Phase Transitions, 85 (2012) 995
Some characteristic phenomena
in a transverse Ising nanotube
T. Kaneyoshi 1,2
1-510, Kurosawadai, Midoriku, Nagoya, 458-0003, Japan
1, E-mail;
2, Prof. Emeritus at Nagoya University, Japan.
Keywords; Nanomagnetic material, Transverse Ising model,
Phase diagrams, Magnetizations, Ferrimagnetism,
Abstract
The phase diagram and magnetizations of a cylindrical nanotube described by the transverse Ising model are investigated by the use of the effective field theory with correlations. Some comparisons between the nanotube and the nanowire have been given for the phase diagrams. In particular, the temperature dependences of longitudinal magnetization in the system with a negative shell-core interaction are investigated. Some characteristic phenomena (new types in ferrimagnetism) which have not been observed in the nanowire as well as similar phenomena are found in the thermal variations, depending on the ratio of the physical parameters in the surface shell and the core. The possibilities of two compensation points and a field induced compensation point in the nanotube are also discussed.
1. Introduction
Recently, there has been growing interest in the magnetic properties of a material with a nanostructure, such as nanoparticle, nanorod, nanotube and nanowire [1,2]. In particular, considerable attention has been directed experimentally to the study of magnetic nanowires, due to their distinctive properties and potential applications, such as ultrahigh density magnetic recording media. But, one should notice that the structure of a nanowire is a little different from that of a nanotube which exhibits a core-free configuration. In a series works [3-6], the phase diagrams, the temperature dependences of magnetization, the initial susceptibility in a Ising nanowire (or nanotube ) are examined by the use of the mean field theory (MFA) and the effective field theory with correlation (EFT) [7,8]. Some similarities between the nanowire and the nanotube have been discussed by using the two theoretical frameworks. Experimentally, an interesting fact for the existence of a negative core-shell coupling has been reported for recent data for La0.67Ca0.33MnO3 (LCMN) nanoparticle [9,10]. The bulk LCMN is a typical ferromagnet, but the magnetic properties of a LCMN nanoparticle have been better described in the framework of a ferrimagnetic structure., namely the core-shell model with a negative core-shell coupling (J1 < 0). For the analyses of the experimental data, the Neel hyperbola [12,13] have been used in the paramagnetic region. But, the Neel hyperbola is only valid for bulk ferrimagnetic materials and it is also derived by the use of the MFA. Whether it is valid for a nano-scaled system and is also valid for the EFT better than the MFA has been also discussed and clarified theoretically by the use of the EFT [4,5]. The hysteresis behaviors in a nanowire with a negative (or a positive) core-shell coupling have also been discussed in [11]. Furthermore, some characteristic behaviors of mixed spin-1/2 and spin-1 Ising nanotube has been obtained in [14]. The spin-1 Ising nanotube has also been discussed in [15] by using the EFT.
The transverse Ising model (TIM) is generally believed to describe the phase transition of order-disorder type ferroelectrics. The model has also been applied successfully to the investigation of a material with a nanostructure. Except the studies of magnetic nanoparticles [16-18] and nano-scaled thin films [19-22], however, only a few related results of the model have been reported for the magnetic properties of a material with a nanostructure. In the previous works [23, 24], the phase diagrams and the temperature dependences of magnetizations in a transverse Ising nanowire have been examined by the use of the two theoretical frameworks of MFA and EFT [5,25]. However, the effects of a negative core-shell coupling on the magnetic properties in a transverse Ising nanotube have not been discussed except the work [26] with a surface dilution, in which the possibility of two compensation points has been discussed. Very recently, the phase diagrams in cylindrical ferroelectric nanotube having a shape different from the above nanotube [26] have been discussed by the EFT [27]. In addition, the magnetic properties of a two-dimension circle nanoparticle described by the TIM, wihich consists of a spin-3/2 core and a spin-5/2 surface shell, are also examined by the EFT [28].
In this work, the phase diagrams and the magnetizations of a cylindrical TIM nanotube with the two (S=2) shells, namely one shell of the surface and one shell in the core, where the center is vacant, are examined by the use of the EFT. The model may be also applied to the magnetic nanotube, such as FePt and Fe3O4 [29], in a magnetic field applied to the transverse direction for the spontaneous magnetization [2]. The EFT corresponds to the Zernike approximation [30], which is superior to the standard MFA, since some parts of spin correlations are automatically included (see the review article [8]). The paper is organized as follows. In section 2, we define the model and give briefly the formulations of the EFT. In section 3, the phase diagrams of the nanotube are examined, in order to clarify the relations between the results of the nanowire in [23] and the present results. In section 4, some characteristic phenomena found in the nanotube are reported, when the exchange interaction between the surface shell and the core shell is negative. Section 5 is devoted to a brief discussion.
2. Formulation
We consider a nanotube, as depicted in Fig.1(A), in which the tube is consisted of the surface shell and one shell in the core. The each site on the figure is occupied by a Ising spin. The each spin is connected to the two nearest neighbor spins on the above and below sections. The surface shell is coupled to the next shell in the core with an exchange interaction J1. In other words, the system is consisted from the two (S=2) shells, namely one shell of the surface and one shell in the core, where the center is vacant.
The Hamiltonian of the system is given by
H = － JS∑σi ZσjZ－J∑σmZ σnZ －J1∑σiZσm Z
(ij) (mn) (im)
−ΩS Σσi X − Ω Σσm X , (1)
(i) (m)
where σi α (α = z, x) is the Pauli spin operator with σiZ = 1. The JS is the exchange interaction between two nearest-neighbor magnetic atoms at the surface shell and the J is the exchange interaction in the core. ΩS and Ω represent the transverse fields at the surface shell and in the core, respectively. The surface exchange interaction JS is often defined as
JS = J (1 + △S ) , (2)
in order to clarify the effects of surface on the physical properties in the system.
For the nanotube depicted in Fig.1(A), there exit two longitudinal magnetizations ( mS1 and mS2 ) on the surface shell and one longitudinal magnetizations mC on the core in the z direction. Within the framework of the EFT, they can be given by
mS1 = [cosh(A) + mS1 sinh(A)]2 [cosh(A) + mS2 sinh(A)]2
[ cosh(B) + mc sinh(B) ] fS(x)|x=0
(3)
mS2 = [cosh(A) + mS2 sinh(A)]2 [cosh(A) + mS1 sinh(A)]2
[ cosh(B) + mc sinh(B) ]2 fS(x)|x=0
(4)
mc= [ cosh(C) + mc sinh(C) ]4
[cosh(B)+mS1sinh(B)][cosh(B)+mS2sinh(B)]2f(x)|x=0 ,
(5)
where A, B and C are defined by A = JS ∇, B = J1∇ and C = J ∇. ∇ = ∂/∂x is the differential operator. Here, the functions fS(x) and f(x) are defined by
fS(x) = ( x / yS ) tanh( β yS )
and (6)
f(x) = ( x / y ) tanh( β y )
with
yS = ( x2 + ΩS2 )1/2
and (7)
y = ( x2 + Ω2 )1/2 ,
where β= 1 / kBT and T is a temperature.
Furthermore, let us define the total longitudinal magnetization per site and the total transverse magnetization per site as follows;
(8)
and
, (9)
where the transverse magnetizations in (9) are also given by the same equations as those of (3) – (5), only replacing the functions fS(x) and f(x) in (3)-(5) by the new functions hS(x) and h(x) respectively. The new functions hS(x) and h(x) are defined by
ｈS(x) = ( ΩS / yS ) tanh( β yS )
and (10)
ｈ(x) = ( Ω / y ) tanh( β y )
In order to obtain the phase diagrams of the present system, we must expand the right-hand side of the three coupled equations (3)- (6). They are obtained as follows;
= 0 , (11)
The transition temperature of the system can be determined from. Here, the coefficients An (n=1-8) in the matrix take the following forms;
A1 = cosh3(C) sinh( C ) cosh3(B) f(x)|x=0
A2 = cosh4(C) sinh(B) cosh2(B) f(x)|x=0
A3 = cosh4( A ) sinh( B ) fS(x)|x=0 (12)
A4 = cosh3( A ) sinh(A) cosh( B ) fS(x)|x=0
A5 = cosh4 ( A ) sinh( B ) cosh (B) fS(x)|x=0
A6 = cosh3( A ) sinh(A) cosh2(B) fS(x)|x=0 ,
where the coefficients An (n=1-6) can be easily calculated by using the mathematical relation exp(αf(x) = f(+x ). Here, one should notice that the phase diagrams obtained fromare independent of the sign of J1 (or B), since the value ofdoes not change for the variation of the sign of J1 (or B).
For the following discussions, let us define the parameters, h, q and r as
h = (13)
q = (14)
and
r = (15)
3. Phase diagrams
In this section, let us investigate the phase diagrams of the nanotube, using the formulation given in section 2 and selecting the value of J1 as positive. That is to say, the spins on the surface shell are aliened to the same direction as the spins on the core. Here, one should notice that the phase diagrams obtained in this section are also valid for the case of r < 0.0, namely for the case that the spins on the surface shell are aliened oppositely to the direction of the spins on the core.
Figure 2 shows the Curie temperature (TC) versus Ω plots for the two nanosystems ( nanotube of Fig.1(A) and nanowire of Fig.1(B) ), when the values of r, q and h are fixed at r = 1.0, q = 1.0 and h = 1.0. The solid and dashed lines represent the results obtained by the EFT and the MFA, respectively. The results noted as ‘wire’ are also equivalent to those obtained in [23]. Such a variation of TC is also observed in the bulk transverse Ising system [31]. Here, one should notice that the EFT results for the both systems are smaller than those of the MFA, which fact indicates that the EFT improves the numerical results to more reasonable direction than the MFA.
Figure 3 shows the dependences of the TC on the surface exchange interaction JS in the two nanosystems ( nanotube of Fig.1(A) and nanowire of Fig.1(B) ) with the fixed values of h= 2.0 and q = 1.0 , when the value of r is changed. The solid and dashed lines also represent the results obtained by the EFT and the MFA, respectively. The EFT results are also smaller than those of the MFA. The TC curves of the both systems may increase generally with the increase of JS. As shown by the curves labeled as r = 0.01, however, when the inter-shell coupling J1 between the surface and the core becomes very weak, the feature of TC curves looks like that normally found in the semi-infinite ferromagnet with a free surface [32]. These features have also been observed for the cylindrical Ising nanowire and nanotube with zero transverse field [3] as well as for the transverse Ising nanowire [23].
In Fig.3, the dotted line is described by the following equation with h = 2.0;
, (16)
which comes from (3) or (4), when the value of r (B) is given by r (B) = 0.0. Accordingly, the TC value described by the dashed line may increase with the increase of △S ( JS ) and it increases linearly with the value of △S, when the value of △S becomes large. On the other hand, the horizontal line labeled ‘Tube (r = 0.01)’ in the figure is approximately given by kBTC/J = 2.477. The value is also given by the equation (16) with h = 2.0. But, the parameter A and the function fS (x) in (16) must be then replaced by C ( = J∇) and f(x), since it comes from (5) with B = 0.0 ( J1 = 0.0). That is to say, the equation is given by
, (17)
which is nothing but the equation for the transition temperature of the bulk transverse Ising system with the coordination number z = 4 [31]. The TC curve reduces to zero at the critical value hC given by hC = 2.751, as has been shown in [31].
In this way, the horizontal part and the linearly increasing part of the curve labeled ‘Tube (r = 0.01)’ in Fig.3 come from the contributions from the core and from the surface shell, respectively. These two contributions must be crossed at △S = 0.0 from the above arguments. In fact, such a result can be seen from the curve labeled ‘Tube (r = 0.01)’. Comparing with the numerical results of the two nano-systems given in Figs.2 and 3, we can understand that they are essentially similar to each other. The main difference in Fig.3 is the values of the horizontal lines. As discussed in [23], the horizontal line for the nanowire is approximately given by kBTC/J = 4.346, which also comes from the core part described by the black points in Fig.1(B).
As shown in [23], the TC vs. △S plots for the nanowire with r = 0.01 and q = 1.0 have exhibited some characteristic behaviors, when the value of h is selected as a value near the hC (hC = 4.32). In Fig.4, the TC vs. △S plots for the nanotube with r = 0.01 and q = 1.0 are depicted by taking some values near the hC (hC = 2.751). For comparison, the MFA result is plotted as a dashed line for the case of h = 2.5. The TC curves labeled h = 2.5 and h = 2.75 take the same form as that labeled r = 0.01 in Fig.3. When the value of h becomes larger than the critical value (hC = 2.751), the feature of the TC curve is clearly different from the above mentioned behavior, such as the curves labeled h = 2.76 and h = 3.0. The reason comes from the fact that the spins of the core are always directed to the x- or y- direction for h > hC. However, we could not find any characteristic features in the TC curve for the nanotube with h = hC, r = 0.01 and q = 1.0, in contrast to the corresponding curve for the nanowire in [23].