new hysteresis in core/shell nanowire with mixed spin ising · 2018. 5. 8. · 2 model and method...

Advanced Studies in Theoretical PhysicsVol. 12, 2018, no. 7, 281 - 299

HIKARI Ltd, www.m-hikari.comhttps://doi.org/10.12988/astp.2018.8730

Hysteresis in Core/Shell Nanowire

with Mixed Spin Ising

M.C. Barrero-Moreno1, E. Restrepo-Parra1 and C.D. Acosta-Medina2

1Departamento de Fisica y Quimica2Departamento de Matemticas y Estadistica

Universidad Nacional de ColombiaSede Manizales, A.A 127 Manizales, Caldas, Colombia

Copyright c© 2018 M.C. Barrero-Moreno, E. Restrepo-Parra and C.D. Acosta-Medina.

This article is distributed under the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original work

is properly cited.

Abstract

In this work, Monte Carlo simulations (MCS) based on Metropolisalgorithm were performed to study the hysteresis loops and coercive fieldof a core/shell nanowire with spins S = ±5

2 ,±32 ,±

12 and σ = ±3

2 ,±12 , re-

spectivility, considering an Ising ferrimagnetic system. The influence ofnearest neighbors exchange interactions and crystal field anisotropy onhysteresis and coercive field behaviors of the system has been analyzed.The calculations were performed using a single-spin flip algorithm. Ineach spin-flip attempt, we randomly choose a site. The results showthat, for a system without anisotropy triple hysteresis loops appear forcertain values. The plateaus in hysteresis loops are due to the valuesof the spins projections. The change of the values for anisotropy influ-ence the shape of the hysteresis loops without present triple hysteresisloops. Also, the effect of the anisotropy in the coercive field was ana-lyzed, where the coercive field present enhanced for certain range of thetemperatures. Finally, for high values of the applied external magneticfield, the Zeeman effect takes control over the system.

Keywords: Monte Carlo method, core/shell nanowire, hysteresis loops,coercive field, ferrimagnetic system

282 M.C. Barrero-Moreno, E. Restrepo-Parra and C.D. Acosta-Medina

1 Introduction

In recent years, magnetic nanostructures as nanoparticles systems, especially,magnetic nanowires, are currently increased the interest of theoretical and ex-perimental researches, because of their fascinating properties, compared withother bulk materials [32]. In particular, the nanowires have been receiving con-siderable attention, experimentally, due to their distinctive properties for beingused as ultrahigh magnetic recording media [20, 6]; they also have potentialbiotechnological applications as [22], [34] tissue targeting or hyperthermia [2],sensors [1] and medical applications [21], among others. On the other hand,core/shell nanotube (nanowire or nonorod) systems have been attracting forseveral technological applications as sensing, batteries and fuel cells. In par-ticular, the properties of these arrays are strongly dependent on the interfacebetween the materials used for building the core and shell [25]. Regardingto magnetic applications, several core/shell nanowires can exhibit differentfunctionalities. For instance, systems formed by combinations of materials asferromagnetic core and antiferromagnetic shell, different ferromagnetic mate-rials for core and shell, ferromagnetic core and antiferromagnetic shell, amongothers, can enhance unidirectional anisotropy, magnetic resonance and gener-ate special properties as exchange bias [18]. Studies of core/shell nanoparticleshave revealed interesting magnetic properties, such as tuning of the blockingtemperature with four-fold increase in the coercivity [30], optimized hyper-thermia [23] and enhanced coercitivy [5].

Theoretical studies have shown that this mixed system has an interestingmagnetic behavior, with several compensation temperatures, tri-critical pointsand reentrant behavior [26], [7]. The possible applications in the design of newmaterials motivate the study of nanoparticles with high spin values. The be-havior of hysteresis has been investigated for cylindrical nanowires within theeffective-field theory (EFT) varying the interaction (ferromagnetic and anti-ferromagnetic) [17]. Also, it has been studied the behavior of hysteresis andcompensation in cylindrical Ising nanotube systems. Kocakaplan et al., [17]observed that areas of hysteresis loops decrease as the temperature increasesand disappears at certain temperatures. Also, showing the different forms thathysteresis loops can exhibit. Using Monte Carlo simulations (MCS) hysteresisloops for core/shell nanoparticles have modeled [15]. A similar work was re-ported by A. Patsopoulos et al [28], that presented MCS for studying Co/CoOcore/shell nanowires. In some cases, with certain conditions, a triple hysteresisloops were observed, in different systems, such cylindrical transverse spin −1Ising nanowire [19] and spin −1/2 cylindrical Ising nanowire with a core/shellstructure [17].

In the literature, other interesting works presenting Monte Carlo simu-lation of core/shell nanostructures can be found; for instance, Eftaxias and

Hysteresis in core/shell nanowire with mixed spin Ising 283

Trohidou [[8]] used Monte Carlo simulations for studying the exchange biasand coercive field of nanoparticles with ferromagnetic and antiferromagneticcore and shell, respectively. They found a strong dependence between theseparameters, and the structure and size of the interface. They also found agood agreement with experimental reports. Zaim and Kerouad [35] performeda Monte Carlo simulation of magnetic properties of spherical nanoparticles,consisting of ferromagnetic core and shell with spins S = ±1/2 (core) and±1.0 and S = ±1/2,±3/2 (shell), with antiferromagnetic interface coupling.They found that, for appropriated values of the system parameters, two com-pensation temperatures can be occurred in the system. Vasilakaki et al [33],employed the Monte Carlo method for studying the exchange bias effect atan atomic level; they also observed a training effect in ferromagnetic/antifloopbehaviors.erromagnetic core/shell nanoparticles. Moreover, they analyzed theeffect of the dipolar interaction between partalso included a study of the hys-teresis loop behaviors.icles, on the exchange bias. Estrader et al [11], presenteda study of magnetic behavior of bi-magnetic hard-soft, core/shell nanoparti-cles. They used experimental techniques as magnetometry, ferromagnetic res-onance and X-ray magnetic circular dichroism. They also performed MonteCarlo simulations for proving the consistency of the AFM coupling. More re-cently, Cansizoglu et al [4], analyzed vertical aligned in core/shell nanowire(nanorod) arrays; they concluded that, this geometry favors the constructionof several nano-scale devices for new energy solar, sensor, detectors and spin-tronic applications. For this reason, they conducted experimental works usingphysical vapor deposition techniques and, as a complement, they carried outMonte Carlo simulations to estimate the best deposition technique for a con-formal shell coating. Finally, Belhaj et al [3], published a study of the magneticbehavior of a nanostructure based on a hexagonal core (spin σ = −3/2) shellgeometry (spin S = 2). Exactly, they are focused on determining the effectof the coupling exchange interactions on the absence/presence of both, anexternal magnetic and crystal fields. As an important result, they found acompensation temperature between the spins σ and S. They also included astudy of the hysteresis loop behaviors.

More recently studies with MCS in hexagonal ferrimagnetic Ising nanowire[12], Blume Capel model [29] and Quantum Monte Carlo [13] show the appear-ance of plateaus and different forms of hysteresis loops. However, mixed spincore/shell systems have been less investigated. For this reason, it is necessaryto devote attention in these kind of systems. In this work, we study the effectof the crystal field and the exchange interactions on the coercive field, pre-senting triple hysteresis loops that disappear at high temperatures. The aimof this paper is to model hysteresis loops behavior of core/shell nanowire withspins ±5

2and ±3

2using Monte Carlo simulations. The outline of this paper is

the following: In Section 2, we give the model and method of the Monte Carlo


simulations. In section 3, we present the results of the hysteresis loops andrelationship coercive field and temperature. Finally, the conclusions are givenin Section 4.

2 Model and Method

We consider a mixed spin Ising system composed by a core/shell nanowirewith spins S = (±5

2,±3

2,±1

2) and σ = (±3

2,±1

2), respectively. The schematic

representation of the cylindrical Ising nanowire is illustrated in Fig 1a. ThisFigure consists of a ferromagnetic core and shell with an anti-ferromagneticcoupling between core and shell, in which, the core is surrounded by the surfaceshell, where r and R are core and shell radii, respectively, with height h . TheFig 1b shows a represent the values of spins by means of arrows. Periodicboundary conditions were imposed in the z direction. To study the magneticbehavior. The Hamiltonian includes terms as exchange interactions, crystalfield anisotropy and the Zeeman effect, described by equation (1):

H = −Jcc∑

Si

〈i,j〉

Sj−Jss∑

σi〈i,j〉

σj−J∑

σi〈i,j〉

Sj−Kv(∑

σ2i

i

+∑

S2j

j

)−H(∑

σii

+∑

Sj

i

)

where Jcc and Jss are the exchange constants between the core-core andshell-shell, J is the constant of exchange representing the anti-ferromagneticcoupling and, for this case, it is defined as −1. S and σ correspond to the spinsof core and shell, respectively. The sum over 〈i, j〉 indicates that interaction isonly taken into account between near neighbors; Kv is the magnetocrystallineanisotropy constant and finally, H is the external magnetic field, that is appliedto the sample in the direction z. Error bars were obtained from averaging anumber of 5 runs for each configuration, using different seeds for the randomnumbers.


a) b)

Figure 1: a) Nanowire Structure and b) Schematic representation of the spins.

The simulations were performed using Metropolis dynamics with 30000Monte Carlo steps; where the first 10000 steps were discarded for the relax-ation. The magnetizations of the two subsystems were found through:

Mc =1

Nc

∑i

Si

Ms =1

Ns

∑i

σi

where Nc and Ns indicate the number of sites of the core and shell, respec-tively. The total magnetization comes given for:

Mt =NcMc +NsMs

Nc +Ns

Hysteresis loops were performed by cycling the magnetic field from H = 25to −25 in steps of 1 by performing 30000 Monte Carlo steps per spin. Thecoercive field is defined as:

Hc =Hright −Hleft

2

where Hrigth and Hleft are the points of the loop that intersect the fieldaxis.

3 Results and Discussion

The results of this work are focused on analyzing the behavior of the hys-teresis loops and coercive field of core/shell nanowires, below and around ofthe critical temperature and the compensation temperature. In a previous


work, the behavior of critical and compensation temperatures of the core-shellnanowires was studied [16], based on this work, the appropriate parameterswere chosen for the simulations, where compensation temperature (Tcomp) wasobserved and there was identified a difference between the compensation andthe critical points.

3.1 Exchange interactions

The magnetic hysteresis loops for Mcore, Mshell and Mtotal at different reducedtemperatures (T = kBT/|J |) below and arround the critical value, in theabsence of the crystal field are calculated and plotted in Figure 2 at differenttemperatures (Te = 1.5, 5.5, 10.5 and 16.0) for Jcc = 0.8, Jss = 4.0 and Kv =0.0. From this result, it is observed that, when the temperature increases,the area of the hysteresis loops decreases, since the paramagnetic phase isreached. In addition, at low temperatures, plateaus appear, and then, theydisappear with the increase of the temperature, close to the critical value.This result is in good agreement with the cylindrical Ising nanowire that hasbeen investigated within the effective-field theory with spin ±1 [17] and thecylindrical nanotube system with spin −3/2 [17]. Figure 2a shows a hysteresisloops at Te = 1.5; for Mcore, triple hysteresis loops appear, while in the case ofMshell and Mtotal, a unique internal area is observed, but including some plateusformation. The plateus represent the values of the possible projections of thespins. The Figure 2b with Te = 5.5 triple hysteresis loops appear for Mtotal;as is observed, the number of plateaus decreases. This behavior is due to,as the temperature increases, less field is required to reach the spins reversionprocess; then, the Zeeman effect dominates the system. In Figure 2c hysteresisloops for Te = 10.5 were obtained, where it is presented a single cycle of thehysteresis for all the magnetizations. Finally, cycles obtained at Te = 16.0 areshown in Figure 2d, where curves exhibit only a unique central loop. For thefour cases, it was obserbed that, Mcore presented more significative changesregarding to the form of the cycle. On the other hand, Mshell remains almostthe same form (square), for all the temperatures analysed here, but varying thearea. The behavior of the shell magnetization resembles a typical ferromagneticmaterial, while Mtotal strongly changes with the increase of the temperature.The changes of Mtotal are mainly affected by Mcore. Calculations were madechanging the values of the projections for the core and shell; and it was foundthat the magnetizations behave the same. The plateaus in Mcore were due todifferents possible projections of the spins. With the change in temperature,the transitions tend to become more gradual, and the cycles are smoothed,until above the system becomes paramagnetic and the hysteresis disappears.Also, a term is added that corresponds to the thermal fluctuations.

In all the Figures the arrows that indicate the changes in the projections


of the spins are shown, which explains the existence of the plateaus in thecycles of hysteresis. As the temperature used for the hysteresis cycle increases,the cycle softens and the plateaus stop appearing. In general, for the coremagnetization, the most notable changes are presented, for which it is con-troled the form of the hysteresis cycle for Mtotal. In the case of the Mshell thechanges of the projections are similar in all cases, this is due to the value ofthe spin that was given to this subsystem, the shape of the cycle resemblesthe behavior of a ferromagnetic material, since the coupling between core-coreions is a ferromagnet. For Mtotal, in all cases, all the possible projectionsof the spins appear, with the increase in temperature changes the numberof plateaus in the cycles. At low temperatures, the number of plateaus isgreater, because the zeeman effect still does not dominate the system, as hap-pens when higher temperature is used. With the increase of the temperature,decreases the area of the cycle. In Figure 2a, for Mcore and Mtotal, all theprojections of the spins appear, because they take all the possible values dueto the temperature that was used. In Figure 2b a triple hysteresis loop withthe central loop appears; similar results appear for the double-walled cubicnanotube [10]. For Figure 2c, the cycle of hysteresis is smoothed and all pro-jections continue to appering, obtaining a simple hysteresis cycle. In Figure 2d,the temperature that is approaching the critical temperature was used and itsarea is shrinking, appearing a central cycle. Multiple hysteresis loops have alsobeen found, theoretically, in a mixed spin−1 and spin −3/2 core/shell cubicnanowire [24] and in CoFeB/Cu,CoNiP/Cu, FeGa/Py and FeGa/CoFeBmultilayered nanowires [31].


a)

b)

c)

d)

Figure 2: The hysteresis loops at different reduced temperatures which arearound the critical temperature with Tc = 16.153, Jcc = 0.8, Jss = 4.0 andKv = 0.0 a) Te = 1.5 b) Te = 5.5 c) Te = 10.5 and d) Te = 16.0.


3.2 Magnetocrystalline Anisotropy

a)

b)

c)

d)

Figure 3: The hysteresis loops at different reduced temperatures which arearound the critical temperature with Tc = 19.6435, Jcc = 0.1, Jss = 4.0 andKv = 15.0 a) Te = 0.5 b) Te = 4.0 c) Te = 7.5 and d) Te = 13.0.


Figure 3 shows the hystersis loops for Mtotal, Mcore and Mshell, respectively, atTe = 1.0, 4.0, 7.5 and 13.0, for Jcc = 0.1, Jss = 4.0 and Kv = 15.0. For all cases,the same behavior that for Kv = 0.0 was observed. Figure 3a shows the cycle atTe = 1.0, where plateaus for Mtotal and Mcore appear. The arrows represent thespin projections. For Mtotal only the projections for spins ±5

2and ±3

2appear,

presenting a triple hysteresis loop. There is no evidence of the presence of spin±1

2, because in this point, the energy of the system is high. Respect to Mcore,

in the hysteresis loops, only the projections for ±52

were observed. Moreover,Mshell only presents values for ±3

2. The presence of plateaus is produced due

to the changes of the hysteresis loops because of the spins projection, that arerepresented by means of arrows. Figure 3b, developed at Te = 4.0, shows thesame projections for the three cases of the magnetizations at a temperature ofTe = 1.0. Regarding Figure 3c, a single cycle with a central hysteresis loop ispresented for Mtotal. In the case of Mcore, the cycle becomes thinner, because itis close to the critical temperature. In Figure 3d, for Mtotal, Mshell and Mcore,the spin projection ±1

2can be observed. In this point, the energy is minimized

by increasing the temperature. In Figure 3c a smoothed loop was observed atTe = 7.5, decreasing the area of the cycle, having the same behavior of thecurve presented in Figure 3a and Figure 3b, where the projections were ±5

2

and ±32. In Figure 3d, for Mcore, two jumps appear; these jumps are due to

the projections ±12

in Mshell. For the case of Mtotal, a central cycle appears,because of the high temperature applied. The effect of the magnetocrystallineanisotropy was a phase transition that happens at higher temperature. Thehystersis loop forms were not affected when Kv is included.

3.3 Coercive Field

The coercive field is the ability of a material to resist the magnetic field withoutdemagnetizing. In this section, the behavior of the coercive field is presentedas a function of the temperature for the core, the shell and the total system,considering variations ofJcc, Jss and Kv.

3.3.1 Shell Coercive Field

Figure 4 shows shell coercive field as a function of the temperature for a)Jss = 4.0 and Kv = 0.0 varying Jcc, b) Jcc = 0.1 and Kv = 0.0 varying Jssand c) Jss = 4.0 and Jcc = 0.1 varying Kv. In Figure 4a, the changes withrespect to Jcc were presented; in the four cases, the curves were overlaped.indicating that Mshell takes the same values of Hc. Therefore, for the coercivefield, the changes in Jcc values do not affect its behavior, because a similarcritical temperature is maintained for the values Jss and Jcc used in thesecases. In Figure 4b, the coercive field curves tend to decrease, exhibting samebehavior than in the case of Figure 4a; nevertheless, in this case the curves


are not overlaped, asJss is increased. An increase in the values of coercivefield is observed, due to Tc moves towards the right, as Jss increases. Thisphenomenon was explained in preliminary works, where it was found that, theincrease in the critical temperature should be maintained periodic with theperiodic variation of Jss [16]. In Figure 4c, as Kvincreases, the same behavioras it obseved in figure 5a appears. As it was previously observed (In section3.2) it does not affect the shape of the cycles; it only imposes more order inthe system.

On the other hand, the coercive field tends to zero, as the temperatureincreases, as is expected for a typical ferromagnetic material. The shell andthe core are considered as independent systems for the analysis of Hc.

a) b)

c)

Figure 4: Shell coercive field in function of the temperature a) Jss = 4.0 andKv = 0.0 varying Jcc, b) Jcc = 0.1 and Kv = 0.0 varying Jss and c) Jss = 4.0and Jcc = 0.1 varying Kv.

3.3.2 Core Coercive Field

Figure 5, shows the core coercive field as a function of the temperature fora)Jss = 4.0 and Kv = 0.0 varying Jcc, b) Jcc = 0.1 and Kv = 0.0 varying


Jss and c) Jss = 4.0 and Jcc = 0.1 varying Kv. Figure 5a presented thecoercive field varying Jcc. it is observed that, for Jcc = 0.2, 04 and 0.6, thesame behavior is presented. At low temperatures, Hc decreases from 0.5 downto 0, reaching a minimum point T < 2; then, as the temperature increases,Hc increases up to values arround 0.5; on the other hand, For Jcc = 0.8, Hc

starts at a high value (in the oder of 2.5), reaching a minimum point, similarto the other cases, but at around at a temperature of 5. This is due to thefact that, for low relations between Jss and Jcc, the behavior of the Hc ofthe core respect is very variable due to the number of ions and the value ofprojections that are taken. In the Figure 5b all behace the same starting at thesame values and reaching the maxiums of Hc, which move to the rigth withthe increase of the Jss. The displacement is due as Jss varies the Tc increases,and for low values the phase transition occurs first. For Figure 5c a behaviorequal to Figure 5b is presented, in this case the behavior is due to the fact thatwhen Kv is included in the system, the phase transition is made at a highertemperature, so the maximum is displaced. Unlike the case of the shell in this,there is as improvement in the coercive field that is due to the combination ofthe values of Jss and Jcc, accompanied by the values of the projections of thespins, which cause the hysteresis loops to vary constantly by increasing thetemperature used for each cycle.


a) b)

c)

Figure 5: Core coercive field in function of the temperature a)Jss = 4.0 andKv = 0.0 varying Jcc, b) Jcc = 0.1 and Kv = 0.0 varying Jss and c) Jss = 4.0and Jcc = 0.1 varying Kv.

3.3.3 Total Coercive Field

Figure 6, shows a total coercive field as a function of the temperature for a)Jss = 4.0 and Kv = 0.0 varying Jcc, b) Jcc = 0.1 and Kv = 0.0 varying Jss andc) Jss = 4.0 and Jcc = 0.1 varying Kv. Figure 4a, shows a similar behavior inall cases, for Jcc = 0.2Hc starts at low values (around 0.1) and then it increases;however, for Jcc = 0.4 ,0.6 and 0.8, HC begins at a high value (around 0.9, 1.8and 3.0, for Jcc = 0.4 ,0.6 and 0.8, respectivility), decreasing and reaching itsmaximum as he temperature increases. Tme maximun value of Hc exhibits ashift towards the right with the temepratue. This point, corresponds to Tcomp

for this combination of Jcc and Jss. In this case of the Figure 4b the oppositecase is presented to Figure 4a, all starts at low values and increases with thestep of temperature, where appear a shift toward the right of the first pointwhere Hc = 0.0, because Tcomp is different for each value of Jss. For figure 4c,at temperatures lower than the critical value, Hc exhibis similar behaviour andvalues; since in this region the effect the order of the system is controled by


the temperature; neverthless, for T > Tc, the magnetocristalline anisotropy,Kv, dominates the system ordering; then, as Kv increases, the transition pointis shifted toward greater temperature values, but remaining the shape of thefour curves similar.

a) b)

c)

Figure 6: Total coercive field in function of the temperature a) Jss = 4.0 andKv = 0.0 varying Jcc, b) Jcc = 0.1 and Kv = 0.0 varying Jss and c) Jss = 4.0and Jcc = 0.1 varying Kv.

Table 1 shows the maximun values of Hc, varying Jcc, Jss and Kv. In figure6a, varying Jcc, as the temperature increases, the maximum of Hc tends toshift toward lower temperature. By varying Jss, the maximum of Hc increases,and it is shifted towardhigher temperatures. On the other hand, varying Kv

the maximum value of Hc remains almost constant, but a litle increase in thetemperature was observed. These changes in the values of Hc and Tc are due tothe combination of Jss and Jcc. The study of the changes in Tc was previouslycarried out, considering different values for Jss and Jcc [16]. In all cases there isa enhancement of Hc, which has been evidenced in previous studies conductedin nanowires by means of Monte Carlo simulations [27, 9, 14].


Table 1: Maximums of the total coercive field.Varying Jcc Varying Jss Varying Kv

Jcc T Hc Jss T Hc Kv T Hc

0.2 7.50132 5.51463 1.0 2.50081 1.0838 5.0 10.9991 4.334860.4 9.4924 4.58372 2.0 4.003 2.72892 10.0 13.0049 4.337720.6 10.501 3.62248 3.0 5.50033 4.36853 15.0 13.001 4.439050.8 12.0092 2.61436 4.0 7.49982 5.97026 20.0 13.5034 4.63014

4 Conclusion

We employed Monte Carlo simulations to study the hysteresis behaviors ofa core/shell ferrimagnetic nanowire with a mixed spin Ising spins −5/2 and−3/2. The influence of the exchange interactions and the crystal field onthe behavior of the hysteresis loops, and coercive field was analysed. Resultsshowed that the shape of the hysteresis cycles depends on the combination ofthe magnetic exchange constants. For the case of Jss = 2.0 and Jcc = 0.4,triple hysteresis loops were presented. Because of the temperature, Hc exhib-ited changes at around Tc. The enhancement of Hc is favored at small Jss andlarge Jcc. When the magnetocrystalline anisotropy was included, a maximumwas observed, and a greater value of this maximum can be found at Kv = 0.0;then, in can be concluded that the anisotropy does not favor the coercive field.

Acknowledgements. The authors gratefully acknowledge the financialsupport provided by Direccin Nacional de Investigaciones of the UniversidadNacional de Colombia, under the project 34705.

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Received: August 3, 2018; Published: September 19, 2018

new hysteresis in core/shell nanowire with mixed spin ising · 2018. 5. 8. · 2 model and method...

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