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Phase diagrams of a transverse Ising superlattice based on two theoretical frameworks
Physica A 291 (2001) 387–398
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Phase diagrams of a transverse Ising superlattice based on two theoretical frameworks T. Kaneyoshi Department of Natural Science Informatics, School of Informatics and Sciences, Nagoya University, 464-8601 Nagoya, Japan Received 23 June 2000
Abstract The phase diagrams of a superlattice composed from two di*erent alternating ferroelectric slabs with the same thickness L are discussed on the basis of the spin-1=2 transverse Ising model. The two theoretical frameworks of the di*erential operator technique are applied to the investigations, namely the Zernike approximation and the decoupling approximation superior to the Bethe–Peierls approximation. The unexpected phenomena obtained in the system with L = 2, such as the reentrant phenomenon due to the competition of the interslab coupling and the transverse 2eld, can be found by the use of both approximations. They seem to be independent c 2001 Elsevier Science B.V. All rights reserved. of the theoretical approximation. Keywords: Transverse Ising model; Ferroelectric superlattice
1. Introduction In recent years, the study of magnetic multilayers has been one of the most rapidly expanding 2elds in basic research and industrial applications. A great number of experimental and theoretical investigations have been devoted to the 2eld. On the other hand, some ferroelectric superlattices have also been fabricated and studied owing to the advances of experimental techniques, such as (BaTiO3 )n =(SrTiO3 )n with perovskite structure [1–3]. Theoretically, the physical properties of ferroelectric superlattices have been discussed on the basis of the Ginzburg–Landau phenomenological theory [4,5]. Afterwards, the transverse Ising model (TIM) has been introduced to the study of E-mail address: [email protected] (T. Kaneyoshi). c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 4 7 8 - 7
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dielectric properties in ferroelectric superlattices [6 –12] as well as the ferroelectric bilayer system [13]. The TIM 2rst proposed by De Gennes for the KDP-like ferroelectrics has been used to study various interesting systems, such as thin 2lms [14 –17] and semi-in2nite system with a surface [18,19]. In the previous works [10 –13] we have discussed the phase diagrams of a ferroelectric superlattice as well as a ferroelectric bilayer system on the basis of the TIM, using the e*ective-2eld theory with correlations (EFT) [20,21]. The EFT is superior to the standard mean 2eld approximation (MFA), but the statistical accuracy corresponds to the Zernike approximation [22]. It has been applied successfully to a great number of interesting physical systems, such as [14 –19]. In Refs. [10 –13], the dependence of the Curie temperature Tc and the critical transverse 2eld c on the interslab interaction R and the slab thickness L has been examined. The c is the critical 2eld at which the Tc versus transverse 2eld plot reduces to zero. In the process, we have found some unexpected phenomena in the phase diagrams, when large values of R and are applied to the ferroelectric superlattice and the ferroelectric bilayer system with a small number of slab thickness L, namely the reentrant phenomenon due to the competition between R and and the reduction of the c versus R plot to zero. As far as we know, however, such phenomena have not been reported. The aim of this work is to study the phase diagram of a spin-1=2 transverse Ising superlattice consisting of two di*erent alternating slabs with the same thickness L, using the two theoretical frameworks in the di*erential operator technique for Ising systems [20,21]. One is the EFT, being equivalent to the Zernike approximation, from which we can derive the same results as those in Refs. [10 –12]. On the other hand, various trials for improving the EFT in a reasonable direction have been conducted in the past [20,21]. Among them, the present author has introduced a new decoupling approximation [23] for the multispin correlation functions appearing in the expansion of Ising spin identities. The decoupling approximation (DA) has proved that the results of Tc in the spin-1=2 Ising systems become better than those of the Zernike and the Bethe–Peierls approximation [24]. In Section 2, the formulation of the ferroelectric superlattices with L = 1 and 2 is discussed on the basis of the DA, in order to compare the critical properties with those of the EFT. The numerical calculations of the critical properties are given in Section 3, using both the formulations. We 2nd that they show in the phase diagrams the characteristic features similar to each other and hence it proves that the characteristic phenomena found in Refs. [10 –13] are independent of the theoretical approximation.
2. Formulation We consider a superlattice composed from two di*erent ferroelectric slabs (A and B) with the same (one or two layer) thickness. Each slab is de2ned on the x–y plane and is stacked alternatively to the z direction. It is with pseudo-spin sites on square
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lattice. The system is described by the Ising Hamiltonian in a transverse 2eld (i = ), i Si x ; (1) Jij Si z Sj z − H =− i
(ij)
where Si z and Si x are the components of the pseudo-spin operator at site i on a slab (A or B) and the 2rst summation runs over all nearest-neighbor pairs. Jij is the exchange interaction constant between pseudo-spins at the ith and jth sites, which takes JA (or JB ) in the slab A (or the slab B) or JAB between the A and B slabs. According to (1) and the di*erential operator technique for the Ising systems [20,21], the average value of an atomic spin at a central site i in the slab ( = A or B) is given by {fi }Si z = {fi } [cosh(Ji+ ∇) + Si z sinh(Ji+ ∇)
+ Si z sinh(Ji+ ∇)] F(x)|x=0
(2)
with Si z = Si z − Si z ;
(3)
where {fi } can take any function of the Ising spin variables as long as it is not a function of the site i, expresses the nearest-neighbor sites and ∇ = @=@x is the di*erential operator. The function F(x) is given by x F(x) = tanh(y) (4) y with y = (x2 + 2 )1=2 ;
(5)
where = 1=kB T . Then, the averaged magnetization of each slab m = Si z (i ∈ A or B) can be obtained by putting {fi } = 1 into (2), when each slab is composed one layer (L = 1) and two layers (L = 2). Expanding the right-hand side of (2), we can obtain many Kuctuating spin correlation functions, such as Si+ z Si+ z and Si+ z Si+ z Si+ z . As discussed in Ref. [23], the DA is to introduce the decoupling approximation Si+ z Si+ z = Si z Si+ z Si z Si+ z Si+ z Si+ z Si+ z = Si z Si z Si+ z Si z Si+ z Si z Si+ z :
(6)
Following such a decoupling procedure, the magnetization mA (or mB ) per site in slab A (or B) includes the two di*erent types of spin correlations de2ned by gA = Si z Si+ z for i and i + ∈ A
(7)
gC = Si z Si+ z for i ∈ A and i + ∈ B
(8)
and
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or gB = Si z Si+ z for i and i + ∈ B
(9)
and gD = Si z Si+ z for i ∈ B and i + ∈ A :
(10)
These correlation functions can be also calculated from (2), substituting {fi } = Si+ z . Physically, gA and gB express the spin correlations in slabs A and B. The gC and gD represent the correlations between slabs A and B. The above statement can be applied to any superlattice with a certain value of L. In order to get the relation from which the phase diagram can be obtained, however, one must 2x the value of L. Depending on the value of the slab thickness (L = 1 or 2), we can get di*erent sets of coupled equations for mA ; mB ; gA ; gB ; gC and gD . So, let us formulate it, depending on whether L = 1 or 2. (a) L = 1: At the transition temperature, the spontaneous magnetizations reduce to zero. Consequently, the correlation functions at T = Tc are given in the form of the following coupled equations, from the equations of gA and gC in slab A gA = A1 [1 + 3(gA )2 ] + A3 (gA )2 [3 + (gA )2 ] + gC [2A2 gA + 6A4 gA {1 + (gA )2 } + 2A6 (gA )3 + A5 gC {1 + 3(gA )2 } + A7 (gA )2 gC {3 + (gA )2 }] ;
(11)
gC = 4gA gC [A1 + A5 + (A3 + A7 )(gA )2 ] + [A2 + 6A4 (gA )2 + A6 (gA )4 ][1 + (gC )2 ]
(12)
and, from the equations of gB and gD in slab B gB = B1 [1 + 3(gB )2 ] + B3 (gB )2 [3 + (gB )2 ] + gD [2B2 gB + 6B4 gB {1 + (gB )2 } + 2A6 (gB )3 + B5 gD {1 + 3(gB )2 } + B7 (gB )2 gD {3 + (gB )2 }] ;
(13)
gD = 4gB gD [B1 + B5 + (B3 + B7 )(gB )2 ] + [B2 + 6B4 (gB )2 + B6 (gB )4 ][1 + (gD )2 ] ;
(14)
where the coeLcients An and Bn (n = 1–7) are de2ned in Appendix A. We are now interested in discussing the Tc of the system. The usual argument that the m ( = A or B) tends to zero in the vicinity of Tc allows us to consider only linear terms of magnetizations. This leads to the matrix equation of the form mA M =0 (15) mB with
M=
U1 ; U3 ;
U2 U4
;
(16)
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where the matrix elements Un (n = 1– 4) are given in Appendix A. Then, the critical information can be determined from det(M ) = 0 by taking the highest solution in it, namely from the relation U1 U4 = U2 U3 :
(17)
Here, one should notice the following facts: When substituting g = 0:0 ( =A–D) into the above formulation (or (17)), we can get the previous formulation based on the EFT (or Zernike approximation) in Refs. [10 –12]. On the other hand, when substituting JAB = 0 into the above formulation, the superlattice system is decomposed into two independent slabs with L=1 de2ned on a square lattice. Then, the correlation functions gC and gD reduce to zero. The correlation functions gA and gB in (11) and (13) are given by the same form, namely g = K1 + 3(K1 + K2 )(g )2 + K2 (g )4 ;
(18)
where =A (or B) and the coeLcients K1 and K2 are given by A1 and A2 (or B1 and B2 ). The transition temperature of each slab can be determined from 1 = 4K1 + 12K2 (g )2 + 4K2 (g )4 :
(19)
Relations (18) and (19) are completely equivalent to those of spin-1=2 Ising square lattice in Ref. [23] when substituting = 0:0 into them. Accordingly, when = 0:0, the Tc of each slab determined from (18) and (19) is given by kB Tc =J = 2:7444, which should be compared to 4.0 for the MFA, 3.0898 for the EFT (Zernike approximation), 2.8854 for the Bethe–Peierls approximation and 2.2692 for the exact (Onsager) solution [25], where we have de2ned J = J in the bulk. (b) L = 2: Following the same procedure as that of the case (a), one can easily get the following set of coupled equations for gA ; gB ; gC and gD : gA = A1 [1 + 4(gA )2 ] + 2A3 (gA )2 [3 + 2(gA )2 ] + A5 (gA )4 + gA gC [A2 + 2A4 {2 + 3(gA )2 } + A6 (gA )2 {4 + (gA )2 }] ; gC [1 − gA {5A1 + 10A3 (gA )2 + A5 (gA )4 }] = A2 + 10A4 (gA )2 + 5A6 (gA )4
(20) (21)
and gB = B1 [1 + 4(gB )2 ] + 2B3 (gB )2 [3 + 2(gB )2 ] + B5 (gB )4 + gB gD [B2 + 2B4 {2 + 3(gB )2 } + B6 (gB )2 {4 + (gB )2 }] ; gD [1 − gB {5B1 + 10B3 (gB )2 + B5 (gB )4 }] = B2 + 10B4 (gB )2 + 5B6 (gB )4 ;
(22) (23)
where the coeLcients An and Bn (n = 1– 6) are de2ned in Appendix B. On the other hand, the critical properties of the system with L = 2 can be determined by taking the same process as that of the case of L = 1, namely from the relation V1 V4 = V2 V3 ;
(24)
where the matrix elements Vn (n =1– 4) are de2ned in Appendix B. The critical information of the superlattice with L = 2 can be obtained by taking the highest solution
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Fig. 1. The phase diagram (k B Tc =J versus JAB =J plot) of the spin-1=2 Ising superlattice consisting of two di*erent alternating slabs with the same thickness L (L = 1 and 2) in zero transverse 2eld (=J = 0:0), when the values of JA =J and JB =J are 2xed at JA =J = 1:0 and JB =J = 0:5. The dashed and solid lines represent, respectively, the results of the EFT (or Zernike approximation) and the DA. The curves labeled A are obtained for the system with L = 1 and the curves are the results of L = 2.
in (24). When substituting the condition of g = 0:0 ( =A–D) into (24), we can also get the critical properties within the formulation of the Zernike approximation. 3. Numerical results In this section, let us examine the phase diagram of the spin-1=2 Ising superlattice with L = 1 (or L = 2) by solving the coupled equations (11) – (14) and (17) for the system with L = 1 (or the coupled equations (20) – (23) and (24) for the system with L=2) numerically. As noted in Section 2, we can easily obtain the results of the Zernike approximation (or EFT) as well as those of the DA from the present formulation. In the following, therefore, the critical properties are studied by the use of both formulations and compared to each other. We de2ne the following parameters: A = JA =J;
B = JB =J
and
R = JAB =J :
(25)
Furthermore, in order to compare with the results in Refs. [10 –13] let us take the 2xed values of A = 1:0 and B = 0:5 for the following numerical calculations. At 2rst, let us show the phase diagram of the spin-1=2 Ising superlattices in zero transverse 2eld ( = 0:0), since Eq. (2) then becomes an exact identity [26] in the Ising spin system, as discussed in [20,21]. Fig. 1 shows a typical result in the phase diagrams of the system. It is depicted as the Tc versus R plot, in order to compare with the following characteristic 2gures as well as those in Refs. [10 –12]. The curves labeled A represent the results of the system with L = 1 and the curves labeled B are
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the results of the system with L = 2. For each case, the dashed line shows the result of the EFT (or Zernike approximation) and the solid line is the result of the DA. As can be seen from the 2gure, the DA improves the result of the EFT in a reasonable direction and the values at R = 0:0; k B Tc =J = 3:0898 and 2.7444 for the curves A, are equivalent to the results of square lattice [23]. Furthermore, the characteristic features of dashed lines A and B, namely the facts that curve A crosses curve B and they increase monotonically with the increase of R, have also been observed in Refs. [6,7,9 –12]. As discussed in Refs. [10 –13], on the other hand, the phase diagram of the superlattice in a 2nite transverse 2eld starts to exhibit many unexpected features, when the large values of and R are selected especially in the system with a small value of L (L¿2). In order to show the facts clearly, let us at 2rst give the phase diagram (Tc versus R plot) of the superlattice with L = 1, as Fig. 2. Fig. 2(A) shows the results of the Zernike approximation (EFT), which are equivalent to those in Refs. [10 –12]. Fig. 2(B) represents the DA results. In each case, the 2ve values of =J are selected; =J = 0:0 for curve A, =J = 1:0 for curve B, =J = 2:0 for curve C, =J = 4:0 for curve D and =J = 5:0 for curve E. In each 2gure, the feature of the curves D and E is clearly di*erent from that of curves A, B and C. The di*erence comes from the existence of the critical transverse 2eld c at which the Tc versus plot reduces to zero. Anyway, when comparing the two 2gures A and B, we can also 2nd that the decoupling approximation (DA) improves the results of the EFT (Zernike approximation) in a reasonable direction. The DA and EFT give results essentially similar to each other. Fig. 3 shows the phase diagram (Tc versus R plot) of the superlattice with L = 2, when the two typical values of =J are selected; =J = 1:0 for curves A and =J = 4:0 for curves B. The dashed and solid lines in the 2gure represent, respectively, the results of the EFT and the DA. In each approximation, the obtained curves exhibit behaviors that are essentially similar to each other and the DA also improves the results of the EFT in a reasonable direction. At this place, we 2nd some unexpected phenomena in the phase diagram: Each curve labeled A at 2rst increases monotonically from the value at R = 0:0 with the increase of R, like the curves B in Fig. 2. However, it shows the reentrant behavior (the double values of Tc ) in a certain region of large R and the curve reduces to zero at a critical value (Rc ) of R. On the other hand, each curve labeled B exhibits the two critical values of R (R1 and R2 ; R2 ¿ R1 ) where the curve reduces to zero and in the vicinity of R2 it also expresses the weak reentrant phenomenon. Thus, the curves are clearly di*erent from the corresponding ones in Fig. 2. Now, in order to understand the big di*erence of the behavior in the Tc curves between Figs. 2 and 3 more clearly, in Fig. 4, the critical value c is plotted as a function of R for the two superlattices with L = 1 and 2. The curves labeled A and B represent the results of L = 1 and 2, respectively. The dashed and solid lines are the results of the EFT and the DA and they take forms that are similar to each other. The DA also improves the result of the EFT in a reasonable direction. For the small value
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Fig. 2. The phase diagram (k B Tc =J versus JAB =J plot) of the spin-1=2 Ising superlattice with L = 1, when the values of JA =J and JB =J are 2xed at JA =J = 1:0 and JB =J = 0:5 and 2ve values of =J are selected; =J = 0:0 for curve A, =J = 1:0 for curve B, =J = 2:0 for curve C, =J = 4:0 for curve D and =J = 5:0 for curve E: (A) shows the results of the system, when the EFT is applied; (B) represents the results of the system, when the DA is used.
of R, the behavior of curves A and B is very similar to that of Fig. 1, as noted in Refs. [10 –12]. When the value of R increases, however, each curve labeled B shows the decrease after expressing a broad maximum and reduces to zero at a critical value of R. The characteristic behavior of the curves B also gives the big di*erences between the results of Fig. 2 for L = 1 and Fig. 3 for L = 2.
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Fig. 3. The phase diagram (k B Tc =J versus JAB =J plot) of the spin-1=2 Ising superlattice with L = 2, when the values of JA =J and JB =J are 2xed at JA =J = 1:0 and JB =J = 0:5 and two typical values of =J are selected; =J = 1:0 for curve A and =J = 4:0 for curve B. The dashed and solid lines represent the results of the EFT and the DA, respectively.
Fig. 4. The critical transverse 2eld c is plotted as a function of R for the two superlattices with L = 1 and 2, when the values of JA =J and JB =J are 2xed at JA =J = 1:0 and JB =J = 0:5. The dashed and solid lines represent the results of the EFT and the DA, respectively.
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4. Conclusions In this work, we have studied the phase diagrams of the spin-1/2 transverse Ising superlattice with the same slab thickness L (L = 1 or 2) on the basis of two theoretical frameworks (the EFT and the DA) in the di*erential operator technique. As shown in Figs. 1– 4, the results of the DA improve those of the EFT in a reasonable direction. They show forms that are essentially similar to those of the EFT. Thus, the present results prove that the characteristic phenomena observed in Figs. 1– 4 are clearly independent of the theoretical approximation. In Fig. 3, the reentrant phenomenon has been observed for the superlattice with L = 2, when large values of R and are selected. In Fig. 4, the c versus R plot for the system with L=2 reduces to zero at a critical value of R. As discussed in Refs. [10 – 13], these phenomena come from the frustration e*ect due to the competition between the large values of R and . With the increase of L, such unexpected phenomena gradually disappear. Finally, the results obtained in this work will be related to the experimental data of a ferroelectric superlattice properly described by the spin-1=2 transverse Ising model, although at present we do not have enough experimental data to compare. It will be worth investigating furthermore from the experimental and theoretical points of view whether they are truly real phenomena or not. Appendix A The coeLcients An and Bn (n = 1–7) in (11) – (14) are de2ned by A1 = cosh3 (a) sinh(a) cosh2 (c) F(x)|x=0 ; A2 = cosh4 (a) sinh(c) cosh(c) F(x)|x=0 ; A3 = cosh(a) sinh3 (a) cosh2 (c) F(x)|x=0 ; A4 = cosh2 (a) sinh2 (a) sinh(c) cosh(c) F(x)|x=0 ; A5 = cosh3 (a) sinh(a) sinh2 (c) F(x)|x=0 ; A6 = sinh4 (a) sinh(c) cosh(c) F(x)|x=0 ; A7 = cosh(a) sinh3 (a) sinh2 (c) F(x)|x=0
(A.1)
and B1 = cosh3 (b) sinh(b) cosh2 (c) F(x)|x=0 ; B2 = cosh4 (b) sinh(c) cosh(c) F(x)|x=0 ; B3 = cosh(b) sinh3 (b) cosh2 (c) F(x)|x=0 ; B4 = cosh2 (b) sinh2 (b) sinh(c) cosh(c) F(x)|x=0 ; B5 = cosh3 (b) sinh(b) sinh2 (c) F(x)|x=0 ; B6 = sinh4 (b) sinh(c) cosh(c) F(x)|x=0 ; B7 = cosh(b) sinh3 (b) sinh2 (c) F(x)|x=0 ;
(A.2)
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where a = JA ∇; b = JB ∇ and c = JAB ∇. These coeLcients can be easily calculated by applying the mathematical relation exp(a∇) f (x) = f (x + a). The matrix elements in (17) are given by U1 = 4A1 − 1 + 12 A3 (gA )2 + 8gA gC [3A4 + A6 (gA )2 ] + 4(gC )2 [A5 + 3A7 (gA )2 ] + 2gA [2A3 (gA )2 + 6A4 gA gC + A6 (gA )3 gC + 2A5 (gA )2 + 2A7 (gA )2 (gC )2 ] ; U2 = 2A2 + 2(gA )2 [6A4 + A6 (gA )2 ] + 8gA gC [A5 + A7 (gA )2 ] ; U3 = 2B2 + 2(gB )2 [6B4 + B6 (gB )2 ] + 8gB gD [B5 + B7 (gB )2 ] ; U4 = 4B1 − 1 + 12B3 (gB )2 + 8gB gD [3B4 + B6 (gB )2 ] + 4(gD )2 [B5 + 3B7 (gB )2 ] + 2gB [2B3 (gB )2 + 6B4 gB gD + B6 (gB )3 gD + 2B5 (gB )2 + 2B7 (gB )2 (gD )2 ] : (A.3) Appendix B The coeLcients An and Bn (n = 1–6) in (20) – (23) are de2ned by A1 = cosh4 (a) sinh(a) cosh(c) F(x)|x=0 ; A2 = cosh5 (a) sinh(c) F(x)|x=0 ; A3 = cosh2 (a) sinh3 (a) cosh(c) F(x)|x=0 ; A4 = cosh3 (a) sinh2 (a) sinh(c) F(x)|x=0 ; A5 = sinh5 (a) cosh(c) F(x)|x=0 ; A6 = sinh4 (a) cosh(a) sinh(c) F(x)|x=0
(A.4)
and B1 = cosh4 (b) sinh(b) cosh(c) F(x)|x=0 ; B2 = cosh5 (b) sinh(c) F(x)|x=0 ; B3 = cosh2 (b) sinh3 (b) cosh(c) F(x)|x=0 ; B4 = cosh3 (b) sinh2 (b) sinh(c) F(x)|x=0 ; B5 = sinh5 (b) cosh(c) F(x)|x=0 ; B6 = sinh4 (b) cosh(b) sinh(c) F(x)|x=0 : The matrix elements in (24) are given by V1 = 4A1 − 1 + 30A3 (gA )2 + 5A5 (gA )4 + 20gA gC [A4 + A6 (gA )2 ] + (gA )2 [10(A3 gA + A4 gC ) + (gA )2 (A5 gA + 5A6 gC )] ; V2 = A2 + 10A4 (gA )2 + 5A6 (gA )4 ; V3 = B2 + 10B4 (gB )2 + 5B6 (gB )4 ;
(A.5)
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V4 = 4B1 − 1 + 30B3 (gB )2 + 5B5 (gB )4 + 20gB gD [B4 + B6 (gB )2 ] + (gB )2 [10(B3 gB + B4 gD ) + (gB )2 (B5 gB + 5B6 gD )] : References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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(A.6)
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Phase diagrams of a transverse Ising superlattice based on two theoretical frameworks T. Kaneyoshi Department of Natural Science Informatics, School of Informatics and Sciences, Nagoya University, 464-8601 Nagoya, Japan Received 23 June 2000
Abstract The phase diagrams of a superlattice composed from two di*erent alternating ferroelectric slabs with the same thickness L are discussed on the basis of the spin-1=2 transverse Ising model. The two theoretical frameworks of the di*erential operator technique are applied to the investigations, namely the Zernike approximation and the decoupling approximation superior to the Bethe–Peierls approximation. The unexpected phenomena obtained in the system with L = 2, such as the reentrant phenomenon due to the competition of the interslab coupling and the transverse 2eld, can be found by the use of both approximations. They seem to be independent c 2001 Elsevier Science B.V. All rights reserved. of the theoretical approximation. Keywords: Transverse Ising model; Ferroelectric superlattice
1. Introduction In recent years, the study of magnetic multilayers has been one of the most rapidly expanding 2elds in basic research and industrial applications. A great number of experimental and theoretical investigations have been devoted to the 2eld. On the other hand, some ferroelectric superlattices have also been fabricated and studied owing to the advances of experimental techniques, such as (BaTiO3 )n =(SrTiO3 )n with perovskite structure [1–3]. Theoretically, the physical properties of ferroelectric superlattices have been discussed on the basis of the Ginzburg–Landau phenomenological theory [4,5]. Afterwards, the transverse Ising model (TIM) has been introduced to the study of E-mail address: [email protected] (T. Kaneyoshi). c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 4 7 8 - 7
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dielectric properties in ferroelectric superlattices [6 –12] as well as the ferroelectric bilayer system [13]. The TIM 2rst proposed by De Gennes for the KDP-like ferroelectrics has been used to study various interesting systems, such as thin 2lms [14 –17] and semi-in2nite system with a surface [18,19]. In the previous works [10 –13] we have discussed the phase diagrams of a ferroelectric superlattice as well as a ferroelectric bilayer system on the basis of the TIM, using the e*ective-2eld theory with correlations (EFT) [20,21]. The EFT is superior to the standard mean 2eld approximation (MFA), but the statistical accuracy corresponds to the Zernike approximation [22]. It has been applied successfully to a great number of interesting physical systems, such as [14 –19]. In Refs. [10 –13], the dependence of the Curie temperature Tc and the critical transverse 2eld c on the interslab interaction R and the slab thickness L has been examined. The c is the critical 2eld at which the Tc versus transverse 2eld plot reduces to zero. In the process, we have found some unexpected phenomena in the phase diagrams, when large values of R and are applied to the ferroelectric superlattice and the ferroelectric bilayer system with a small number of slab thickness L, namely the reentrant phenomenon due to the competition between R and and the reduction of the c versus R plot to zero. As far as we know, however, such phenomena have not been reported. The aim of this work is to study the phase diagram of a spin-1=2 transverse Ising superlattice consisting of two di*erent alternating slabs with the same thickness L, using the two theoretical frameworks in the di*erential operator technique for Ising systems [20,21]. One is the EFT, being equivalent to the Zernike approximation, from which we can derive the same results as those in Refs. [10 –12]. On the other hand, various trials for improving the EFT in a reasonable direction have been conducted in the past [20,21]. Among them, the present author has introduced a new decoupling approximation [23] for the multispin correlation functions appearing in the expansion of Ising spin identities. The decoupling approximation (DA) has proved that the results of Tc in the spin-1=2 Ising systems become better than those of the Zernike and the Bethe–Peierls approximation [24]. In Section 2, the formulation of the ferroelectric superlattices with L = 1 and 2 is discussed on the basis of the DA, in order to compare the critical properties with those of the EFT. The numerical calculations of the critical properties are given in Section 3, using both the formulations. We 2nd that they show in the phase diagrams the characteristic features similar to each other and hence it proves that the characteristic phenomena found in Refs. [10 –13] are independent of the theoretical approximation.
2. Formulation We consider a superlattice composed from two di*erent ferroelectric slabs (A and B) with the same (one or two layer) thickness. Each slab is de2ned on the x–y plane and is stacked alternatively to the z direction. It is with pseudo-spin sites on square
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lattice. The system is described by the Ising Hamiltonian in a transverse 2eld (i = ), i Si x ; (1) Jij Si z Sj z − H =− i
(ij)
where Si z and Si x are the components of the pseudo-spin operator at site i on a slab (A or B) and the 2rst summation runs over all nearest-neighbor pairs. Jij is the exchange interaction constant between pseudo-spins at the ith and jth sites, which takes JA (or JB ) in the slab A (or the slab B) or JAB between the A and B slabs. According to (1) and the di*erential operator technique for the Ising systems [20,21], the average value of an atomic spin at a central site i in the slab ( = A or B) is given by {fi }Si z = {fi } [cosh(Ji+ ∇) + Si z sinh(Ji+ ∇)
+ Si z sinh(Ji+ ∇)] F(x)|x=0
(2)
with Si z = Si z − Si z ;
(3)
where {fi } can take any function of the Ising spin variables as long as it is not a function of the site i, expresses the nearest-neighbor sites and ∇ = @=@x is the di*erential operator. The function F(x) is given by x F(x) = tanh(y) (4) y with y = (x2 + 2 )1=2 ;
(5)
where = 1=kB T . Then, the averaged magnetization of each slab m = Si z (i ∈ A or B) can be obtained by putting {fi } = 1 into (2), when each slab is composed one layer (L = 1) and two layers (L = 2). Expanding the right-hand side of (2), we can obtain many Kuctuating spin correlation functions, such as Si+ z Si+ z and Si+ z Si+ z Si+ z . As discussed in Ref. [23], the DA is to introduce the decoupling approximation Si+ z Si+ z = Si z Si+ z Si z Si+ z Si+ z Si+ z Si+ z = Si z Si z Si+ z Si z Si+ z Si z Si+ z :
(6)
Following such a decoupling procedure, the magnetization mA (or mB ) per site in slab A (or B) includes the two di*erent types of spin correlations de2ned by gA = Si z Si+ z for i and i + ∈ A
(7)
gC = Si z Si+ z for i ∈ A and i + ∈ B
(8)
and
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or gB = Si z Si+ z for i and i + ∈ B
(9)
and gD = Si z Si+ z for i ∈ B and i + ∈ A :
(10)
These correlation functions can be also calculated from (2), substituting {fi } = Si+ z . Physically, gA and gB express the spin correlations in slabs A and B. The gC and gD represent the correlations between slabs A and B. The above statement can be applied to any superlattice with a certain value of L. In order to get the relation from which the phase diagram can be obtained, however, one must 2x the value of L. Depending on the value of the slab thickness (L = 1 or 2), we can get di*erent sets of coupled equations for mA ; mB ; gA ; gB ; gC and gD . So, let us formulate it, depending on whether L = 1 or 2. (a) L = 1: At the transition temperature, the spontaneous magnetizations reduce to zero. Consequently, the correlation functions at T = Tc are given in the form of the following coupled equations, from the equations of gA and gC in slab A gA = A1 [1 + 3(gA )2 ] + A3 (gA )2 [3 + (gA )2 ] + gC [2A2 gA + 6A4 gA {1 + (gA )2 } + 2A6 (gA )3 + A5 gC {1 + 3(gA )2 } + A7 (gA )2 gC {3 + (gA )2 }] ;
(11)
gC = 4gA gC [A1 + A5 + (A3 + A7 )(gA )2 ] + [A2 + 6A4 (gA )2 + A6 (gA )4 ][1 + (gC )2 ]
(12)
and, from the equations of gB and gD in slab B gB = B1 [1 + 3(gB )2 ] + B3 (gB )2 [3 + (gB )2 ] + gD [2B2 gB + 6B4 gB {1 + (gB )2 } + 2A6 (gB )3 + B5 gD {1 + 3(gB )2 } + B7 (gB )2 gD {3 + (gB )2 }] ;
(13)
gD = 4gB gD [B1 + B5 + (B3 + B7 )(gB )2 ] + [B2 + 6B4 (gB )2 + B6 (gB )4 ][1 + (gD )2 ] ;
(14)
where the coeLcients An and Bn (n = 1–7) are de2ned in Appendix A. We are now interested in discussing the Tc of the system. The usual argument that the m ( = A or B) tends to zero in the vicinity of Tc allows us to consider only linear terms of magnetizations. This leads to the matrix equation of the form mA M =0 (15) mB with
M=
U1 ; U3 ;
U2 U4
;
(16)
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where the matrix elements Un (n = 1– 4) are given in Appendix A. Then, the critical information can be determined from det(M ) = 0 by taking the highest solution in it, namely from the relation U1 U4 = U2 U3 :
(17)
Here, one should notice the following facts: When substituting g = 0:0 ( =A–D) into the above formulation (or (17)), we can get the previous formulation based on the EFT (or Zernike approximation) in Refs. [10 –12]. On the other hand, when substituting JAB = 0 into the above formulation, the superlattice system is decomposed into two independent slabs with L=1 de2ned on a square lattice. Then, the correlation functions gC and gD reduce to zero. The correlation functions gA and gB in (11) and (13) are given by the same form, namely g = K1 + 3(K1 + K2 )(g )2 + K2 (g )4 ;
(18)
where =A (or B) and the coeLcients K1 and K2 are given by A1 and A2 (or B1 and B2 ). The transition temperature of each slab can be determined from 1 = 4K1 + 12K2 (g )2 + 4K2 (g )4 :
(19)
Relations (18) and (19) are completely equivalent to those of spin-1=2 Ising square lattice in Ref. [23] when substituting = 0:0 into them. Accordingly, when = 0:0, the Tc of each slab determined from (18) and (19) is given by kB Tc =J = 2:7444, which should be compared to 4.0 for the MFA, 3.0898 for the EFT (Zernike approximation), 2.8854 for the Bethe–Peierls approximation and 2.2692 for the exact (Onsager) solution [25], where we have de2ned J = J in the bulk. (b) L = 2: Following the same procedure as that of the case (a), one can easily get the following set of coupled equations for gA ; gB ; gC and gD : gA = A1 [1 + 4(gA )2 ] + 2A3 (gA )2 [3 + 2(gA )2 ] + A5 (gA )4 + gA gC [A2 + 2A4 {2 + 3(gA )2 } + A6 (gA )2 {4 + (gA )2 }] ; gC [1 − gA {5A1 + 10A3 (gA )2 + A5 (gA )4 }] = A2 + 10A4 (gA )2 + 5A6 (gA )4
(20) (21)
and gB = B1 [1 + 4(gB )2 ] + 2B3 (gB )2 [3 + 2(gB )2 ] + B5 (gB )4 + gB gD [B2 + 2B4 {2 + 3(gB )2 } + B6 (gB )2 {4 + (gB )2 }] ; gD [1 − gB {5B1 + 10B3 (gB )2 + B5 (gB )4 }] = B2 + 10B4 (gB )2 + 5B6 (gB )4 ;
(22) (23)
where the coeLcients An and Bn (n = 1– 6) are de2ned in Appendix B. On the other hand, the critical properties of the system with L = 2 can be determined by taking the same process as that of the case of L = 1, namely from the relation V1 V4 = V2 V3 ;
(24)
where the matrix elements Vn (n =1– 4) are de2ned in Appendix B. The critical information of the superlattice with L = 2 can be obtained by taking the highest solution
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Fig. 1. The phase diagram (k B Tc =J versus JAB =J plot) of the spin-1=2 Ising superlattice consisting of two di*erent alternating slabs with the same thickness L (L = 1 and 2) in zero transverse 2eld (=J = 0:0), when the values of JA =J and JB =J are 2xed at JA =J = 1:0 and JB =J = 0:5. The dashed and solid lines represent, respectively, the results of the EFT (or Zernike approximation) and the DA. The curves labeled A are obtained for the system with L = 1 and the curves are the results of L = 2.
in (24). When substituting the condition of g = 0:0 ( =A–D) into (24), we can also get the critical properties within the formulation of the Zernike approximation. 3. Numerical results In this section, let us examine the phase diagram of the spin-1=2 Ising superlattice with L = 1 (or L = 2) by solving the coupled equations (11) – (14) and (17) for the system with L = 1 (or the coupled equations (20) – (23) and (24) for the system with L=2) numerically. As noted in Section 2, we can easily obtain the results of the Zernike approximation (or EFT) as well as those of the DA from the present formulation. In the following, therefore, the critical properties are studied by the use of both formulations and compared to each other. We de2ne the following parameters: A = JA =J;
B = JB =J
and
R = JAB =J :
(25)
Furthermore, in order to compare with the results in Refs. [10 –13] let us take the 2xed values of A = 1:0 and B = 0:5 for the following numerical calculations. At 2rst, let us show the phase diagram of the spin-1=2 Ising superlattices in zero transverse 2eld ( = 0:0), since Eq. (2) then becomes an exact identity [26] in the Ising spin system, as discussed in [20,21]. Fig. 1 shows a typical result in the phase diagrams of the system. It is depicted as the Tc versus R plot, in order to compare with the following characteristic 2gures as well as those in Refs. [10 –12]. The curves labeled A represent the results of the system with L = 1 and the curves labeled B are
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the results of the system with L = 2. For each case, the dashed line shows the result of the EFT (or Zernike approximation) and the solid line is the result of the DA. As can be seen from the 2gure, the DA improves the result of the EFT in a reasonable direction and the values at R = 0:0; k B Tc =J = 3:0898 and 2.7444 for the curves A, are equivalent to the results of square lattice [23]. Furthermore, the characteristic features of dashed lines A and B, namely the facts that curve A crosses curve B and they increase monotonically with the increase of R, have also been observed in Refs. [6,7,9 –12]. As discussed in Refs. [10 –13], on the other hand, the phase diagram of the superlattice in a 2nite transverse 2eld starts to exhibit many unexpected features, when the large values of and R are selected especially in the system with a small value of L (L¿2). In order to show the facts clearly, let us at 2rst give the phase diagram (Tc versus R plot) of the superlattice with L = 1, as Fig. 2. Fig. 2(A) shows the results of the Zernike approximation (EFT), which are equivalent to those in Refs. [10 –12]. Fig. 2(B) represents the DA results. In each case, the 2ve values of =J are selected; =J = 0:0 for curve A, =J = 1:0 for curve B, =J = 2:0 for curve C, =J = 4:0 for curve D and =J = 5:0 for curve E. In each 2gure, the feature of the curves D and E is clearly di*erent from that of curves A, B and C. The di*erence comes from the existence of the critical transverse 2eld c at which the Tc versus plot reduces to zero. Anyway, when comparing the two 2gures A and B, we can also 2nd that the decoupling approximation (DA) improves the results of the EFT (Zernike approximation) in a reasonable direction. The DA and EFT give results essentially similar to each other. Fig. 3 shows the phase diagram (Tc versus R plot) of the superlattice with L = 2, when the two typical values of =J are selected; =J = 1:0 for curves A and =J = 4:0 for curves B. The dashed and solid lines in the 2gure represent, respectively, the results of the EFT and the DA. In each approximation, the obtained curves exhibit behaviors that are essentially similar to each other and the DA also improves the results of the EFT in a reasonable direction. At this place, we 2nd some unexpected phenomena in the phase diagram: Each curve labeled A at 2rst increases monotonically from the value at R = 0:0 with the increase of R, like the curves B in Fig. 2. However, it shows the reentrant behavior (the double values of Tc ) in a certain region of large R and the curve reduces to zero at a critical value (Rc ) of R. On the other hand, each curve labeled B exhibits the two critical values of R (R1 and R2 ; R2 ¿ R1 ) where the curve reduces to zero and in the vicinity of R2 it also expresses the weak reentrant phenomenon. Thus, the curves are clearly di*erent from the corresponding ones in Fig. 2. Now, in order to understand the big di*erence of the behavior in the Tc curves between Figs. 2 and 3 more clearly, in Fig. 4, the critical value c is plotted as a function of R for the two superlattices with L = 1 and 2. The curves labeled A and B represent the results of L = 1 and 2, respectively. The dashed and solid lines are the results of the EFT and the DA and they take forms that are similar to each other. The DA also improves the result of the EFT in a reasonable direction. For the small value
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Fig. 2. The phase diagram (k B Tc =J versus JAB =J plot) of the spin-1=2 Ising superlattice with L = 1, when the values of JA =J and JB =J are 2xed at JA =J = 1:0 and JB =J = 0:5 and 2ve values of =J are selected; =J = 0:0 for curve A, =J = 1:0 for curve B, =J = 2:0 for curve C, =J = 4:0 for curve D and =J = 5:0 for curve E: (A) shows the results of the system, when the EFT is applied; (B) represents the results of the system, when the DA is used.
of R, the behavior of curves A and B is very similar to that of Fig. 1, as noted in Refs. [10 –12]. When the value of R increases, however, each curve labeled B shows the decrease after expressing a broad maximum and reduces to zero at a critical value of R. The characteristic behavior of the curves B also gives the big di*erences between the results of Fig. 2 for L = 1 and Fig. 3 for L = 2.
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Fig. 3. The phase diagram (k B Tc =J versus JAB =J plot) of the spin-1=2 Ising superlattice with L = 2, when the values of JA =J and JB =J are 2xed at JA =J = 1:0 and JB =J = 0:5 and two typical values of =J are selected; =J = 1:0 for curve A and =J = 4:0 for curve B. The dashed and solid lines represent the results of the EFT and the DA, respectively.
Fig. 4. The critical transverse 2eld c is plotted as a function of R for the two superlattices with L = 1 and 2, when the values of JA =J and JB =J are 2xed at JA =J = 1:0 and JB =J = 0:5. The dashed and solid lines represent the results of the EFT and the DA, respectively.
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4. Conclusions In this work, we have studied the phase diagrams of the spin-1/2 transverse Ising superlattice with the same slab thickness L (L = 1 or 2) on the basis of two theoretical frameworks (the EFT and the DA) in the di*erential operator technique. As shown in Figs. 1– 4, the results of the DA improve those of the EFT in a reasonable direction. They show forms that are essentially similar to those of the EFT. Thus, the present results prove that the characteristic phenomena observed in Figs. 1– 4 are clearly independent of the theoretical approximation. In Fig. 3, the reentrant phenomenon has been observed for the superlattice with L = 2, when large values of R and are selected. In Fig. 4, the c versus R plot for the system with L=2 reduces to zero at a critical value of R. As discussed in Refs. [10 – 13], these phenomena come from the frustration e*ect due to the competition between the large values of R and . With the increase of L, such unexpected phenomena gradually disappear. Finally, the results obtained in this work will be related to the experimental data of a ferroelectric superlattice properly described by the spin-1=2 transverse Ising model, although at present we do not have enough experimental data to compare. It will be worth investigating furthermore from the experimental and theoretical points of view whether they are truly real phenomena or not. Appendix A The coeLcients An and Bn (n = 1–7) in (11) – (14) are de2ned by A1 = cosh3 (a) sinh(a) cosh2 (c) F(x)|x=0 ; A2 = cosh4 (a) sinh(c) cosh(c) F(x)|x=0 ; A3 = cosh(a) sinh3 (a) cosh2 (c) F(x)|x=0 ; A4 = cosh2 (a) sinh2 (a) sinh(c) cosh(c) F(x)|x=0 ; A5 = cosh3 (a) sinh(a) sinh2 (c) F(x)|x=0 ; A6 = sinh4 (a) sinh(c) cosh(c) F(x)|x=0 ; A7 = cosh(a) sinh3 (a) sinh2 (c) F(x)|x=0
(A.1)
and B1 = cosh3 (b) sinh(b) cosh2 (c) F(x)|x=0 ; B2 = cosh4 (b) sinh(c) cosh(c) F(x)|x=0 ; B3 = cosh(b) sinh3 (b) cosh2 (c) F(x)|x=0 ; B4 = cosh2 (b) sinh2 (b) sinh(c) cosh(c) F(x)|x=0 ; B5 = cosh3 (b) sinh(b) sinh2 (c) F(x)|x=0 ; B6 = sinh4 (b) sinh(c) cosh(c) F(x)|x=0 ; B7 = cosh(b) sinh3 (b) sinh2 (c) F(x)|x=0 ;
(A.2)
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where a = JA ∇; b = JB ∇ and c = JAB ∇. These coeLcients can be easily calculated by applying the mathematical relation exp(a∇) f (x) = f (x + a). The matrix elements in (17) are given by U1 = 4A1 − 1 + 12 A3 (gA )2 + 8gA gC [3A4 + A6 (gA )2 ] + 4(gC )2 [A5 + 3A7 (gA )2 ] + 2gA [2A3 (gA )2 + 6A4 gA gC + A6 (gA )3 gC + 2A5 (gA )2 + 2A7 (gA )2 (gC )2 ] ; U2 = 2A2 + 2(gA )2 [6A4 + A6 (gA )2 ] + 8gA gC [A5 + A7 (gA )2 ] ; U3 = 2B2 + 2(gB )2 [6B4 + B6 (gB )2 ] + 8gB gD [B5 + B7 (gB )2 ] ; U4 = 4B1 − 1 + 12B3 (gB )2 + 8gB gD [3B4 + B6 (gB )2 ] + 4(gD )2 [B5 + 3B7 (gB )2 ] + 2gB [2B3 (gB )2 + 6B4 gB gD + B6 (gB )3 gD + 2B5 (gB )2 + 2B7 (gB )2 (gD )2 ] : (A.3) Appendix B The coeLcients An and Bn (n = 1–6) in (20) – (23) are de2ned by A1 = cosh4 (a) sinh(a) cosh(c) F(x)|x=0 ; A2 = cosh5 (a) sinh(c) F(x)|x=0 ; A3 = cosh2 (a) sinh3 (a) cosh(c) F(x)|x=0 ; A4 = cosh3 (a) sinh2 (a) sinh(c) F(x)|x=0 ; A5 = sinh5 (a) cosh(c) F(x)|x=0 ; A6 = sinh4 (a) cosh(a) sinh(c) F(x)|x=0
(A.4)
and B1 = cosh4 (b) sinh(b) cosh(c) F(x)|x=0 ; B2 = cosh5 (b) sinh(c) F(x)|x=0 ; B3 = cosh2 (b) sinh3 (b) cosh(c) F(x)|x=0 ; B4 = cosh3 (b) sinh2 (b) sinh(c) F(x)|x=0 ; B5 = sinh5 (b) cosh(c) F(x)|x=0 ; B6 = sinh4 (b) cosh(b) sinh(c) F(x)|x=0 : The matrix elements in (24) are given by V1 = 4A1 − 1 + 30A3 (gA )2 + 5A5 (gA )4 + 20gA gC [A4 + A6 (gA )2 ] + (gA )2 [10(A3 gA + A4 gC ) + (gA )2 (A5 gA + 5A6 gC )] ; V2 = A2 + 10A4 (gA )2 + 5A6 (gA )4 ; V3 = B2 + 10B4 (gB )2 + 5B6 (gB )4 ;
(A.5)
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V4 = 4B1 − 1 + 30B3 (gB )2 + 5B5 (gB )4 + 20gB gD [B4 + B6 (gB )2 ] + (gB )2 [10(B3 gB + B4 gD ) + (gB )2 (B5 gB + 5B6 gD )] : References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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(A.6)