The hysteresis behavior of an Ising nanowire with core/shellmorphology: Monte Carlo treatment

B. Boughazi, M. Boughrara n, M. KerouadLaboratoire 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 3 November 2013Received in revised form3 March 2014Available online 15 March 2014

Keywords:Monte Carlo simulationNanowireHysteresis behavior

a b s t r a c t

We have used Monte Carlo Simulations (MCS) to study the hysteresis behavior of the magnetic nanowirewith core/shell morphology described by the spin 1

2 Ising particles in the core and the spin 32 Ising

particles in the surface shell. The hysteresis curves are obtained for different temperatures. We find thatthe hysteresis loop areas decrease when the temperature increases and the hysteresis loops disappear atcertain temperatures. Barkhausen jumps are observed for the ferromagnetic nanowire system. Anunusual form of triple hysteresis behaviors is observed for the ferrimagnetic nanowire system. Thethermal behaviors of the coercivity and the remanent magnetization are also investigated.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

In recent years, magnetic nanoparticle systems have been thesubject of a large number of experimental and theoretical studies,because of their great potential for technological applications, such asthe high-density magneto-optical recording [1,2], the ultrahigh-density magnetic storage devices [3], sensors [4], permanent magnets[5], and medical applications [6]. These systems have been studied bya variety of techniques, including variational cumulant expansion(VCE) method [7], mean-field approximation and effective-field theory[8], Green's function technique [9] and Monte Carlo simulation [10].

Hysteresis is well known in ferromagnetic materials. When anexternal magnetic field is applied to a ferromagnet, a ferromag-netic material absorbs some of the external field. Even when theexternal field is removed, the magnet will retain some field.Magnetic hysteresis and magnetic relaxation are two relatedphenomena, they are present in all stages of the development ofmagnetism and in different branches of technology. We should alsomention that the hysteresis properties (hysteresis area, coercivityand remanence) are very important in magnetic recording media[11]. The hysteresis behaviors of maghemite nanoparticles have beenstudied by Monte Carlo Simulations [12]. Dynamics and scaling oflow-frequency hysteresis loops in nanomagnets are investigated bynumerically solving the Landau–Lifshitz–Gilbert equation [13]. Hys-teresis loops for spin-1 Ising model of noninteracting nanoparticleshave been studied with the pair approximation method [14]. Thehysteresis behaviors of the cylindrical Ising nanowire for both

ferromagnetic and antiferromagnetic interactions between the shelland the core have been investigated by the use of the EFT, as well asthe temperature dependence of the coercivities and remanentmagnetizations [15]. The authors have observed that the results of

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Journal of Magnetism and Magnetic Materials

Fig. 1. Schematic representation of a cross-section of the nanowire. Open circlesindicate the magnetic Ising particles at the surface shell and solid circles are themagnetic Ising particles constituting the core.

http://dx.doi.org/10.1016/j.jmmm.2014.03.0370304-8853/& 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 363 (2014) 26–33

the hysteresis behaviors of the nanowires are in good agreementwith both theoretical and experimental results as well as the thermalbehavior of coercivities and remanent magnetizations. In Ref. [16],the authors have studied the hysteresis loops and the susceptibilityof a ferromagnetic or ferrimagnetic bilayer system consisting of twomagnetic monolayers (A and B) with different spins ðSA ¼ 1

2 andSB ¼ 3

2Þ by using the effective field theory. They have shown that thetype of the hysteresis loops can be changed depending on thetemperature and the sign of the exchange interaction between twonearest-neighbors magnetic atoms in the sublattices A and B. In an

other work, Zaim et al. [17] have investigated the effects of the shellcoupling and the antiferromagnetic interface coupling on the beha-vior of the hysteresis loops and the compensation temperatures ofan Ising ferrimagnetic core/shell nanocube by the use of the MonteCarlo simulations. It is shown that as increasing the absolute ratio ofthe core and interface coupling; the hysteresis curve changes fromone central loop to triple loops.

In this work, we are interested in studying the influence of thecrystal field and the exchange interaction on the behavior of thehysteresis loops and the compensation temperatures of a smallparticle on a hexagonal lattice. In our analysis, we use Monte CarloSimulations (MCS) according to the heat bath algorithm [18].

The paper is organized as follows: In Section 2, we define themodel and we give a brief definition of the magnetization per siteand the initial susceptibility of our system. The numerical resultsand discussions are presented in Section 3. Section 4 is devoted toa brief conclusion.

2. Model and formalism

We consider a nanowire Ising system, consisting of twomagnetic subsystems core and shell with different spins S¼ 1

2and S¼ 3

2 in a longitudinal magnetic field and in the presence ofthe crystal field in the shell, a cross section of the system is shownin Fig. 1. The sites of the core are occupied by the spins si ¼ 1

2,while those of the shell are occupied by the spins Si ¼ 3

2. TheHamiltonian of the system is given by

H¼ � J∑⟨ij⟩sisj� JS∑

⟨ij⟩SiSj� J1∑

⟨ij⟩Sisj�D∑

⟨i⟩S2i �h ∑

⟨i⟩Siþ∑

⟨i⟩si

!ð1Þ

The first three sums are carried out only over nearest-neighborspairs. J, JS and J1 are the exchange interactions constants betweentwo nearest-neighbors magnetic Ising particles in the core, theshell and between the core and the shell, respectively. D isthe crystal field acting on the shell Ising particles. h representsthe external longitudinal magnetic field.

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

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.80.00

0.02

0.04

0.06

0.08

Fig. 2. The magnetic susceptibility versus T=J for R1 ¼ 0:3, RS ¼ 1:0, D=J ¼ �2:5 andfor different lengths of the nanowire.

0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

1.5

0.0 0.4 0.8 1.20.0

0.5

1.0

1.5

Fig. 3. The temperature dependence of the core, the shell and the total magnetizations for RS ¼ 1 and R1 ¼ 0:3; (a) D=J ¼ �1:5 and (b) D=J ¼ �2:5.

B. Boughazi et al. / Journal of Magnetism and Magnetic Materials 363 (2014) 26–33 27

spins �32, and the core contains NCore spins �1

2. The total numberof spins in the wire is NTotal ¼NCoreþNShell. NCore ¼ 7� L, NShell ¼12� L and L¼500. NCore and NShell are the spin numbers of thecore and of the surface, respectively, and L denotes the wirelengths. We use the Monte Carlo Simulation and we flip the spinsonce a time, according to the heat bath algorithm [18]. 4�104

Monte Carlo steps were used to obtain each data point in the

system, after discarding the first 2�104 steps. The magnetizationM of a configuration is defined by the sum over all the spin valuesof the lattice sites.

The total magnetization per site is given by

MTotal ¼1

NTotalðNCoreMCoreþNShellMShellÞ ð2Þ

-3 -2 -1 0 1 2 3-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-2 -1 0 1 2-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-2 -1 0 1 2-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-2 -1 0 1 2-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-2 -1 0 1 2

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Fig. 4. The hysteresis loops of the nanowire for D=J ¼ �2:5, RS ¼ 1, R1 ¼ 0:3 and for different values of the temperature. (a) T=J ¼ 0:2, (b) T=J ¼ 0:3, (c) T=J ¼ 0:4,(d) T=J ¼ 0:6 and (e) T=J ¼ 0:8.

B. Boughazi et al. / Journal of Magnetism and Magnetic Materials 363 (2014) 26–3328

whereMCore ¼ 1=NCore∑NCorei si andMShell ¼ 1=NShell∑

NShellj Sj. and the

total susceptibility is defined as

χTotal ¼ βNTotalð⟨MTotal⟩2þ ⟨M2

Total⟩Þ ð3Þ

with β¼ 1=KBT

3. Results and discussions

In order to investigate the hysteresis behaviors of the cylind-rical Ising nanowire at different temperatures, we first examinethe behavior of the magnetizations as a function of the tempera-ture for several values of D. These investigations enable us tocharacterize the nature (first or second order) of the transitionsand obtain the transition temperatures. Then, we study thehysteresis behaviors of our system as well as the thermal beha-viors of the coercivity and the remanent magnetization. In thiswork, our investigations focus on both ferromagnetic ðJ140Þ andferrimagnetic ðJ1o0Þ cases.

-4 -2 0 2 4

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Fig. 5. The hysteresis loops of the nanowire at the first order critical temperatureðTC=J ¼ 0:08Þ for D=J ¼ �2:5, RS ¼ 1 and R1 ¼ 0:3.

-2 -1 0 1 2-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Fig. 6. The hysteresis loops of the nanowire for D=J ¼ �1:5, RS ¼ 1, R1 ¼ 0:3 and fordifferent values of the temperature.

0.8 1.2 1.6 2.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.8 1.2 1.6 2.00.0

0.2

0.4

0.6

0.8

1.0

1.2

Fig. 7. Temperature-dependent coercivity and remanent magnetization forD=J ¼ �1:5, RS ¼ 1 and R1 ¼ 0:3: (a) coercivity and (b) remanent magnetization.

B. Boughazi et al. / Journal of Magnetism and Magnetic Materials 363 (2014) 26–33 29

3.1. The ferromagnetic case

To investigate the size effect of the system, we have plotted thetotal susceptibility as a function of the temperature for R1 ¼0:3; RS ¼ 1:0 and D=J ¼ �2:5 and for different lengths of thenanowire (L¼400, 500 and 600) (Fig. 2). We can see, from thisfigure, that the value of the critical temperature is independent ofthe size of the system when LZ500.

In Fig. 3, we present the total ðMTotalÞ, the core ðMCoreÞ and theshell ðMShellÞ magnetizations per spin as a function of the temperatureðT=JÞ for two values of the crystal field (D=J ¼ �1:5 (Fig. 3(a)) andD=J ¼ �2:5 (Fig. 3(b))) and for RS ¼ JS=J ¼ 1 and R1 ¼ J1=J ¼ 0:3. ForD=J ¼ �1:5, it is seen that, when we increase the temperature; thecore, the shell and the total magnetizations decrease continuously andvanish at a second-order transition ðTC=J ¼ 2:12Þ, while whenD=J ¼ �2:5, we can remark that the shell magnetization undertakesa jump from the spin-32 state to the spin-12 state. Then, the totalmagnetization presents a discontinuity at T=J ¼ 0:08 (first ordertransition) and vanishes at a second order transition ðTC=J ¼ 0:92Þ.

In order to study the hysteresis behaviors of the ferromagneticcylindrical Ising nanowire, we have plotted the magnetic hyster-esis loops at different temperatures, which are below and aroundthe second order critical temperature for D=J ¼ �2:5, RS ¼ 1 andR1 ¼ 0:3 (Fig. 4). In Fig. 4(a), we recover the behavior of theisotropic case presenting three sectors, with tendency to closethe loop at intermediate values of the magnetic field where themost significant Barkhausen jumps occur ðh=J ¼ �1; h=J ¼ þ1Þ.We can also see that we have only one ferromagnetic hysteresisloop and that with increasing the temperature, the loop break intosub loops (Fig. 4(b) and (c)) and the external loops disappear first(Fig. 4(d)) as well as the central one (Fig. 4(e)) when thetemperature approaches its critical value. To see what happenswhen the system undergoes a first order transition from the ð32; 12Þordered phase to the ð12; 12Þ ordered phase, we have shown thehysteresis loop behavior at the temperature TC=J ¼ 0:08 (Fig. 5), itis clear that we have only one hysteresis cycle. In the case ofD=J ¼ �1:5, RS ¼ 1 and R1 ¼ 0:3, the influence of the temperatureon the hysteresis behavior is shown in Fig. 6. We can remarkthat the shape of the hysteresis loops becomes narrower as the

temperature increases below the transition temperature and thehysteresis loops disappear for a temperature above the transitionone ðTC=J ¼ 2:12Þ.

In Fig. 7, we show the thermal behaviors of the coercivity(Fig. 7(a)) and the remanent magnetization (Fig. 7(b)) forD=J ¼ �1:5, RS ¼ 1 and R1 ¼ 0:3. We can see that the coercivityand the remanent magnetization decrease as the temperatureincreases and vanish at T=J � 2:0. Our results are similar to thoseobtained in Refs. [19] and [15]. Mounkachi et al. [20] have

0.0 0.5 1.0 1.5 2.0 2.5

-1.5

-1.0

-0.5

0.0

0.5

0.0 0.4 0.8 1.2

-1.5

-1.0

-0.5

0.0

0.5

Fig. 8. The temperature dependence of the core, the shell and the total magnetizations for RS ¼ 1 and R1 ¼ 0:3; (a) D=J ¼ �1:5 and (b) D=J ¼ �2:5.

0.2 0.4 0.6 0.8 1.0

-0.18

-0.12

-0.06

0.00

0.06

0.12

0.18

Fig. 9. The magnetization versus the temperature for D=J ¼ �2:5, RS ¼ 0:08 and fordifferent values of R1.

B. Boughazi et al. / Journal of Magnetism and Magnetic Materials 363 (2014) 26–3330

-4 -2 0 2 4-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

-4 -2 0 2 4-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

-4 -2 0 2 4-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

-4 -2 0 2 4-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Fig. 10. The hysteresis loops for RS ¼ 0:08, D=J ¼ �2:5, R1 ¼ �0:5 and for different temperatures.

-4 -2 0 2 4-0.9

-0.6

-0.3

0.0

0.3

0.6

-4 -2 0 2 4-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-0.3 -0.2 -0.1 0.0 0.1 0.2

-0.4-0.20.00.20.4

Magnetization

h/J

Fig. 11. The hysteresis loops of the nanowire for D=J ¼ �2:5, RS ¼ 1, R1 ¼ �0:3 and for different values of the temperature. (a) T=J ¼ 0:08 and (b) T=J ¼ 0:29. The figure insidepresents a zoom of the central loop of the hysteresis cycle.

B. Boughazi et al. / Journal of Magnetism and Magnetic Materials 363 (2014) 26–33 31

synthesized Zn0:8ðFe0:1;Co0:1ÞO diluted magnetic semiconductors,they have shown that with the increase of temperature, themagnetization and corresponding coercive field are decreasing;these results are similar to ours obtained and shown in Fig. 7(a).

3.2. The ferrimagnetic case

In this subsection, we present the behaviors of the longitudinalmagnetizations (MCore, MSell and MTotal) as a function of the tem-perature, for RS ¼ 1, R1 ¼ �0:3 and for two values of D=J (Fig. 8).Fig. 8(a) shows the temperature dependence of magnetizations,obtained for D=J ¼ �1:5. It is seen that MCore decreases and MSell

increases continuously and they vanish at the critical temperatureðTC=J ¼ 2:15Þ; therefore, a second-order phase transition occurs.Fig. 8(b) illustrates the thermal variation of the magnetizations forD=J ¼ �2:5, we can remark that MCore decreases continuously andvanish at a second-order phase transition ðTC=J ¼ 0:92Þ, while MShell

and MTotal undertake a jump, at TC=J ¼ 0:08 (first order transition).In order to study the effect of the exchange interaction between

spins which are located in the core and in the shell ðR1Þ on thecompensation phenomenon, we have plotted the magnetization as afunction of the temperature for RS ¼ 0:08, D=J ¼ �2:5 and fordifferent values of R1 (Fig. 9). It is clear that the system exhibitsthe compensation phenomenon and the values of the compensationpoint ðTk=JÞ increases as increasing jR1j (for example, for jR1j ¼ 0:1,we have Tk=J ¼ 0:1 while for jR1j ¼ 0:5, we have Tk=J ¼ 0:33). Wecan also see that the critical temperature increases as increasing jR1j.

Fig. 10 shows the hysteresis behavior for D=J ¼ �2:5, RS ¼ 0:08,R1 ¼ �0:5 and for different temperatures which are below andaround the second order temperature. We can see that the systemexhibits the triple hysteresis loops (Fig. 10(a)) and when weincrease the temperature, this triple loop turns to one central loop(Fig. 10(b)) and disappears for the high values of the temperature(Fig. 10(c) and (d)). The triple hysteresis loop behaviors have alsobeen seen for different systems, such as in molecular-basedmagnetic materials [21]. Multiple hysteresis loop behaviors have

been observed experimentally in Py/Cu, CoFeB/Cu, CoNiP/Cu, FeGa/Py and FeGa/CoFeB multilayered nanowires [22]. More investiga-tions on the hysteresis behaviors of our system have been done atand around the first order critical temperature for D=J ¼ �2:5,RS ¼ 1 and R1 ¼ �0:3 (Fig. 11). One hysteresis loop has been seenin Fig. 11(a), while in Fig. 11(b), we recover the behavior of theisotropic case presenting three hysteresis cycles, with tendency toclose the external loops at intermediate values of the magnetic fieldwhere the most significant Barkhausen jumps occur ðh=J ¼ �2;h=J ¼ þ2Þ. In Fig. 12, we have also studied the effect of thetemperature on the hysteresis behavior for D=J ¼ �1:5, RS ¼ 1 andfor R1 ¼ �0:3. It is obvious that the shape of the hysteresis loopsbecomes narrower as the temperature increases and disappearwhen the temperature is higher than the transition one.

-2 -1 0 1 2

-1.0

-0.5

0.0

0.5

1.0

Fig. 12. The hysteresis loops of the nanowire for D=J ¼ �1:5, RS ¼ 1, R1 ¼ �0:3 andfor different temperatures.

0.8 1.0 1.2 1.4 1.6 1.8 2.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.8 1.2 1.6 2.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fig. 13. Temperature-dependent coercivity and remanent magnetization forD=J ¼ �1:5, RS ¼ 1 and R1 ¼ �0:3: (a) coercivity and (b) remanent magnetization.

B. Boughazi et al. / Journal of Magnetism and Magnetic Materials 363 (2014) 26–3332

The thermal behavior of the coercivity and the remanentmagnetization is plotted in Fig. 13(a) and (b), respectively. Onecan notice that the coercivity and the remanent magnetizationdecrease as increasing the temperature and vanish at T=J � 2:0.

4. Conclusion

In summary, by the use of Monte Carlo Simulations based onthe heat bath algorithm, we have investigated the crystal field, theinterface and the surface interactions effects on the hysteresisbehaviors and the compensation temperature of a magneticnanowire with core/shell morphology. We have shown that, theanisotropy produces irregularities in the low-temperature hyster-esis curve due to Barkhausen spin avalanches, these irregularitiesoccur at different values of the magnetic field. For the ferrimag-netic case, it is clear that the system presents the triple hysteresisloops. It is also shown that the coercivity and the remanentmagnetization decrease with increasing the temperature.

Acknowledgments

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

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