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shell spin-1 Ising nanowire
Solid State Communications 152 (2012) 354–359
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Two distinct magnetic susceptibility peaks and magnetic reversal events in a cylindrical core/shell spin-1 Ising nanowire Numan Şarlı a , Mustafa Keskin a,b,∗ a
Institute of Science, Erciyes University, 38039 Kayseri, Turkey
b
Department of Physics, Erciyes University, 38039 Kayseri, Turkey
article
info
Article history: Received 16 November 2011 Received in revised form 8 December 2011 Accepted 9 December 2011 by G. E. W. Bauer Available online 16 December 2011 Keywords: A. Nanowires D. Magnetic susceptibility D. Phase transition E. Effective-field theory
abstract The magnetization and susceptibility of a cylindrical core/shell spin-1 Ising nanowire are investigated within the effective-field theory with correlations for both ferromagnetic and antiferromagnetic exchange interactions between the shell and the core. We find that the nanowire system exhibits two distinct susceptibility peaks and two successive phase transitions; either both of them are second-order transitions or one is a second-order transition and the other is a first-order transition for a small exchange interaction. The susceptibility versus switching field and the hysteresis behavior are investigated for different temperatures. It is found that the magnetization reversal events appear as peaks in the susceptibility versus switching field curve, the positions of which define the coercive field points of the nanowire system; the distance between the two susceptibility peaks decreases with increasing temperature. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Recently, there has been considerable interest in theoretical and experimental studies of the magnetic properties of core/shell nanostructures [1–7], because they have a wide possibility of technological applications, such as ultrahigh-density magnetic storage devices [8], sensors [9], permanent magnets [10], and medical applications [11]. The magnetic properties of nanoparticles are a scientifically interesting research area since their magnetic properties are quite different from those of the bulk and are greatly affected by the particle size [12]. Magnetization reversal events and magnetic susceptibility of the multilayered core/shell nanowires have been investigated [13]. The magnetic properties of cylindrical core/shell spin-1/2 Ising nanowires have been investigated using effective-field theory (EFT) with correlations [14–17]. Moreover, the magnetic properties of the nanostructures have been studied by using Monte Carlo (MC) simulations [18–22]. The magnetization dynamics in nanoparticle systems have been studied using Langevin dynamics [23]. Nanomagnets have been investigated by Landau–Lifshitz–Gilbert equations [24]. Recently, some characteristic behavior of a spin-1 Ising nanotube has also been studied [25]. On the other hand, nanowires and nanotubes have also been investigated experimentally [26–31]. The theoretical studies, except for
references [18,19,22,25], are concerned cylindrical core/shell spin1/2 Ising nanowire or nanotube systems. In this paper, the magnetizations and susceptibilities in a cylindrical core/shell spin-1 Ising nanowire are reported within the EFT with correlations. The reason we use a spin-1 nanowire is to investigate the effects of the crystal field (single-ion anisotropy) on the nanowire system. We also study the magnetic susceptibility behavior versus switching field and the hysteresis behavior for different reduced temperatures. The outline of this paper is as follows. In Section 2, we present the model and formalism. In Section 3, numerical results and discussions are given, followed by a brief summary. 2. Model and formulation The schematic representation of a cylindrical core/shell spin1 Ising nanowire is illustrated in Fig. 1. Each site on the figure is occupied by a spin-1 Ising particle. The Hamiltonian of the model is H = −JShell
⟨ij⟩
0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.12.015
Sm Sn − JInt
⟨mn⟩
Si Sm
⟨im⟩
− DShell
i
∗ Corresponding author at: Institute of Science, Erciyes University, 38039 Kayseri, Turkey. E-mail address: [email protected] (M. Keskin).
Si Sj − JCore
Si2
− DCore
m
2 Sm
−H
i
Si +
Sm
,
(1)
m
where the JShell is the exchange interaction parameter between two nearest-neighbor magnetic particles at the surface shell; the JCore is the exchange interaction parameter in the core; and the surface
N. Şarlı, M. Keskin / Solid State Communications 152 (2012) 354–359
355
shell is coupled to the next shell in the core with an exchange interaction JInt . DShell and DCore are the crystal fields. The spin Si takes the values ±1.0 and 0.0. The summation indices ⟨ij⟩, ⟨mn⟩ and ⟨im⟩ denote a summation over all pairs of neighboring spins on the shell surface, core and between the shell surface and core shell, respectively. The surface exchange interaction is defined as JShell = JCore (1 + ∆s )[14–16,32]. The longitudinal magnetizations on the shell surface ms1 and ms2 , and on the core mc1 and mc2 , within the EFT with correlations, can be obtained as ms1 = [S1]2 × [S2]2 × [C 2] FShell (x)|x=0 , ms2 = [S2]2 × [S1]2 × [C 2]2 FShell (x)|x=0 , mc2 = [C 2]4 × [C 1] × [S1] × [S2]2 FCore (x)|x=0 ,
(2)
mc1 = [C 1]2 × [C 2]6 FCore (x)|x=0 . The quadrupolar moments (qi = m2i ) qs1 , qs2 , qc1 , and qc2 can be obtained as qs1 = [S1]2 × [S2]2 × [C 2] GShell (x)|x=0 , qs2 = [S2]2 × [S1]2 × [C 2]2 GShell (x)|x=0 , qc2 = [C 2]4 × [C 1] × [S1] × [S2]2 GCore (x)|x=0 ,
(3)
Fig. 1. Schematic representation of a cylindrical core/shell spin-1 Ising nanowire. Open circles indicate magnetic atoms at the surface shell and solid circles are magnetic atoms constituting the core.
C 1 = 1 + mc1 sinh(C ) + m2c1 (cosh(C ) − 1) ,
∂ FShell [x] , ∂H ∂ FShell [x] , χs2 = b1 χs1 + b2 χs2 +b3 χc2 + b4 ∂H ∂ FCore [x] χc2 = c1 χs1 + c2 χs2 +c 3 χc2 +c 4 χc1 + c5 , ∂H ∂ FCore [x] . χc1 = d1 χc1 + d2 χc2 + d3 ∂H
where A, B, and C are defined as A = JShell ∇ , B = JInt ∇ , and C = JCore ∇ . ∇ = ∂/∂ x is the differential operator. The F (x) and G(x) functions are defined for a spin-1 Ising nanowire system as follows [33]:
χShell =
χs1 = a1 χs1 + a2 χs2 +a3 χc2 + a4
qc1 = [C 1]2 × [C 2]6 GCore (x)|x=0 . For spin-1 Ising systems, S1, S2, C 2, and C 1 are defined as S1 = 1 + ms1 sinh(A) + m2s1 (cosh(A) − 1) , S2 = 1 + ms2 sinh(A) +
m2s2
(cosh(A) − 1) ,
C 2 = 1 + mc2 sinh(B) +
m2c2
(cosh(B) − 1) ,
FShell [x] =
2Sinh[β(x + H )]
2Cosh[β(x + H )] + Exp[−β DCore ]
GShell [x] = GCore [x] =
χCore =
,
χT = (5)
2Cosh[β(x + H )] 2Cosh[β(x + H )] + Exp[−β DShell ] 2Cosh[β(x + H )] 2Cosh[β(x + H )] + Exp[−β DCore ]
,
MCore = MT =
1 2 1 7
.
1 19
(6)
(6(ms1 + ms2 ) + 6mc2 + mc1 ) .
Now, by differentiating each side of the longitudinal magnetization mi (i = s1, s2, c1, c2) in Eq. (2) with H, we get the initial susceptibility of the χi as follows:
∂ mi . H →0 ∂ H
χi = lim
7
1 19
(χs1 + χs2 ), (6χc2 + χc1 ),
(9)
(6(χs1 + χs2 ) + 6χc2 + χc1 ) .
We first examine the behavior of the magnetization and susceptibility as a function of reduced temperature for both ferromagnetic (r > 0) and antiferromagnetic (r < 0) interactions. Then, we study the magnetic reversal events in the nanowire system. For these studies, the interaction parameter r is defined as r = J Int /JCore .
(ms1 + ms2 ), (6mc2 + mc1 ),
1 2 1
3. Numerical results and discussions
Eqs. (2)–(5) can be used to investigate the magnetization of the shell surface MShell , core MCore , and total nanowire system MT as follows: MShell =
The coefficients ai , bi , ci , and di have complicated and long expressions, so they will not be given here. Using Eq. (8), χShell , χCore , and χT can be defined as follows:
,
2Cosh[β(x + H )] + Exp[−β DShell ] 2Sinh[β(x + H )]
FCore [x] =
(4)
(8)
(7)
Using Eqs. (2) and (7), we can easily obtain the susceptibilities χs1 , χs2 , χc1 , and χc2 as follows:
(10)
3.1. The ferromagnetic case The temperature dependence of the magnetizations and susceptibilities are shown in Fig. 2(a)–(c) for the ferromagnetic exchange interaction parameter r (r > 0) between the shell and core. In Fig. 2(a), the system undergoes a second-order phase transition from the ferromagnetic phase to the paramagnetic phase and the magnetizations become zero at Tc = 0.455. There exists a distinct peak in the susceptibility–temperature curves corresponding to magnetizations of the shell, the core, and the total nanowire system at Tc . The susceptibilities diverge as the temperature approaches the critical temperature. In Fig. 2(b), a second-order phase transition and susceptibility peaks occur at Tc = 0.375.
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N. Şarlı, M. Keskin / Solid State Communications 152 (2012) 354–359
Fig. 2. Temperature dependence of the magnetizations and susceptibilities for ferromagnetic interaction and DShell = DCore = 0, exhibiting a second-order phase transition. (a) r = 1.0 and ∆s = 0.0, (b) r = 0.1 and ∆s = 0.0, (c) r = 0.001 and ∆s = 0.0.
The susceptibility behavior of the core and the total nanowire system is similar to that in Fig. 2(a), except that χShell and χT form a shoulder before the critical temperature. In Fig. 2(c), the system undergoes two successive second-order phase transitions, namely Tc1 = 0.262 for the shell, and Tc2 = 0.372 for the core and the total nanowire system; therefore, two distinct peaks appear in the susceptibility–temperature curves. Thus, the magnetizations and susceptibilities of the shell and core behave independently of each other for a low value of the exchange interaction parameter
Fig. 3. The same as Fig. 2, but (a) exhibiting a first-order phase transition: r = 1.0 ∆s = 0.0 and DShell = DCore = −0.332; (b) exhibiting two successive first-order phase transitions: r = 0.1 ∆s = 0.0 and DShell = DCore = −0.269; (c) exhibiting two successive phase transitions, the first one being a first-order phase transition and the second one being a second-order phase transition: r = 0.001 ∆s = 0.0 and DShell = DCore = −0.204.
r (0.001). Our results are in a good agreement with some other theoretical [18,22,23,32,34–37] and experimental results [38,39]. The magnetizations and susceptibilities as a function of the temperature are shown in Fig. 3(a)–(c). In Fig. 3(a), it is found that the system undergoes a first-order phase transition from the ferromagnetic phase to the paramagnetic phase, because the magnetizations go to zero discontinuously as the temperature
N. Şarlı, M. Keskin / Solid State Communications 152 (2012) 354–359
Fig. 4. The same as Fig. 2, but for antiferromagnetic interaction and DShell = DCore = 0, exhibiting a second-order phase transition. (a) r = −1.0 and ∆s = 0.5, (b) r = −0.1 and ∆s = −0.5, (c) r = −0.001 and δs = −0.5.
increases. The first-order phase transition occurs at Tt = 0.146. Moreover, in the vicinity of Tt , the susceptibilities rapidly increase at T < Tt and suddenly decrease at Tt . At temperatures T > Tt , the susceptibilities exhibit a broad maximum. In Fig. 3(b), it is found that the shell has two first-order phase transition temperature points, namely Tt1 = 0.04 and Tt2 = 0.134. The susceptibility of the shell has a smaller peak at Tt1 than that of the core at Tt2 . In Fig. 3(c), the nanowire system has both second-order and first-order phase transition temperature points; hence the system undergoes two successive phase transitions. At the first-order phase transition temperature, Tt = 0.075, a discontinuity occurs
357
Fig. 5. The same as Fig. 3, but for antiferromagnetic interaction. (a) r = −1.0 ∆s = 0.5 and DShell = DCore = −0.339, (b) r = −0.1 ∆s = 0.0 and DShell = DCore = −0.269, (c) r = −0.001 ∆s = 0.0 and DShell = DCore = −0.204.
in the magnetization and susceptibility curves for the shell and the total nanowire system. The second-order phase transition and susceptibility peaks occur at Tc = 0.277 for the core and the total nanowire system. Similar results were obtained in some other theoretical [40,41] results and experimental work [42]. 3.2. The ferrimagnetic case The magnetizations and susceptibilities for the antiferromagnetic exchange interaction parameter r (r < 0) between the shell
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N. Şarlı, M. Keskin / Solid State Communications 152 (2012) 354–359
Fig. 6. The susceptibilities and magnetizations versus switching field for ferromagnetic (a)–(c) r = 1.0, ∆s = 0.0, and antiferromagnetic (d)–(f) exchange interaction r = −1.0, ∆s = 0.5, at different constant temperatures.
and core as a function of temperature are shown in Fig. 4(a)–(c). In Fig. 4(a), it is found that the second-order phase transition and susceptibility peak occur at Tc = 0.511. The susceptibilities diverge as the temperature approaches the critical temperature. For the antiferromagnetic exchange interaction, the susceptibility behavior of the shell and core is similar to that in the ferromagnetic case, except that χShell has a maximum, χCore has a minimum, and χT has a small maximum at Tc . In Fig. 4(b), χCore and χT have a maximum, and χShell has a small maximum at Tc . χShell and χT form a shoulder before Tc . Moreover, the second-order phase transition is observed at Tc = 0.375. In Fig. 4(c), the system undergoes two successive second-order phase transitions, namely at Tc1 = 0.131 for the shell and at Tc2 = 0.372 for the core; hence two distinct peaks
appear in the susceptibility–temperature curves. Therefore, for low values of the antiferromagnetic exchange interactions parameter r, the shell and core of the nanowire system behave separately from each other just as in the ferromagnetic case. Moreover, χCore has a maximum at Tc = 0.372 and χShell has a minimum at Tc = 0.131. The thermal behavior of the magnetizations and susceptibilities is shown in Fig. 5(a)–(c). In Fig. 5(a), the magnetizations and susceptibilities exhibit a discontinuity characteristic at Tt . The firstorder phase transitions occur at Tt = 0.146. The susceptibility behavior of the core is similar to that of the shell and the total susceptibility, but in the opposite direction. In Fig. 5(b), the system undergoes two successive first-order phase transitions, namely at Tt1 = 0.04 and Tt2 = 0.134. For the shell and the total nanowire,
N. Şarlı, M. Keskin / Solid State Communications 152 (2012) 354–359
the first-order phase transition is observed at Tt = 0.04 and a first-order phase transition takes place at Tt = 0.134 for the core and the total nanowire. In Fig. 5(c), the system undergoes two successive phase transitions, the first one is a first-order phase transition at Tt = 0.075 and the second one is a second-order phase transition at Tc = 0.277. On the other hand, for the shell and the total nanowire, a first-order phase transition takes place at Tt = 0.075 and a second-order phase transition occurs at Tc = 0.277 for the core and the total nanowire. Moreover, the susceptibilities of the core and the total nanowire have a peak at Tc = 0.277. The susceptibilities of the shell and total nanowire display a discontinuity at Tt = 0.075.
359
the hysteresis behavior for ferromagnetic and antiferromagnetic interactions. We found that the magnetization reversal events appear as peaks in the susceptibility versus switching field curve, the positions of which define the coercive field points of the nanowire system; the distance between the two susceptibility peaks decreases as the temperature increases. Acknowledgment This work was supported by Erciyes University Research Funds, Grant No: FBD-10-3350. References
3.3. The magnetic reversal events The magnetization and susceptibility behavior versus switching field is presented for ferromagnetic interaction (r = 1 and ∆s = 0) in Fig. 6(a)–(c) and for the antiferromagnetic case (r = −1 and ∆s = 0.5) in Fig. 6(d)–(f) at T = 0.4, 0.425, and 0.45. In Fig. 6(a), the susceptibilities have peaks at the magnetic reversal event points of each branch of the hysteresis loops, the positions of which define the right and left points of Hc , namely, Hc ± = ±0.016 at T = 0.4. Subsequently, the susceptibilities decrease with increasing switching field. In Fig. 6(b), the magnetization reversal events appear as susceptibility peaks at Hc ± = ±0.007 at T = 0.425. In Fig. 6(c), the susceptibility peaks are observed at Hc ± = ±0.001 at T = 0.45. In Fig. 6(a)–(c), it is observed that the distance between the two susceptibility peaks decreases as the temperature increases. In Fig. 6(d), the switching field dependence of the magnetizations and susceptibilities are displayed for antiferromagnetic interaction at T = 0.4. The susceptibilities have a peak at Hc ± = ±0.098 similar to ferromagnetic case. The main difference is that there is an extra broad maximum before Hc , because of the changes in the magnetization versus switching field curve. Moreover, one sees in Fig. 6(d) that when the core switches, the shell is affected as well, which makes sense intuitively but has not been observed experimentally [13]. In Fig. 6(e), the susceptibility peaks and magnetic reversal events occur at Hc ± = ±0.079 at T = 0.425. In Fig. 6(f), the peaks are at Hc ± = ±0.059 at T = 0.45. The distance between two susceptibility peaks decreases as the temperature increases, similarly to the ferromagnetic case. Tripled hysteresis behavior is not observed, whereas it is observed in the spin-1/2 system [17]. Our results are in good agreement with the experimental results of Chong et al. [13] and the theoretical results of Konstantinova [43]. Moreover, the hysteresis properties (hysteresis loops, remanence, and coercivity) decrease as the temperature increases. These results are consistent with some experimental [26–31] and theoretical results [17–19,21,36]. In summary, we have investigated the thermal behavior of the magnetization and susceptibilities of a cylindrical core/shell spin1 Ising nanowire using EFT. It is observed that the susceptibilities diverge as the temperature approaches the critical temperature for a second-order transition, whereas the susceptibilities rapidly increase at T < Tt and suddenly decrease at Tt . At temperatures T > Tt , the susceptibilities exhibit a broad maximum. For a small exchange interaction, the nanowire system has two distinct susceptibility peaks and two successive phase transitions: either both of them are second-order transitions or one is a secondorder transition and the other is a first-order transition. We also studied the magnetic susceptibilities versus switching field and
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Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Two distinct magnetic susceptibility peaks and magnetic reversal events in a cylindrical core/shell spin-1 Ising nanowire Numan Şarlı a , Mustafa Keskin a,b,∗ a
Institute of Science, Erciyes University, 38039 Kayseri, Turkey
b
Department of Physics, Erciyes University, 38039 Kayseri, Turkey
article
info
Article history: Received 16 November 2011 Received in revised form 8 December 2011 Accepted 9 December 2011 by G. E. W. Bauer Available online 16 December 2011 Keywords: A. Nanowires D. Magnetic susceptibility D. Phase transition E. Effective-field theory
abstract The magnetization and susceptibility of a cylindrical core/shell spin-1 Ising nanowire are investigated within the effective-field theory with correlations for both ferromagnetic and antiferromagnetic exchange interactions between the shell and the core. We find that the nanowire system exhibits two distinct susceptibility peaks and two successive phase transitions; either both of them are second-order transitions or one is a second-order transition and the other is a first-order transition for a small exchange interaction. The susceptibility versus switching field and the hysteresis behavior are investigated for different temperatures. It is found that the magnetization reversal events appear as peaks in the susceptibility versus switching field curve, the positions of which define the coercive field points of the nanowire system; the distance between the two susceptibility peaks decreases with increasing temperature. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Recently, there has been considerable interest in theoretical and experimental studies of the magnetic properties of core/shell nanostructures [1–7], because they have a wide possibility of technological applications, such as ultrahigh-density magnetic storage devices [8], sensors [9], permanent magnets [10], and medical applications [11]. The magnetic properties of nanoparticles are a scientifically interesting research area since their magnetic properties are quite different from those of the bulk and are greatly affected by the particle size [12]. Magnetization reversal events and magnetic susceptibility of the multilayered core/shell nanowires have been investigated [13]. The magnetic properties of cylindrical core/shell spin-1/2 Ising nanowires have been investigated using effective-field theory (EFT) with correlations [14–17]. Moreover, the magnetic properties of the nanostructures have been studied by using Monte Carlo (MC) simulations [18–22]. The magnetization dynamics in nanoparticle systems have been studied using Langevin dynamics [23]. Nanomagnets have been investigated by Landau–Lifshitz–Gilbert equations [24]. Recently, some characteristic behavior of a spin-1 Ising nanotube has also been studied [25]. On the other hand, nanowires and nanotubes have also been investigated experimentally [26–31]. The theoretical studies, except for
references [18,19,22,25], are concerned cylindrical core/shell spin1/2 Ising nanowire or nanotube systems. In this paper, the magnetizations and susceptibilities in a cylindrical core/shell spin-1 Ising nanowire are reported within the EFT with correlations. The reason we use a spin-1 nanowire is to investigate the effects of the crystal field (single-ion anisotropy) on the nanowire system. We also study the magnetic susceptibility behavior versus switching field and the hysteresis behavior for different reduced temperatures. The outline of this paper is as follows. In Section 2, we present the model and formalism. In Section 3, numerical results and discussions are given, followed by a brief summary. 2. Model and formulation The schematic representation of a cylindrical core/shell spin1 Ising nanowire is illustrated in Fig. 1. Each site on the figure is occupied by a spin-1 Ising particle. The Hamiltonian of the model is H = −JShell
⟨ij⟩
0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.12.015
Sm Sn − JInt
⟨mn⟩
Si Sm
⟨im⟩
− DShell
i
∗ Corresponding author at: Institute of Science, Erciyes University, 38039 Kayseri, Turkey. E-mail address: [email protected] (M. Keskin).
Si Sj − JCore
Si2
− DCore
m
2 Sm
−H
i
Si +
Sm
,
(1)
m
where the JShell is the exchange interaction parameter between two nearest-neighbor magnetic particles at the surface shell; the JCore is the exchange interaction parameter in the core; and the surface
N. Şarlı, M. Keskin / Solid State Communications 152 (2012) 354–359
355
shell is coupled to the next shell in the core with an exchange interaction JInt . DShell and DCore are the crystal fields. The spin Si takes the values ±1.0 and 0.0. The summation indices ⟨ij⟩, ⟨mn⟩ and ⟨im⟩ denote a summation over all pairs of neighboring spins on the shell surface, core and between the shell surface and core shell, respectively. The surface exchange interaction is defined as JShell = JCore (1 + ∆s )[14–16,32]. The longitudinal magnetizations on the shell surface ms1 and ms2 , and on the core mc1 and mc2 , within the EFT with correlations, can be obtained as ms1 = [S1]2 × [S2]2 × [C 2] FShell (x)|x=0 , ms2 = [S2]2 × [S1]2 × [C 2]2 FShell (x)|x=0 , mc2 = [C 2]4 × [C 1] × [S1] × [S2]2 FCore (x)|x=0 ,
(2)
mc1 = [C 1]2 × [C 2]6 FCore (x)|x=0 . The quadrupolar moments (qi = m2i ) qs1 , qs2 , qc1 , and qc2 can be obtained as qs1 = [S1]2 × [S2]2 × [C 2] GShell (x)|x=0 , qs2 = [S2]2 × [S1]2 × [C 2]2 GShell (x)|x=0 , qc2 = [C 2]4 × [C 1] × [S1] × [S2]2 GCore (x)|x=0 ,
(3)
Fig. 1. Schematic representation of a cylindrical core/shell spin-1 Ising nanowire. Open circles indicate magnetic atoms at the surface shell and solid circles are magnetic atoms constituting the core.
C 1 = 1 + mc1 sinh(C ) + m2c1 (cosh(C ) − 1) ,
∂ FShell [x] , ∂H ∂ FShell [x] , χs2 = b1 χs1 + b2 χs2 +b3 χc2 + b4 ∂H ∂ FCore [x] χc2 = c1 χs1 + c2 χs2 +c 3 χc2 +c 4 χc1 + c5 , ∂H ∂ FCore [x] . χc1 = d1 χc1 + d2 χc2 + d3 ∂H
where A, B, and C are defined as A = JShell ∇ , B = JInt ∇ , and C = JCore ∇ . ∇ = ∂/∂ x is the differential operator. The F (x) and G(x) functions are defined for a spin-1 Ising nanowire system as follows [33]:
χShell =
χs1 = a1 χs1 + a2 χs2 +a3 χc2 + a4
qc1 = [C 1]2 × [C 2]6 GCore (x)|x=0 . For spin-1 Ising systems, S1, S2, C 2, and C 1 are defined as S1 = 1 + ms1 sinh(A) + m2s1 (cosh(A) − 1) , S2 = 1 + ms2 sinh(A) +
m2s2
(cosh(A) − 1) ,
C 2 = 1 + mc2 sinh(B) +
m2c2
(cosh(B) − 1) ,
FShell [x] =
2Sinh[β(x + H )]
2Cosh[β(x + H )] + Exp[−β DCore ]
GShell [x] = GCore [x] =
χCore =
,
χT = (5)
2Cosh[β(x + H )] 2Cosh[β(x + H )] + Exp[−β DShell ] 2Cosh[β(x + H )] 2Cosh[β(x + H )] + Exp[−β DCore ]
,
MCore = MT =
1 2 1 7
.
1 19
(6)
(6(ms1 + ms2 ) + 6mc2 + mc1 ) .
Now, by differentiating each side of the longitudinal magnetization mi (i = s1, s2, c1, c2) in Eq. (2) with H, we get the initial susceptibility of the χi as follows:
∂ mi . H →0 ∂ H
χi = lim
7
1 19
(χs1 + χs2 ), (6χc2 + χc1 ),
(9)
(6(χs1 + χs2 ) + 6χc2 + χc1 ) .
We first examine the behavior of the magnetization and susceptibility as a function of reduced temperature for both ferromagnetic (r > 0) and antiferromagnetic (r < 0) interactions. Then, we study the magnetic reversal events in the nanowire system. For these studies, the interaction parameter r is defined as r = J Int /JCore .
(ms1 + ms2 ), (6mc2 + mc1 ),
1 2 1
3. Numerical results and discussions
Eqs. (2)–(5) can be used to investigate the magnetization of the shell surface MShell , core MCore , and total nanowire system MT as follows: MShell =
The coefficients ai , bi , ci , and di have complicated and long expressions, so they will not be given here. Using Eq. (8), χShell , χCore , and χT can be defined as follows:
,
2Cosh[β(x + H )] + Exp[−β DShell ] 2Sinh[β(x + H )]
FCore [x] =
(4)
(8)
(7)
Using Eqs. (2) and (7), we can easily obtain the susceptibilities χs1 , χs2 , χc1 , and χc2 as follows:
(10)
3.1. The ferromagnetic case The temperature dependence of the magnetizations and susceptibilities are shown in Fig. 2(a)–(c) for the ferromagnetic exchange interaction parameter r (r > 0) between the shell and core. In Fig. 2(a), the system undergoes a second-order phase transition from the ferromagnetic phase to the paramagnetic phase and the magnetizations become zero at Tc = 0.455. There exists a distinct peak in the susceptibility–temperature curves corresponding to magnetizations of the shell, the core, and the total nanowire system at Tc . The susceptibilities diverge as the temperature approaches the critical temperature. In Fig. 2(b), a second-order phase transition and susceptibility peaks occur at Tc = 0.375.
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Fig. 2. Temperature dependence of the magnetizations and susceptibilities for ferromagnetic interaction and DShell = DCore = 0, exhibiting a second-order phase transition. (a) r = 1.0 and ∆s = 0.0, (b) r = 0.1 and ∆s = 0.0, (c) r = 0.001 and ∆s = 0.0.
The susceptibility behavior of the core and the total nanowire system is similar to that in Fig. 2(a), except that χShell and χT form a shoulder before the critical temperature. In Fig. 2(c), the system undergoes two successive second-order phase transitions, namely Tc1 = 0.262 for the shell, and Tc2 = 0.372 for the core and the total nanowire system; therefore, two distinct peaks appear in the susceptibility–temperature curves. Thus, the magnetizations and susceptibilities of the shell and core behave independently of each other for a low value of the exchange interaction parameter
Fig. 3. The same as Fig. 2, but (a) exhibiting a first-order phase transition: r = 1.0 ∆s = 0.0 and DShell = DCore = −0.332; (b) exhibiting two successive first-order phase transitions: r = 0.1 ∆s = 0.0 and DShell = DCore = −0.269; (c) exhibiting two successive phase transitions, the first one being a first-order phase transition and the second one being a second-order phase transition: r = 0.001 ∆s = 0.0 and DShell = DCore = −0.204.
r (0.001). Our results are in a good agreement with some other theoretical [18,22,23,32,34–37] and experimental results [38,39]. The magnetizations and susceptibilities as a function of the temperature are shown in Fig. 3(a)–(c). In Fig. 3(a), it is found that the system undergoes a first-order phase transition from the ferromagnetic phase to the paramagnetic phase, because the magnetizations go to zero discontinuously as the temperature
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Fig. 4. The same as Fig. 2, but for antiferromagnetic interaction and DShell = DCore = 0, exhibiting a second-order phase transition. (a) r = −1.0 and ∆s = 0.5, (b) r = −0.1 and ∆s = −0.5, (c) r = −0.001 and δs = −0.5.
increases. The first-order phase transition occurs at Tt = 0.146. Moreover, in the vicinity of Tt , the susceptibilities rapidly increase at T < Tt and suddenly decrease at Tt . At temperatures T > Tt , the susceptibilities exhibit a broad maximum. In Fig. 3(b), it is found that the shell has two first-order phase transition temperature points, namely Tt1 = 0.04 and Tt2 = 0.134. The susceptibility of the shell has a smaller peak at Tt1 than that of the core at Tt2 . In Fig. 3(c), the nanowire system has both second-order and first-order phase transition temperature points; hence the system undergoes two successive phase transitions. At the first-order phase transition temperature, Tt = 0.075, a discontinuity occurs
357
Fig. 5. The same as Fig. 3, but for antiferromagnetic interaction. (a) r = −1.0 ∆s = 0.5 and DShell = DCore = −0.339, (b) r = −0.1 ∆s = 0.0 and DShell = DCore = −0.269, (c) r = −0.001 ∆s = 0.0 and DShell = DCore = −0.204.
in the magnetization and susceptibility curves for the shell and the total nanowire system. The second-order phase transition and susceptibility peaks occur at Tc = 0.277 for the core and the total nanowire system. Similar results were obtained in some other theoretical [40,41] results and experimental work [42]. 3.2. The ferrimagnetic case The magnetizations and susceptibilities for the antiferromagnetic exchange interaction parameter r (r < 0) between the shell
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Fig. 6. The susceptibilities and magnetizations versus switching field for ferromagnetic (a)–(c) r = 1.0, ∆s = 0.0, and antiferromagnetic (d)–(f) exchange interaction r = −1.0, ∆s = 0.5, at different constant temperatures.
and core as a function of temperature are shown in Fig. 4(a)–(c). In Fig. 4(a), it is found that the second-order phase transition and susceptibility peak occur at Tc = 0.511. The susceptibilities diverge as the temperature approaches the critical temperature. For the antiferromagnetic exchange interaction, the susceptibility behavior of the shell and core is similar to that in the ferromagnetic case, except that χShell has a maximum, χCore has a minimum, and χT has a small maximum at Tc . In Fig. 4(b), χCore and χT have a maximum, and χShell has a small maximum at Tc . χShell and χT form a shoulder before Tc . Moreover, the second-order phase transition is observed at Tc = 0.375. In Fig. 4(c), the system undergoes two successive second-order phase transitions, namely at Tc1 = 0.131 for the shell and at Tc2 = 0.372 for the core; hence two distinct peaks
appear in the susceptibility–temperature curves. Therefore, for low values of the antiferromagnetic exchange interactions parameter r, the shell and core of the nanowire system behave separately from each other just as in the ferromagnetic case. Moreover, χCore has a maximum at Tc = 0.372 and χShell has a minimum at Tc = 0.131. The thermal behavior of the magnetizations and susceptibilities is shown in Fig. 5(a)–(c). In Fig. 5(a), the magnetizations and susceptibilities exhibit a discontinuity characteristic at Tt . The firstorder phase transitions occur at Tt = 0.146. The susceptibility behavior of the core is similar to that of the shell and the total susceptibility, but in the opposite direction. In Fig. 5(b), the system undergoes two successive first-order phase transitions, namely at Tt1 = 0.04 and Tt2 = 0.134. For the shell and the total nanowire,
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the first-order phase transition is observed at Tt = 0.04 and a first-order phase transition takes place at Tt = 0.134 for the core and the total nanowire. In Fig. 5(c), the system undergoes two successive phase transitions, the first one is a first-order phase transition at Tt = 0.075 and the second one is a second-order phase transition at Tc = 0.277. On the other hand, for the shell and the total nanowire, a first-order phase transition takes place at Tt = 0.075 and a second-order phase transition occurs at Tc = 0.277 for the core and the total nanowire. Moreover, the susceptibilities of the core and the total nanowire have a peak at Tc = 0.277. The susceptibilities of the shell and total nanowire display a discontinuity at Tt = 0.075.
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the hysteresis behavior for ferromagnetic and antiferromagnetic interactions. We found that the magnetization reversal events appear as peaks in the susceptibility versus switching field curve, the positions of which define the coercive field points of the nanowire system; the distance between the two susceptibility peaks decreases as the temperature increases. Acknowledgment This work was supported by Erciyes University Research Funds, Grant No: FBD-10-3350. References
3.3. The magnetic reversal events The magnetization and susceptibility behavior versus switching field is presented for ferromagnetic interaction (r = 1 and ∆s = 0) in Fig. 6(a)–(c) and for the antiferromagnetic case (r = −1 and ∆s = 0.5) in Fig. 6(d)–(f) at T = 0.4, 0.425, and 0.45. In Fig. 6(a), the susceptibilities have peaks at the magnetic reversal event points of each branch of the hysteresis loops, the positions of which define the right and left points of Hc , namely, Hc ± = ±0.016 at T = 0.4. Subsequently, the susceptibilities decrease with increasing switching field. In Fig. 6(b), the magnetization reversal events appear as susceptibility peaks at Hc ± = ±0.007 at T = 0.425. In Fig. 6(c), the susceptibility peaks are observed at Hc ± = ±0.001 at T = 0.45. In Fig. 6(a)–(c), it is observed that the distance between the two susceptibility peaks decreases as the temperature increases. In Fig. 6(d), the switching field dependence of the magnetizations and susceptibilities are displayed for antiferromagnetic interaction at T = 0.4. The susceptibilities have a peak at Hc ± = ±0.098 similar to ferromagnetic case. The main difference is that there is an extra broad maximum before Hc , because of the changes in the magnetization versus switching field curve. Moreover, one sees in Fig. 6(d) that when the core switches, the shell is affected as well, which makes sense intuitively but has not been observed experimentally [13]. In Fig. 6(e), the susceptibility peaks and magnetic reversal events occur at Hc ± = ±0.079 at T = 0.425. In Fig. 6(f), the peaks are at Hc ± = ±0.059 at T = 0.45. The distance between two susceptibility peaks decreases as the temperature increases, similarly to the ferromagnetic case. Tripled hysteresis behavior is not observed, whereas it is observed in the spin-1/2 system [17]. Our results are in good agreement with the experimental results of Chong et al. [13] and the theoretical results of Konstantinova [43]. Moreover, the hysteresis properties (hysteresis loops, remanence, and coercivity) decrease as the temperature increases. These results are consistent with some experimental [26–31] and theoretical results [17–19,21,36]. In summary, we have investigated the thermal behavior of the magnetization and susceptibilities of a cylindrical core/shell spin1 Ising nanowire using EFT. It is observed that the susceptibilities diverge as the temperature approaches the critical temperature for a second-order transition, whereas the susceptibilities rapidly increase at T < Tt and suddenly decrease at Tt . At temperatures T > Tt , the susceptibilities exhibit a broad maximum. For a small exchange interaction, the nanowire system has two distinct susceptibility peaks and two successive phase transitions: either both of them are second-order transitions or one is a secondorder transition and the other is a first-order transition. We also studied the magnetic susceptibilities versus switching field and
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