The phase diagrams and the magnetic properties of a ferrimagneticmixed spin 1/2 and spin 1 Ising nanowire

M. Boughrara n, M. Kerouad, A. ZaimLaboratoire 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 13 April 2013Received in revised form4 December 2013Available online 19 February 2014

Keywords:NanowireCompensation pointIsing modelMonte Carlo simulationEffective field theory

a b s t r a c t

In this work, we have used Monte Carlo Simulation technique (MCS) and effective field theory (EFT) tostudy the critical and the compensation behaviors of a ferrimagnetic cylindrical nanowire. The systemconsists of a ferromagnetic spin SA¼1/2 core and a ferromagnetic spin SB¼1 surface shell coupled withan antiferromagnetic interlayer coupling J1 to the core. The effects of the uniaxial anisotropy, the shellcoupling and the interface negative coupling on both the critical and compensation temperatures areinvestigated.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

During the last few years, one could observe a growing interestin the experimental and theoretical investigations of various newstructures at nano-scale [1–4]. This is motivated by numerouspossibilities of their applications in nanotechnology [5–8]. Thesestructures include different geometric configurations such as full-erenes, nanotubes and nanowires. The exploration of differentproperties of these objects opens wide perspectives for applica-tions. Due to their potential application in high density magneticrecording media, high attention is paid to the magnetic nanowireand nanotube based on the transition metals, such as Co–Pt,Co–Pd, Fe–Pt and Fe–Pd alloys [9–13]. Besides technologicalapplications, the magnetic properties of nanoparticles are scienti-fically interesting research areas since their magnetic propertiesare quite different from those of the bulk and greatly affected bythe particle size [14].

Theoretically, the core–shell model has been accepted to explainmany characteristic phenomena in nanoparticle magnetism[15–19]. The same concept has been applied to the investigationof magnetic nanowires and nanotubes [20–22]. In particular, themagnetic properties of a nanocube [23], which consists of aferromagnetic spin 1/2 core and a ferromagnetic spin 1 shellcoupled with an antiferromagnetic interlayer coupling Jint to thecore, have been investigated by the use of Monte Carlo simulation

(MCS). Some characteristic feature have been obtained in it.The system consists of Lc layers in the core and two layers in thespin 1 shell, so that the total number of layers L is given byL¼ Lcþ4. The authors have examined the effects of shell couplingand interface coupling on both the compensation and magnetiza-tion profiles. They have observed that as the shell thicknessincreases, both critical and compensation temperatures of thesystem increase and reach a saturation values for high values ofthe thickness. The magnetic properties of the ferromagnetic (FM)-antiferromgnetic (AFM) core–shell morphology were studied byusing MC Metropolis method [24]. Kaneyoshi has investigatedphase diagrams [25] and magnetizations [26] of the transverseIsing nanowire by using the effective field theory with correlation(EFT). He has found that the magnetic properties are stronglyinfluenced by surface effects and finite size. Keskin et al. [27] havestudied the hysteresis behaviors of the cylindrical Ising nanowire byEFT. They have obtained phase transition temperatures and foundthat the results of hysteresis behaviors of the nanowires are in goodagreement with both theoretical and experimental results. Inanother work, Zaim and Kerouad [28] have simulated a sphericalparticle consisting of a ferromagnetic spin 1/2 core and a ferro-magnetic spin 1 or 3/2 shell with antiferromagnetic interfacecoupling. They have focused on the effect of the shell and theinterface coupling and found that two compensation temperaturescan occur when the sites of the shell sublattice are occupied byS¼3/2 spins. In a series of recent works [29–33], the hysteresisbehavior and the susceptibility of the nanowire have been investi-gated by using EFT [29,30], the effect of the diluted surface on thephase diagrams and the magnetic properties of the nanowire and

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Journal of Magnetism and Magnetic Materials

http://dx.doi.org/10.1016/j.jmmm.2014.02.0430304-8853 & 2014 Elsevier B.V. All rights reserved.

n Corresponding author.E-mail address: boughrara_mourad@yahoo.fr (M. Boughrara).

Journal of Magnetism and Magnetic Materials 360 (2014) 222–228

nanotube have been also studied [31–33]. Beside these, higher spinnanowire or nanotube have been investigated, e.g. spin-1 nanowire[34,35] and nanotube [36], mixed spin 1/2, 1 nanotube [37].

The aim of this work is to study the effects of the crystal field,the shell and the interface coupling on the phase diagrams and themagnetic properties of a cylindrical ferrimagnetic nanowire with aspin �1/2 core surrounded by a spin �1 shell layer. In our analysiswe use Monte Carlo (MC) technique according to the heat bathalgorithm [38] and compare the simulation results with those ofthe effective field theory.

The outline of this paper is as follow: In Section 2, we definethe model and give briefly the formulation of magnetic propertieswithin the Monte Carlo simulation and the effective field theory.The results and discussions are presented in Section 3, and finallySection 4 is devoted to the conclusion.

2. Model and formalism

We consider a ferrimagnetic cylindrical nanowire consisting ofa spin 1/2 ferromagnetic core which is surrounded by a spin �1ferromagnetic shell layer. At the interface, we have an antiferro-magnetic interaction between core and shell spins. A cross-sectionof the wire is depicted in Fig. 1. The Hamiltonian of the system isgiven by

H ¼ � Js∑i;jSzi S

zj � J∑

m;nszmsz

n� J1∑i;mszmS

zi �D∑

iðSzi Þ2 ð1Þ

where Js is the exchange interaction between two nearest neigh-bor magnetic atoms at the surface shell, J is the exchangeinteraction in the core and J1 is the exchange interaction betweenthe spins Szi in the surface shell and the spins sz

m in the next shellin the core. D represents the single ion anisotropy terms of thesurface shell sublattice.

Our system consists of three shells, namely one shell of thesurface and two shells in the core; the surface shell contains Ns � Lspins �1, and the core contains Nc � L spins �1/2. The totalnumber of spins in the wire is NT ¼ ðNsþNcÞL. Ns¼12, Nc¼7 andL¼500. Ns and Nc are the spin numbers of the nanowire cross-section, of the surface and of the core, respectively. L denotes thewire length's. We use the Monte Carlo Simulation and we flip thespins once a time, according to the heat bath algorithm [38]. 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 [40] by taking all the measurements and grouping them in20 blocks. This error bar is negligible, so it does not appear inour plots.

The sublattice magnetizations per site in the core and in theshell surface are defined by

M1=2 ¼1

NcL∑NcL

m ¼ 1szm ð2Þ

and

M1 ¼1NsL

∑NsL

i ¼ 1Szi ð3Þ

The total magnetization per site is defined by

MT ¼119

ð12M1þ7M1=2Þ ð4Þ

The total susceptibility χT is defined by

χT ¼ βNT ð⟨M2T ⟩� ⟨MT ⟩

2Þ ð5Þwith β¼ 1=KBT .

On the other hand, in the framework of the well knowneffective field theory, based on the use of a probability distributiontechnique [39], the longitudinal site order parameters are given by

For the central site:

ms1 ¼ ∑N2

i1 ¼ 0∑N4

i2 ¼ 0CN2i1 CN4

i2 ð1=2�ms1Þi1ð1=2þms1ÞN2� i1ð1=2�ms2Þi2

�ð1=2þms2ÞN4� i2Fð0:5ððN2þN4Þ�2ði1þ i2ÞÞ; TÞ ð6ÞFor the first shell of the core:

ms2 ¼ ∑N3

i1 ¼ 0∑N1

i2 ¼ 0∑N1

i3 ¼ 0∑

N1� i3

j3 ¼ 0∑N2

i4 ¼ 0∑

N2� i4

j4 ¼ 0CN3i1 CN1

i2 CN1i3 CN1� i3

j3 CN2i4 CN2� i4

j4 ð1=2�ms2Þi1

�ð1=2þms2ÞN3� i1ð1=2�ms1Þi2ð1=2þms1ÞN1� i2ð1�q1Þi3ðq1�m1Þj3

�ðq1þm1ÞN1� i3� j3ð1�q2Þi4ðq2�m2Þj4ðq2þm2ÞN2� i4� j4

�Fð0:5ðN3þN1�2ði1þ i2ÞÞþR1ðN1þN2� i3� i4�2ðj3þ j4ÞÞ; TÞ ð7ÞFor the surface shell:

m1 ¼ ∑N2

i1 ¼ 0∑

N2�1

j1 ¼ 0∑N2

i2 ¼ 0∑

N2� i2

j2 ¼ 0∑N1

i3 ¼ 0CN2i1 CN2� i1

j1 CN2i2 CN2� i2

j2 CN1i3 ð1�q2Þi1ðq2�m2Þj1

�ðq2þm2ÞN2� i1� j1ð1�q1Þi2ðq1�m1Þj2ðq1þm1ÞN2� i2� j2ð1=2�ms2Þi3

�ð1=2þms2ÞN1� i3G1ðRSð2N2� i1�2j1� i2�2j2ÞþðR1=2ÞðN1�2i3Þ;D; TÞð8Þ

q1 ¼ ∑N2

i1 ¼ 0∑

N2�1

j1 ¼ 0∑N2

i2 ¼ 0∑

N2� i2

j2 ¼ 0∑N1

i3 ¼ 0CN2i1 CN2� i1

j1 CN2i2 CN2� i2

j2 CN1i3 ð1�q2Þi1ðq2�m2Þj1

�ðq2þm2ÞN2� i1� j1ð1�q1Þi2ðq1�m1Þj2ðq1þm1ÞN2� i2� j2ð1=2�ms2Þi3

�ð1=2þms2ÞN1� i3G2ðRSð2N2� i1�2j1� i2�2j2ÞþðR1=2ÞðN1�2i3Þ;D; TÞð9Þ

m2 ¼ ∑N2

i1 ¼ 0∑

N2�1

j1 ¼ 0∑N2

i2 ¼ 0∑

N2� i2

j2 ¼ 0∑N2

i3 ¼ 0CN2i1 CN2� i1

j1 CN2i2 CN2� i2

j2 CN2i3 ð1�q1Þi1ðq1�m1Þj1

�ðq1þm1ÞN2� i1� j1ð1�q2Þi2ðq2�m2Þj2ðq2þm2ÞN2� i2� j2ð1=2�ms2Þi3

�ð1=2þms2ÞN2� i3G1ðRSð2N2� i1�2j1� i2�2j2ÞþðR1=2ÞðN2�2i3Þ;D; TÞð10Þ

q2 ¼ ∑N2

i1 ¼ 0∑

N2�1

j1 ¼ 0∑N2

i2 ¼ 0∑

N2� i2

j2 ¼ 0∑N2

i3 ¼ 0CN2i1 CN2� i1

j1 CN2i2 CN2� i2

j2 CN2i3 ð1�q1Þi1ðq1�m1Þj1

�ðq1þm1ÞN2� i1� j1ð1�q2Þi2ðq2�m2Þj2ðq2þm2ÞN2� i2� j2ð1=2�ms2Þi3

�ð1=2þms2ÞN2� i3G2ðRSð2N2� i1�2j1� i2�2j2ÞþðR1=2ÞðN2�2i3Þ;D; TÞð11Þ

where q1 and q2 are the quadrupolar moments, R1 ¼ J1=J andRs ¼ Js=J. N1¼1, N2¼2, N3¼4 and N4¼6 denote respectively the

Fig. 1. Schematic representation of a cross section of a cylindrical nanowire. Solidcircles indicate spin �1 atoms at the surface shell and open circle are spin �1/2atoms constituting the core.

M. Boughrara et al. / Journal of Magnetism and Magnetic Materials 360 (2014) 222–228 223

coordination numbers.

G1ðX;D; TÞ ¼2 sinh

XKBT

� �

2 coshð XKBT

Þþexp � DKBT

� � ð12Þ

G2ðX;D; TÞ ¼2cosh

XKBT

� �

2 coshX

KBT

� �þexp � D

KBT

� � ð13Þ

FðX; TÞ ¼ 12tanh

XKBT

� �ð14Þ

KB is the Boltzman constant.The averaged total magnetization M of the system is given by

MT ¼119

ðms1þ6ðms2þm1þm2ÞÞ ð15Þ

In the vicinity of the transition temperature (Tc), the layerquadrupolar moments qi-q0i such as q0i is the solution ofEqs. (9) and (11) for msi-0 and mi-0. To obtain the criticaltemperature, we expand the right hand sides of the equationsgiving the magnetizations of different shells of the nanowire andwe consider only the linear terms. This leads to a matrix equationof the type: AM¼M where M¼ ðms1;ms2;m1;m2Þ

The phase transition temperature Tc/J is obtained from theequation: det(A - I)¼0

On the other hand, the compensation temperature Tk/J, if itdoes exist in the system, can be obtained by introducing thecondition: M¼0 into Eqs. (4) and (15).

The first order transition is obtained from the magnetizationcurves. At this temperature, these curves present a jump discon-tinuities.

It should be mentioned that the point at which the beginningof the second and the end of the first order transition line connectto each other is called the tricritical point (○), the point at whichtwo first order lines emerge from the end of the second orderphase transition line is called the critical end point ðΔÞ.

The second order transitions, the first order transitions and thecompensation points are presented by the squares with solid line,the black points with solid line and the stars with solid line,respectively.

3. Results and discussions

In this section, we examine the phase diagrams and the tem-perature dependencies of the magnetic properties of the system forsome selected values of Hamiltonian parameters. The phase diagramsare examined only for the case of R1o0 (ferrimagnetic case).

Fig. 2 represents the variation of the critical and compensationtemperatures on the antiferromagnetic interface couplingbetween core and shell spins (R1) of the nanowire for D=J ¼ 0:0and RS¼0.2 (a) obtained by MCS and (b) obtained by EFT.We observe that in both cases, the critical and compensationtemperatures increase while increasing jR1j. The compensationtemperature exists only in a certain range of jR1j. The range of jR1jwhere we have this phenomenon is 0o jR1jo0:5 for the resultsobtained by EFT and 0o jR1jo0:18 for those obtained by MCS. It isobvious to notice that this range is much smaller with MCS.

In Fig. 3, we have plotted the phase diagrams in the (T=J, RS)plane for R1¼�0.05 and D=J ¼ 0:0. Fig. 3(a) and (b) gives MCS andEFT results, respectively. In both cases, we observe that we have thesame topology of the phase diagrams (curie temperature); that is,when we increase RS, the critical temperature remains constantbelow a critical value of RS (RSC¼0.32 for MCS and RSC¼0.5 for EFT)and then increases linearly with RS. Concerning the compensationtemperature, it is seen that the system can exhibit a compensationphenomenon in a certain range of RS below its critical value. We canalso see that the compensation temperature increases linearly withRS, and hence the difference between the results of the MCS and theEFT is in the range of RS where we can have the compensationphenomenon (0r jRSjo0:31 for MCS and 0r jRSjo0:5 for EFT).It is important to notice that a similar behavior has been found forthe ferrimagnetic superlattice with disordered interface [41].

Let us examine the influence of the uniaxial anisotropy onthe phase diagrams of the system. We have plotted the variationsof the critical and the compensation temperatures versus the

0.00.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

0.0-0.4 -0.8 -1.2 -1.6 -2.0 -0.4 -0.8 -1.2 -1.6 -2.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

Fig. 2. The phase diagram in the ðT=J;R1Þ plane for RS¼0.2 and D=J ¼ 0:0 (a) obtained by MCS and (b) obtained by EFT.

M. Boughrara et al. / Journal of Magnetism and Magnetic Materials 360 (2014) 222–228224

uniaxial anisotropy for RS¼0.5 and R1¼�0.05 (Fig. 4) (a) obtainedby MCS and (b) obtained by EFT. In both cases, we can remark thatthere are two types of transitions lines. The first one is the secondorder transition line where the critical temperature remainsconstant below a critical values of D/J and then increases with D/J to reach a saturation value for large positive values of D/J. Thesecond one is the first order transition line separating the twoordered phases designated by (�1/2; 1) and (�1/2; 0); it starts

from T/J¼0.0 for a value of D/J(D/J¼�1.64 for both methods), andthen increases speedily with increasing the anisotropy to termi-nate at a point which is connected to the compensation tempera-ture line. The coordinates of the point, which is the connectionbetween the first order line and the compensation one, are (T/J¼0.35, D/J¼�1.02) for the results obtained by MCS and (T/J¼0.5, D/J¼�1.02) for those obtained by EFT. Concerning the compensa-tion phenomenon, we can observe that the line of the compensation

0

1

2

3

4

0.0 0.4 0.8 1.2 1.6 2.0 0.0 0.4 0.8 1.2 1.6 2.00

1

2

3

4

Fig. 3. The phase diagram in the ðT=J;RSÞ plane for R1¼�0.05 and D=J ¼ 0:0 (a) obtained by MCS and (b) obtained by EFT.

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

-8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Fig. 4. The phase diagram in the ðT=J;D=JÞ plane for RS¼0.5 and R1¼�0.05 (a) obtained by MCS and (b) obtained by EFT.

M. Boughrara et al. / Journal of Magnetism and Magnetic Materials 360 (2014) 222–228 225

point starts from the end of the first order line and extends to connectto the second order transition one. The results obtained, then, by EFTand MCS are qualitatively the same.

In order to confirm the existence of the compensation phe-nomenon and the first order transition, we have plotted the total,the surface and the bulk magnetizations versus the temperaturefor RS¼0.5, R1¼�0.05 and for D/J¼�0.95 (Fig. 5) and D/J¼�1.1

(Fig. 6). Fig. 6(a) is obtained by MCS and (b) by EFT. It is clear fromFig. 5 that the system exhibits a compensation phenomenon. Fig. 6shows that the system presents a first order transition, due to thejump discontinuity presented by the surface magnetization.

Finally, we have studied the effect of the uniaxial anisotropyðD=JÞ and the surface shell coupling (RS) on the behavior of thesystem. In Fig. 7, we have presented the variation of the critical

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 5. The profile of the total, the bulk and the surface magnetizations as a function of the temperature for R1¼�0.05, RS¼0.5 and D=J ¼ �0:95 (a) obtained by MCSand (b) obtained by EFT.

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 6. The profile of the total, the bulk and the surface magnetizations as a function of the temperature for R1¼�0.05, RS¼0.5 and D=J ¼ �1:1 (a) obtained by MCSand (b) obtained by EFT.

M. Boughrara et al. / Journal of Magnetism and Magnetic Materials 360 (2014) 222–228226

and the compensation temperatures versus the uniaxial aniso-tropy for R1¼�0.2 and for different values of RS. Fig. 7(a) isobtained by MCS and Fig. 7(b) by EFT. For RS¼0.2, concerning bothmethods, the critical temperature is slightly influenced by thechange of D/J, and we notice that the critical temperature remainsconstant below a critical values of D/J (DC/J¼0.0) and thenincreases with a weak slope to reach rapidly a saturation valuefor D=J41; this slope is more important for the results obtainedby MCS. We can also see that we have a first order transition lineseparating the (�1/2,1) and (�1/2,0) states at low temperatureregion. The first order transition line starts from zero forD/J¼�0.88 and extends to connect to the compensation line.Concerning the compensation phenomenon, it is remarked thatthe system exhibits the compensation phenomenon for a certainrange of D/J (for EFT we have the compensation phenomenon forD=J4�0:62 and �0:62oD=Jo�0:05 is the range where we havethe compensation temperature for MCS method). With increasingthe values of RS, for both methods, it is observed that the criticaltemperature remains constant (TCcons¼0.71 for MCS andTCcons¼1.17 for EFT) below a critical value of D/J (DC/J) whichdecreases when we increase RS. For D=J4DC=J, the critical tem-perature increases with increasing D/J to reach saturation valuesfor large positive values of D/J, this saturation value increases withRS. Concerning the first order transition and the compensationphenomenon, we can see that the range of D/J, where we have thefirst order transition, increases with increasing RS and those wherewe have the compensation phenomenon decrease to disappear forRS41. It is also noticed that the values of D/J, at which the firstorder transition line starts from T/J¼0.0, decreases as increasingRS. For RS¼1.5, we have two second order lines, one separating the(�1/2; 1) region and the disordered phase in the high tempera-ture region and the second separating the (�1/2; 0) phase fromthe disordered one in the low temperature region and a first orderline separating in its high temperature region the (�1/2; 1) phasefrom the disordered one and in its low temperature region the twoordered phases (�1/2; 1) and (�1/2; 0). The first second orderline is connected to the first order one, at the tricritical point whilethe second one is connected to the first order line at the critical

end point. It is worthwhile to mention that the phase diagramshave the same topology for both methods. In order to clarify theexistence of the critical end point and the two segments of thesecond order transition line, we have plotted in the inset of Fig. 7(a) and (b) the phase diagrams only for RS¼1.5. It is clear that wehave two segments of second order transition line and the criticalend point is depicted at the end of the second order transition lineseparating the (�1/2; 0) and the disordered phase.

4. Conclusion

In this work, we have applied the Monte Carlo simulation andthe effective field theory to study the magnetic properties and thephase diagrams of a ferrimagnetic cylindrical nanowire. We havestudied the effect of the uniaxial anisotropy, the surface and theinterface coupling on the critical and the compensation behaviors.We can see that the results obtained by the two methods have thesame topology and those obtained by the MCS are smaller thanthose of the EFT. We have shown that, depending on the values ofRS, R1 and D/J, the system can exhibit a compensation point. It wasfound that the system presents very rich critical behaviors, whichincludes the first and second order phase transitions. Thus thetricritical point and the critical end point are also observed.

Acknowledgments

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

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0.0

0.5

1.0

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2.0

2.5

3.0

3.5

4.0

-8 -6 -4 -2 4 6 8 -8 -6 -4 -20 2 0 2 4 6 8 100

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M. Boughrara et al. / Journal of Magnetism and Magnetic Materials 360 (2014) 222–228 227

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