Solid State Communications 151 (2011) 1025–1030

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Solid State Communications

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

Hysteresis behaviors in a cylindrical Ising nanowireMustafa Keskin a,b,∗, Numan Şarlı b, Bayram Deviren c

a Department of Physics, Erciyes University, 38039 Kayseri, Turkeyb Institute of Science, Erciyes University, 38039 Kayseri, Turkeyc Department of Physics, Nevsehir University, 50300 Nevsehir, Turkey

a r t i c l e i n f o

Article history:Received 8 April 2011Accepted 25 April 2011by G.E.W. BauerAvailable online 5 May 2011

Keywords:A. NanowiresD. Hysteresis loopD. Phase transitionE. Effective-field theory

a b s t r a c t

The hysteresis behaviors of the cylindrical Ising nanowire are investigated within the effective-fieldtheorywith correlations at temperatures below, around and above the critical temperature. The hysteresiscurves are obtained for different reduced temperatures both ferromagnetic and antiferromagneticinteractions between the shell and the core. We find that the hysteresis loop areas decrease when thereduced temperatures increase, and the hysteresis loops disappear at certain reduced temperatures.Moreover, for the antiferromagnetic nanowire the hysteresis loop areas disappear earlier than theferromagnetic case as the reduced temperature increases. An unusual form of triple hysteresis behaviorsis observed for the antiferromagnetic nanowire system. The thermal behaviors of the coercivitiesand remanent magnetizations are also investigated. The results are in good agreement with someexperimental and theoretical results.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, magnetic nanoparticle systems have been veryintensively studied both experimentally and theoretically (see[1–7] and references therein). Besides technological applications,such as the ultrahigh-density magnetic storage devices [8], sen-sors [9], permanent magnets [10], medical applications [11], etc.,the magnetic properties of nanoparticles are scientifically inter-esting research areas since their magnetic properties are quitedifferent from those of the bulk and greatly affected by the par-ticle sizes [12]. Theoretical studies on the magnetic properties ofcylindrical Ising nanowires have also been investigated using theeffective-field theory (EFT) with correlations [13–15]. Moreover,magnetic properties of the ferrimagnetic core/shell nanocube Isingmodel were studied by the Monte Carlo (MC) simulations [16].The compensation and critical behaviors of the ferromagneticcore/shell nanoparticle Ising model were investigated by the useof MC simulations [17]. Magnetic properties of the ferromagnetic(FM)–antiferromagnetic (AFM) core-shell morphology were stud-ied by using theMCMetropolismethod [18]. On the other hand, thecharacterization of multiferroic MnWO4 nanowires and nanowirearrays has been performed using the X-ray diffraction (XRD),the scanning electron microscopy (SEM), the transmission elec-tron microscopy (TEM), the high resolution TEM, and the energy-

∗ Corresponding author at: Department of Physics, Erciyes University, 38039Kayseri, Turkey. Tel.: +90 352 4374938; fax: +90 352 4374931.

E-mail address: keskin@erciyes.edu.tr (M. Keskin).

0038-1098/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2011.04.019

dispersive X-ray spectroscopy [19]. Magnetic and optical prop-erties of cobalt nanowires fabricated in polycarbonate ion-tracktemplates were studied by the XRD and the SEM analysis [20].Magnetic properties of core-shell catalyst nanoparticles for the car-bon nanotube growth [21] and surface magnetization processes insoft magnetic nanowires [22] have also been investigated experi-mentally.

Despite all these theoretical studies, as far as we know, thehysteresis behaviors of Ising nanowire have not been investigated.Therefore, the purpose of this study is to investigate the hysteresisbehaviors of the cylindrical Ising nanowire for both ferromagneticand antiferromagnetic interactions between the shell and thecore by the use of the EFT with correlations at temperaturesbelow, around and above the critical temperature. We also studythe temperature dependence of the coercivities and remanentmagnetizations.

We should also mention that the hysteresis properties (hys-teresis area, coercivity and remanence) are very important inmagnetic recording media [23]. The hysteresis behaviors ofmaghemite nanoparticles have been studied by the MC simu-lations [24]. Dynamics and scaling of low-frequency hysteresisloops in nanomagnets are investigated by numerically solving theLandau–Lifshitz–Gilbert equation [25]. Hysteresis loops for spin-1 Ising model of noninteracting nanoparticles have been studiedwithin the pair approximation method [26].

The outline of the rest of the paper is as follows. In Section 2, themodel and its formulation are briefly given. The detailed numericalresults and discussions are presented in Section 3, followed by abrief summary.

1026 M. Keskin et al. / Solid State Communications 151 (2011) 1025–1030

2. Model and formulation

Since the cylindrical Ising nanowire was defined and its formu-lation was given by Kaneyoshi [13–15,27], within the theoreticalframework of the EFT with correlations, we shall follow his proce-dure.

The schematic representation of the cylindrical Ising nanowireis illustrated in Fig. 1, and each site on the figure is occupied by aspin-1/2 Ising particle. The figure consists of the surface shell andthe core in which the core is surrounded by the surface shell. Thecore is formed a core shell with a spin-1/2 particle at the centerof the core as seen in Fig. 1; hence we have a cylindrical Isingnanowire system. The Hamiltonian of the model is given by

H = −JShell−⟨ij⟩

SiSj − JCore−⟨mn⟩

SmSn − JInt

×

−⟨im⟩

SiSm − H

−i

Si +−m

Sm

, (1)

where the JShell and JCore are the exchange interaction parametersbetween two nearest-neighbor magnetic atoms at the surfaceshell and core, respectively, and JInt is the interaction parametersbetween two nearest-neighbor magnetic atoms at the surfaceshell and the core shell. Si spins take the values ±1 at eachsite i of the cylindrical Ising nanowire system and a summationindex ⟨ij⟩, ⟨mn⟩ and ⟨im⟩ denote a summation over all pairs ofneighboring spins at the shell surface, core and between shelland core, respectively. The surface exchange interaction is oftendefined as JShell = JCore(1 + ∆s) in order to clarify the effects ofsurface on the physical properties in the nanosystem [13–15,27].Two longitudinal magnetizations on the shell surface, namely ms1andms2, and two longitudinal magnetizations on the core, namelymc1 and mc2, within the framework of the EFT with correlations,can be obtained as,

ms1 = [cosh(A) + ms1 sinh(A)]2x[cosh(A) + ms2 sinh(A)]2

x[cosh(B) + mc2 sinh(B)]F1/2(x)|x=0,

ms2 = [cosh(A) + ms2 sinh(A)]2x[cosh(A) + ms1 sinh(A)]2

x[cosh(B) + mc2 sinh(B)]2F1/2(x)|x=0,

mc2 = [cosh(C) + mc2 sinh(C)]4x[cosh(C) + mc1 sinh(C)]

x[cosh(B) + ms1 sinh(B)]

x[cosh(B) + ms2 sinh(B)]2F1/2(x)|x=0,

mc1 = [cosh(C) + mc1 sinh(C)]2

x[cosh(C) + mc2 sinh(C)]6F1/2(x)|x=0,

(2)

where A, B and C are defined by A = JShell∇, B = JInt∇ andC = JCore∇ . ∇ = ∂/∂x is the differential operator and F1/2(x) isdefined by as follow:

F1/2(x) = tanh[β(x + H)], (3)

where β = 1/kBTA, TA is the absolute temperature and kB is theBoltzmann factor.

Furthermore, let us define the longitudinal magnetizations persite of the shell surface (MShell), the core (MCore), and the totallongitudinal magnetization of per site (MT ) as follows:

MShell =12(ms1 + ms2),

MCore =17(6mc2 + mc1),

MT =119

(6(ms1 + ms2) + 6mc2 + mc1).

(4)

Fig. 1. Schematic representation of a cylindrical nanowire. Open circles indicatemagnetic atoms at the surface shell and solid circles are magnetic atomsconstituting the core.

a

b

c

Fig. 2. The reduced temperature dependence of magnetizations for the ferromag-netic interaction, MShell (solid lines), MCore (dotted lines) and MT (dashed lines): (a)r = 1.0 and ∆s = 0.0, (b) r = 1.0 and ∆s = 0.5, (c) r = 0.1 and ∆s = −0.5.

M. Keskin et al. / Solid State Communications 151 (2011) 1025–1030 1027

a b c

Fig. 3. The hysteresis loops at different reduced temperatures which are aroundthe critical temperature Tc = 5.922 for the ferromagnetic interaction and r =

1.0, ∆s = 0.0: (a) for the shell, (b) for the core, (c) for the total system.

3. Numerical results and discussions

In order to investigate the hysteresis behaviors of the cylindricalIsing nanowire at temperatures below, around and above thecritical temperature, we first examine the behaviors of themagnetizations as a function of the reduced temperature. Theseinvestigations enable us to characterize the nature (first or secondorder) of the transitions and obtain the transition temperatures.Then, we have studied the hysteresis behaviors of the cylindricalIsing nanowire as well as the thermal behaviors of the coercivitiesand remanent magnetizations. For these studies, let us here definethe parameter r , which is called the core-shell coupling, as

r = JInt/JCore. (5)

Our investigations in this work are for both ferromagnetic andferrimagnetic cases. Hence, r > 0 (positive core-shell coupling)and r < 0 (negative core-shell coupling) correspond to theferromagnetic and antiferromagnetic interactions, respectively.

3.1. The ferromagnetic case

We investigate the thermal behavior of the longitudinal mag-netizations, namely MShell,MCore and MT as a function of the re-duced temperature (T = kBTA/JCore) for several values of r and ∆sin the absence of the external magnetic field (H = 0) by solving

a

b

Fig. 4. Reduced temperature-dependent coercivities and remanentmagnetizationsobtained from the hysteresis curves of r = 1.0 and ∆s = 0.0 for the ferromagneticinteraction: (a) coercivities, (b) remanent magnetizations. The inset figures showthe coercivities and remanent magnetizations for r = 0.1 and ∆s = −0.5.

Eq. (4), numerically. A few explanatory and interesting examplesare displayed in Fig. 2. Fig. 2(a) is obtained for r = 1.0 and ∆s =

0,MShell = MCore = MT = 1 at zero temperature, and they de-crease to zero continuously as the reduced temperature increases;hence a second-order phase transition occurs at Tc = 5.922. Thetransition is from the ferromagnetic (F ) phase to the paramagnetic(P) phase. Fig. 2(b) presents the thermal variation of the magneti-zations for r = 1.0 and ∆s = 0.5. The behavior of this figure issimilar to Fig. 2(a), but following differences have been found: (1)The values of MShell,MCore and MT are very close each other. (2) Tcoccurs at higher temperature, namely Tc = 6.637. Fig. 2(c) is plot-ted for r = 0.1 and ∆s = −0.5; the behavior ofMCore in this figureis similar to Fig. 2(a) and (b) but only differs from Fig. 2(a) and (b) inwhich theMShell andMT rapidly decrease continuously and becomezero at Tc = 4.756; hence Tc occurs at lower temperature. We alsoinvestigate the influence of r and ∆s on the magnetizations. Forhigh values of r and ∆s, the behaviors of the magnetizations aresimilar with Fig. 2(a) and (b), but the critical temperatures occur athigh temperatures.

Now, we study the hysteresis behaviors of a cylindrical Isingnanowire for ferromagnetic interaction. The magnetic hysteresisloops for MShell,MCore and MT at different reduced tempera-tures, which are below, around and above the critical tempera-ture, are calculated and plotted in Fig. 3. Fig. 3(a)–(c) show thehysteresis loops for MShell,MCore and MT , respectively, at T =

5.5, 6.0, 6.5, 7.0 and 7.5 for r = 1.0 and ∆s = 0.0; Tc =

5.922. From these figures, one can see that the hysteresis loopareas decrease as the temperature increases. This result is ingood agreement with the experimental results for the ferromag-netism in Mn+-implanted Si nanowires [28], in ferromagnetic

1028 M. Keskin et al. / Solid State Communications 151 (2011) 1025–1030

a

b

Fig. 5. The reduced temperature dependence of magnetizations for the antiferro-magnetic interaction,MShell (solid lines),MCore (dotted lines) andMT (dashed lines):(a) r = −1.0 and ∆s = 0.0, (b) r = −0.1 and ∆s = −0.5.

nanowires [29], in La2/3Sr1/3MnO3 nanoparticle assembled nan-otubes [30] as well as for carbon nanotubes [21,31–34], and withthe theoretical result for the ferromagnetic single-walled nan-otubes [35]. We should also mention that the similar behavior hasbeen found for maghemite nanoparticles [24] and noninteractingnanoparticles [26].Moreover, the changing of r and∆s only changethe shape of the hysteresis loop areas.

We also calculate the thermal behaviors of the coercivities andremanent magnetizations for r = 1, ∆s = 0.0, and differentreduced temperatures from 5.5 to 7.5 and also plotted in Fig. 4(a)and (b), respectively. Moreover, the insets of Fig. 4(a) and (b)show the coercivities and remanent magnetizations for r =

0.1, ∆s = −0.5 at the reduced temperatures from 4.0 to 6.0.We observe that the coercivities and remanent magnetizationsdecrease as the temperature increases. While the coercivities areoverlapping for r = 1 and ∆s = 0.0, they breakaway from eachothers for r = 0, 1 and ∆s = −0.5. The similar behavior hasbeen seen experimentally in carbon nanotubes produced usingzeolite Co-beta [21,31,33,34], in La2/3Sr1/3MnO3 nanoparticleassembled nanotubes [30], in nanostructures composed of aPt/Co multilayer exchange coupled with an antiferromagnetFeMn [36] and Fe3O4 nanowire arrays [37], and theoreticallyin ferromagnetic–antiferromagnetic core-shell morphology bythe MC Metropolis method [18] and in antiferromagnetic NiOnanoparticle within a Green function technique [38].

3.2. The ferrimagnetic case

In this subsection we present the behaviors of longitudinalmagnetizations, namely MShell,MCore and MT as a function of thereduced temperature (T = kBTA/JCore), for several values ofr < 0 which correspond to the antiferromagnetic interaction, ∆s

a

b

c

Fig. 6. The hysteresis loops at different reduced temperatures which are aroundthe critical temperature Tc = 5.922 for the antiferromagnetic interaction andr = −1.0, ∆s = 0.0: (a) for the shell, (b) for the core, (c) for the total systemand the inset figures at T = 6.0 and T = 6.5.

and H = 0. In this case, following two interesting behaviors havebeen found and displayed in Fig. 5. Fig. 5(a) shows the temperaturedependence ofmagnetizations, obtained for r = −1.0 and∆s = 0.In this figure, MShell = 1.0,MCore = −1.0 and MT = 0.263 at zerotemperature. MShell decreases and MCore increases continuouslyand they become zero at the critical temperature, Tc = 5.922,as the reduced temperature increases; therefore, a second-orderphase transition occurs. The transition is from the ferrimagnetic (I)phase to the paramagnetic (P) phase. Moreover, from the behaviorof MT , one can see that the Q -type behavior occurs accordingto the classification in Néel theory [39]. Fig. 5(b) illustrates thethermal variation of the magnetizations for r = −0.1 and ∆s =

−0.5. MShell = −1.0,MCore = 1.0 and MT = −0.263 at zerotemperature andMShell increases andMCore decreases continuouslyand they become zero at the critical temperature, Tc = 4.756;hence a second-order phase transition occurs and the transitionis from the ferrimagnetic (I) phase to the paramagnetic (P) phase.In this figure, MShell rapidly increases with the increasing thevalues of reduced temperature. Moreover, MT illustrates the N-type behavior according to the classification in Néel theory [39].

Fig. 6(a)–(c) show the hysteresis loops for MShell,MCore andMT , respectively, for different reduced temperatures from 5.5to 6.5 that are around the critical temperature, Tc = 5.922,

M. Keskin et al. / Solid State Communications 151 (2011) 1025–1030 1029

a

b

Fig. 7. Reduced temperature-dependent coercivities and remanentmagnetizationsobtained from the hysteresis curves of r = −1.0 and ∆s = 0.0 for theantiferromagnetic interaction: (a) coercivities, (b) remanent magnetizations. Theinset figures are calculated for r = −0.1 and ∆s = −0.5.

for r = −1.0 and ∆s = 0.0. The hysteresis behaviorsof Fig. 6(a) and (b) are similar to the hysteresis behaviors ofFig. 3(a)–(c), but following two differences have been found: (1)The shape of the loop areas is different. (2) The loop areas disappearearlier than the ferromagnetic case as the reduced temperatureincreases. Fig. 6(c) also exhibits the triple hysteresis loops for thelow values of the reduced temperatures that are different fromFig. 3(a)–(c) and Fig. 6(a) and (b). Moreover, these triple loopareas disappear earlier as the reduced temperature increases as inthe case of Fig. 6(a) and (b). The triple hysteresis loop behaviorshave also been seen for different systems, such as in singlechain magnets with antiferromagnetic interchain coupling [40]and in molecular-based magnetic materials [41]. More than triplehysteresis loop behaviors have been also seen experimentallyin CoFeB/Cu, CoNiP/Cu, FeGa/Py and FeGa/CoFeB multilayerednanowires [22]. We should also mention that for∆s = 0, the triplehysteresis loop behavior disappears and changing the values of rand ∆s effects only on the shape of the hysteresis loop areas.

We plotted the thermal behavior of the coercivities and rema-nent magnetizations obtained for r = −1, ∆s = 0.0 and differentreduced temperatures from 5.5 to 7.5 in Fig. 7(a) and (b), respec-tively.Moreover, the insets of Fig. 7(a) and (b) show the coercivitiesand remanent magnetizations obtained for r = −0.1, ∆s = −0.5,and the reduced temperatures from 4.0 to 6.0. These behaviors aresimilar to Fig. 4(a) and (b) in the ferromagnetic case except thatcoercivities and remanent magnetizations for ferrimagnetic casebecome zero earlier than ferromagnetic case. The similar behaviorhas also been observed experimentally [21,30,31,33,34,36,37] andtheoretically [18,38].

In summary, we have investigated hysteresis behaviors ofthe cylindrical Ising nanowire defined by Kaneyoshi, within theeffective-field theory with correlations, for the reduced tempera-tures below, around and above the critical temperature for bothferromagnetic and antiferromagnetic interactions between theshell and core. We first study the behaviors of the magnetizationsas a function of the reduced temperature in order to character-ize the nature (first or second order) of the transitions and obtainthe transition temperatures, and then we examine the hysteresisloops. We have observed that the results of hysteresis behaviorsof nanowires are in good agreement with both theoretical and ex-perimental results. We have also studied the thermal behavior ofcoercivities and remanent magnetizations and found that our re-sults are in agreement with both theoretical and experimental re-sults. We hope that this study will stimulate theoretical physiciststo explore more complicated nanosystems. Moreover, we expectthis study to give some light into theoretical and experimental re-search for the investigation of magnetic properties of nanosystemsas well as in the research of magnetism.

Acknowledgment

This work was supported by Erciyes University Research Funds,Grant No. FBD-10-3350.

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