J Supercond Nov Magn (2013) 26:201–211 DOI 10.1007/s10948-012-1713-z
O R I G I N A L PA P E R
Magnetic Properties of Diluted Magnetic Nanowire S. Bouhou · I. Essaoudi · A. Ainane · F. Dujardin · R. Ahuja · M. Saber
Received: 11 May 2012 / Accepted: 25 June 2012 / Published online: 12 July 2012 © Springer Science+Business Media, LLC 2012
Abstract A magnetic diluted nanowire with cylindrical structure described by the Ising model is investigated. Using the effective field theory with a probability distribution technique, the influence of the dilution on the phase diagrams, susceptibility and the hysteresis loops are discussed in detail. Novel features are obtained for the thermal variations of longitudinal susceptibility and longitudinal magnetization. We have investigated the magnetic reversal of the system and have found the existence of triple hysteresis loops patterns, affected by the concentration of magnetic atoms, the temperature, and the exchange interaction between the core and the surface shell. Keywords Nanowire core/shell · Hysteresis loops · Susceptibility
S. Bouhou · I. Essaoudi · A. Ainane () · M. Saber Laboratoire de Physique des Matériaux et Modélisation des Systèmes, (LP2MS), Unité Associée au CNRST-URAC 08, Physics Department, Faculty of Sciences, University of Moulay Ismail, B.P. 11201, Meknes, Morocco e-mail:
[email protected] A. Ainane · M. Saber Max-Planck-Institut für Physik Complexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany A. Ainane · F. Dujardin Laboratoire de Physique des Milieux Denses (LPMD), Institut de Chimie, Physique et Matériaux (ICPM), 1 Bd. Arago, 57070, Metz, France R. Ahuja Condensed Matter Theory Group, Department of Physics and Astronomy, Uppsala University, 75120 Uppsala, Sweden
1 Introduction The magnetic nanoparticles have important potential technological and biomedical [1] applications as magnetic nanosensors recording media and magnetodielectrics for microwave devices, color imaging, ferrofluids, etc. [2–4]. With decreasing the size of the magnetic nanoparticle, the transition temperature and the spontaneous magnetization of nanopotential may decrease [5–7]. In different matrices and magnetic fields, nanoparticles may exhibit some new thermal behaviors, then the experimental and theoretical investigation are very important and essential in both pure and doped cases. In particular, patterned nanowire have received considerable attention. Quasi onedimensional magnetic nanowire have drawn a lot of research due to their unique physical properties. Recently, it was reported that Ni nanowire can be used in bioseparation and have higher yields compared with magnetic polymer microspheres [8, 9]. This provides a new chance for magnetic nanowire applied in biomedical fields. Theoretically, the core shell model has been accepted to explain many characteristic phenomena in nanoparticle magnetism [10–16]. The same concept has been applied to the investigation of magnetic nanowires and nanotubes. Recently, Trohido et al. [19] have investigated numerically the exchange bias properties of a complex system composed of a ferromagnetic core surrounded by a magnetically disordered shell. The same concept has been applied to the investigation of magnetic nanowires and nanotubes. In particular, the magnetic properties of a cubic Ising nanowire [20], which consists of a ferromagnetic spin-1/2 core and a ferromagnetic spin-1 shell coupled with an antiferromagnetic interlayer coupling Jinter to the core, have been investigated by the use of the Monte Carlo method.
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During the last few years, one could observe a growing interest of diluted magnetic systems in nanometric scale or low dimension structure because it is normally difficult to fabricate a pure magnetic material [21–24]. The authors explained the behaviors of these systems. In low dimension, Oubelkacem et al. [22] have studied the effect of the concentration c of magnetic atoms on the phase diagrams and the order parameters of a diluted superlattice with antiferromagnetic interface coupling. In the same concept, Htoutou et al. [25] have studied the temperature dependence of the magnetic properties such as the order parameters as function of site dilution c, using the transverse spin-1 Ising model with crystal field by within effective field theory with a probability distribution technique that accounts for single site kinematic relations. In the nanoscale case, in recent publication [14–18], the phase diagrams and temperature dependencies of magnetization in an Ising nanowire (or nanotube) with diluted surface have been examined using the core-shell concept and two theoretical frameworks, namely the mean field theory (MFT) and the effective field theory (EFT) [26, 27]. The same similarities between the nanowire and nanotube in the phase diagrams have been discussed using these two theoretical frameworks. The new experimental works which have been done recently in this area opened the way for the creation of diluted nanowire core-shell based on the composite molecules that contain metal atoms [29–31]. The authors have shown that the properties of the diluted systems are different from those of the corresponding pure systems. The aim of this paper is to study the magnetic and thermodynamic properties of the diluted Ising nanowire, consisting of core and surface shell coupling by the exchange interaction Jcs within the framework of the EFT with the probability distribution technique. The important magnetic properties such as magnetic susceptibilities, internal energy, free energy, entropy, and specific heat of the system are investigated. The outline of this work is as follow. In Sect. 2, we outline the formalism. In Sect. 3, the theory is applied to discuss the effects of the concentration c of magnetic atoms, the exchange interaction core/shell Jcs , and the external longitudinal magnetic field on the magnetic properties of the system. The last Sect. 4 is devoted to a brief conclusion.
Fig. 1 A transverse coup of nanowire, the black circles represent magnetic atoms in the surface shell. The empty circles are magnetic atoms constituting the core
− Jcs
ci cm σiz σmz
im
z z −h ci σ i + cm σm , (1) i
i
where σiz denotes the z component of a quantum spin opera→ tor − σ of magnitude σ = ±1/2 at site i. The first three sums are out only over nearest-neighbors pairs. Js , Jc , and Jcs are the exchange interactions constants between two nearestneighbors magnetic spins in the surface shell, the core, and between the core and surface shell, respectively. h represents the applied longitudinal magnetic field. ci is the occupation number of the site i, ci = 1 if the site is occupied by a magnetic atom and zero otherwise. Each spin is connected to the two nearest neighbor spins on above and below sections. The purpose of this paper is to study the magnetic properties of a such system using the EFT with a probability distribution technique that accounts for the single-site kinematic relations [32]. The longitudinal magnetization can be calculated without introducing mathematical complexities. For the system under consideration, the application of this method leads to the site longitudinal magnetizations: For the nanowire center site c1 : mzc1 = c σcz1 c = c fz Jc1 j cj σjz + h , (2) j
2 Model and Formalism We consider a magnetic nanowire consisting of surface shell and core where each site is occupied by an Ising spin as depicted in Fig. 1. The Hamiltonian of the system is given by H = −Js ci cj σiz σjz − Jc cm cn σmz σnz ij
mn
for the spins around the center: z z z Jc2 j cj σ j + h , mc2 = c σc2 c = c fz
(3)
j
for the spins of type-1 surface shell nanowire: z z z Js 1 j cj σ j + h , ms1 = c σs1 c = c fz j
(4)
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for spins of type-2 surface shell nanowire: Js2 j cj σjz + h . mzs2 = c σsz2 c = c fz
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(5)
j
· · ·c indicates the usual canonical ensemble thermal average for a given configuration and the sum is over all the nearest neighbors of the site i and · · · is the spacial configurational average. Using the EFT with a probability distribution technique [33], the function fz is given by fz (x) = tanh(βx),
(6)
with β = 1/kB T . To perform thermal averaging on the right-hand side of Eqs. (2)–(5) and we introduce the probability distribution of the spin variables:
1
P cn , σnz = (1 − c)δ(cn ) δ σnz + 1 2
1 + δ σnz − 1 + δ(cn − 1) 2
× c − mzn δ σnz + 1
+ c + mzn δ σnz − 1 .
1 z m + 6mzc2 . 7 c1 The magnetization of the surface sell is defined by
Within the framework of the EFT, Kaneyoshi et al. [35–37] have analyzed the internal energy and specific heat for pure or diluted magnetic systems. Recently, Canko et al. and Yuksel et al. [38–40] have extended this study to the low-dimensional systems by calculating the internal energy, specific heat, entropy, and free energy. From Eq. (14), we express the internal energy of the system by
1 1 U =− uc1 + 6uc2 + 6 us1 + us2 2 19
1 z mc1 + 6mzc2 + 6 mzs1 + mzs2 , (15) −h 19 where uc1 = c
Jc1 j cj σjz fz
j
(7)
The explicit self consistent equations for the site longitudinal magnetizations are given in detail in the Appendix. The longitudinal magnetization of the core is defined by mzc =
We obtain the internal energy U by using the approximated spin correlation identities introduced by Sá Barreto et al. [34]: T ri σi exp(−βHi ) . (14) fi σi = fi T ri exp(−βHi )
(8)
1 z ms1 + mzs2 . (9) 2 The longitudinal magnetization of the nanowire is defined by mzs =
1 z 6 ms1 + mzs2 + mzc1 + 6mzc2 . (10) 19 It is interesting to study the behavior of the longitudinal susceptibility of each site of the nanowire which is defined by ∂mt , (11) χt = ∂h h=0 mt =
uc2 = c
Jc2 j cj σjz fz
j
Js1 j cj σjz fz
j
us2 = c
Jc1 j cj σjz + h ,
(16)
+h ,
(17)
Js1 j cj σjz + h ,
(18)
Js2 j cj σjz + h .
(19)
j
j
us1 = c
j
Js2 j cj σjz fz
j
Jc2 j cj σjz
j
The specific heat of the system is obtained from ∂U Ch = . ∂T h
(20)
The entropy of the system is obtained numerically by the following relationship:
T Ch dt. (21) S= t 0 The free energy of the system is defined as F = U − T S.
(22)
where the susceptibilities of the core and the shell are given by
3 Results and Discussions
1 χc = (χc1 + 6χc2 ), (12) 7 and 1 χs = (χs1 + χs2 ). (13) 2 The details of the calculus of each site longitudinal susceptibility are given in the Appendix.
The magnetic properties are important in practice. In particular, the longitudinal susceptibilities are interesting physical quantities, which describe the characteristics of the charge of the longitudinal magnetizations with the fields and can show the phase transition properties, particularly its critical temperature. The phase transitions are usually predicted by the abnormal behavior of the longitudinal susceptibility at the critical temperature.
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Fig. 2a Temperature dependence of the total magnetization mt and the susceptibility χt of ferrimagnetic nanowire for few values of the concentration c, rcs = −0.5, rs = 0.5, and Jhc = 0.0
Fig. 2b Temperature dependence of the total magnetization mt and the susceptibility χt of ferrimagnetic nanowire for few values of the concentration c, rcs = −1.0, rs = 0.5, and Jhc = 0.0
In first, we have studied the longitudinal susceptibility and the longitudinal magnetization in the absence of the external longitudinal magnetic field. We take Jc as the unit of energy and we introduce the reduced exchange interactions (rcs = Jcs /Jc , rs = Js /Jc ). In Figs. 2a, 2b, 2c we depict the temperature dependence of the total longitudinal magnetization and susceptibility with different values of concentration c. In the ferrimagnetic case, Fig. 2a shows the effect of the concentration on the magnetization and susceptibility, when the parameters rcs and rs are fixed at rcs = −0.5 and rs = 0.5. The critical temperature, which corresponds to topic of the susceptibility decreases with the decrease of concentration c (see also the phase diagram in the same figure). The shape of the magnetization versus temperature exhibits N-type behavior. This shape shows a compensation point below its transition temperature (kB Tcomp /Jc = 2.66 for c = 1.0, kB Tcomp /Jc = 2.04 for c = 0.8 and kB Tcomp /Jc = 1.42 for c = 0.6). When the exchange interaction between the core and the surface shell is rcs = −1.0 (Fig. 2b), we see the same behavior of the magnetization in previous figure, but the transition temperature is larger than the first case. In fact, for the same concentration, the phase diagrams of this figure show that the critical temperature is greater than that of the previous cases. In the pure case (c = 1.0), our results are in qualitative agreement with those obtained by Kaneyoshi in pure and diluted surface shell [14, 15]. For a ferromagnetic coupling between
Fig. 2c Temperature dependence of the total magnetization mt and the susceptibility χt of ferromagnetic nanowire for few values of the concentration c, rcs = 1.0, rs = 0.5, and Jhc = 0.0
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the core and the surface shell spins (Fig. 2c), the total magnetization exhibits Q-type behavior and the critical temperature decreases with the decrease of c (the transition temperature is independent of the sign of the exchange interaction between the core and the surface shell). In Figs. 3a, 3b, 3c, 3d we study the effect of the concentration c on the hysteresis loops of magnetic diluted nanowire. In Fig. 3a, we show the dependence of the hysteresis loops on the dilution of the site in the core and the surface shell, with antiferromagnetic coupling between the core and the surface shell spins. We select the parameters of the system as follow: rcs = −1.0, rs = 1.0 and kB T /Jc = 0.2. It is shown that as c decreases the hysteresis curve changes from one large central loop with three areas (c = 0.7, 0.9) to triple loops for c = 0.5 and for a small central loop for the concentration c = 0.4. When the value of rcs is selected as a positive value (Fig. 3b), the system exhibits a ferromagnetic spin configuration. From this figure, we observe a normal large hysteresis loop and will becomes narrower by decreasing the value of the concentration c, which means a reduction of the coercive field. Now, let us discuss the effect of the temperature on the hysteresis loops. We take the parameters rcs = −1.0 and rs = 1.0 and c = 0.8 (Fig. 3c). We see a great hysteresis loop with three areas for the temperature kB T /Jc = 0.2, which change into three loops for kB T /Jc = 2.0 and will become smaller for kB T /Jc = 4.0 to disappear at the temperature kB T /Jc = 6.0. This behavior is similar to that found by M. Keskin et al. [28] for the same system in pure case. In Fig. 3d, we investigate the influence of the exchange interaction rcs on the hysteresis loops of the diluted nanowire when the parameters are selected as: rs = 1.0, kB T /Jc = 0.2, and c = 0.8. We see one hysteresis loop with three areas for each value of rcs . This hysteresis loop changes shape for the central region and becomes narrower with increasing rcs from −1 to −0.5. However, the opposite behavior is observed in the two extremes regions. In Figs. 4a, 4b, 4c, 4d we present the temperature dependence of internal energy, free energy, entropy, and specific heat of the system in the absence of the external magnetic field. Figure 4a shows the variation of the internal energy as a function of the temperature for different values of the concentration of the magnetic sites (c = 0.6, 0.7, 0.8, and 0.9) for a fixed values of the parameters rcs = 1.0 and rs = 0.5. At low temperatures, the entropy gives a minor contribution to the free energy. Therefore, the free energy equals to the internal energy at the ground state (Fig. 4b). This quantity decreases with increasing the concentration of magnetic sites. Figure 4c shows the entropy of the system for the same fixed parameters. We remark that with increasing the temperature, the system wants to maximize its entropy in order to minimize its free energy. For these parameters, the system presents only a second order transition because the entropy
205
Fig. 3a Hysteresis loops of ferrimagnetic nanowire for few values of the concentration c
Fig. 3b Hysteresis loops of ferromagnetic nanowire for few values of the concentration c
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Fig. 3c Hysteresis loops of ferrimagnetic nanowire for the concentration c = 0.8 and for few values of reduced temperature kB T /Jc
Fig. 4a Internal energy per site of ferromagnetic nanowire for a few values of the concentration c, rcs = 1.0, rs = 0.5, and Jhc = 0.0
Fig. 3d Hysteresis loops of ferrimagnetic nanowire for the concentration c = 0.8 and for few values of core shell exchange interaction
Fig. 4b Free energy of ferromagnetic nanowire for a few values of the concentration c, rcs = 1.0, rs = 0.5, and Jhc = 0.0
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and the free energy curves do not exhibit a discontinuous behavior. In Fig. 4d, we plot the numerical results of specific heat of the system for the same parameters in previous figures. We note that these curves exhibit a peak at the critical transition and rapidly decrease as the temperature increases. The specific heat is higher in the ordered state than in the disordered one.
4 Conclusion
Fig. 4c Entropy of ferromagnetic nanowire for a few values of the concentration c, rcs = 1.0, rs = 0.5, and Jhc = 0.0
In this work, we have used the EFT with a probability distribution technique to investigate the magnetic thermodynamic properties of a diluted magnetic nanowire in the presence of the external longitudinal magnetic field. Firstly, we have studied the total susceptibility and magnetization for negative and positive core/shell exchange interaction and for few values of the concentration c of magnetic atoms. We have observed that the transition temperature decreases when the concentration decreases. In ferrimagnetic case, the compensation point is observed for larger concentrations with strong exchange interaction. Secondly, we have investigated the hysteresis behavior of the system in particular we note the appearance of triple hysteresis loop which may have potential applications in multistate memory devices. Furthermore, we have completed our study by calculating internal energy, free energy, entropy, and specific heat of the system in ferromagnetic case. Acknowledgements This work has been initiated with the support of URAC: 08, the project RS:02 (CNRST) and the Swedish Research Links programme dnr-348-2011-7264 and completed during a visit of A.A. and M.S. at the Max Planck Institut für Physik Komplexer Systeme Dresden, Germany. The authors would like to thank all the organizations.
Appendix For the nanowire center spin c1 : mzc1 = c
Fig. 4d Specific heat of ferromagnetic nanowire for a few values of the concentration c, rcs = 1.0, rs = 0.5, and Jhc = 0.0
1 2N2 +N4
N2 N4 N 2 −μ1 N 4 −μ2 μ1 =0 μ2 =0 ν1 =0
ν2 =0
2(μ1 +μ2 ) CμN12 CμN24
× CνN12 −μ1 CνN24 −μ2
ν
N −(μ +ν ) × (1 − c)μ1 +μ2 c − mzc1 1 c + mzc1 2 1 1
ν
N −(μ +ν ) × c − mzc2 2 c + mzc2 4 2 2
× fz Jc N2 + N4 − 2(μ1 + μ2 ) + h (23) for the spins around the center:
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mzc2 = c
N3 N1 N1 N2 N 3 −μ1 N 1 −μ2 N 1 −μ3 N 2 −μ4
1 2N2 +2N1 +N3
μ1 =0 μ2 =0 μ3 =0 μ4 =0 ν1 =0
ν2 =0
ν3 =0
ν4 =0
2(μ1 +μ2 +μ3 ) CμN13 CμN21 CμN31 CμN42
ν
N −(μ +ν ) × CνN13 −μ1 CνN21 −μ2 CνN31 −μ3 CνN42 −μ4 (1 − c)μ1 +μ2 +μ3 +μ4 c − mzc2 1 c + mzc2 3 1 1
ν
N −(μ +ν )
ν
N −(μ +ν ) × c − mzc1 2 c + mzc1 1 2 2 c − mzs1 3 c + mzs1 1 3 3
ν
N −(μ +ν ) × c − mzs2 4 c + mzs2 2 4 4
× fz Jc N3 + N1 − 2(μ1 + μ2 ) + Jcs N1 + N2 − 2(μ3 + μ4 ) + h
(24)
for spins of type-1 surface shell nanowire: mzs1
=c
N2 N2 N1 N 2 −μ1 N 2 −μ2 N 1 −μ3
1 22N2 +N1
μ1 =0 μ2 =0 μ3 =0 ν1 =0
ν2 =0
ν3 =0
2(μ1 +μ2 +μ3 ) CμN12 CμN22 CμN31 CνN12 −μ1 CνN22 −μ2
ν
N −(μ +ν ) × CνN31 −μ3 (1 − c)μ1 +μ2 +μ3 c − mzs1 1 c + mzs1 2 1 1
ν
N −(μ +ν
ν
N −(μ +ν ) × c − mzs2 2 c + mzs2 2 2 2 c − mzc2 3 c + mzc2 1 3 3
× f z Js 2N2 − 2(μ1 + μ2 ) + Jcs (N1 − 2μ3 ) + h
(25)
for spins of type-2 surface shell nanowire: mzs2 = c
N2 N2 N2 N 2 −μ1 N 2 −μ2 N 2 −μ3
1 23N2
μ1 =0 μ2 =0 μ3 =0 ν1 =0
ν2 =0
ν3 =0
2(μ3 +μ2 +μ3 ) CμN12 CμN22 CμN32 CνN12 −μ1 CνN22 −μ2 CνN32 −μ3
ν
N −(μ +ν ) c − mzs2 1 c + mzs2 2 1 1 × (1 − c)
ν
N −(μ +ν )
ν
N −(μ +ν ) × c − mzs1 2 c + mzs2 2 2 2 c − mzc2 3 c + mzc2 2 3 3
× fz Js 2N2 − 2(μ1 + μ2 ) + Jcs (N2 − 2μ3 ) + h . μ1 +μ2 +μ3
Susceptibility of the nanowire: ∂mzc1 ∂mzc1 = A1,1 + A1,2 ∂h h=0 ∂h h=0 ∂mzc1 ∂mzc2 = A2,1 + A2,2 ∂h h=0 ∂h h=0 ∂mzs1 ∂mzc2 = A3,2 + A3,3 ∂h h=0 ∂h h=0 ∂mzc2 ∂mzs2 = A4,2 + A4,3 ∂h h=0 ∂h h=0
∂mzc2 ∂h ∂mzc2 ∂h ∂mzs1 ∂h ∂mzs1 ∂h
(26)
+ B1
h=0
h=0
+ A2,3 + A3,4
h=0
+ A4,4 h=0
(27) ∂mzs1 ∂h ∂mzs2 ∂h ∂mzs2 ∂h
+ A2,4
h=0
∂mzs2 ∂h
+ B2
(28)
h=0
+ B3
(29)
+ B4 .
(30)
h=0
h=0
With N1 = 1, N2 = 2, N3 = 4, and N4 = 6 denote respectively the coordination numbers and Ckl are the binomial coefficient l! Ckl = k!(l−k)! . The elements Ai,j and Bi are given by: A1,1 = c
1 2N2 +N4
N2 N4 N ν1 N2 −(μ 2 −μ1 N 4 −μ2 1 +ν1 ) μ1 =0 μ2 =0 ν1 =0
ν2 =0 i=0
j =0
2(μ1 +μ2 ) CμN12 CμN24 CνN12 −μ1 CνN24 −μ2
i+j −1 N −(μ +ν ) × Ciν1 Cj 2 1 1 (−1)i (i + j )cN2 −(μ1 +i+j ) mc1 (1 − c)μ1 +μ2
ν
N −(μ +ν )
× c − mzc2 2 c + mzc2 4 2 2 fz Jc N2 + N4 − 2(μ1 + μ2 ) + h ,
A1,2 = c
1 2N2 +N4
N2 N4 N ν1 N2 −(μ 2 −μ1 N 4 −μ2 1 +ν1 ) μ1 =0 μ2 =0 ν1 =0
ν2 =0 i=0
j =0
(31)
2(μ1 +μ2 ) CμN12 CμN24 CνN12 −μ1
i+j −1 N −(μ +ν ) × CνN24 −μ2 Ciν2 Cj 4 2 1 cN4 −(μ2 +i+j ) (−1)i (i + j )mc2 (1 − c)μ1 +μ2
ν
N −(μ +ν )
× c − mzc1 1 c + mzc1 2 1 1 fz Jc N2 + N4 − 2(μ1 + μ2 ) + h ,
(32)
J Supercond Nov Magn (2013) 26:201–211
B1 = c
N2 N4 N 2 −μ1 N 4 −μ2
1 2N1 +N4
μ1 =0 μ2 =0 ν1 =0
ν2 =0
209
2(μ1 +μ2 ) CμN12 CμN24 CνN12 −μ1 CνN24 −μ2
ν
N −(μ +ν )
ν
N −(μ +ν ) c − mzc1 1 c + mzc1 2 1 1 c − mzc2 2 c + mzc2 4 2 2 × (1 − c)
∂fz Jc N2 + N4 − 2(μ1 + μ2 ) + h , × ∂h μ1 +μ2
A2,1 = c2
−(2N1 +N2 +N3 )
N3 N1 N1 N2 N ν2 N1 −(μ 3 −μ1 N 1 −μ2 N 1 −μ3 N 2 −μ4 2 +ν2 ) μ1 =0 μ2 =0 μ3 =0 μ4 =0 ν1 =0
ν2 =0
ν3 =0
ν4 =0 i=0
(33)
2(μ1 +μ2 +μ3 +μ4 )
j =0
N −(μ +ν ) × CμN13 CμN21 CμN31 CμN42 CνN13 −μ1 CνN21 −μ2 CνN31 −μ3 CνN42 −μ4 Ciν2 Cj 1 2 2 (1 − c)μ1 +μ2 +μ3 +μ4
ν
N −(μ +ν ) i+j −1 c − mzc2 1 c + mzc2 3 1 1 × cN1 −(μ2 +i+j ) (−1)i (i + j )mc1
ν
N −(μ +ν )
ν
N −(μ +ν ) × c − mzs1 3 c + mzs1 1 3 3 c − mzs2 4 c + mzs2 2 4 4
× fz Jc N3 + N1 − 2(μ1 + μ2 ) + Jcs N1 + N2 − 2(μ3 + μ4 ) + h , A2,2 = c2−(2N1 +N2 +N3 )
N3 N1 N1 N2 N ν1 N3 −(μ 3 −μ1 N 1 −μ2 N 1 −μ3 N 2 −μ4 1 +ν1 ) μ1 =0 μ2 =0 μ3 =0 μ4 =0 ν1 =0
ν2 =0
ν3 =0
ν4 =0 i=0
(34) 2(μ1 +μ2 +μ3 +μ4 )
j =0
N −(μ +ν ) × CμN13 CμN21 CμN31 CμN42 CνN13 −μ1 CνN21 −μ2 CνN31 −μ3 CνN42 −μ4 Ciν1 Cj 3 1 1 (1 − c)μ1 +μ2 +μ3 +μ4
ν
N −(μ +ν ) i+j −1 c − mzc1 2 c + mzc1 1 2 2 × cN3 −(μ1 +i+j ) (−1)i (i + j )mc2
ν
N −(μ +ν )
ν
N −(μ +ν ) × c − mzs1 3 c + mzs1 1 3 3 c − mzs2 4 c + mzs2 2 4 4
× f z Jc N3 + N1 − 2(μ1 + μ2 ) + Jcs N1 + N2 − 2(μ3 + μ4 ) + h ,
A2,3 = c2−(2N1 +N2 +N3 )
N3 ν3 N1 −(μ N1 N1 N2 N 3 −μ1 N 1 −μ2 N 1 −μ3 N 2 −μ4 3 +ν3 ) μ1 =0 μ2 =0 μ3 =0 μ4 =0 ν1 =0
ν2 =0
ν3 =0
ν4 =0 i=0
2(μ1 +μ2 +μ3 +μ4 )
j =0
N −(μ +ν ) ν × CμN13 CμN21 CμN31 CμN42 CνN13 −μ1 CνN21 −μ2 CνN31 −μ3 CνN42 −μ4 Ci 3 Cj 1 3 3 (1 − c)μ1 +μ2 +μ3 +μ4
ν
N −(μ +ν ) i+j −1 c − mzc2 1 c + mzc2 3 1 1 × (−1)i cN1 −(μ3 +i+j ) (i + j )ms1
ν
N −(μ +ν )
ν
N −(μ +ν ) × c − mzc1 2 c + mzc1 1 2 2 c − mzs2 4 c + mzs2 2 4 4
× f z Jc N3 + N1 − 2(μ1 + μ2 ) + Jcs N1 + N2 − 2(μ3 + μ4 ) + h ,
A2,4 = c2−(2N1 +N2 +N3 )
N3 N1 N1 N2 N ν4 N2 −(μ 3 −μ1 N 1 −μ2 N 1 −μ3 N 2 −μ4 4 +ν4 ) μ1 =0 μ2 =0 μ3 =0 μ4 =0 ν1 =0
ν2 =0
ν3 =0
ν4 =0 i=0
B2 = c
2N2 +2N1 +N3
N3 N1 N1 N2 N 3 −μ1 N 1 −μ2 N 1 −μ3 N 2 −μ4 μ1 =0 μ2 =0 μ3 =0 μ4 =0 ν1 =0
ν2 =0
ν3 =0
ν4 =0
(36)
2(μ1 +μ2 +μ3 +μ4 )
j =0
N −(μ +ν ) × CμN13 CμN21 CμN31 CμN42 CνN13 −μ1 CνN21 −μ2 CνN31 −μ3 CνN42 −μ4 Ciν4 Cj 2 4 4 (1 − c)μ1 +μ2 +μ3 +μ4
ν
N −(μ +ν ) i+j −1 c − mzc2 1 c + mzc2 3 1 1 × cN2 −(μ4 +i+j ) (−1)i (i + j )ms2
ν
N −(μ +ν )
ν
N −(μ +ν ) × c − mzs1 3 c + mzs1 1 3 3 c − mzc1 2 c + mzc1 1 2 2
× f z Jc N3 + N1 − 2(μ1 + μ2 ) + Jcs N1 + N2 − 2(μ3 + μ4 ) + h ,
1
(35)
2(μ1 +μ2 +μ3 +μ4 ) CμN13 CμN21 CμN31 CμN42
ν
N −(μ +ν ) × CνN13 −μ1 CνN21 −μ2 CνN31 −μ3 CνN42 −μ4 (1 − c)μ1 +μ2 +μ3 +μ4 c − mzc2 1 c + mzc2 3 1 1
ν
N −(μ +ν )
ν
N −(μ +ν ) × c − mzc1 2 c + mzc1 1 2 2 c − mzs1 3 c + mzs1 1 3 3
ν
N −(μ +ν ) × c − mzs2 4 c + mzs2 2 4 4
(37)
210
J Supercond Nov Magn (2013) 26:201–211
×
A3,2 = c
∂fz Jc N3 + N1 − 2(μ1 + μ2 ) + Jcs N1 + N2 − 2(μ3 + μ4 ) + h , ∂h ν3 N1 −(μ N2 N2 N1 N 2 −μ1 N 2 −μ2 N 1 −μ3 3 +ν3 )
1 22N2 +N1
μ1 =0 μ2 =0 μ3 =0 ν1 =0
ν2 =0
ν3 =0
i
(38)
2μ1 +μ2 +μ3 CμN12 CμN22 CμN31 CνN12 −μ1
j
N −(μ +ν ) ν × CνN22 −μ2 CνN31 −μ3 Ci 3 Cj 1 3 3 (1 − c)μ1 +μ2 +μ3
ν
N −(μ +ν ) i+j −1 c − mzs1 1 c + mzs1 2 1 1 × cN1 −(μ3 +i+j ) (−1)i (i + j )mc2
ν
N −(μ +ν ) × c − mzs2 2 c + mzs2 2 2 2
×fz Js 2N2 − 2(μ1 + μ2 ) + Jcs (N1 − 2μ3 ) + h , A3,3 = c
N2 N2 N1 N ν1 N2 −(μ 2 −μ1 N 2 −μ2 N 1 −μ3 1 +ν1 )
1 22N2 +N1
μ1 =0 μ2 =0 μ3 =0 ν1 =0
ν2 =0
ν3 =0
i
(39)
2μ1 +μ2 +μ3 CμN12 CμN22 CμN31 CνN12 −μ1
j
N −(μ +ν ) × CνN22 −μ2 CνN31 −μ3 Ciν1 Cj 2 1 1 (1 − c)μ1 +μ2 +μ3
ν
N −(μ +ν ) i+j −1 c − mzs2 2 c + mzs2 2 2 2 × cN2 −(μ1 +i+j ) (−1)i (i + j )ms1
ν
N −(μ +ν ) × c − mzc2 3 c + mzc2 1 3 3
× f z Js 2N2 − 2(μ1 + μ2 ) + Jcs (N1 − 2μ3 ) + h , A3,4 = c
N2 N2 N1 N ν2 N2 −(μ 2 −μ1 N 2 −μ2 N 1 −μ3 2 +ν2 )
1 22N2 +N1
μ1 =0 μ2 =0 μ3 =0 ν1 =0
ν2 =0
ν3 =0
i
(40)
2μ1 +μ2 +μ3 CμN12 CμN22 CμN31 CνN12 −μ1
j
N −(μ +ν ) × CνN22 −μ2 CνN31 −μ3 Ciν2 Cj 2 2 2 (1 − c)μ1 +μ2 +μ3
ν
N −(μ +ν ) i+j −1 c − mzs1 1 c + mzs1 2 1 1 × cN2 −(μ2 +i+j ) (−1)i (i + j )ms2
ν
N −(μ +ν ) × c − mz2 3 c + mzc2 1 3 3
× f z Js 2N2 − 2(μ1 + μ2 ) + Jcs (N1 − 2μ3 ) + h , B3 = c
N2 N2 N1 N 2 −μ1 N 2 −μ2 N 1 −μ3
1 22N2 +N1
μ1 =0 μ2 =0 μ3 =0 ν1 =0
ν2 =0
ν3 =0
(41)
2μ1 +μ2 +μ3 CμN12 CμN22 CμN31 CνN12 −μ1 CνN22 −μ2
ν
N −(μ +ν ) × CνN31 −μ3 (1 − c)μ1 +μ2 +μ3 c − mzs1 1 c + mzs1 2 1 1
ν
N −(μ +ν )
ν
N −(μ +ν ) × c − mzs2 2 c + mzs2 2 2 2 c − mzc2 3 c + mzc2 1 3 3 ×
A4,2 = c
∂fz Js 2N2 − 2(μ1 + μ2 ) + Jcs (N1 − 2μ3 ) + h , ∂h 1
23N2
ν3 N2 −(μ N2 N2 N2 N 2 −μ1 N 2 −μ2 N 2 −μ3 3 +ν3 ) μ1 =0 μ2 =0 μ3 =0 ν1 =0
ν2 =0
ν3 =0 i=0
j =0
(42)
2μ1 +μ2 +μ3 CμN12 CμN22 CμN32 CνN12 −μ1
× CνN22 −μ2 CνN32 −μ3 (1 − c)μ1 +μ2 +μ3 cN2 −(μ3 +i+j ) (−1)i (i + j )
ν
N −(μ +ν )
ν
N −(μ +ν ) (i+j −1) c − mzs2 1 c + mzs2 2 1 1 c − mzs1 2 c + mzs2 2 2 2 × mc2
× f z Js 2N2 − 2(μ1 + μ2 ) + Jcs (N2 − 2μ3 ) + h , A4,3 = c
1 23N2
N2 N2 N2 N ν2 N2 −(μ 2 −μ1 N 2 −μ2 N 2 −μ3 2 +ν2 ) μ1 =0 μ2 =0 μ3 =0 ν1 =0
ν2 =0
ν3 =0 i=0
j =0
(43)
2μ1 +μ2 +μ3 CμN12 CμN22 CμN32 CνN12 −μ1 CνN22 −μ2
N −(μ +ν ) × CνN32 −μ3 Ciν2 Cj 2 2 2 (1 − c)μ1 +μ2 +μ3 cN2 −(μ2 +i+j ) (−1)i (i
(i+j −1)
+ j )ms1
c − mzs2
ν1
J Supercond Nov Magn (2013) 26:201–211
211
N −(μ +ν )
ν
N −(μ +ν ) × c + mzs2 2 1 1 c − mzc2 3 c + mzc2 2 3 3
× f z Js 2N2 − 2(μ1 + μ2 ) + Jcs (N2 − 2μ3 ) + h , A4,4 = c
N2 N2 N2 N ν1 N2 −(μ 2 −μ1 N 2 −μ2 N 2 −μ3 1 +ν1 )
1 23N2
μ1 =0 μ2 =0 μ3 =0 ν1 =0
ν2 =0
ν3 =0 i=0
j =0
(44)
2μ1 +μ2 +μ3 CμN12 CμN22 CμN32 CνN12 −μ1
N −(μ +ν ) × CνN22 −μ2 CνN32 −μ3 Ciν1 Cj 2 1 1 (1 − c)μ1 +μ2 +μ3 cN2 −(μ1 +i+j ) (−1)i (i
ν
N −(μ +ν )
ν
N −(μ +ν ) × c − mzs1 2 c + mzs1 2 2 2 c − mzc2 3 c + mzc2 2 3 3
(i+j −1)
+ j )ms2
× fz Js 2N2 − 2(μ1 + μ2 ) + Jcs (N2 − 2μ3 ) + h , B4 = c
1 23N2
N2 N2 N2 N 2 −μ1 N 2 −μ2 N 2 −μ3 μ1 =0 μ2 =0 μ3 =0 ν1 =0
ν2 =0
ν3 =0
(45)
2μ1 +μ2 +μ3 CμN12 CμN22 CμN32 CνN12 −μ1 CνN22 −μ2 CνN32 −μ3
ν
N −(μ +ν ) × (1 − c)μ1 +μ2 +μ3 c − mzs2 1 c + mzs2 2 1 1
ν
N −(μ +ν )
ν
N −(μ +ν ) × c − mzs1 2 c + mzs2 2 2 2 c − mzc2 3 c + mzc2 2 3 3
∂fz Js 2N2 − 2(μ1 + μ2 ) + Jcs (N2 − 2μ3 ) + h , × ∂h
∂fz = β 1 − tanh2 (βy) . ∂h h=0
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
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