Monte Carlo simulation of the magnetic properties of a spin-1Blume–Capel nanowire

H. Magoussi, A. Zaim n, M. KerouadLaboratoire de Physique des Matériaux et Modélisation des Systèmes (LP2MS), Unité Associée au CNRST-URAC: 08, University of Moulay Ismail,Faculté des Sciences, B.P. 11201 Zitoune, Meknes, Morocco

a r t i c l e i n f o

Article history:Received 7 March 2014Received in revised form24 July 2014Accepted 2 September 2014by C. LacroixAvailable online 26 September 2014

Keywords:B. Ising modelC. NanowireD. Hysteresis behaviorE. Monte Carlo simulation

a b s t r a c t

Monte Carlo simulation has been used to study the magnetic properties and hysteresis loops of a Blume–Capel nanowire, consisting of a ferromagnetic core of spin-1 atoms surrounded by a ferromagnetic shellof spin-1 atoms with ferromagnetic or anti-ferromagnetic interfacial coupling. We have examined theinfluence of the crystal field, the temperature, and the interfacial coupling on the hysteresis behavior,susceptibility, specific heat and internal energy. The remanent magnetization and the coercive field as afunction of the temperature have also been investigated. We have found that the system exhibits the firstorder phase transition, one or double hysteresis loops in the ferromagnetic case, and one or triplehysteresis loops in the ferrimagnetic case.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Materials with a nanostructure, such as nanoparticles, nanotubeand nanowires, have attracted considerable attention because of notonly their academic interest, but also their technological [1,2] andbiomedical [3–5] applications, namely in the areas of magneticrecording media, spin electronics, optics, sensors and thermoelec-tronics devices [6]. In the experimental area, the magnetizationof certain nanomaterials such as γ-Fe2O3 nanoparticles has beenmeasured [7]. The magnetic nanowires have been studied, and theirmagnetic properties have been investigated, especially Fe–Co [8],Co–P [9], Ni [10], Ga1xCuxN [11], etc. The magnetic nanowires andnanotubes such as ZnO [12], FePt, and Fe3O4 [13] can be synthesizedby various experimental techniques and they are utilized as rawmaterials in fabrication of ultrahigh density magnetic recordingmedia [14–16]. Theoretically, these systems have been studied by awide variety of techniques; these include mean field theory (MFT)[17–19], effective field theory (EFT) [20–28] Green functions form-alism (GF) [29], variational cumulant expansion (VCE) [30,31], andMonte Carlo simulations (MCS) [32–35].

Moreover, many interesting studies have been devoted to theBlume–Capel (BC) model [36,37], initially introduced for the study offirst order magnetic phase transition. It is a spin-1 Ising model with asingle ion anisotropy. The BC model has been studied by different

techniques using the mean-field approximation [38], effective fieldtheory [39], Beth approximation [40], series expansion methods [41],renormalization group theory [42], Monte carlo simulation [43,44],finite cluster approximation [45], constant-coupling approximation[46] and the cluster-variational method [47]. Most of these approx-imation schemes predict in the BC model the existence of a tricritical

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Solid State Communications

ZL

Mc

Msh Jc JInt

Js

Fig. 1. Schematic representation of a nanowire with a length L, formed of twoshells in the core surrounded by a surface shell.

http://dx.doi.org/10.1016/j.ssc.2014.09.0030038-1098/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail addresses: ah_zaim@yahoo.fr (A. Zaim),

kerouad@fs-umi.ac.ma (M. Kerouad).

Solid State Communications 200 (2014) 32–41

point at which the phase transition changes from second order to firstorder when the value of anisotropy becomes sufficiently negative.

Furthermore the core–shell concept can be successfully appliedin nanomagnetism since it is capable to explain various character-istic behaviors observed in nanoparticle magnetism. Jian et al. [48]have examined the magnetization of an hexagonal nanowireconsisting of a ferromagnetic spin-3/2 core and spin-1 outer shellcoupled with ferromagnetic interlayer coupling. They have foundthat the compensation temperature can appear for appropriatevalues of the system parameters. Using MCS, Zaim et al. havestudied the critical and compensation behaviors of a ferrimagneticnanocube, consisting of a ferromagnetic core surrounded by aferromagnetic shell coupled antiferromagnetically [49]. They haveshown that the compensation temperature exists only belowcritical values of the shell and interface couplings, and they havealso investigated in Ref. [50] the possibility of two compensation

points of a ferrimagnetic core/shell nanoparticle. Recently, wehave examined the influence of the trimodal random longitudinalfield, on the magnetic properties and the phase diagram of a spin-1 nanotube, the results show that the system can exhibit the firstorder phase transition, tricritical point, reentrant and double reentrantphenomena [51]. Using the EFT, the dynamic phase transitions andmagnetic properties of the hexagonal Ising nanowire have beenexamined in Refs. [52–54]. A number of interesting properties havebeen found in the dynamic phase diagrams, namely many dynamiccritical points (tricritical point, double critical end point, critical endpoint, zero temperature critical point, multicritical point, tetracriticalpoint, and triple point) as well as reentrant phenomena.

On the other hand, the hysteresis properties (hysteresis area,coercivity and remanent) are very important in the magneticrecording media [55]. Real magnetic recording media quality testand their relationship to the hysteresis based methods can be

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MCMshMt

| D/Jc|

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χ t Cv

E

Js /Jc= JInt /Jc=1T/Jc =0.5

Fig. 2. The crystal field dependence of the total magnetization (a), internal energy (b), longitudinal susceptibility (c) and specific heat (d), for Js=Jc ¼ JInt=Jc ¼ 1 and T=Jc ¼ 0:5.

H. Magoussi et al. / Solid State Communications 200 (2014) 32–41 33

found in Ref. [56]. Based on Monte Carlo simulation, the effects ofsize and surface anisotropy on hysteresis loops of a small sphericalparticle have been investigated [57]. These simulations show thatthe hysteresis loops and coercivity are strongly influenced by theparticle size and the thickness of the surface layer with largeanisotropy. Keskin et al. [58] have investigated hysteresis loops of

the cylindrical Ising nanowire for the temperatures below, around andabove the critical temperature. They have found that the results are ingood agreement with both theoretical and experimental results.Recently, hysteresis behavior of the Blume–Capel model on a cylind-rical Ising nanotube has been studied by using the effective fieldtheory with correlations [59]. A number of characteristic behaviors are

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D/Jc =- 0.8

JInt /J c =1 Js /J c =1T/Jc =0.5

D/Jc =- 2 D/Jc =- 3.3

Mt

ECv

χ t

H/Jc H/Jc H/Jc

Fig. 3. The magnetic properties of the ferromagnetic nanowire (total magnetization Mt, longitudinal susceptibility χt, specific heat Cv and internal energy E) versus theexternal field for D=Jc4Dc=Jc (D=Jc ¼ �0:8, �2, and �3.3) and for Js=Jc ¼ JInt=Jc ¼ 1 and T=Jc ¼ 0:5.

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obtained, especially for the double and triple hysteresis loop patternsfor ferromagnetic and anti-ferromagnetic interactions, respectively.

The magnetic properties and the hysteresis behaviors of thenanowire system with spin-1 atoms in the presence of the crystalfield are not studied by using the Monte Carlo simulation accord-ingly based on heat bath algorithm. Since, as we know less attention

has been paid to the double hysteresis loops near the first-orderphase transition in the literature, in this paper to investigate thehysteresis loops, we propose a spin-1 Blume–Capel nanowire withcore–shell structure. In particular, thermal behaviors of the system,total magnetization, susceptibility, internal energy and specific heat,are examined for both ferro- and anti-ferromagnetic interfacial

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JInt / Jc =1 Js / Jc =1 T/Jc =0.5

EMt

Cv

D/Jc = - 3.47 D/Jc = - 3.65 D/Jc = - 4.1

χ t

H/Jc H/Jc H/JcFig. 4. The magnetic properties of the ferromagnetic nanowire (total magnetization Mt, longitudinal susceptibility χt, specific heat Cv and internal energy E) versus theexternal field for D=JcrDc=Jc (D=Jc ¼ �3:47, �3.65, and �4.1) and for Js=Jc ¼ JInt=Jc ¼ 1 and T=Jc ¼ 0:5.

H. Magoussi et al. / Solid State Communications 200 (2014) 32–41 35

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T/Jc =1.5 T/Jc = 2.1 T/Jc = 2.6 T/Jc = 3.8

Cv

χ tE

JInt / Jc =1 Js / Jc =1 D/Jc =1

H/Jc H/Jc H/Jc H/Jc

Mt

Fig. 5. The magnetic properties of the ferromagnetic nanowire (total magnetization Mt, longitudinal susceptibility χt, specific heat Cv and internal energy E) versus theexternal field for different values of the temperature (T=Jc ¼ 1:5, 2.1, 2.6, and 3.8) and for Js=Jc ¼ JInt=Jc ¼ 1 and D=Jc ¼ 1.

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coupling cases. We also investigate the hysteresis properties (hys-teresis curves, coercivity and remanent) which are very importantin the magnetic recording media.

The outline of this paper is as follows. In Section 2, we givethe model and formalism upon which the MCS with heat-bathalgorithm is based. In Section 3, we present the numerical resultsand discussions, followed by a brief conclusion.

2. Model and formalism

We consider an Ising nanowire model as depicted in Fig. 1,consisting of a spin-1 ferromagnetic core with Nc ¼ 7� L magneticatoms, which is surrounded by a spin-1 ferromagnetic surfaceshell with Nsh ¼ 12� Lmagnetic atoms. In Monte Carlo simulation,based on heat-bath algorithm [60], the results are reported for thesystem size L¼200. A number of additional simulations wereperformed for L¼300, 400 and 500, but no significant differ-ences have been found from the obtained results. We apply freeboundary condition in OX and OY directions, and periodic bound-ary condition in OZ direction. The Hamiltonian of the model isgiven by

H¼ �∑⟨i;j⟩

JijSiSj�D∑iðSiÞ2�H∑

iSi ð1Þ

where Si is the usual Ising variable taking the values 71, 0 at eachsite i of the nanowire, Jij is the exchange interaction between thespins at nearest-neighbor site i and j, D is the crystal field, and H isthe longitudinal magnetic field.

We assume that Jij ¼ Js if both spins belong to the surface shell,Jij ¼ JInt between the core and the surface shell, and Jij ¼ Jc in thecore. At each temperature, 4�104 MCS steps per site have beenused for computing averages of thermodynamic quantities afterdiscarding the first 2�104 initial MCS steps. The error bars werecalculated with a Jackknife method [60] by taking all the measure-ment and grouping them into 10 blocks.

The total magnetization Mt per site is given by

Mt ¼1Nt

ðNcMcþNshMshÞ ð2Þ

where Mc and Msh are, respectively, the magnetization of the coreand the surface shell defined by

Mc ¼ 1Nc

∑Nc

i ¼ 1Si

and

Msh ¼1Nsh

∑Nsh

i ¼ 1Si

The total susceptibility χt of the system is given by

χt ¼ βNtð⟨M2t ⟩� ⟨Mt⟩

2Þ ð3ÞThe internal energy E and the specific heat Cv of the total systemwere evaluated according to the following relations [61]:

E¼ ⟨H⟩

Ntð4Þ

Cv ¼ Nt

KBT2½⟨E

2⟩�⟨E⟩2� ð5Þ

β¼ 1=KBT with KB being the Boltzmann constant and T theabsolute temperature; Nt denotes the total magnetic atoms ofthe nanoparticle with Nt ¼ ðNcþNshÞ

The remanent magnetization and the coercive field are defined,respectively by

MR ¼jMR2 �MR1 j

2

Hc ¼jHcr �Hcl j

2

where Hcl and Hcr are the left and right coercive fields, respectivelyand MR1 and MR2 are the top and low remanent magnetizations,respectively.

3. Results and discussion

In this section, we are interested in investigating the hysteresisloops and the thermal behaviors of a spin-1 Ising Blume–Capelnanowire for both ferromagnetic and ferrimagnetic cases.

3.1. The ferromagnetic case

We have investigated, in Fig. 2, the magnetization Mt, theinternal energy E, the susceptibility χt and the specific heat Cv ofthe Blumpe–Capel nanowire, as a function of the crystal field D=Jcfor Js=Jc ¼ JInt=Jc ¼ 1, T=Jc ¼ 0:5 and for H=Jc ¼ 0. It is shown that, inFig. 2a, the first-order transition from the ferromagnetic (Mta0)to the paramagnetic (Mt ¼ 0) phase occurs at a critical value ofD=Jc (Dc=Jc ¼ �3:47). This behavior is confirmed by the internalenergy curve (Fig. 2b) which undertakes a jump at Dc=Jc . It is also

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MR

D/Jc = 1D/Jc = 0D/Jc = -1

JInt / Jc =1 Js / Jc =1

Hc

T /JC

Fig. 6. The temperature dependence of the remanent magnetization MR, andcoercive field Hc for different values of D=Jc (D=Jc ¼ �1, 0, and 1) and forJs=Jc ¼ JInt=Jc ¼ 1.

H. Magoussi et al. / Solid State Communications 200 (2014) 32–41 37

seen that the susceptibility (Fig. 2c) and the specific heat (Fig. 2d)present a peak at Dc=Jc .

In order to investigate the influence of the crystal field on themagnetic properties of a ferromagnetic Blume–Capel nanowire, forD=JcrDc=Jc and D=Jc4Dc=Jc , we have examined the thermal andhysteresis behaviors for the parameter values, Js=Jc ¼ JInt=Jc ¼ 1 andT=Jc ¼ 0:5. In Fig. 3, we plot the magnetization, the susceptibility,the specific heat and the internal energy versus the applied fieldH=Jc for D=Jc4Dc=Jc. We can observe that for D=Jc ¼ �0:8, we haveonly a normal hysteresis loop with a coercive field point Hc¼74.1.The susceptibility, the specific heat and the internal energy havetwo distinct peaks at the coercive field points. It is noted thatwhen D=Jc decreases, the area of the hysteresis loop decreases, andthe zone between the peaks of the susceptibility, the specific heatand the internal energy also decreases. We can remark that whenthe crystal field D=Jc approaches its critical value, two steps appearin the curve of the magnetization, two distinct broad minimaappear, and two distinct peaks (corresponding to the two steps)are obtained in the curves of the susceptibility and the specific

heat. However the observed peaks in the curve of the internalenergy become two distinct broad maxima.

For D=JcrDc=Jc (Fig. 4), it is observed that the shapes of thehysteresis loop change from a ferromagnetic hysteresis loop todouble hysteresis loops at Dc=Jc ¼ �3:47. The double hyste-resis loops have been seen theoretically in Refs. [62,63] andexperimentally in different systems, for example in Cu-dopedK0:5Na0:5NbO3 (KNN) ceramics [64] and in Fe3O4=Mn3O3 super-lattices [65]. It is shown that the anisotropy produces irregularitiesin the low-temperature hysteresis curve due to Barkhausen spinavalanches. These irregularities occur at different values of themagnetic field. It is also noticed that a broad minimum and a centralpeak take place in the curves of the susceptibility and the specificheat. The curve of the internal energy presents a broad maximum(larger than those observed in Fig. 3). When D=Jc decreases, thedouble hysteresis loops stretch further out horizontally. In the curvesof the susceptibility and the specific heat, two distinct broad minimaappear and the central peaks become two distinct peaks. It is seenthat two distinct broad maxima appear in the curve of the internal

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ECv

χ tΜt

H/Jc H/Jc H/Jc

Fig. 7. The magnetic properties of the ferrimagnetic nanowire (total magnetization Mt, longitudinal susceptibility χt, specific heat Cv and internal energy E) versus theexternal field for different values of the interfacial coupling (JInt=Jc ¼ �0:03, �1 and �1.5) and for T=Jc ¼ 1:4, Js=Jc ¼ 1:2 and D=Jc ¼ 0.

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energy. The peaks of the χt and Cv, and the broad maximum of the Estretch further out horizontally with the decreasing D=Jc .

The dependence of the magnetic properties on the temperature forJs=Jc ¼ JInt=Jc ¼ 1 and D=Jc ¼ 1 is shown in Fig. 5. From this figure, wecan observe that the magnetization is symmetric for both positive andnegative values of the external magnetic field, and the curves of thesusceptibility, specific heat, and internal energy present two peaks atthe coercive field points Hc¼72.05. It is also seen that with theincreasing temperature, the hysteresis loops gets more compact andgoes lower, and the zone between the peaks of the susceptibility,specific heat, and internal energy decreases. When T=Jc4Tc=Jc(Tc=Jc ¼ 3:47), the hysteresis loops disappear and only a central peakis observed in the curves of the susceptibility, specific heat, andinternal energy. Similar behaviors of hysteresis loops have beenobserved in the Blume–Capel model on a cylindrical nanotube [59].Furthermore, with the increasing temperature, the hysteresis loopsdecrease and the results are qualitatively similar to the experimentalresults obtained for the FePt=Fe3O4 and FePt=CoFe2O4 core/shellstructure nanoparticles [66], the core–shell type nanoparticles (Fe–C

or Fe–carbosiloxane polymer) [67], the ferromagnetism in Mnþ-imp-lanted Si nanowire [68] and the Ni particles in carbon nanotube [69].

In Fig. 6, we display the temperature dependence of the rem-anent magnetization MR and the coercive magnetic field Hc fordifferent values of the crystal fields D=Jc (D=Jc ¼ �1, 0 and 1), andfor Js=Jc ¼ JInt=Jc ¼ 1. It is noticed that the remanent magnetizationdecreases with the increasing temperature from its saturationvalue at low temperature region, and vanishes at the criticaltemperature which depends on the value of D=Jc . It is alsoremarked that the coercive field decreases with the increasingtemperature and vanishes at Tc=Jc .

3.2. The ferrimagnetic case

In this case, we have examined the influence of the tempera-ture and the interfacial coupling on the hysteresis and thermalbehaviors of a ferrimagnetic nanowire.

In Fig. 7, we have plotted the total magnetization Mt, suscept-ibility χt, specific heat Cv and internal energy E versus applied field,

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T/ J = 6T/ J = 1.5

ECv

χt

H/Jc H/Jc H/Jc

Fig. 8. The magnetic properties of the ferrimagnetic nanowire (total magnetization Mt, longitudinal susceptibility χt, specific heat Cv and internal energy E) versus theexternal field for different values of the temperature (T=Jc ¼ 1:5, 2 and 6) and for JInt=Jc ¼ �1:2, Js=Jc ¼ 1:2 and D=Jc ¼ 1:2.

H. Magoussi et al. / Solid State Communications 200 (2014) 32–41 39

for selected value of the interfacial coupling (Jint=Jc ¼ �0:03, �1and �1.5), and for fixed parameters T=Jc ¼ �1:4, Js=Jc ¼ 1:2 andD=Jc ¼ 0. We remark that for Jint=Jc ¼ �0:03, the system presentsonly a central loop with a coercive field points Hc ¼ 70:15, whichinvolves two steps, due to different ferromagnetic properties ofthe core and surface shell of the system. We can remark that thecurves of the susceptibility and specific heat have four distinctpeaks, but the curve of the internal energy presents only two distinctpeaks corresponding to the coercive field points Hc ¼ 70:15 of thehysteresis loop. It is seen that for Jint=Jc ¼ �1 the hysteresis loopchanges from one central loop to triple loop. The larger the anti-ferromagnetic coupling constant is, the more difficult it is for theapplied magnetic field to change the direction of the magnetizationat the interfacial layers in the core and the shell. This triple hysteresisloop behavior has been seen theoretically in ferromagnetic or ferri-magnetic core/shell nanotube [59,70] and experimentally in CoFeB/Cu, CoNip/Cu, FeGa/py, and FeGa/CoFeB multilayered nanowires [71].It is noted that the susceptibility and the specific heat have sixdistinct peaks, two central corresponding to the central loop, and theouter peaks corresponding to the outer loops. Nevertheless, the curveof the internal energy presents only two distinct peaks correspond-ing to the central loop and two outer broad maxima correspondingto the outer loops. The outer loops, the outer peaks observed in thecurves of the susceptibility and specific heat, and the outer broadmaximum obtained in the curve of the internal energy stretchfurther out horizontally with the decreasing Jint=Jc . It is implied thatthe behaviors of the system are ferromagnetic link for smaller jJint j=Jcand antiferromagnetic-like for larger jJint j=Jc .

To study the effect of the temperature on the magnetic proper-ties of the system, we present in Fig. 8 the total magnetization,susceptibility, specific heat, and internal energy versus appliedfield, for selected values of the temperature T=Jc (T=Jc ¼ 1:5, 2, 6),and for fixed parameters Jint=Jc ¼ �1:2, Js=Jc ¼ 1:2 and D=Jc ¼ 1:2.As shown in this figure, the size of the central loop reduces and theouter loops disappear when we increase the value of the tem-perature. We can also notice that the zone between the centralpeaks observed in the curves of the susceptibility specific heat andinternal energy decreases, the outer peaks of the susceptibility andspecific heat have been confounded, but the two broad maxima ofthe internal energy disappear. When T=Jc is very important, thehysteresis loop, and the peaks of the susceptibility and the specificheat disappear; however, the internal energy has a broad centralmaximum.

4. Conclusion

In conclusion, we have studied the magnetic properties of theBlume–Capel nanowire. We have investigated for both ferromag-netic and ferrimagnetic cases the influence of the crystal field, thetemperature, and the interfacial coupling on the thermal and thehysteresis behaviors of the system. It has been found that for theferromagnetic case, the hysteresis loop changes from one centralloop to double hysteresis loops, and the double peaks observed inthe curves of the susceptibility and specific heat become onecentral peak. It has also been found that the remanent magnetiza-tion and coercive field decrease with the increasing temperature.For the ferrimagnetic case, it has been shown that the triplehysteresis loop occurs for the larger antiferromagnetic couplingand that when T increases, the hysteresis loops disappear.

Acknowledgments

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

References

[1] T.Y. Kim, Y. Yamazaki, T. Hirano, Phys. Status Solidi B 241 (2004) 1601.[2] R.H. Kodama, J. Magn. Magn. Mater. 200 (1999) 359.[3] Q.A. Pankhurst, J. Connolly, S.K. Jones, J. Dobson, J. Phys. D: Appl. Phys. 36

(2003) R167.[4] A.H. Habib, C.L. Ondeck, P. Chaudhary, M.R. Bockstaller, M.E. McHenry, J. Appl.

Phys. 103 (2008) 07A307.[5] N. Sounderya, Y. Zhang, Recent Pat. Biomed. Eng. 1 (2008) 34.[6] J. Maller, K.Y. Zhang, C.L. Chien, T.S. Eagleton, P.C. Searson, Appl. Phys. Lett. 84

(2004) 3900;C.M. Lieber, Z.L. Wang, G. Editors, MRS Bull. 32 (2007) 99;S.S.P. Parkin, M. Hayashi, L. Thomas, Science 320 (2008) 190;H. Zhang, A. Hoffmann, R. Divan, P. Wang, Appl. Phys. Lett. 95 (2009) 232503.

[7] B. Martínez, X. Obradors, L1. Balcells, A. Rouanet, C. Monty, Phys. Rev. Lett. 80(1998) 181.

[8] H.L. Su, G.B. Ji, S.L. Tang, Z. Li, B.X. Gu, Y.W. Du, Nanotechnology 16 (2005) 429;X. Lin, G. Ji, T. Gao, X. Chang, Y. Liu, H. Zhang, Y. Du, Solid State Commun. 151(2011) 1708.

[9] W. Chen, Z. Li, G.B. Ji, S.L. Tang, M. Lu, Y.W. Du, Solid State Commun. 133 (2005)235;J.Y. Chen, H.R. Liu, N. Ahmad, Y.L. Li, Z.Y. Chen, W.P. Zhou, X.F. Han, J. Appl. Phys.109 (2011) 07E157.

[10] H. Pan, B.H. Liu, J.B. Yi, C. Poh, S. Lim, J. Ding, Y.P. Feng, C. Huan, J.Y. Lin, J. Phys.Chem. B 109 (2005) 3094.

[11] H.-K. Seong, J.-Y. kim, J.-J. Kim, S.-C. Lee, S.-R. Kim, U. Kim, T.-E. Park, H.-J. Choi,Nano Lett. 7 (2007) 3366.

[12] Z. Fan, J.G. Lu, Int. J. Hi, Speed Electron. Syst. 16 (2006) 883.[13] Y.C. Sui, R. Skomski, K.D. Sorge, D.J. Sellmyer, Appl. Phys. Lett. 84 (2004) 1525.[14] J.E. Wegrowe, D. Kelly, Y. Jaccard, P. Guittienne, J.P. Ansermet, Europhys. Lett.

45 (1999) 626.[15] A. Fert, L. Piraux, J. Magn. Magn. Mater. 200 (1999) 338.[16] S.D. Bader, Rev. Mod. Phys. 78 (2006) 1.[17] T. Kaneyoshi, J. Magn. Magn. Mater. 322 (2010) 3014.[18] T. Kaneyoshi, Phys. Status Solidi B 248 (2011) 250.[19] T. Kaneyoshi, Phase Transit. 85 (2012) 264.[20] C.D. Wang, Z.Z. Lu, W.X. Yuan, S.Y. Kwok, B.H. Teng, Phys. Lett. A 375 (2011)

3405.[21] A. Zaim, M. Kerouad, M. Boughrara, EPJ Web Confer. 29 (2012).[22] T. Kaneyoshi, Physica A 391 (2012) 3616.[23] N. Şarlı, M. Keskin, Solid State Commun. 152 (2012) 354.[24] O. Canko, A. Erdinç, F. Taşkın, A.F. Yıldırım, J. Magn. Magn. Mater. 324 (2012)

508.[25] Wei Jiang, Hong-yu Guan, Zan Wang, An-bang Guo, Physica B 407 (2012) 378.[26] T. Kaneyoshi, Solid State Commun. 152 (2012) 883.[27] A. Zaim, M. Kerouad, M. Boughrara, A. Ainane, J.J. de Miguel, J. Supercond. Nov.

Magn. 25 (2012) 2407.[28] A. Zaim, M. Kerouad, M. Boughrara, J. Magn. Magn. Mater. 331 (2013) 37.[29] D.A. Garanin, H. Kachkachi, Phys. Rev. Lett. 90 (2003) 65504.[30] H.-Y. Wang, Y.-S. Zhou, E. Wang, D.L. Lin, Chin. J. Phys. 39 (2001) 85.[31] H. Wang, Y. Zhou, D.L. Lin, C. Wang, Phys. Stat. Sol. (b) 232 (2002) 254.[32] Y. Yüksel, E. Aydıner, H. Polat, J. Magn. Magn. Mater. 323 (2011) 3168.[33] O. Iglesias, A. Labarta, X. Batlle, J. Nanosci. Nanotechnol. 8 (2008) 2761.[34] Y. Hu, A. Du, J. Appl. Phys. 102 (2007) 113911.[35] A.J. Bennett, J.M. Xu, Appl. Phys. Lett. 82 (2003) 3304.[36] M. Blume, Phys. Rev. 141 (1966) 517.[37] H.W. Capel, Physica 32 (1966) 966.[38] M. Blume, V.J. Emery, R.B. Grifffiths, Phys. Rev. A 4 (1971) 1071.[39] M. Kerouad, M. Saber, J.W. Tucker, Physica A 210 (1994) 424;

T. Kaneyoshi, J. Magn. Magn. Mater. 98 (2000) 201;A.L. De Lima, B.D. Stošić, I.F. Fittipaldi, J. Magn. Magn. Mater. 226 (2001) 635;E. Kantar, B. Deviren, M. Keskin, Eur. Phys. J. B 86 (2013) 1;E. Kantar, M. Ertaş, Solid State Commun. 188 (2014) 71.

[40] Y. Tanaka, N. Uryû, J. Phys. Soc. Jpn. 50 (1981) 1140.[41] J.G. Brankov, J. Przystawa, E. Praveczki, J. Phys. C: Solid State Phys. 5 (1972)

3387.[42] T.W. Burkhardt, H.J.P. Knops, Phys. Rev. B 15 (1977) 1602.[43] B.L. Arora, D.P. Landau, in: Proceedings of AIP, vol. 5, 1972, p. 352.[44] A. Zaim, Y. El Amraoui, M. Kerouad, H. Arhchoui, J. Magn. Magn. Mater. 320

(2008) 1030.[45] B. Laaboudi, M. Kerouad, Physica A 241 (1997) 729.[46] M. Tanaka, K. Takahashi, Phys. Stat. Sol. (b) 93 (1979) K85.[47] Wai Man Ng, Jeremiah H. Barry, Phys. Rev. B 17 (1978) 3675;

T. Balcerzak, J.W. Tucker, J. Magn. Magn. Mater. 278 (2004) 87.[48] Wei Jiang, Fan Zhang, Xiao-Xi Li, Hong-Yu Guan, An-Bang Guo, Zan Wang,

Physica E 47 (2013) 95.[49] A. Zaim, M. Kerouad, Y. El Amraoui, J. Magn. Magn. Mater. 321 (2009) 1077.[50] A. Zaim, M. Kerouad, Physica A 389 (2010) 3435.[51] H. Magoussi, A. Zaim, M. Kerouad, J. Magn. Magn. Mater. 344 (2013) 109.[52] E. Kantar, M. Ertaş, M. Keskin, J. Magn. Magn. Mater. 361 (2014) 61.[53] M. Ertaş, Y. Kocakaplan, Phys. Lett. A 378 (2014) 845.[54] B. Deviren, M. Ertaş, M. Keskin, Phys. Scr. 85 (2012) 055001.[55] J.-M. Liu, H.L.W. Chan, C.L. Choy, C.K. Ong, Phys. Rev. B 65 (2001) 014416.[56] A. Berger, N. Supper, Y. Ikeda, B. Lengsfield, A. Moser, E.E. Ful-lerton, Appl.

Phys. Lett. 93 (2008) 122502.

H. Magoussi et al. / Solid State Communications 200 (2014) 32–4140

[57] Z. Huang, Z. Chen, S. Li, Q. Feng, F. Zhang, Y. Du, Eur. Phys. J. B 51 (2006) 65.[58] M. Keskin, N. Şarlı, B. Deviren, Solid State Commun. 151 (2011) 1025.[59] O. Canko, F. Taskin, K. Argin, A. Erdinç, Solid State Commun. 183 (2014) 35.[60] M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Statistical Physics,

Clarendon Press, Oxford, 1999.[61] V.T. Ngo, H.V. Nguyen, H.T. Diep, V.L. Nguyen, Phys. Rev. B 69 (2004) 134429.[62] K. Okada, H. Sugie, Phys. Lett. 37A (1971) 337.[63] B. Yang, D.-M. Zhang, C.-D. Zheng, J. Wang, J. Yu, Appl. Phys. 40 (2007) 5696.[64] Dunmin Lin, K.W. Kwok, H.L.W. Chan, Appl. Phys. Lett. 90 (2007) 232903.

[65] G. Chern, Lance Horng, W.K. Shieh, T.C. Wu, Phys. Rev. B 63 (2001) 094421.[66] H. Zeng, S. Sun, J. Li, Z.L. Wang, J.P. Liu, Appl. Phys. Lett. 85 (2004) 792.[67] C.T. Fleaca, I. Morjan, R. Alexandrescu, F. Dumitrache, I. Soare, L. Gavrila-Florescu,

F. Le Normand, A. Derory, Appl. Surf. Sci. 255 (2009) 5386.[68] H.W. Wu, C.J. Tsai, L.J. Chen, Appl. Phys. Lett. 90 (2007) 043121.[69] J.A. García, E. Bertran, L. Elbaile, J. García-Céspedes, A. Svalov, Phys. Status

Solidi C 7 (2010) 2679.[70] H. Magoussi, A. Zaim, M. Kerouad, Chin. Phys. B 22 (2013) 116401.[71] N. Lupu, L. Lostun, H. Chiriac, J. Appl. Phys. 107 (2010) 09E315.

H. Magoussi et al. / Solid State Communications 200 (2014) 32–41 41