Advances in Condensed Matter Physics

Volume 2012 (2012), Article ID 714625, 10 pages

doi:10.1155/2012/714625

Review Article

Ferroelectric Instability and Dimensionality in Bi-Layered Perovskites and Thin Films

Akira Onodera, Masanori Fukunaga, and Masaki Takesada

Department of Physics, Hokkaido University, Sapporo 060-0810, Japan

Received 25 May 2012; Accepted 3 August 2012

Academic Editor: Biljana Stojanovic

Copyright © 2012 Akira Onodera et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The dielectric and thermal properties of Bi (bismuth)-layered perovskite SrBi₂Ta₂O₉ (SBT) are discussed in comparison with ferroelectric thin BaTiO₃ films. Although these two perovskites exhibit quite a different nature, the dielectric properties of BaTiO₃ thin film are similar to those in bulk SBT. The dielectric properties and pseudo-two-dimensional structure between SBT and thin film suggest that the bulk layered ferroelectric SBT is a good model of ultra-thin ferroelectric film with two perovskite layers, free from any misfit lattice strain with substrate and surface charge at the interface with electrodes. Based on the mechanism of ferroelectric phase transition of SBT, it seems plausible that the ferroelectric interaction is still prominent but shows a crossover from ferroelectric to antiferroelectric interaction in perovskite ultra-thin films along the tetragonal axis.

1. Introduction

Barium titanate (BaTiO₃) is a well-known ferroelectric with perovskite structure, which has been considered to be a prototype for many oxide ferroelectrics. The dielectric constant () follows a well-known Curie-Weiss law, , and shows a large dielectric anomaly at (409 K), which has been explained by the concept of soft mode by Cochran [1, 2]. Recently, however, it has been recognized that the dielectric constant begins to decrease with decreasing the sample thickness, and shows broad and non-Curie-Weiss behavior. This evidence indicates that the well-established soft mode may be modified in the treatment of lattice dynamics of thin films of BaTiO₃, if the LST (Lyddane-Sachs-Teller) relation

holds even in thin films as well as in bulk crystals. Furthermore, the anomalous part of specific heat () changes to a very broad and unclear one in thin BaTiO₃ films, while it is sharp and large at in bulk crystals. This suggests that the temperature dependence of an order parameter (the spontaneous polarization in the case of ordinary ferroelectrics) plays somewhat in different manners as the usual mean-field behavior known in typical ferroelectrics, because is related to in the mean field theory as

As pointed out by many researches, these novel ferroelectric properties are considered to be essentially due to the two-dimensionality in thin films.

On the other hand, the similar novel dielectric behavior has been found in layered perovskite compounds. SrBi₂Ta₂O₉ (SBT) has been extensively studied as a candidate for ferroelectric nonvolatile memory devices [3–5]. SBT has a pseudo-two-dimensional crystal structure with two TaO₆ perovskite groups sandwiched by Bi₂O₂ semiconducting layers along the pseudotetragonal axis within the unit cell. SBT undergoes a ferroelectric phase transition at 608 K, while the dielectric anomaly and anomalous specific heat are small in comparison with BaTiO₃. This similarity of crystal structures and dielectric properties between BaTiO₃ thin films and layered perovskite SBT shows that both materials have a common mechanism resulted in dielectric nature as discussed in the following section. Moreover, it suggests that SBT may be a good model for an ideal ferroelectric ultra-thin film with two perovskite unit cells, though SBT itself is a bulk crystal. In this paper, we will discuss the ferroelectric nature and stability in ultra-thin films on the basis of ferroelectric properties of Bi-layered perovskites.

2. Dielectric and Thermal Properties in BaTiO₃ and SrBi₂Ta₂O₉

The temperature dependence of of BaTiO₃ and SBT is compared in Figure 1 as a function of for simplicity, where is a ferroelectric phase transition temperature of each perovskite, SBT, and BaTiO₃. The large Curie-Weiss dielectric anomaly associated with ferroelectric phase transition was noticed in BaTiO₃ at , while a broad non-Curie-Weiss behavior was observed in layered perovskite SBT [6, 8–14]. The half temperature widths of dielectric anomalies are 10 K for BaTiO₃ and about 400 K for SBT, respectively. The spontaneous polarization () of SBT is relatively large ( μC/cm² along the axis at room temperature), while  μC/cm² in BaTiO₃ along the tetragonal axis. As in SBT is the same order as those in order-disorder ferroelectrics, we expect that obeys the Curie-Weiss law and the peak value of is the order of several thousands. However, we could only observe a round peak of about 300 and weak temperature dependency around 608 K () in SBT.

(a)

(b)

Figure 1: Temperature dependence of dielectric constant of (a) BaTiO₃ (black line) and SrBi₂Ta₂O₉ (red line) and (b) that of an expanded figure in SrBi₂Ta₂O₉ [6].

It is interesting to point out that similar diffuse and round dielectric anomalies were observed in ferroelectric thin films. In BaTiO₃ epitaxial thin films, diffuse dielectric anomaly was reported by Yoneda et al. [7, 15–18]. Thin films of BaTiO₃ showed a small and broad dielectric anomaly as shown in Figure 2. The peak value of is 1400 for 4000 Å thick film, and only 370 for 1000 Å thick film. This similarity of dielectric properties suggests the existence of some common mechanism between SBT bulk crystal and BaTiO₃ thin film. Recently, Tagantsev et al. discussed a two-dimensional strain effect on dielectric constant in perovskite ferroelectric thin films and reported a diffuse dielectric anomaly [19].

Figure 2: Temperature dependence of dielectric constant of SBT and BaTiO₃ epitaxial thin films on SrTiO₃ with 4000 Å and 1000 Å [7].

Thermal anomaly associated with the phase transition is also diffuse and widely spread in SBT as shown in Figure 3 [8, 10], while a sharp and large anomaly was found at in bulk BaTiO₃ (Figure 4) [20]. However, the specific heat of BaTiO₃ epitaxial films shows diffuse temperature dependence (Figure 5) just as those in SBT [21]. With decreasing thickness of epitaxial films of BaTiO₃, the shape of anomalous part of specific heat becomes round and diffuse and exhibits thermal hysteresis in those with 2000 Å and 60 Å. In BaTiO₃ polycrystalline thin films, a rather sharp anomaly was reported [22, 23]. The diffuse dielectric and thermal behavior observed around may be attributed to the two-dimensional effect in SBT and BaTiO₃ epitaxial thin films.

Figure 3: Temperature dependence of specific heat of SrBi₂Ta₂O₉ single crystal [8].

Figure 4: Temperature dependence of specific heat of BaTiO₃ single crystal around the paraelectric-ferroelectric phase transition point [20].

(a)

(b)

Figure 5: Temperature dependence of specific heat of BaTiO₃ epitaxial thin films on SrTiO₃ substrate; (a) 2000 Å thick and (b) 60 Å thick. The arrows show the direction of heating and cooling. The sharp anomalous specific heat in the single crystal changes to diffuse one in thin films [21].

3. Structural Analogy: Pseudo-Two-Dimensionality

BaTiO₃ has a prototype cubic crystal structure of perovskite in the paraelectric high-temperature phase (space group Pm3 m, with lattice parameter  Å at 393 K). It changes to tetragonal (space group P4 mm, ,  Å at r. t.) associated with the paraelectric-ferroelectric phase transition. The unit cell contains one TiO₆ octahedron at the center and Ba ions locate at the corners. The TiO₆ octahedron is tightly bonded with each other and generally considered as a hard unit, which has a regular octahedral form in the paraelectric phase. According to Slater, the correlation along the -O-Ti-O-Ti-O- chain is very strong along the tetragonal axis and the small octahedral distortion results in ferroelectricity below [26]. Epitaxial BaTiO₃ films were grown by on SrTiO₃ and MgO substrates using the activated reaction evaporation method, which were characterized by X-ray diffraction as shown in Figure 6 [24, 27]. Only the (0 0 l) reflections were observed. The submaximum peaks of the Laue function were observed around the fundamental peaks of the (0 0 1), (0 0 2), (0 0 3), and (0 0 4). The strong peaks denoted as “Sub” in Figure 6 mean the (0 0 l) reflections from the SrTiO₃ substrate. This evidence indicates that this film is an epitaxial crystal with a perovskite structure. Two films with thickness of 2000 Å and 60 Å were used for the specific heat measurements. The heat capacity was measured using an a. c. calorimeter.

Figure 6: X-ray diffraction profiles along the [0 0 l] direction of the BaTiO₃ thin films with 60 Å on SrTiO₃ substrate (monochromatized CuKα). Subscript “Sub” means reflections from SrTiO₃ substrate [24].

SrBi₂Ta₂O₉ is a Bi-layered perovskite oxide with a highly anisotropic structure, where perovskite-type [SrTa₂O₇]²⁻ groups and semiconducting [Bi₂O₂]²⁺ layers are stacked alternately along the pseudo-tetragonal axis as shown in Figure 7. The general chemical formula of Bi-layered perovskite oxides with stacking number () of octahedral BO₆ perovskite groups is given as (Bi₂O₂)2+ ()²⁻, where , K⁺, Ba²⁺, Ca²⁺, Pb²⁺, Sr²⁺, Bi³⁺, , Ti⁴⁺, Nb⁵⁺, Ta⁵⁺, W⁶⁺, and to 6. Many of these compounds are ferroelectric with high transition temperatures. Ferroelectricity in SBT with was discovered below 608 K by Smolenskii et al. in 1961 [6]. The crystal structure of SBT is orthorhombic at r. t. with space group A2₁am (, ,  Å). The lattice is highly anisotropic and pseudo-two-dimensional, and the -O-Ta-O-Ta-O- linkage along the pseudo-tetragonal axis is interrupted by the existence of semiconducting Bi₂O₂ layers in SBT [11, 14, 25]. SBT undergoes two successive phase transitions at 608 K and 850 K. The high-temperature paraelectric phase is tetragonal with space group I4/mmm, , and  Å at 1000 K, where the TaO₆ octahedra take antiparallel arrangements along the tetragonal axis. The crystal structures of the ferroelectric (300 K) and paraelectric phases (1000 K) were determined by X-ray diffraction as summarized in Tables 1 and 2. X-ray diffraction profile was given in Figure 8 [25].

Table 1: Positional and thermal parameters of ferroelectric phase (A2₁am) of SrBi₂Ta₂O₉ at 300 K [25].

Atom

Sr
0
0.2610
0
0.69 (1)

Bi
0.4833 (14)
0.7738 (4)
0.2011 (1)
1.77 (1)

Ta
0.5177 (16)
0.7468 (6)
0.4143 (1)
0.37 (1)

O(1)
0.5365 (89)
0.3335 (67)
0
1.04 (5)

O(2)
0.4904 (72)
0.7316 (49)
0.3402 (4)
1.55 (4)

O(3)
0.7057 (76)
0.0146 (130)
0.2623 (5)
0.85 (4)

O(4)
0.7523 (94)
0.0002 (101)
0.0730 (6)
0.62 (5)

O(5)
0.7854 (87)
0.9744 (93)
0.5850 (6)
0.91 (4)

Table 2: Positional and thermal parameters of the paraelectric phase (I4/mmm) SrBi₂Ta₂O₉ at 1000 K [25].

Atom

Sr
0
0
0
1.09 (1)

Bi
0
0
0.2007 (1)
3.41 (1)

Ta
0
0
0.4140 (1)
0.05 (1)

O(1)
0
0.5
0.25
0.8 (6)

O(2)
0
0
0.5
1.41 (5)

O(3)
0
0.5
0.0756 (3)
0.75 (4)

O(4)
0
0
0.3479 (5)
1.17 (4)

(a)

(b)

Figure 7: Crystal structure of (a) fundamental perovskite and (b) Bi-layered perovskite SrBi₂Ta₂O₉ (upper half of the unit cell) [11, 25].

Figure 8: X-ray diffraction patterns of SrBi₂Ta₂O₉ (CuKα) at 1000 K [25].

The shape of TaO₆ octahedron is not perfect and is still distorted even in the paraelectric phase along the axis. The Ta-O bond length owned commonly by adjacent octahedra normal to the pseudo-tetragonal axis is a little bit short and that of O directed to Bi₂O₂ layer is long. In the ferroelectric phase, this crystal favors canted octahedral arrangements below , which results in the net spontaneous polarization along the -axis.

4. Ferroelectricity

Ferroelectricity in Bi-layered perovskite SrBi₂Ta₂O₉ (SBT) does not appear along the pseudo-tetragonal axis but along the axis normal to the pseudo-tetragonal axis, while along the tetragonal axis in BaTiO₃. The crystal structure is highly anisotropic along the pseudo-tetragonal axis and the single crystal of SBT has easily cleaved along the pseudo-tetragonal axis, as the semiconducting Bi₂O₂ layer connects weakly upper and lower perovskite groups by van der Waals interaction. Crystal structure analysis of SBT shows that TaO₆ octahedra are still distorted along the pseudo-tetragonal axis, and locate in antiparallel way even in the high-temperature paraelectric phase above 850 K. Although the -O-Ta-O-Ta-O- chain along the pseudo-tetragonal axis is interrupted by the existence of semiconducting Bi₂O₂ layers, this evidence indicates the strong interaction to displace along the axis exists and the strongly correlated -O-Ta-O-Ta-O- chain should play an essential role for the appearance of ferroelectricity in SBT as those in BaTiO₃ [26]. In the high-temperature paraelectric phase, total polarization cancels by the antiparallel arrangement of octahedra, since TaO₆ is not perfect octahedron but elongates along the axis; this indicates SBT is antiferroelectric above 850 K. In the ferroelectric phase, this crystal favors canted octahedral arrangements below , which results in the net spontaneous polarization along the axis. As this situation is just the same as those reported in the case of weak ferromagnetism in magnetic materials, we suggest canted ferroelectricity (weak ferroelectricity) for SBT.

On the other hand, this pseudo-two-dimensional character of crystal structure is just suitable for fabrication of thin films. The spontaneous polarization () is relatively large ( μC/cm² along the axis at room temperature), while  μC/cm² in a representative perovskite ferroelectric BaTiO₃ along the tetragonal axis.

It should be noticed that these two perovskite oxides exhibit quite different dielectric nature. However, it is very interesting to point out the similarity in dielectric properties between BaTiO₃ thin films and SBT bulk crystals, since BaTiO₃ thin ferroelectric films exhibit diffuse dielectric anomaly.

5. Size Effect in Ferroelectric Thin Films

Ferroelectricity appears due to a delicate balance between long-range dipole-dipole interaction along the polar axis and short-range interaction. The typical dipolar correlation lengths for many ferroelectrics are  Å along the polar axis and  Å normal to the polar axis [28]. The needle-shaped correlation region is sketched in Figure 9, where is several unit cells, of the same order as the thickness of a 180° domain wall, and is about  unit cells in the case of BaTiO₃. Therefore, it should be reasonable to consider that the stability of the ferroelectricity may be suppressed by decreasing the thickness of the thin films.

Figure 9: Ferroelectric correlation region in dipolar materials [28].

Measurements of thickness dependence of ferroelectricity are generally not so easy, because of difficulties in preparation of good quality of thin films, and additional surface effects such as depolarization fields, space-charge effects, and misfit strain between film and substrate. These effects essentially influence the ferroelectric behavior in thin films. Many attempts have been reported to study the thickness dependence of ferroelectric properties of ferroelectric thin films prepared by vacuum evaporation, sputtering techniques, chemical vapor deposition, and so on.

Tybell, Ahn, and Triscone have examined the possibility of the existence of a critical thickness and showed the detection of the ferroelectricity in perovskite Pb(Zr0.2Ti0.8)₃ films down to a thickness of 10 unit cells (40 Å) [37]. Bune et al. reported ferroelectric activity in ferroelectric polymer films with thickness of two layers (10 Å) and the near absence of finite-size effects in these two-dimensional ferroelectrics which may be generated by coupling only within the plane of the film [38]. Recent Ab initio studies have confirmed the possibility of retaining the ferroelectricity in ultra-thin films, and suggested the absence of the critical size effect [39, 40].

On the other hand, Junquera and Ghosez reported first-principle calculations on a realistic model of perovskite thin films with metallic electrodes. They showed that BaTiO₃ films with SrRuO₃ electrodes have the critical thickness of 6 unit cells (~24 Å) and lose the ferroelectricity below this thickness, due to the depolarization field effect at the ferroelectric-metal interfaces (Figure 10) [29].

Figure 10: First-principles calculations of the energy of BaTiO₃ ferroelectric films with unit cells as a function of the soft-mode distortion [29].

Recent Raman scattering studies showed that ultra-thin BaTiO₃ films grown commensurately on SrTiO₃ substrates have a spontaneous polarization as thin as 4 unit cells (16 Å) [30]. The ferroelectric phase transition temperature are shown as a function of BaTiO₃ film thickness in Figure 11. Recent experimental and theoretical works showed the critical thickness is much smaller than those previously reported.

Figure 11: The plot of the ferroelectric phase transition temperature versus film thickness of BaTiO₃ films grown on SrTiO₃ substrates, observed by Raman scattering [30].

6. Mechanism of Phase Transition in SBT

It should be pointed out the close analogy of ferroelectric behavior between Bi-layered perovskite SrBi₂Ta₂O₉ (SBT) and thin film of the so-called typical ferroelectric perovskite BaTiO₃. The crystal structures of the high-temperature paraelectric phase at 1000 K and the ferroelectric phase at r. t. are schematically shown in Figure 10 [25]. The TaO₆ octahedra are distorted in both phases. Even in the high-temperature paraelectric phase, the octahedra are elongated along the pseudo-tetragonal axis and locate in an antiparallel way. The distortion parameter of one TaO₆ octahedron, , defined as

is estimated to be 1.365 debye along the axis as shown in Table 3, where and are charge and position of each constituent ion, and is the gravity center of TaO₆. The parameter is obtained as (0, 0, 1.365) in debye unit in the high-temperature paraelectric phase. In the ferroelectric phase, the absolute value is 1.682 debye, which is almost the same as that of the high-temperature paraelectric phase within errors. If the distortion of octahedron corresponds to a dipole moment, it should be probable that SBT prefers an antiferroelectric structure along the -axis, when the distorted octahedra are mainly responsible for the ferroelectric activity. Furthermore, these distorted octahedra have a canted arrangement in the ferroelectric phase as shown in Figure 12.

Table 3: Estimated dipole moment of TaO₆ octahedron of the ferroelectric phase at 300 K (A2₁am) and the high-temperature paraelectric phase (I4/mmm) at 1000 K [25].

A2₁am
I4/mmm

Present study
Miura [34]
Shimakawa et al. [35]
Rae et al. [36]
Present study

0.035
0.564
0.425
0.386
0

−0.702
−0.074
0.001
0.003
0

1.528
1.747
1.169
1.117
1.365

1.682
1.837
1.244
1.182
1.365

(a)

(b)

Figure 12: Crystal structures of (a) the paraelectric and (b) the ferroelectric phases of layered perovskite SrBi₂Ta₂O₉. Arrows show estimated dipole arrangements of TaO₆ octahedra [25].

These structural arrangements of TaO₆ octahedra in SBT remind us of a canted-ferromagnet (or weak ferromagnet) discussed by Moriya [41] and in layered canted-ferromagnets by de Gennes [42]. The free energy is given in terms of sublattice polarizations , , and an angle between them as

where the third symmetrical term is a well-known exchange interaction and the last term is an antisymmetric Dzialoshinski-Moriya interaction. If and , the above expression is just the same as that for antiferroelectrics proposed by Kittel [43]. The last term favors the canted arrangement of dipoles rather than antiferroelectric one. From the condition, , we get the following relation:

As the observed value of is 154°, the coefficient of the antisymmetric term is about half of the symmetric one . From the relations

we have the expression for the dielectric susceptibility as

where

when we assume . It is evident that the dielectric susceptibility shows small anomaly just like antiferroelectric one, which may be smeared by two-dimensional surface strain as pointed out by Tagantsev et al. [19]. Because of the term of the Dzialoshinski-Moriya interaction, the ferroelectric phase transition will be realized in SBT, and the characteristic weak dielectric properties may be explained. The asymmetric interaction may be attributed to the two-dimensional effect in SBT and BaTiO₃ epitaxial thin films.

The decrease and nonclear behavior of dielectric constant were also observed around with decreasing film thickness in (Ba, Sr)TiO₃ thin films. In this case, Hwang analyzed this decrease by considering the effects of a finite charge-screening length of metal electrodes and an intrinsic dead layer of the surface [44]. The layered ferroelectric SBT is a bulk crystal itself, and is considered to be an example of ultra-thin ferroelectric model with two perovskite layers, free from any misfit lattice strain and interface charge layer with electrodes.

It should be pointed out that the ferroelectricity of perovskite thin films with two unit cells may be stable even in those with , although its polarization does not appear along the tetragonal axis but along the axis normal to the tetragonal axis (within layer). Therefore, there is a crossover of ferroelectric to antiferroelectric interaction along the tetragonal axis, with decreasing the stacking number of perovskite units. Furthermore, the ultra-thin films prefer an antiferroelectric structure rather than a ferroelectric structure in thin films.

7. Soft Mode in Bi-Layered Perovskite

Measurements of the soft mode of SBT have been reported by Raman scattering as shown in Figure 13 [31, 32, 45]. The square of soft mode frequency decreases toward the highest phase transition temperature (850 K) and shows a clear anomaly at 610 K (). The soft mode exists in the ferroelectric phase and shows highly overdamped behavior near , although dielectric constant does not diverse at . Another anomaly was detected in the share strain c₄₄ at 850 K [46], which corresponds to the high-temperature phase transition at 850 K.

Figure 13: Temperature dependence of soft mode frequency (square of Raman frequency shift) and a normalized damping factor Γ(T)/Γ() in SrBi₂Ta₂O₉ [31–33].

A series of Bi-layered perovskites compounds Bi2mAn−mBnO3(n+m) were synthesized by Aurivillius in 1949 [47]. As Sr₂Bi₄Ti₅O₁₈ is a compound with five perovskite layers with Bi₂O₂ semiconducting layers as shown in Figure 14, it is considered as a model of perovskite thin film with five unit cells. This crystal undergoes a ferroelectric phase transition at 558 K, although it has been reported that the ferroelectricity of thin films may be suppressed below six unit cells after Junquera and Ghosez [29]. In this compound, similar dielectric properties as those observed in SBT have been reported. The spontaneous polarization appears along the axis and the dielectric constant shows rather weak temperature dependence. In Sr₂Bi₄Ti₅O₁₈, similar temperature dependence of soft mode was observed by Raman scattering [33]. Raman spectra showed two anomalies at 551 K and at 730 K. The soft mode frequency decreases toward 730 K, although it shows overdamped behavior over 450 K. Furthermore, two additional relaxation modes have been observed, which show an anomaly at 551 K. Even in the case of SBT () and Sr₂Bi₄Ti₅O₁₈ (), the soft mode still exists, although it becomes highly overdamped near .

Figure 14: Crystal structure of Bi-layered perovskite A₂Bi₄Ti₅O₁₈ with .

8. Summary and Discussion

Bi-layered perovskites have been much attracted as a promising material for ferroelectric integrated devices. According to the calculations of local field by Luttinger and Tisza [48], the strong ferroelectric interaction between Ti (or Ta) ion and O ions appears along the tetragonal axis. Although the strongly correlated -O-Ti-O-Ti-O- chain plays an important role for the appearance of ferroelectricity in bulk BaTiO₃ [26], the -O-Ta-O-Ta-O- chain along the pseudo-tetragonal axis is interrupted by the existence of semiconductor Bi₂O₂ layers in SBT. The strength of interaction between Ti ion and adjacent O ions is half, but antiferroelectric in the - plane. However, the configuration of distortion parameter in Figure 12 suggests us that the interaction is strong but antiferroelectric along the pseudotetragonal axis, and ferroelectric along the axis in SBT. It is expected that a crossover of ferroelectric to antiferroelectric interaction will be realized even in thin films of BaTiO₃, considering the similarity of dielectric properties between SBT bulk crystal and thin BaTiO₃ films mentioned above. In this sense, SBT bulk crystals may be considered as a model of ferroelectric thin films. It might be expected that the cant of dipole moments may be induced even in thin BaTiO₃ films by the two-dimensionality.

The suppression of ferroelectricity has been reported for nano-particles of BaTiO₃ and PbTiO₃, where the critical size is  unit cells [49]. However, the dipole-dipole interactions within plane are retained and contribute to ferroelectric activity in the case of thin films.

The dielectric and structural analogy between SrBi₂Ta₂O₉ and thin film suggests that this bulk layered ferroelectric is a good model of ultra-thin ferroelectric films with two layers of TaO₆ perovskite units, free from any misfit lattice strain with substrate and surface charge at the interface with electrodes. In perovskite ultra-thin films, ferroelectric interactions are still prominent and the octahedra prefer an antiferroelectric arrangement rather than ferroelectric one along the tetragonal axis (normal to the plane) in the case of two layers. The soft mode exists but changes to be highly over damped near the ferroelectric phase transition temperature. Moreover, the mechanism mentioned above might be applicable for Sr₂Bi₄Ti₅O₁₈ with five perovskite groups (), because the almost same dielectric behavior has been observed in Sr₂Bi₄Ti₅O₁₈ which has perovskite units less than six unit cells. Further experiments will be necessary both in layered oxide ferroelectrics and perovskite thin films.

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