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Phase diagrams and magnetic properties of a ferrimagnetic cylindrical core/shell spin-1 Ising nanowire B. Boughazi, M. Boughrara n , M. Kerouad Laboratoire de Physique des Matériaux et Modélisation des Systèmes (LP2MS), unité associée au CNRST-URAC 08, Physics Department, Faculty of Sciences, University of Moulay Ismail, B.P.11201, Meknes, Morocco article info Article history: Received 5 June 2013 Received in revised form 7 October 2013 Available online 6 November 2013 Keywords: Ising nanowire Phase diagram Magnetic properties Monte Carlo simulation abstract A Monte Carlo Simulation has been used to study the critical behavior of the magnetic spin-1 Ising nanowire on an hexagonal lattice structure. Our system consists of a ferromagnetic core with two shells, coupled antiferromagnetically with a surface shell. The effects of the crystal eld, the shell and the core/ shell interactions on the behavior of the system are examined. We found that the system exhibits one or two compensation points, the rst and the second order transitions. Moreover, we have obtained that the critical point depends on the shell coupling. & 2013 Elsevier B.V. All rights reserved. 1. Introduction Several years ago, the spin-1 Ising model under crystal eld (BlumeCapel model (BC)) [14], has been the aim of many interesting researches. This model is the subject of a lot of studies because of the two important terms in physics: the anisotropy (crystal eld) and the exchange interaction. Since its introduction, the BC model has been largely studied not only in terms of the theoretical importance but also because of its application to describe several systems, for instance: ternary uid [5,6], the solid liquid gas mixture and the binary uid [7,8], microemulsions [9,10], the semiconductor alloys [11,12] and the conduction electrons model [13]. The effect of the uniaxial anisotropy (D) on the critical behavior of the BC model, has been studied by different techniques: mean eld approximation (MFA) [14], effective eld theory (EFT) [1517], Beth approximation [18], series expansion methods [19], renormalization group theory [20], Monte Carlo simulation (MCS) [21], cluster approximation [22], constant cou- pling approximation [23] and the cluster variational approxima- tion [24,25]. Most of these studies show the existence of the tricritical point when the value of D is sufciently negative. The study of nanoparticle systems has attracted attention not only because of their interest but also because of their technolo- gical applications. The investigation and the explanation of the magnetic properties of magnetic nanowires are extremely impor- tant due to their potential wide range applications, such as high coercivity and perpendicular magnetic anisotropy, ultrahigh den- sity magnetic recording media, sensors, etc. [2631]. Magnetic nanowires have been fabricated by various methods; including electron-beam lithography [32], electroplating into anodized alu- minum [33], polycarbonate track-etched membranes [34], inter- ference lithography [35] and physical deposition into selectively etched semiconducting wafers [36]. On the other hand, theoretical studies of magnetic properties of nanoparticles have been investigated using the effective-eld theory (EFT) with correlations and the Monte Carlo study. The magnetic properties of the two-dimensional antiferromagnetic small particle have been investigated [37]. The results obtained by both methods are qualitatively the same, it was shown that these properties are strongly related to the degree of the particle surface disorder. Zaim et al. have studied the magnetic properties and the hysteresis loops of a single nanocube [38], consisting of a ferromagnetic spin- 1 2 in the core surrounded by a ferromagnetic spin-1 on the shell with antiferromagnetic interface coupling. They have shown that the existence of the compensation point is strongly inuenced by the surface and the interface coupling. Various nanostructures, such as nanowire and nanotube can be modeled by coreshell models and these models can also be solved by EFT, MFA and MCS. Kaneyoshi [39] has studied a cylindrical nanowire composed of two layers in the core and a single layer on the surface. The phase diagrams were examined by using the effective eld theory and the mean eld theory. The effects of the exchange interactions and the transverse magnetic eld on the phase diagrams were examined. It was shown that, the phase diagram of the system is strongly affected by the surface situations. Some characteristic phenomena are found in the phase Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jmmm Journal of Magnetism and Magnetic Materials 0304-8853/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2013.10.052 n Corresponding author. Tel.: þ212 6750 2425 3. E-mail address: [email protected] (M. Boughrara). Journal of Magnetism and Magnetic Materials 354 (2014) 173177

Phase diagrams and magnetic properties of a ferrimagnetic cylindrical core/shell spin-1 Ising nanowire

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Phase diagrams and magnetic properties of a ferrimagnetic cylindricalcore/shell spin-1 Ising nanowire

B. Boughazi, M. Boughrara n, M. KerouadLaboratoire de Physique des Matériaux et Modélisation des Systèmes (LP2MS), unité associée au CNRST-URAC 08, Physics Department, Faculty of Sciences,University of Moulay Ismail, B.P. 11201, Meknes, Morocco

a r t i c l e i n f o

Article history:Received 5 June 2013Received in revised form7 October 2013Available online 6 November 2013

Keywords:Ising nanowirePhase diagramMagnetic propertiesMonte Carlo simulation

a b s t r a c t

A Monte Carlo Simulation has been used to study the critical behavior of the magnetic spin-1 Isingnanowire on an hexagonal lattice structure. Our system consists of a ferromagnetic core with two shells,coupled antiferromagnetically with a surface shell. The effects of the crystal field, the shell and the core/shell interactions on the behavior of the system are examined. We found that the system exhibits one ortwo compensation points, the first and the second order transitions. Moreover, we have obtained that thecritical point depends on the shell coupling.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Several years ago, the spin-1 Ising model under crystal field(Blume–Capel model (BC)) [1–4], has been the aim of manyinteresting researches. This model is the subject of a lot of studiesbecause of the two important terms in physics: the anisotropy(crystal field) and the exchange interaction. Since its introduction,the BC model has been largely studied not only in terms of thetheoretical importance but also because of its application todescribe several systems, for instance: ternary fluid [5,6], the solidliquid gas mixture and the binary fluid [7,8], microemulsions[9,10], the semiconductor alloys [11,12] and the conductionelectrons model [13]. The effect of the uniaxial anisotropy (D) onthe critical behavior of the BC model, has been studied by differenttechniques: mean field approximation (MFA) [14], effective fieldtheory (EFT) [15–17], Beth approximation [18], series expansionmethods [19], renormalization group theory [20], Monte Carlosimulation (MCS) [21], cluster approximation [22], constant cou-pling approximation [23] and the cluster variational approxima-tion [24,25]. Most of these studies show the existence of thetricritical point when the value of D is sufficiently negative.

The study of nanoparticle systems has attracted attention notonly because of their interest but also because of their technolo-gical applications. The investigation and the explanation of themagnetic properties of magnetic nanowires are extremely impor-tant due to their potential wide range applications, such as high

coercivity and perpendicular magnetic anisotropy, ultrahigh den-sity magnetic recording media, sensors, etc. [26–31]. Magneticnanowires have been fabricated by various methods; includingelectron-beam lithography [32], electroplating into anodized alu-minum [33], polycarbonate track-etched membranes [34], inter-ference lithography [35] and physical deposition into selectivelyetched semiconducting wafers [36].

On the other hand, theoretical studies of magnetic properties ofnanoparticles have been investigated using the effective-fieldtheory (EFT) with correlations and the Monte Carlo study. Themagnetic properties of the two-dimensional antiferromagneticsmall particle have been investigated [37]. The results obtainedby both methods are qualitatively the same, it was shown thatthese properties are strongly related to the degree of the particlesurface disorder. Zaim et al. have studied the magnetic propertiesand the hysteresis loops of a single nanocube [38], consisting of aferromagnetic spin-12 in the core surrounded by a ferromagneticspin-1 on the shell with antiferromagnetic interface coupling.They have shown that the existence of the compensation pointis strongly influenced by the surface and the interface coupling.Various nanostructures, such as nanowire and nanotube can bemodeled by core–shell models and these models can also besolved by EFT, MFA and MCS. Kaneyoshi [39] has studied acylindrical nanowire composed of two layers in the core and asingle layer on the surface. The phase diagrams were examined byusing the effective field theory and the mean field theory. Theeffects of the exchange interactions and the transverse magneticfield on the phase diagrams were examined. It was shown that, thephase diagram of the system is strongly affected by the surfacesituations. Some characteristic phenomena are found in the phase

Contents lists available at ScienceDirect

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

Journal of Magnetism and Magnetic Materials

0304-8853/$ - see front matter & 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.jmmm.2013.10.052

n Corresponding author. Tel.: þ212 6750 2425 3.E-mail address: [email protected] (M. Boughrara).

Journal of Magnetism and Magnetic Materials 354 (2014) 173–177

diagram, depending on the ratio of the physical parameters in thesurface shell and the core. In a series of recent works [40–47],Kaneyoshi has also studied the phase diagrams and the magneticproperties of the magnetic nanostructure, such as nanowire,nanotube and nanoparticle by using the effective field theory. Hehas found that the magnetic properties in nanoscaled Isingmaterials are strongly influenced by finite-size, dilution and sur-face effects. Besides these works, higher spin nanowire or nano-tube have been investigated, e.g. spin-1 nanowire [48] andnanotube [49], mixed spin 1/2, 1 nanotube [50]. In the experi-mental area, the magnetic nanowires, such as β�FeOOH, Fe3O4,Co–Cu, Ni–Zn and Co–Pb, have been synthesized and their mag-netic properties have been investigated [51–55]. The authors havemeasured the magnetic moment as a function of the temperaturefor different diameters at a fixed applied fields. It was found thatthe magnetic properties are strongly depending on the diameter ofnanowires and that the Neel transition temperatures are muchlower than that of bulk material.

In this paper, we are interested in studying the influence of thecrystal field and the exchange interaction on the behavior of thespin-1 Ising nanowire on an hexagonal structure. In our analysis,we use Monte Carlo technique according to the heat bath algo-rithm [56].

The paper is organized as follows: In Section 2, we define themodel and we give briefly the definition of the magnetization persite and the initial susceptibility of the cylindrical Ising nanowire.In Section 3, we present the numerical results. Section 4 is devotedto a brief conclusion.

2. Model and formalism

We consider a cylindrical spin-1 Ising nanowire, as depicted inFig. 1, in which the wire consists of a core (the blue circles)surrounded by a surface shell (the black circles). Each site in thefigure is occupied by a spin-1 atom.

The Hamiltonian of the system is given by

H¼ �∑⟨ij⟩JijSiSj�D∑

iS2i ð1Þ

where Si is the spin 1 Ising with Si ¼ 0; 71. Jij ¼ JS is the exchangeinteraction between two nearest-neighbor magnetic atoms at thesurface shell, Jij ¼ J1 is the exchange interaction between the spinsin the surface and the next shell in the core, and Jij ¼ J is theexchange interaction in the core. D is the crystal field.

Our system consists of three shells, namely one shell of thesurface and two shells in the core, the surface shell contains NShell

spins �1, and the core contains NCore spins �1. The total number

of spins in the wire is NT ¼ ðNShellþNCoreÞ.NShell ¼ 12� L;NCore ¼ 7� L, we take L¼300. L denotes the wirelength. We use the Monte Carlo Simulation and we flip the spinsonce a time, according to the heat bath algorithm [56]. 4� 104

Monte Carlo steps were used to obtain each data point in thesystem, after discarding the first 104 steps. The magnetization M ofa configuration is defined by the sum over all the spin values of thelattice sites, the critical temperature is determined from the peakof the susceptibility. The error bars are calculated with a Jackknifemethod [57] by taking all the measurements and grouping them in20 blocks. This error bar is negligible, so it does not appear inour plots.

The total magnetization per site is giving by

MT ¼1NT

ðNCoreMCoreþNShellMShellÞ ð2Þ

Where MCore ¼ 1=NCore� �

∑NCorei Si and MShell ¼ 1=NShell

� �∑NShell

j Sj.The total susceptibility χT is defined by

χT ¼ βNT /M2TS�/MTS

2� �

ð3Þ

with β¼ 1=kBT

3. Results and discussions

In this study, we have examined the phase diagrams (criticaland compensation temperatures) and the magnetic properties ofthe ferrimagnetic cylindrical spin-1 Ising nanowire.

In order to investigate the finite size effect when the systemexhibits the second and the first order transitions, we have plottedthe susceptibility as a function of the temperature for differentlengths of the wire (L¼100, 200, 300, 400 and 500) and forD=J ¼ 0:0, RS ¼ 0:1 and R1 ¼ �1:0 (Fig. 2(a)) and for D=J ¼ �10:0,RS ¼ 3:5 and R1 ¼ �1:0 (Fig. 2(b)). We can see from these figures,that the values of the critical temperature (second (Fig. 2(a)) orfirst order transition (Fig. 2(b))) are independent of the size of thesystem when LZ300. We can notice that the first order transitionis characterized by the discontinuity of the order-parameter, and isthe fact of the presence of the longitudinal anisotropy in thehamiltonian.

Let us examine the influence of the crystal field on the phasediagrams of the system. We have plotted the phase diagrams ofthe nanowire in the (T=J, D=J) plane, when R1 ¼ J1=J ¼ �1 andRS ¼ JS

J ¼ 0:1; 0:5; 1; 1:5; 2:5; 3:5 and 5 (Fig. 3). We show that theordered and disordered phases are separated by a second ordertransition line (square points) for the large positive values of D=Jand by a first order transition line (black points) for the largenegative values of D=J. The second order line is connected to thefirst one at the tricritical point (tricritical points are shown bystars), hence the tricritical line separates the first-order phasetransition regions from the second-order ones. We have alsoshown that, the tricritical line (stars with dashed line) decreasesfirst and then increases with rising RS. It is also seen that, thecritical temperature increases from its minimum value at negativevalues of D=J, which depends on RS, to reach saturation value forlarge positive values of D=J. The critical temperature and thesaturation value increase with RS. In order to confirm the existenceof the first order transition, we have plotted the temperaturedependence of the magnetizations (MCore (square points), MShell

(sphere points) and MTotal (star points)) for RS ¼ 1:5, R1 ¼ �1:0 andD=J ¼ �4:5 (Fig. 4). It is clear that, at low temperatures, themagnetizations of the core, the shell and the total system areMCore ¼ �1:0, MShell ¼ 1:0 and MTotal ¼ 0:263, respectively, andwhen we increase the temperature, the magnetizations undergoa first order transition at TC=J ¼ 0:45.

Fig. 1. Schematic representation of a cylindrical nanowire. The black circles denotethe spin-1 surface shell atoms. The blue circles are the spin-1 atoms constitutingthe core. L denotes the length of the wire. (For interpretation of the references tocolor in this figure caption, the reader is referred to the web version of this article.)

B. Boughazi et al. / Journal of Magnetism and Magnetic Materials 354 (2014) 173–177174

In order to investigate the compensation phenomenon of thesystem, we have plotted in Fig. 5, the variations of the critical andcompensation temperatures versus D=J for the same value of R1 asin Fig. 3 and for two values of RS, namely RS ¼ 0:1 and 0:5.For RS ¼ 0:1, we can see that the compensation point (triangleswith solid line) exists only for D=J4�1:6 and that we can havetwo compensation points in a certain range of D=J(�1:6rD=Jr�1:2). When we increase RS (RS ¼ 0:5), the intervalof D=J where we have a compensation behavior decreases (ForRS ¼ 0:1, we have a compensation phenomenon for D=J4�1:6while for RS ¼ 0:5 a compensation point exists only for�2:5rD

J r�0:5.). The existence of two compensation points,obtained for RS ¼ 0:1, is confirmed in Fig. 6 where we have plottedthe core (square points), the shell (sphere points) and the total(star points) magnetizations of the system versus the temperaturefor RS ¼ 0:1, R1 ¼ �1:0 and D=J ¼ �1:4. It is clear from this figure

that the system exhibits two compensation points (Tk1=J ¼ 0:09and Tk2=J ¼ 0:77).

Fig. 2. The susceptibility versus T=J for different lengths of the wire. RS ¼ 0:1, R1 ¼ �1:0 and D/J¼0 ((a)) and RS ¼ 3:5, R1 ¼ �1:0 and D=J ¼ �10 ((b)).

Fig. 3. The critical temperature versus the crystal field D=J for R1 ¼ �1 and fordifferent values of RS (0.1, 0.5, 1, 1.5, 2.5, 3.5 and 5).

Fig. 4. Temperature dependence of magnetizations for RS ¼ 1:5, R1 ¼ �1:0 andD=J ¼ �4:5.

Fig. 5. The phase diagram in the ðT=J;D=JÞ plane for R1 ¼ �1:0 and for RS ¼ 0:1and 0.5.

B. Boughazi et al. / Journal of Magnetism and Magnetic Materials 354 (2014) 173–177 175

In Fig. 7, we have presented the phase diagrams in the (T=J, RS)plane for D=J ¼ 0:0 and for different values of R1. The criticaltemperature is presented by the black points with solid line andthe compensation one (Tk=J ) by the black squares with solid line.We can remark that Tk exists only for a certain range of RS,and that this range decreases as decreasing R1, for example for

R1 ¼ �0:7, Tk exists for 0oRSo0:63, while for R1 ¼ �1:3, the rangewhere we can have the compensation phenomenon is0oRSo0:27. Concerning the critical temperature, it is seen thatTC=J increases with Rs and jR1j. In order to confirm the existence ofthe compensation phenomenon, we have plotted the magnetiza-tion as a function of the temperature for RS ¼ 0:2, D=J ¼ 0 and fordifferent values of R1 (Fig. 8). From this figure, it is clear that thesystem presents the compensation phenomenon and the values ofthe compensation temperature increases as increasing jR1j.

4. Conclusion

This work was focused on the study of the phase transitionsand the magnetic properties of a spin-1 Ising nanowire with theanisotropy D=J.

We have applied Monte Carlo Simulation based on the heatbath algorithm to study the critical behavior of the system. Theeffects of the parameters RS, R1 and D=J on the phase diagramswere studied. We noted that the ferromagnetic phase and theparamagnetic one are separated by a line of second order transi-tion for large positive values of the uniaxial anisotropy, and a lineof the first order transition for large negative values of D=J.We have shown that the critical temperature increases with increas-ing RS, jR1j and D=J. We have also shown that the tricritical pointsdepend on the exchange interactions RS and the uniaxial anisotropy.Concerning the compensation behavior, we can conclude that; thecompensation temperature increases with increasing jR1j and RS;and that depending on the values of the system parameters, we canhave, zero, one or two compensation points.

Acknowledgment

This work has been supported by the URAC:08, the RS02 of theCNRT Morocco, and the Project No:A/030519/10 financed by A.E.C.I.

References

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0.5 1.0 1.5 2.0 2.5 3.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

Fig. 6. The core, the shell and the total magnetizations versus the temperature forD=J ¼ �1:4, RS ¼ 0:1 and for R1 ¼ �1.

Fig. 7. Variations of the critical and compensation temperatures versus RS forD=J ¼ 0 and for R1 ¼ �0:7, �1.0 and �1.3.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

R1=-0.7

R1= -1

M

T/J

RS=0.2D/J=0.0R1= -1.3

Fig. 8. The magnetization versus the temperature for D=J ¼ 0, RS ¼ 0:2 and forR1 ¼ �1:3, �1 and �0.7.

B. Boughazi et al. / Journal of Magnetism and Magnetic Materials 354 (2014) 173–177176

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