Monte Carlo study of a mixed spin (1, 3/2) ferrimagnetic nanowire with core/shell morphology

Monte Carlo study of a mixed spin (1, 3/2) ferrimagnetic nanowirewith core/shell morphology

A. Feraoun, A. Zaim n, M. KerouadQ1

Laboratoire Physique des Matériaux et Modélisation des Systèmes (LP2MS), Unité Associée au CNRST-URAC: 08, Faculty of Sciences, University Moulay Ismail,B.P. 11201, Zitoune, Meknes, Morocco

a r t i c l e i n f o

Article history:Received 22 December 2013Received in revised form22 February 2014Accepted 25 March 2014

Keywords:Ising modelMonte Carlo simulationCompensation temperatureCritical temperature

a b s t r a c t

In this work, Monte Carlo simulation based on Metropolis algorithm was used to study the magneticbehavior of a ferrimagnetic nanowire on a hexagonal lattice with a spin-3/2 core surrounded by a spin-1shell layer with antiferromagnetic interface coupling in the presence of the crystal field interactions. Theinfluences of the crystal field interactions, the interfacial and core couplings on the critical andcompensation behaviors of the nanowire, are investigated. The results present rich critical behavior,which includes the first-and second-order phase transitions, the tricritical and critical end points. Inaddition, the compensation points can appear for appropriate values of the system parameters.

& 2014 Published by Elsevier B.V.

1. Introduction

Nowadays, growing interest is continuously directed towardsthe magnetic properties of nanomaterials. Among these nanoma-terials, core/shell nanowires and nanotubes such as ZnO [1], FePt,and Fe3O4 [2] are of considerable interest for both theoretical andexperimental studies because of their potential technologicalapplication such as ultrahigh-density recording, biology and med-icine [3,4]. From the theoretical point of view, these systems havebeen studied by a wide variety of techniques such as mean fieldtheory (MFT) [5,6], effective field theory (EFT) [7,8], Green func-tions formalism (GF) [9], variational cumulant expansion (VCE)[10,11], and Monte Carlo simulations (MCS) [12–17]. Some studiesshow that we can find in these systems a very rich critical behaviorand many interesting phenomena. Recently, Magoussi et al. [18]have investigated the effect of the trimodal longitudinal field onthe critical behavior of a spin-1 nanotube. It has been found thatthe system exhibits tricritical point and reentrant or doublereentrant phenomenon. Zaim et al. [19] have investigated themagnetic properties of a cubic Ising nanocube which consists of aferromagnetic spin-1/2 core and a ferromagnetic spin-1 shellcoupled with an antiferromagnetic interlayer coupling JInt to thecore by the use of Monte Carlo method. A number of characteristicphenomena are found, in particular, compensation temperaturemay occurs in this system.

The occurrence of a compensation point is of great technolo-gical importance, since at this point only a small driving field isrequired to change the sign of the resultant magnetization. Thisproperty is very useful in thermomagnetic recording. Canko et al.[20] have investigated the crystal field dependence of the mag-netic properties of the mixed spin-(1/2,1) Ising nanotube using theEFT. Some characteristic phenomena are found such as thetricritical points. Akıncı et al. [21,22] have studied the effects ofa randomly distributed magnetic field on the phase diagrams ofIsing nanowires and the dynamic behavior of a site diluted Isingferromagnet in the presence of a periodically oscillating magneticfield using EFT Q2. It has been shown that the system exhibitsreentrant phenomena, as well as a dynamic tricritical point whichdisappears for sufficiently weak dilution.

In recent years, much attention has been paid to the study ofthe mixed spin nanoparticles. The phase diagrams of a ferrimag-netic cubic nanoparticle with spin-3/2 core and spin-1 shellstructure have been investigated in Refs. [23,24]. It was observedthat the occupation of the sites of the particle core by the spin-3/2plays an important role on the shape of the phase diagrams. Otherwork studied by using Monte Carlo simulation [25], the dynamicphase transition properties of a single spherical ferrimagneticcore–shell nanoparticle. It has been found that the dynamic phaseboundaries depend strongly on the Hamiltonian parameters suchas the high amplitude and the period of the external field. InRefs. [26,28], the magnetic properties of an hexagonal nanowire,cubic nanowire and multisublattice cubic nanowire with mixedspin (1, 3/2) have been examined by the EFT. Two compensationpoints can exist for certain values of the system parameters. Zaimet al. [29] have studied the magnetic behavior of a mixed spin (1, 3/2)

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100

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journal homepage: www.elsevier.com/locate/physb

Physica B

http://dx.doi.org/10.1016/j.physb.2014.03.0710921-4526/& 2014 Published by Elsevier B.V.

n Corresponding author.E-mail addresses: [email protected] (A. Zaim),

[email protected] (M. Kerouad).

Please cite this article as: A. Feraoun, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.03.071i

Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

www.sciencedirect.com/science/journal/09214526

www.elsevier.com/locate/physb

http://dx.doi.org/10.1016/j.physb.2014.03.071



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ferrimagnetic spherical nanoparticle using Monte Carlo simula-tions. The results present rich critical behavior, which includes thefirst-and second-order phase transitions, thus also the tricriticaland critical end points. In Ref. [30] the authors have used theeffective field theory to study the hysteresis loops of a mixed spin(1, 3/2) cubic nanowire in the presence of the crystal field and thetransverse field. The triple, pentamerous and heptamerous hyster-esis loops have been observed at low temperature. In other recentwork, Jiang et al. [31] have investigated the compensation beha-vior, and magnetic properties of a ferrimagnetic nanotube, whichincludes ferromagnetic spin-3/2 inner shell and spin-1 outer layerwith the ferrimagnetic interlayer coupling. Two compen-sation points have been found for certain values of the systemparameters.

Despite these studies, as far as we know, the phase diagramsand the compensation behavior of a mixed spin (1, 3/2) hexagonalIsing nanowire have not been investigated. Therefore, in this

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Fig. 1. Schematic representation of core/shell nanowire (two and three dimen-sional) with a spin-3/2 core and a spin-1 shell.

0 1 2 3 4 5-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

T/Jsh

MT

Jc /Jsh =0.2D/Jsh =0.5

J Int /Jsh=-0.2 -0.6-1.0

0 1 2 3 4 5

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6 JInt/Jsh =-0.2 -0.6-1.0

M

T/Jsh

Jc /Jsh =0.2D/Jsh =0.5

core

shell

0 1 2 3 4 50

10

20

30

40

50 JInt /Jsh=-0.2 -0.6-1.0

Jc/Jsh =0.2D/Jsh=0.5

Tc=3.53

Tc=3.24

Tc=2.9

χ T

T/Jsh

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

T

JInt /Jsh

Tcomp; D/Jsh=-0.5 Tc; D/Jsh =-0.5 Tcomp; D/Jsh=0.5 Tc; D/Jsh=0.5

Jc /Jsh =0.2

Fig. 2. The temperature dependencies of (a) total magnetizations, (b) core and shell magnetizations Mc, Msh and (c) total susceptibility for Jc=Jsh¼ 0:2 and D=Jsh ¼ 0:5 and fordifferent values of JInt=Jsh , (d) phase diagram of the system in ðT ; JInt=JshÞ plan for Jc=Jsh¼ 0:2, D=Jsh ¼ 0:5, and D=Jsh ¼ �0:5.

A. Feraoun et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎2

paper, the effects of the core/shell exchange interfacial coupling,the exchange interaction coupling in the core, and the crystal fieldon the phase diagrams and the compensation behavior of themixed spin (1, 3/2) are discussed within the framework of theMonte Carlo simulations.

The purpose of the present work is, by using Monte Carlosimulations, to investigate the phase diagrams of a ferrimagneticnanowire with a spin-s¼ 3=2 core surrounded by a spin-S¼ 1shell layer. The outline of the paper is organized as follows: inSection 2, we briefly present our model and the related formula-tion. The results and discussions are presented in Section 3, andfinally Section 4 is devoted to our conclusions.

2. Model and formalism

We consider a ferrimagnetic nanowire composed of spin-3/2ferromagnetic core which is surrounded by a spin-1 ferromagneticshell coupled with ferrimagnetic interface coupling. As we can seein Fig. 1, the core of the system consists of four layers surroundedby two layers of the shell.

The Hamiltonian describing our model can be written as

H¼ � Jsh∑⟨ij⟩SiSj� Jc∑

⟨kl⟩sksl� JInt∑

⟨ik⟩Sisk�Dsh∑

iS2i �Dc∑

ks2k ; ð1Þ

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0 1 2 3 4

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

JInt /Jsh=-0.1D/Jsh=0.0

MT

T/Jsh

Jc /Jsh =0.20.30.4

0 1 2 3 4

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6 JInt /Jsh=-0.1D/Jsh =0.0

T/Jsh

Jc/Jsh=0.20.30.4

core

shell

M

0 1 2 3 40

10

20

30

40

50

60

70

80

90

Tc=3.0

T/Jsh

Jc /Jsh=0.20.30.4


Tc=2.85

T

0.1 0.2 0.3 0.4 0.5 0.60.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

T

J c /Jsh

Tcomp

Tc


Jc/Jsh =0.38

Fig. 3. The temperature dependencies of (a) total magnetizations, (b) core and shell magnetizations Mc, Msh and (c) total susceptibility for JInt=Jsh ¼ �0:1 and D=Jsh ¼ 0:0 andfor different values of Jc=Jsh , (d) phase diagram of the system in ðT ; Jc=JshÞ plan for JInt=Jsh¼ �0:1 and D=Jsh ¼ 0:0.

A. Feraoun et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3





where Si ¼ 71;0 and sk ¼ 73=2; 71=2 denote the usual Isingvariables. JInt, Jc and Jsh define interface, core and shell exchangeinteractions, respectively. The summation index ⟨ij⟩, ⟨kl⟩ and ⟨ik⟩denote a summation over all pairs of neighboring spins S�S, s�s,and S�s respectively. Dc and Dsh represent single ion anisotropyterms of the core and the shell, respectively. For simplicity, we alsoassume Dc¼Dsh¼D. Using Monte Carlo simulation based onMetropolis [32] algorithm, we apply periodic boundary conditionsin the z-direction, free boundary conditions are applied in thex- and y-directions. Data were generated over 20–40 realizationsby using 30 000 Monte Carlo steps per site after discarding thefirst 10 000 steps. Our program calculates the following para-meters, namely, the sublattice magnetizations per site defined by

Msh ¼1

NshL∑iðSiÞ

* +; ð2Þ

and

Mc ¼1NcL

∑iðsiÞ

* +: ð3Þ

The total magnetization per site is

MT ¼ðNc �McÞþðNsh �MshÞ

NcþNsh: ð4Þ

The magnetic susceptibility of the nanowire

χT ¼ β � Nð⟨M2T ⟩� ⟨MT ⟩

2Þ; ð5Þ

and we have also calculated the internal energy per site

E¼ 1ðNcþNshÞL

⟨H⟩; ð6Þ

where β¼ 1=kBT , T is the absolute temperature and kB is theBoltzmann factor (here kB¼1). Nc and Nsh denote the number ofspins in the core and the shell, respectively. L ðL¼ 250Þ denote thelength of the nanowire. A number of additional simulations wereperformed for L¼200 and L¼300, but no significant differenceswere found from the results presented here.

To determine the compensation temperature Tcomp from thecomputed magnetization data, the intersection point of the abso-lute value of the core and shell magnetizations was found usingthe following equations:

∣Nc �McðTcompÞ∣¼ ∣Nsh �MshðTcomp∣; ð7Þ

signðNc �McðTcompÞÞ ¼ �signðNsh �MshðTcompÞÞ; ð8Þwith TcompoTc , Tc is the critical temperature i.e. Néel temperature[33]. Thus, the compensation temperature is the temperaturewhere the resultant magnetization vanishes below the criticalone. Eqs. (7) and (8) indicate that the sign of the sublatticemagnetizations is different, however, absolute values of them areequal to each other at the compensation point. The second-orderphase transitions are determined from the maxima of the suscept-ibility curves and the first-order phase transitions are obtainedby locating the discontinuities of the magnetization and internalenergy curves.

3. Results and discussions

In this section, we have firstly studied the magnetizationbehaviors of the system for some selected values of the Hamilto-nian parameters.

Fig. 2, represents the effects of the thermal variations of themagnetic properties of the nanowire for Jc=Jsh ¼ 0:2, D=Jsh ¼ 0:5and selected values of JInt=Jsh (JInt=Jsh ¼ �0:2, �0:4, and �1:0).In Fig. 2(a) we plot the total magnetization versus reducedtemperature T=Jsh. As seen in this figure, there are two zeros ofmagnetization curves for different JInt=Jsh values. The first oneindicates the temperature value at which the total magnetizationof the nanowire reduces to zero, it corresponds to the compensa-tion temperature whereas the sublattice ones Msh and Mc aredifferent from zero. The second zero denotes the temperaturevalue at which the magnetizations MT, Msh and Mc depress to zero,it corresponds to the critical temperature of the system. Besides,both compensation and critical temperatures of the systemincrease as the absolute value of JInt=Jsh increases and the total

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-5 -4 -3 -2 -1 0 1 2 3 40.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

T

D/Jsh

Tcomp

Tc(SO)

Tc(FO)

Jc/Jsh=0.2

JInt /Jsh=-0.1

-5 -4 -3 -2 -1 0 1 2 3 40

1

2

3

4

5

6

Tc(SO)

Tcomp

Tc(FO)

Jc/Jsh=0.6

JInt /Jsh =-0.1

T

D/Jsh

Fig. 4. The phase diagrams of the system in the T ;D=Jsh plane for JInt=Jsh¼ �0:1 and for Jc=Jsh ¼ 0:2 (a) and for Jc=Jsh ¼ 0:6 (b).






magnetization curves exhibit an apparent minimum. Fig. 2(b)shows the variation of the core and shell magnetizations of thesystem with temperature for the same selected values as in Fig. 2(a).We can clearly observe that as the temperature increases, themagnetizations of the nanowire core and shell approach to zeroand vanish at the critical temperature. Fig. 2(c) shows the tempera-ture dependence of the total susceptibility χT for the same selectedvalues as in Fig. 2(a). It is clear that the susceptibility–temperature χTcurve presents a peak at Tc and as increasing the absolute value ofJInt=Jsh, the position of the peak shifts to higher temperature. Thismeans that the critical temperature is enhanced. In order toinvestigate the influence of JInt=Jsh on both critical and compensationtemperatures of the nanowire, we plot the phase diagram of the

system in ðT ; JInt=JshÞ plane (Fig. 2(d)). As it is seen, as the absolutevalue of JInt=Jsh increases, compensation temperature Tcomp of thenanowire increases to reach a saturation value which depends onD=Jsh and the critical temperature Tc of the system increases linearly.In addition, Tcomp and Tc increase with D=Jsh.

Next, in Fig. 3, we represent the influence of the ferromagneticexchange interaction Jc=Jsh on the thermal and magnetic proper-ties of the nanowire for some selected values of Jc=Jsh (0.2, 0.3, and0:4Þ with JInt=Jsh ¼ �0:1 and D=Jsh ¼ 0:0. In Fig. 3(a), the totalmagnetization versus T=Jsh curves are plotted. As it is seen in thisfigure, for Jc=Jsh ¼ 0:2 and 0.3, the magnetization curves exhibittwo successive zeros. Namely, the first one which emerges atlower temperatures corresponds to the compensation point, and

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-5 -4 -3 -2 -1 0 1

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Mc

Msh

T=1.6Jc/Jsh=0.2JInt /Jsh=-0.1

M

D/Jsh

-5 -4 -3 -2 -1 0 10.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6 Mc

Msh

T=0.6Jc /Jsh=0.2JInt /Jsh=-0.1

M

D/Jsh

compensationpoint

-5 -4 -3 -2 -1 0 1

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

T=0.15Jc/Jsh =0.2JInt /Jsh =-0.1

M

D/Jsh

Mc

Msh

Fig. 5. The variation of the nanowire magnetizations with D=Jsh for the same parameters as in Fig. 4(a) and for selected values of the temperature in (a) ðT=Jsh ¼ 1:6Þ, (b)ðT=Jsh ¼ 0:6Þ and (c) ðT=Jsh ¼ 0:15Þ.






the second one occurs at the critical temperature of the system. Onthe other hand, for Jc=Jsh ¼ 0:4, the total magnetization of thenanowire exhibits one zero at the critical temperature of thesystem. In order to make a reasonable explanation of this behavior,we should investigate the variations of the core and shell magne-tizations of the system with temperature which are depicted inFig. 3(b) for selected values of Jc=Jsh with JInt=Jsh ¼ �0:1 andD=Jsh ¼ 0:0. This figure shows that as the temperature increasesthe magnetizations of the two sublattices approach to zero andvanish at the critical temperature. In addition, shell magnetiza-tions are less affected by this circumstance. The temperaturedependence of the susceptibility of the nanowire is plotted in

Fig. 3(c) for some selected values of Jc=Jsh with JInt=Jsh ¼ �0:1 andD=Jsh ¼ 0:0. In this figure, we observe that the susceptibilityexhibits a peak at the transition temperature. Moreover, locationof these peak is the same for Jc=Jsh ¼ 0:2 and 0.3. To examine theeffect of Jc=Jsh on the critical and compensation temperature, weplot in Fig. 3(d) the phase diagram of the system in ðT ; Jc=JshÞ plane.It is clearly seen that as Jc=Jsh increases, the critical temperature ofthe system remains constant (Tc¼2.85), and the compensation oneincreases linearly up to a threshold value of Jc=Jsh. If Jc=Jsh exceedsthis threshold value the compensation temperature disappearsand the critical one increases linearly. On the other hand, when,Jc=Jsh40:38, the effect of the ferromagnetic core interactions

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-5 -4 -3 -2 -1 0 1

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

T/Jsh=1.6Jc /Jsh=0.2JInt /Jsh=-0.1

E

D/Jsh

-5 -4 -3 -2 -1 0 1-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

T/Jsh=0.6Jc /Jsh=0.2JInt /Jsh =-0.1

E

D/Jsh

-5 -4 -3 -2 -1 0 1-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

T/Jsh=0.15Jc /Jsh=0.2JInt /Jsh=-0.1

E

D/Jsh

Fig. 6. The variation of the internal energy with D=Jsh for the same parameters as in Fig. 5.






becomes dominant, and hence, the linearly increasing part of Tcline originates from the transition temperature of the nanowirecore. These results are similar to those obtained in the case of thesemi-infinite systems [34].

Fig. 4 presents the variations of the critical and compensationtemperatures Tc and Tcomp, respectively, with the crystal field D=Jshfor JInt=Jsh ¼ �0:1 and for Jc=Jsh ¼ 0:2 (Fig. 4(a)) and Jc=Jsh ¼ 0:6(Fig. 4(b)). It is observed that the phase diagram is very rich; itexhibits second-order and first-order phase transition tempera-tures, tricritical point ð■Þ, critical end point ð□Þ, isolated criticalpoint ð○Þ , and compensation temperature. The phase diagramsshow a tricritical point where the second-order phase transitionline turns to a first-order one. In addition, at low temperatureregion and for a certain range of D=Jsh, this line of first-order isreminiscent of separating the transition from the ferri (71=2; 71)region to the ferri (71=2;0) region. The system exhibits an otherfirst-order transition line between the phases (73=2; 71) and(71=2;0) which extends from D=Jsh ¼ �0:77 for Fig. 4(a) andD=Jsh ¼ �2:25 for Fig. 4(b) at zero temperature and terminates byan isolated critical point ðT=JshC0:25;D=JshC�0:77Þ for Fig. 4(a)and ðT=JshC0:64;D=JshC�2:25Þ for Fig. 4(b). Beyond this pointa continuous passage occur. Another line of second-order phasetransition appears in negative region of D=Jsh with a constantcritical temperature (Tc¼0.26), this line terminates a critical endpoint located at ðT=JshC0:26;D=JshC�3:7Þ for Fig. 4(a) andðT=JshC0:8;D=JshC�3:13Þ for Fig. 4(b) and separates the para-magnetic phase from the ferrimagnetic one. Notice that theisolated critical point is not found in Ref. [27]. We can remarkthat some results obtained in this paper such as first-order phasetransition, tricritical point and critical end point have beenobserved in various systems [35–38]. Concerning the compensa-tion behavior, it is found that the system exhibits a compensationpoint in a certain range of D=Jsh. The range of D=Jsh where we havea compensation behavior decreases when we increase Jc=Jsh.In order to complete the discussion of the above phase diagrams,we plot in Fig. 5(a)–(c) the magnetizations as a function of D=Jsh,for JInt=Jsh ¼ �0:1, Jc=Jsh ¼ 0:2 and for several values of the tem-perature (T=Jsh ¼ 1:6;0:6;0:15). In the case of T=Jsh ¼ 1:6 (Fig. 5(a)),the system presents a continuous passage between the ferrimag-netic phase and the paramagnetic one at D=Jsh ¼ �2:8. In Fig. 5(b)(T=Jsh ¼ 0:6), it is observed that the compensation point appearsat D=Jsh ¼ �0:8 when the core and the shell magnetizations areequal. Since the core and shell magnetizations have opposite signsat this value. It is also clear that the system exhibits a discontinuityof the core and the shell magnetizations at a first-order point(D=Jsh ¼ �3:2) between the ferrimagnetic phase and the para-magnetic one. In Fig. 5(c) for T=Jsh ¼ 0:15, we found two disconti-nuities in the magnetization curves between three ferrimagneticphases, the first one from (3=2; �1) to (1=2; �1) at D=Jsh ¼ �0:74and the second one from (1=2; �1) to (1=2;0) at D=Jsh ¼ �3:95.On the other hand, in order to confirm the discussion above, werepresent in Fig. 6(a), Fig. 6(b) and Fig. 6(c) the internal energy Eversus D=Jsh for the same system parameters as in Fig. 5(a), Fig. 5(b)and Fig. 5(c), respectively. In the case of T=Jsh ¼ 1:6 (Fig. 6(a)),we can see that the system presents a continuous passage.When T=Jsh ¼ 0:6 (Fig. 6(b)), it is clear that the system exhibitsa discontinuity at D=Jsh ¼ �3:17. In Fig. 6(c) for T=Jsh ¼ 0:15the system presents two discontinuities at D=Jsh ¼ �0:74 andD=Jsh ¼ �3:95.

4. Conclusions

Using Monte Carlo simulations with Metropolis algorithm, wehave investigated the phase diagrams, thermal and magnetic

properties of a ferrimagnetic nanowire with a ferromagneticspin-3/2 core which is coupled antiferromagnetically witha ferromagnetic spin-1 shell layer. We have investigated the effectsof the crystal field interactions, the interfacial and core couplingson the critical and compensation behaviors of the nanowire. Theresults are present in particular rich varieties of phase diagramssuch as first-order and second-order transitions. It is observed thattwo line of first-order transitions occur in this system, each ofthese lines separating two distinct ferrimagnetic phases and one ofthose terminating a isolated critical point. It is also found that thecompensation points can appear when D=Jsh becomes larger thana critical value ðD=JshÞc . The compensation points [39] of theferrimagnetic materials have potential use in magnetic recordingmaterials. We hope that the results on phase diagrams andmagnetic properties of this nanowire studied may provide somepotential useful information to the experimental nanomaterials inthe future.

Acknowledgement

This work has been initiated with the support of URAC: 08, theProject RS: 02 (CNRST) and the swedish research Links Programdnr-348-2011-7264.

References

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