008) 34. [6] T. Kaneyoshi, Journal of Magnetism and Magnetic Materials 321 (2009) 3430. [7] T. Kaneyoshi, Physica Status Solidi B 242 (2005) 2938. [8] T. Kaneyoshi, Journal of Magnetism and Magnetic Materials 323 (2011) 2483. [9] T. Kaneyoshi, Physica Status Solidi B 248 (2011) 250. [10] D.A. Garanin, H. Kachkachi, Physical Review Letters 90 (2003) 65504. [11] H. Wang, Y. Zhou, E. Wang, D.L. Lin, Chinese Journal of Physics 39 (2001) 85. [12] H. Wang, Y. Zhou, D.L. Lin, C. Wang, Physica Status Solidi B 232 (2002) 254. [13] E. Eftaxias, K.N. Trohidou, Physical Review B 71 (2005) 134406.
3175
Iglesias, X. Batlle, A. Labarta, Physical Review B 72 (2005) 212401. [14] O. [15] D. Kechrakos, K.N. Trohidou, M. Vasilakaki, Journal of Magnetism and Magnetic Materials 316 (2007) e291. [16] Y. Hu, A. Du, Journal of Applied Physics 102 (2007) 113911. [17] K.N. Trohidou, M. Vasilakaki, L. Del Bianco, D. Fiorani, A.M. Testa, Journal of Magnetism and Magnetic Materials 316 (2007) e82. [18] M.H. Wu, Q.C. Li, J.M. Liu, Journal of Physics Condensed Matter 19 (2007) 186202. Iglesias, X. Battle, A. Labarta, Journal of Physics Condensed Matter [19] O. 19 (2007) 406232. Iglesias, X. Battle, A. Labarta, Journal of Nanoscience and Nanotechnology [20] O. 8 (2008) 2761. [21] M. Vasilakaki, E. Eftaxias, K.N. Trohidou, Physica Status Solidi A 205 (2008) 1865. [22] M. Vasilakaki, K.N. Trohidou, Physical Review B 79 (2009) 144402. [23] Y. Hu, A. Du, Physica Status Solidi B 246 (2009) 2384. [24] Y. Hu, L. Liu, A. Du, Physica Status Solidi B 247 (2010) 972. [25] A. Zaim, M. Kerouad, Y. El Amraoui, Journal of Magnetism and Magnetic Materials 321 (2009) 1083. [26] A. Zaim, M. Kerouad, Physica A 389 (2010) 3435. [27] L. Jiang, J. Zhang, Z. Chen, Q. Feng, Z. Huang, Physica B 405 (2010) 420. [28] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, Journal of Chemical Physics 21 (1953) 1087. [29] K. Binder, in: K. Binder (Ed.), Monte Carlo Methods in Statistical Physics, Springer, Berlin, 1979. [30] M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Statistical Physics, Clarendon Press, Oxford, 2001. [31] T. Kaneyoshi, Introduction to Surface Magnetism, CRC Press, Boca Raton, FL, 1991. [32] L. Ne´el, Annales de Physique 3 (1948) 137. [33] J. Strecˇka, Physica A 360 (2006) 379 and references therein.