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

Solid State Communications 200 (2014) 32–41

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Monte Carlo simulation of the magnetic properties of a spin-1 Blume–Capel nanowire H. Magoussi, A. Zaim n, M. Kerouad Laboratoire 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

art ic l e i nf o

a b s t r a c t

Article history: Received 7 March 2014 Received in revised form 24 July 2014 Accepted 2 September 2014 by C. Lacroix Available online 26 September 2014

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 shell of spin-1 atoms with ferromagnetic or anti-ferromagnetic interfacial coupling. We have examined the influence 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 a function of the temperature have also been investigated. We have found that the system exhibits the first order phase transition, one or double hysteresis loops in the ferromagnetic case, and one or triple hysteresis loops in the ferrimagnetic case. & 2014 Elsevier Ltd. All rights reserved.

Keywords: B. Ising model C. Nanowire D. Hysteresis behavior E. Monte Carlo simulation

1. Introduction Materials with a nanostructure, such as nanoparticles, nanotube and nanowires, have attracted considerable attention because of not only their academic interest, but also their technological [1,2] and biomedical [3–5] applications, namely in the areas of magnetic recording media, spin electronics, optics, sensors and thermoelectronics devices [6]. In the experimental area, the magnetization of certain nanomaterials such as γ-Fe2 O3 nanoparticles has been measured [7]. The magnetic nanowires have been studied, and their magnetic properties have been investigated, especially Fe–Co [8], Co–P [9], Ni [10], Ga1x Cux N [11], etc. The magnetic nanowires and nanotubes such as ZnO [12], FePt, and Fe3 O4 [13] can be synthesized by various experimental techniques and they are utilized as raw materials in fabrication of ultrahigh density magnetic recording media [14–16]. Theoretically, these systems have been studied by a wide variety of techniques; these include mean field theory (MFT) [17–19], effective field theory (EFT) [20–28] Green functions formalism (GF) [29], variational cumulant expansion (VCE) [30,31], and Monte Carlo simulations (MCS) [32–35]. Moreover, many interesting studies have been devoted to the Blume–Capel (BC) model [36,37], initially introduced for the study of first order magnetic phase transition. It is a spin-1 Ising model with a single ion anisotropy. The BC model has been studied by different

techniques using the mean-field approximation [38], effective field theory [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 approximation schemes predict in the BC model the existence of a tricritical

Msh Mc

L

Jc

JInt Js

Z

n

Corresponding author. E-mail addresses: [email protected] (A. Zaim), [email protected] (M. Kerouad). http://dx.doi.org/10.1016/j.ssc.2014.09.003 0038-1098/& 2014 Elsevier Ltd. All rights reserved.

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

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

point at which the phase transition changes from second order to first order when the value of anisotropy becomes sufficiently negative. Furthermore the core–shell concept can be successfully applied in nanomagnetism since it is capable to explain various characteristic behaviors observed in nanoparticle magnetism. Jian et al. [48] have examined the magnetization of an hexagonal nanowire consisting of a ferromagnetic spin-3/2 core and spin-1 outer shell coupled with ferromagnetic interlayer coupling. They have found that the compensation temperature can appear for appropriate values of the system parameters. Using MCS, Zaim et al. have studied the critical and compensation behaviors of a ferrimagnetic nanocube, consisting of a ferromagnetic core surrounded by a ferromagnetic shell coupled antiferromagnetically [49]. They have shown that the compensation temperature exists only below critical values of the shell and interface couplings, and they have also investigated in Ref. [50] the possibility of two compensation

33

points of a ferrimagnetic core/shell nanoparticle. Recently, we have examined the influence of the trimodal random longitudinal field, on the magnetic properties and the phase diagram of a spin1 nanotube, the results show that the system can exhibit the first order phase transition, tricritical point, reentrant and double reentrant phenomena [51]. Using the EFT, the dynamic phase transitions and magnetic properties of the hexagonal Ising nanowire have been examined in Refs. [52–54]. A number of interesting properties have been found in the dynamic phase diagrams, namely many dynamic critical points (tricritical point, double critical end point, critical end point, zero temperature critical point, multicritical point, tetracritical point, 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 magnetic recording media [55]. Real magnetic recording media quality test and their relationship to the hysteresis based methods can be

Js /Jc= JInt /Jc=1 T / Jc =0.5 0

1.0

-1 0.8 -2

MC Msh

-3

E

0.6

Mt

0.4

-4

-5 0.2 -6

0.0 0

1

2

3

4

-7

5

0

1

2

3

4

5

| D/Jc|

0.3 8

6

χt

Cv

0.2

4

0.1 2

0.0

0 0

1

2

3

| D/Jc|

4

5

0

2

4

| D/Jc|

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

34

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

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

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

JInt /J c =1

Js /J c =1

T/Jc =0.5 1

1

1.0

D/Jc =- 0.8

D/Jc =- 3.3

D/Jc =- 2 0.5

0

0

Mt

0.0 -0.5

-1

-1.0

-1 -4

-2

0

2

4

-4

-2

0

2

4

-4

-2

0

2

4

-4

-2

0

2

4

-4

-2

0

2

4

-4

-2

0

2

4

0.2

0.05 0.01

χt

0.1

0.00 -4

-2

0

2

4

1.5

0.00

0.0 -4

-2

0

2

4 5

4

4

3

1.0

Cv

3 2 2

0.5 1

0.0 -4

-2

0

2

4

0

-4.30

-5.57

-4.32

-5.58

-4.34

-5.59

-4.36

E

-5.56

-5.60

1

-4

-2

0

2

4

0

-1

-2

-4.38

-5.61

0

-4.40

-5.62

-3

-4.42 -4

-2

0

H/Jc

2

4

-4

-2

0

H/Jc

2

4

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 the external field for D=J c 4Dc =J c (D=J c ¼  0:8,  2, and  3.3) and for J s =J c ¼ J Int =J c ¼ 1 and T=J c ¼ 0:5.

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

obtained, especially for the double and triple hysteresis loop patterns for ferromagnetic and anti-ferromagnetic interactions, respectively. The magnetic properties and the hysteresis behaviors of the nanowire system with spin-1 atoms in the presence of the crystal field are not studied by using the Monte Carlo simulation accordingly based on heat bath algorithm. Since, as we know less attention

35

has been paid to the double hysteresis loops near the first-order phase transition in the literature, in this paper to investigate the hysteresis loops, we propose a spin-1 Blume–Capel nanowire with core–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

JInt / Jc =1

Js / Jc =1

T/Jc =0.5 1

D/Jc = - 3.47

1

D/Jc = - 3.65

D/Jc = - 4.1

Mt

1

0

0

0

-1

-1

-1

-4

-2

0

2

-4

4

-2

0

2

4

-4

-2

0

2

4

-4

-2

0

2

4

-4

-2

0

2

4

-4

-2

0

2

4

0.2

0.3

0.2

0.1

χt

0.1 0.1

0.0

0.0 -4

-2

0

2

4

-4

-2

0

2

4

0.0

4

4

3

3

2

2

1

1

Cv

6

3

0

-4

-2

0

2

4

0

0

-4

-2

0

2

4

0

0

0.0 -0.5

-1

E

-1.0

-1

-1.5 -2

-2.0 -2

-2.5 -3

-4

-2

0

H/Jc

2

4

-4

-2

0

H/Jc

2

4

H/Jc

Fig. 4. The magnetic properties of the ferromagnetic nanowire (total magnetization Mt, longitudinal susceptibility χt, specific heat Cv and internal energy E) versus the external field for D=J c r Dc =J c (D=J c ¼  3:47,  3.65, and  4.1) and for J s =J c ¼ J Int =J c ¼ 1 and T=J c ¼ 0:5.

36

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

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

T/Jc =1.5

1.0

T/Jc = 2.1

T/Jc = 2.6

T/Jc = 3.8

1

1

0

0

-1

-1

Mt

0.5 0.0

0

-0.5 -1.0

-1 -4

-2

0

2

4

-4

-2

0

2

-4

4

-2

0

2

4

-4

-2

0

2

4

-4

-2

0

2

4

-4

-2

0

2

4

-4

-2

0

2

4

3

0.10 0.1 0.02

χt

2 0.05

0.01

0.00

1

-4

-2

0

2

4

0.00 3

-4

-2

0

2

4

0.0

-4

-2

0

2

4

0

2 1.0

2

Cv

2 1 0.5

0.0

1

1

-4

-2

0

2

4

0

-4

-2

0

2

4

0

-4

-2

0

2

4

0

-6

-7.1

-3 -7.2 E

-4

-5

-7

-7.3

-7 -6

-7.4 -4

-2

0 H/Jc

2

4

-4

-2

0 H/Jc

2

4

-4

-2

0 H/Jc

2

4

H/Jc

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

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

coupling cases. We also investigate the hysteresis properties (hysteresis curves, coercivity and remanent) which are very important in the magnetic recording media. The outline of this paper is as follows. In Section 2, we give the model and formalism upon which the MCS with heat-bath algorithm is based. In Section 3, we present the numerical results and discussions, followed by a brief conclusion.

Hc ¼

37

jH cr  H cl j 2

where H cl and H cr are the left and right coercive fields, respectively and M R1 and M R2 are the top and low remanent magnetizations, respectively.

3. Results and discussion

We consider an Ising nanowire model as depicted in Fig. 1, consisting of a spin-1 ferromagnetic core with N c ¼ 7  L magnetic atoms, which is surrounded by a spin-1 ferromagnetic surface shell with Nsh ¼ 12  L magnetic atoms. In Monte Carlo simulation, based on heat-bath algorithm [60], the results are reported for the system size L¼200. A number of additional simulations were performed for L ¼300, 400 and 500, but no significant differences have been found from the obtained results. We apply free boundary condition in OX and OY directions, and periodic boundary condition in OZ direction. The Hamiltonian of the model is given by H ¼  ∑ J ij Si Sj D∑ðSi Þ2  H∑Si 〈i;j〉

i

ð1Þ

i

where Si is the usual Ising variable taking the values 7 1, 0 at each site i of the nanowire, Jij is the exchange interaction between the spins at nearest-neighbor site i and j, D is the crystal field, and H is the longitudinal magnetic field. We assume that J ij ¼ J s if both spins belong to the surface shell, J ij ¼ J Int between the core and the surface shell, and J ij ¼ J c in the core. At each temperature, 4  104 MCS steps per site have been used for computing averages of thermodynamic quantities after discarding the first 2  104 initial MCS steps. The error bars were calculated with a Jackknife method [60] by taking all the measurement and grouping them into 10 blocks. The total magnetization Mt per site is given by Mt ¼

1 ðN c M c þ N sh M sh Þ Nt

In this section, we are interested in investigating the hysteresis loops and the thermal behaviors of a spin-1 Ising Blume–Capel nanowire for both ferromagnetic and ferrimagnetic cases. 3.1. The ferromagnetic case We have investigated, in Fig. 2, the magnetization Mt, the internal energy E, the susceptibility χt and the specific heat Cv of the Blumpe–Capel nanowire, as a function of the crystal field D=J c for J s =J c ¼ J Int =J c ¼ 1, T=J c ¼ 0:5 and for H=J c ¼ 0. It is shown that, in Fig. 2a, the first-order transition from the ferromagnetic (M t a 0) to the paramagnetic (M t ¼ 0) phase occurs at a critical value of D=J c (Dc =J c ¼  3:47). This behavior is confirmed by the internal energy curve (Fig. 2b) which undertakes a jump at Dc =J c . It is also

JInt / Jc =1

1.0

Js / Jc =1 D/J c = 1

D/Jc = 0

0.8

D/Jc = -1 0.6

MR

2. Model and formalism

0.4

ð2Þ

0.2

where Mc and Msh are, respectively, the magnetization of the core and the surface shell defined by 0.0

1 Nc ∑ S Mc ¼ Nc i ¼ 1 i

2.0

2.5

3.0

3.5

4.0

T /JC

and

2.0

M sh ¼

JInt / Jc =1

Nsh

1 ∑ S N sh i ¼ 1 i

The total susceptibility

χ t ¼ βN t ð〈M 2t 〉  〈Mt 〉2 Þ

χt of the system is given by

D/Jc = 0

ð3Þ

D/Jc = -1

ð4Þ

Nt K BT

2

½〈E2 〉 〈E〉2 

ð5Þ

1.0

Hc

〈H〉 E¼ Nt

Js / Jc =1 D/J c = 1

1.5

The internal energy E and the specific heat Cv of the total system were evaluated according to the following relations [61]:

Cv ¼

1.5

0.5

β ¼ 1=K B T with K B being the Boltzmann constant and T the absolute temperature; Nt denotes the total magnetic atoms of the nanoparticle with Nt ¼ ðN c þ N sh Þ The remanent magnetization and the coercive field are defined, respectively by MR ¼

jM R2  M R1 j 2

0.0 1.5

2.0

2.5

3.0

3.5

4.0

T /JC Fig. 6. The temperature dependence of the remanent magnetization MR, and coercive field Hc for different values of D=J c (D=J c ¼  1, 0, and 1) and for J s =J c ¼ J Int =J c ¼ 1.

38

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

seen that the susceptibility (Fig. 2c) and the specific heat (Fig. 2d) present a peak at Dc =J c . In order to investigate the influence of the crystal field on the magnetic properties of a ferromagnetic Blume–Capel nanowire, for D=J c rDc =J c and D=J c 4 Dc =J c , we have examined the thermal and hysteresis behaviors for the parameter values, J s =J c ¼ J Int =J c ¼ 1 and T=J c ¼ 0:5. In Fig. 3, we plot the magnetization, the susceptibility, the specific heat and the internal energy versus the applied field H=J c for D=J c 4Dc =J c . We can observe that for D=J c ¼  0:8, we have only a normal hysteresis loop with a coercive field point Hc ¼ 7 4.1. The susceptibility, the specific heat and the internal energy have two distinct peaks at the coercive field points. It is noted that when D=J c decreases, the area of the hysteresis loop decreases, and the zone between the peaks of the susceptibility, the specific heat and the internal energy also decreases. We can remark that when the crystal field D=J c approaches its critical value, two steps appear in the curve of the magnetization, two distinct broad minima appear, 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 internal energy become two distinct broad maxima. For D=J c rDc =J c (Fig. 4), it is observed that the shapes of the hysteresis loop change from a ferromagnetic hysteresis loop to double hysteresis loops at Dc =J c ¼  3:47. The double hysteresis loops have been seen theoretically in Refs. [62,63] and experimentally in different systems, for example in Cu-doped K0:5 Na0:5 NbO3 (KNN) ceramics [64] and in Fe3 O4 =Mn3 O3 superlattices [65]. It is shown that the anisotropy produces irregularities in the low-temperature hysteresis curve due to Barkhausen spin avalanches. These irregularities occur at different values of the magnetic field. It is also noticed that a broad minimum and a central peak take place in the curves of the susceptibility and the specific heat. The curve of the internal energy presents a broad maximum (larger than those observed in Fig. 3). When D=J c decreases, the double hysteresis loops stretch further out horizontally. In the curves of the susceptibility and the specific heat, two distinct broad minima appear and the central peaks become two distinct peaks. It is seen that two distinct broad maxima appear in the curve of the internal

T/Jc =1.4 Js / Jc =1.2 D/Jc =0.0

Μt

1

1

JInt / Jc= -0.03

1

JInt / Jc= -1

0

0

0

-1

-1

-1

-4

-2

0

2

4

-5

0

5

0.5

12

JInt / Jc= -1.5

-5

0

5

-5

0

5

-5

0

5

0

5

0.3

0.4 8

0.2

χt

0.3 0.2

4

0.1

0.1 0

-4

-2

0

2

4

5

0.0

-5

0

5

6

0.0 4

4 4

3

Cv

2 2

2

1 0

-4

-2

0

2

4

0

-3.6

-2

-4.0

-4

-5

0

5

0

E

-3

-6

-4.4

-6 -4

-2

0

H/Jc

2

4

-5

0

H/Jc

5

-5

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 the external field for different values of the interfacial coupling (J Int =J c ¼  0:03,  1 and  1.5) and for T=J c ¼ 1:4, J s =J c ¼ 1:2 and D=J c ¼ 0.

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

energy. The peaks of the χt and Cv, and the broad maximum of the E stretch further out horizontally with the decreasing D=J c . The dependence of the magnetic properties on the temperature for J s =J c ¼ J Int =J c ¼ 1 and D=J c ¼ 1 is shown in Fig. 5. From this figure, we can observe that the magnetization is symmetric for both positive and negative values of the external magnetic field, and the curves of the susceptibility, specific heat, and internal energy present two peaks at the coercive field points Hc ¼ 72.05. It is also seen that with the increasing temperature, the hysteresis loops gets more compact and goes lower, and the zone between the peaks of the susceptibility, specific heat, and internal energy decreases. When T=J c 4 T c =J c (T c =J c ¼ 3:47), the hysteresis loops disappear and only a central peak is observed in the curves of the susceptibility, specific heat, and internal energy. Similar behaviors of hysteresis loops have been observed in the Blume–Capel model on a cylindrical nanotube [59]. Furthermore, with the increasing temperature, the hysteresis loops decrease and the results are qualitatively similar to the experimental results obtained for the FePt=Fe3 O4 and FePt=CoFe2 O4 core/shell structure nanoparticles [66], the core–shell type nanoparticles (Fe–C

39

or Fe–carbosiloxane polymer) [67], the ferromagnetism in Mn þ -implanted Si nanowire [68] and the Ni particles in carbon nanotube [69]. In Fig. 6, we display the temperature dependence of the remanent magnetization MR and the coercive magnetic field Hc for different values of the crystal fields D=J c (D=J c ¼  1, 0 and 1), and for J s =J c ¼ J Int =J c ¼ 1. It is noticed that the remanent magnetization decreases with the increasing temperature from its saturation value at low temperature region, and vanishes at the critical temperature which depends on the value of D=J c . It is also remarked that the coercive field decreases with the increasing temperature and vanishes at T c =J c . 3.2. The ferrimagnetic case In this case, we have examined the influence of the temperature and the interfacial coupling on the hysteresis and thermal behaviors of a ferrimagnetic nanowire. In Fig. 7, we have plotted the total magnetization Mt, susceptibility χt, specific heat Cv and internal energy E versus applied field,

JInt / Jc =-1.2

Js / Jc =1.2

DJc =1.2

Μt

1

T/ J =

T/ J =2.0

1

1.5

0

0

0

-1

-1 -10

-5

0

5

T/ J = 6

1

-1

10

-10

0.3

0.3

0.2

0.2

0.1

0.1

-5

0

5

10

-10

-5

0

5

10

-5

0

5

10

-5

0

5

10

0

5

10

χ

t

0.5

0.0

-10

-5

0

5

10

0.0

-10

-5

0

5

10

8

0.0 -10 1

Cv

3

2

4

1

0

0 -10

-5

0

5

10

-10

-5

0

5

10

0

-10

-1 -4

-4

E

-5 -6

-6

-7 -2

-8

-8 -10

-5

0

H/Jc

5

10

-10

-5

0

H/Jc

5

10

-10

-5

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 the external field for different values of the temperature (T=J c ¼ 1:5, 2 and 6) and for J Int =J c ¼  1:2, J s =J c ¼ 1:2 and D=J c ¼ 1:2.

40

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

for selected value of the interfacial coupling (J int =J c ¼ 0:03,  1 and  1.5), and for fixed parameters T=J c ¼  1:4, J s =J c ¼ 1:2 and D=J c ¼ 0. We remark that for J int =J c ¼  0:03, the system presents only a central loop with a coercive field points H c ¼ 7 0:15, which involves two steps, due to different ferromagnetic properties of the core and surface shell of the system. We can remark that the curves of the susceptibility and specific heat have four distinct peaks, but the curve of the internal energy presents only two distinct peaks corresponding to the coercive field points H c ¼ 7 0:15 of the hysteresis loop. It is seen that for J int =J c ¼  1 the hysteresis loop changes from one central loop to triple loop. The larger the antiferromagnetic coupling constant is, the more difficult it is for the applied magnetic field to change the direction of the magnetization at the interfacial layers in the core and the shell. This triple hysteresis loop behavior has been seen theoretically in ferromagnetic or ferrimagnetic 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 six distinct peaks, two central corresponding to the central loop, and the outer peaks corresponding to the outer loops. Nevertheless, the curve of the internal energy presents only two distinct peaks corresponding to the central loop and two outer broad maxima corresponding to the outer loops. The outer loops, the outer peaks observed in the curves of the susceptibility and specific heat, and the outer broad maximum obtained in the curve of the internal energy stretch further out horizontally with the decreasing J int =J c . It is implied that the behaviors of the system are ferromagnetic link for smaller jJ int j=J c and antiferromagnetic-like for larger jJ int j=J c . To study the effect of the temperature on the magnetic properties of the system, we present in Fig. 8 the total magnetization, susceptibility, specific heat, and internal energy versus applied field, for selected values of the temperature T=J c (T=J c ¼ 1:5, 2, 6), and for fixed parameters J int =J c ¼  1:2, J s =J c ¼ 1:2 and D=J c ¼ 1:2. As shown in this figure, the size of the central loop reduces and the outer loops disappear when we increase the value of the temperature. We can also notice that the zone between the central peaks observed in the curves of the susceptibility specific heat and internal energy decreases, the outer peaks of the susceptibility and specific heat have been confounded, but the two broad maxima of the internal energy disappear. When T=J c is very important, the hysteresis loop, and the peaks of the susceptibility and the specific heat disappear; however, the internal energy has a broad central maximum. 4. Conclusion In conclusion, we have studied the magnetic properties of the Blume–Capel nanowire. We have investigated for both ferromagnetic and ferrimagnetic cases the influence of the crystal field, the temperature, and the interfacial coupling on the thermal and the hysteresis behaviors of the system. It has been found that for the ferromagnetic case, the hysteresis loop changes from one central loop to double hysteresis loops, and the double peaks observed in the curves of the susceptibility and specific heat become one central peak. It has also been found that the remanent magnetization and coercive field decrease with the increasing temperature. For the ferrimagnetic case, it has been shown that the triple hysteresis loop occurs for the larger antiferromagnetic coupling and that when T increases, the hysteresis loops disappear.

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

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