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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 Matriaux et Modlisation des Systmes (LP2MS), Unit Associe 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 eld theory

a b s t r a c t

In this work, we have used Monte Carlo Simulation technique (MCS) and effective eld theory (EFT) tostudy the critical and the compensation behaviors of a ferrimagnetic cylindrical nanowire. The systemconsists of a ferromagnetic spin SA1/2 core and a ferromagnetic spin SB1 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 [14]. This is motivated by numerouspossibilities of their applications in nanotechnology [58]. Thesestructures include different geometric congurations 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 CoPt,CoPd, FePt and FePd alloys [913]. Besides technologicalapplications, the magnetic properties of nanoparticles are scienti-cally interesting research areas since their magnetic propertiesare quite different from those of the bulk and greatly affected bythe particle size [14].

Theoretically, the coreshell model has been accepted to explainmany characteristic phenomena in nanoparticle magnetism[1519]. The same concept has been applied to the investigationof magnetic nanowires and nanotubes [2022]. 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 byL Lc4. The authors have examined the effects of shell couplingand interface coupling on both the compensation and magnetiza-tion proles. 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) coreshell morphology were studied byusing MC Metropolis method [24]. Kaneyoshi has investigatedphase diagrams [25] and magnetizations [26] of the transverseIsing nanowire by using the effective eld theory with correlation(EFT). He has found that the magnetic properties are stronglyinuenced by surface effects and nite 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 byS3/2 spins. In a series of recent works [2933], 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

Contents lists available at ScienceDirect

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

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) 222228

nanotube have been also studied [3133]. 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 eld,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 eld theory.

The outline of this paper is as follow: In Section 2, we denethe model and give briey the formulation of magnetic propertieswithin the Monte Carlo simulation and the effective eld theory.The results and discussions are presented in Section 3, and nallySection 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 Jsi;jSzi S

zj J

m;nszmszn J1

i;mszmS

zi D

iSzi 2 1

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 szm 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 NsNcL. Ns12, Nc7 andL500. 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 ip thespins once a time, according to the heat bath algorithm [38]. 4104 Monte Carlo steps were used to obtain each data point in thesystem, after discarding the rst 104 steps. The magnetization M ofa conguration is dened 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 dened by

M1=2 1

NcLNcL

m 1szm 2

and

M1 1NsL

NsL

i 1Szi 3

The total magnetization per site is dened by

MT 11912M17M1=2 4

The total susceptibility T is dened by

T NT M2T MT 2 5with 1=KBT .

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

For the central site:

ms1 N2

i1 0N4

i2 0CN2i1 C

N4i2 1=2ms1i11=2ms1N2 i11=2ms2i2

1=2ms2N4 i2F0:5N2N42i1 i2; T 6For the rst shell of the core:

ms2 N3

i1 0N1

i2 0N1

i3 0

N1 i3

j3 0N2

i4 0

N2 i4

j4 0CN3i1 C

N1i2 C

N1i3 C

N1 i3j3 C

N2i4 C

N2 i4j4 1=2ms2i1

1=2ms2N3 i11=2ms1i21=2ms1N1 i21q1i3q1m1j3

q1m1N1 i3 j31q2i4q2m2j4q2m2N2 i4 j4F0:5N3N12i1 i2R1N1N2 i3 i42j3 j4; T 7

For the surface shell:

m1 N2

i1 0

N21

j1 0N2

i2 0

N2 i2

j2 0N1

i3 0CN2i1 C

N2 i1j1 C

N2i2 C

N2 i2j2 C

N1i3 1q2i1q2m2j1

q2m2N2 i1 j11q1i2q1m1j2q1m1N2 i2 j21=2ms2i3

1=2ms2N1 i3G1RS2N2 i12j1 i22j2R1=2N12i3;D; T8

q1 N2

i1 0

N21

j1 0N2

i2 0

N2 i2

j2 0N1

i3 0CN2i1 C

N2 i1j1 C

N2i2 C

N2 i2j2 C

N1i3 1q2i1q2m2j1

q2m2N2 i1 j11q1i2q1m1j2q1m1N2 i2 j21=2ms2i3

1=2ms2N1 i3G2RS2N2 i12j1 i22j2R1=2N12i3;D; T9

m2 N2

i1 0

N21

j1 0N2

i2 0

N2 i2

j2 0N2

i3 0CN2i1 C

N2 i1j1 C

N2i2 C

N2 i2j2 C

N2i3 1q1i1q1m1j1

q1m1N2 i1 j11q2i2q2m2j2q2m2N2 i2 j21=2ms2i3

1=2ms2N2 i3G1RS2N2 i12j1 i22j2R1=2N22i3;D; T10

q2 N2

i1 0

N21

j1 0N2

i2 0

N2 i2

j2 0N2

i3 0CN2i1 C

N2 i1j1 C

N2i2 C

N2 i2j2 C

N2i3 1q1i1q1m1j1

q1m1N2 i1 j11q2i2q2m2j2q2m2N2 i2 j21=2ms2i3

1=2ms2N2 i3G2RS2N2 i12j1 i22j2R1=2N22i3;D; T11

where q1 and q2 are the quadrupolar moments, R1 J1=J andRs Js=J. N11, N22, N34 and N46 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) 222228 223

coordination numbers.

G1X;D; T 2 sinh

XKBT

2 cosh XKBT

exp DKBT

12

G2X;D; T 2cosh

XKBT

2 coshX

KBT

exp D

KBT

13

FX; T 12tanh

XKBT

14

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

MT 119ms16ms2m1m2 15

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: AMM where M ms1;ms2;m1;m2

The phase transition temperature Tc/J is obtained from theequation: det(A - I)0

On the other hand, the compensation temperature Tk/J, if itdoes exist in the system, can be obtained by introducing thecondition: M0 into Eqs. (4) and (15).

The rst 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 rst order transition line connectto each other is called the tricritical point (), the point at whichtwo rst order lines emerge from the end of the second orderphase transition line is called the critical end point .

The second order transitions, the rst 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 0:0and RS0.2 (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 R10.05 and D=J 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 (RSC0.32 for MCS and RSC0.5 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 inuence 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 T=J;R1 plane for RS0.2 and D=J 0:0 (a) obtained by MCS and (b) obtained by EFT.

M. Boughrara et al. / Journal of Magnetism and Magnetic Materials 360 (2014) 222228224

uniaxial anisotropy for RS0.5 and R10.05 (Fig. 4) (a) obtainedby MCS and (b) obtained by EFT. In both cases, we can remark thatthere are two types of transitions lines. The rst 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 rst order transition line separating the twoordered phases designated by (1/2; 1) and (1/2; 0); it starts

from T/J0.0 for a value of D/J(D/J1.64 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 rst order line and the compensation one, are (T/J0.35, D/J1.02) for the results obtained by MCS and (T/J0.5, D/J1.02) 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 T=J;RS plane for R10.05 and D=J 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 T=J;D=J plane for RS0.5 and R10.05 (a) obtained by MCS and (b) obtained by EFT.

M. Boughrara et al. / Journal of Magnetism and Magnetic Materials 360 (2014) 222228 225

point starts from the end of the rst 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 conrm the existence of the compensation phe-nomenon and the rst order transition, we have plotted the total,the surface and the bulk magnetizations versus the temperaturefor RS0.5, R10.05 and for D/J0.95 (Fig. 5) and D/J1.1

(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 rst order transition, due to thejump discontinuity presented by the surface magnetization.

Finally, we have studied the effect of the uniaxial anisotropyD=J 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 prole of the total, the bulk and the surface magnetizations as a function of the temperature for R10.05, RS0.5 and D=J 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 prole of the total, the bulk and the surface magnetizations as a function of the temperature for R10.05, RS0.5 and D=J 1:1 (a) obtained by MCSand (b) obtained by EFT.

M. Boughrara et al. / Journal of Magnetism and Magnetic Materials 360 (2014) 222228226

and the compensation temperatures versus the uniaxial aniso-tropy for R10.2 and for different values of RS. Fig. 7(a) isobtained by MCS and Fig. 7(b) by EFT. For RS0.2, concerning bothmethods, the critical temperature is slightly inuenced by thechange of D/J, and we notice that the critical temperature remainsconstant below a critical values of D/J (DC/J0.0) 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 rst order transition lineseparating the (1/2,1) and (1/2,0) states at low temperatureregion. The rst order transition line starts from zero forD/J0.88 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=J40:62 and 0:62oD=Jo0:05 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 (TCcons0.71 for MCS andTCcons1.17 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 rst order transition and the compensationphenomenon, we can see that the range of D/J, where we have therst 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 rstorder transition line starts from T/J0.0, decreases as increasingRS. For RS1.5, 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 rst 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 rst second orderline is connected to the rst order one, at the tricritical point whilethe second one is connected to the rst 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 RS1.5. 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 eld 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 rst 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 nanced by A. E. C. I.

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0.0

0.5

1.0

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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) 222228 227

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M. Boughrara et al. / Journal of Magnetism and Magnetic Materials 360 (2014) 222228228

The phase diagrams and the magnetic properties of a ferrimagnetic mixed spin 1/2 and spin 1 Ising nanowireIntroductionModel and formalismResults and discussionsConclusionAcknowledgmentsReferences