Ferroelectric compounds exhibit many attractive physical properties such as piezoelectricity, pyroelectricity, high-dielectric constant, non-linear optical effects and bistable nature. Therefore various works have been focused on integration of ferroelectric thin films into devices such as ferroelectric nonvolatile memory (FRAM), metal-oxide-semiconductor field-effect transistors (MOSFETs) and sensor devices [1] - [5] . Although ferroelectrics were embedded as a functional element in the form of thin films, our concern is whether the ferroelectricty can persist even in ultra-thin films or not. Since the ferroelectricty originates fundamentally from electrostatic long-range force in dielectric media, it is important to know the exact size effect of thin films on ferroelectricity and the critical thickness where the ferroelectricity may disappear. However, various experiments of ferroelectric thin films have not been enough investigated because of its difficulty to grow a good quality of ultra-thin films and to estimate the effects of misfit strain due to their substrate. It is of fundamental interest in two- dimensional structure of ferroelectric thin films experimentally and theoretically since early times [6] . There have been many attempts to study the size effect on ferroelectricity, leading to a rich literature of both experimental and theoretical works [7] [8] . Especially, studies of ferroelectric perovskites are activated and are promised as one of the best sources of ferroelectric thin films since the perovskite oxides show prominent high dielectric constants and piezoelectric properties.

The similarity of dielectric and structural features has been observed between BaTiO₃ thin films and Bi- layered perovskite compounds. This means that both materials have a common mechanism for ferroelectric activity. Among Bi-layered perovskites, SrBi₂Ta₂O₉ (abbreviated as SBT) has been extensively studied as a candidate for ferroelectric non-volatile memory devices. Based on dielectric and structural similarities, it may be reasonable to consider that SBT is a good model for an ideal ferroelectric thin film, free from any misfit lattice strain. The close analogy between Bi-layered perovskites and ferroelectric thin films was pointed out simply in a previous paper [9] . In this review, we will discuss the ferroelectric stability in ultra-thin films in detail from the viewpoint of the nature of Bi-layered perovskites.

We will simply introduce fundamentals of barium titanate, BaTiO₃, in the form of bulk crystal and thin films. Perovskite oxides have a basic structure of ABO₃, where A and B are cations. This structure has been considered to be a prototype for many ferroelectrics just as for ferromagnetic, semiconducting and superconducting materials, depending on the composition. The first perovskite oxide showed ferroelectricity was BaTiO₃, which is cubic with space group Pm3m in the high-temperature paraelectric phase. It undergoes a ferroelectric phase transition at 409.5 K (T_c) and has a tetragonal structure (P4mm) at room temperature. The characteristic features associated with this phase transition are known such as a large dielectric anomaly. The dielectric constant (ε) is over 14,000 at T_c (Figure 1) and follows the Curie-Weiss law, ε = C/(T − T_c) with C = 14,000 K [10] . The saturated spontaneous polarization (P_s) of the tetragonal phase is about 26 μC/cm² along the tetragonal c-axis.

This phase transition has been explained by the so-called “soft mode theory” after Cochran where a transverse optic (TO) mode softens [11] [12] . The decrease in the frequency of TO mode (ω_TO) towards T_c results in the large dielectric anomaly as we should remember the following LST (Lyddane-Sachs-Teller) relation,

where ε, ε_∞ are dielectric constant of static and at the frequency f = ∞. The ω_LO and ω_TO are frequencies of longitudinal optic and transverse optic modes of the j-th branch, respectively.

Recently, however, it has been recognized that the dielectric constant begins to decrease with decreasing its sample thickness below 2000 Å (Figure 2) [13] - [19] . The peak value of dielectric anomaly is only 380 at 1000 Å. Moreover, the dielectric constant shows broad and non Cure-Weiss behavior as in Figure 3 [13] - [19] . This evidence indicates that the well-established soft mode may be modified in the sense of lattice dynamics of ultra- thin films of BaTiO₃.

The specific heat (C_p) shows a sharp and clear λ-type anomaly at T_c in bulk BaTiO₃ [20] . In the case of thin films, it changes to a broad, small and characteristic one with decreasing film thickness [21] . The spontaneous polarization (P_s) is related to the anomalous specific heat (ΔC_p) in the mean field theory as

This suggests that the temperature dependence of an order parameter (spontaneous polarization P_s) looks to be weak and linear. The upper and lower temperatures of anomalous specific part are shown by short arrows in Figure 5(a). Following the above thermodynamic relation, Equation (2), T_c should be understood as the upper temperature, but it is rather difficult to determine T_c definitely for this film. For film with 60 Å thick, large hysteresis phenomena are observed as shown in Figure 5(b). The sharp anomalous specific heat observed in the single bulk crystal changes to diffuse one in thin films. It plays somewhat in different and diffusive manners from the usual mean-field behavior (P_s ~ (T − T_c)^1/2). As pointed out by many researches, these novel ferroelectric properties are essentially due to the two-dimensionality in thin films.

The similar weak dielectric nature has been pointed out in Bi-layered perovskite compounds [9] SrBi₂Ta₂O₉ (SBT), which has been extensively studied as a candidate for ferroelectric non-volatile memory devices. As discussed in the next section, this similarity of crystal structure and dielectric properties between thin films of BaTiO₃ and Bi-layered perovskites shows that both materials have a common mechanism resulted in dielectric nature. Moreover it suggests us that SBT may be a good model for an ideal ferroelectric thin film, though SBT itself is a bulk crystal.

A series of layered perovskite compounds were discovered by Aurivillius in 1949 [22] . Among these compounds, the ferroelectric activities of Bi-layered perovskites were first discovered by Smolenskii et al. [23] . The general chemical formula of Bi-layered perovskites is A_m−1Bi₂B_mO_3m+3, which is rewritten as [Bi₂O₂]²⁺∙[A_m−1B_mO_3m+1]²⁻. It consists of the regular staking of semiconducting [Bi₂O₂] slabs and perovskite-like octahedral [BO₆] groups, as described by Haeni et al. [24] . Various A and B cations are allowed for these compounds such as A = Na⁺, K⁺, Ba²⁺, Ca²⁺, Pb²⁺, Sr²⁺, Bi³⁺, and B = Fe³⁺, Ti⁴⁺, Nb⁵⁺, Ta²⁺, W⁶⁺. For example, Bi₂WO₆ (m = 1), SrBi₂Ta₂O₉ (m = 2) and Bi₄Ti₃O₁₂ (m = 3) are well studied. Additional members can be generated by allowing the number of stacking (m) of perovskite-like group. The family of Bi-layered perovskites has the face-centered orthorhombic unit cell as shown in Figure 6.

The reported Bi-layered perovskites are summarized in Table 1 [25] [26] . The unit cell is highly anisotropic and pseudo two-dimensional as given in Table 2. The -O-B-O-B-O- linkage along the pseudo-tetragonal unique c-axis is interrupted by the existence of semiconducting Bi₂O₂ layers. The perovskite BO₆ layers and Bi₂O₂ layers are linked by weak van der Waals interaction. This structure is similar to that of thin perovskite films with

m-layers. Among these compounds, Bi₄Ti₃O₁₂ with m = 3 has a monoclinic structure (Pc) and a polarization along the a- and c-axis, which indicates that this crystal has an relatively strong interaction along the c-axis and shows the three-dimensional character. Therefore, compounds with m = 3 (Bi₄Ti₃O₁₂, BaBi₃Ti₂NbO₁₂, and PbBi₃Ti₂NbO₁₂) are not included in the following discussion on two-dimensional nature, although Bi₄Ti₃O₁₂ is an interesting ferroelectric compound for thin film application.

Strontium bismuth tantalate SrBi₂Ta₂O₉ is the most important material in the application for FRAM, because of its low fatigue, low coercive field and Pb-free compound [1] [2] . The room-temperature structure is orthorhombic A2₁am (a = 5.531, b = 5.534, c = 25.984 Å), with 28 atoms in the unit cell. It consists of perovskite-type [SrTa₂O₇]²⁻ groups (two layers of TaO₆ octahedra) and semiconducting [Bi₂O₂]²⁺ layers. They are stacked alternately along the pseudo- tetragonal c-axis as shown in Figure 7 [27] - [33] . Ferroelectricity in SrBi₂Ta₂O₉ with m = 2 was discovered below 608 K (T_c) by Smolenskii et al. in 1961 [23] . SrBi₂Ta₂O₉ undergoes two phase transitions at 608 K and 850 K. This crystal shows small and broad dielectric anomaly around T_c where the maximum value is only 260 (Figure 8). A broad anomaly in specific heat has been observed as shown in Figure 9. This non Curie-Weiss like dielectric anomaly is observed not only in SrBi₂Ta₂O₉ but also in other Bi-layered perovskites commonly.

The spontaneous polarization (P_s) is relatively large and 5.8 ~ 10 μC/cm² along the a-axis (not along the pseudo-tetragonal c-axis) at room temperature [25] , while P_s = 26 μC/cm² along the tetragonal c-axis in BaTiO₃ [34] .

The high-temperature paraelectric phase is tetragonal with space group I4/mmm (a = 3.927, and c = 25.142 Å at 1000 K), where the TaO₆ octahedra take antiparallel arrangements along the tetragonal c-axis [30] . The shape of TaO₆ octahedron is not perfect and is elongated along the c-axis even in the paraelectric tetragonal phase, which means the TaO₆ octahedon has a dipole moment. The Ta-O bond lengths owned commonly by adjacent octahedra along the tetragonal c-axis are a little bit short, and chemical bonds directed to Bi₂O₂-layer are long. In the ferroelectric phase, this crystal favors canted octahedral TaO₆ arrangements below T_c. This structure results in the net spontaneous polarization along the a-axis and no polarization along the pseudo-tetragonal c-axis.

The crystal structures of the high-temperature paraelectric phase at 1000 K and the ferroelectric phase at room temperature are schematically shown in Figure 10 [33] [35] - [38] . The TaO₆ octahedra are distorted in both phases. Even in the high-temperature paraelectric phase, the octahedra are elongated along the pseudo-tetragonal c axis and locate in an antiparallel way. The distortion parameter of one TaO₆ octahedron, p, defined as

is estimated to be 1.365 debye along the c-axis as shown in Table 3, where q_i and r_i and are charge and position of each constituent ion, and r_G is the gravity center of TaO₆ [30] .

The parameter p is calculated 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 SrBi₂Ta₂O₉ prefers an antiferroelectric structure along the c-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 10(a).

The structural arrangement of TaO₆ octahedra in SBT reminds us canted-ferromagnets (or weak ferromagnets) discussed by Moriya [39] and in layered canted-ferromagnets by de Gennes [40] . The free energy is given in terms of sublattice polarizations P₁, P₂ and an angle θ between them as

where the third symmetrical term is a well-known exchange interaction, which prefers an antiparallel configuration. The last term is an antisymmetric Dzialoshinski-Moriya interaction, which forces dipoles to tilt from the c-axis. These two terms are considered to be induced by the two-dimensional size effect of thin films. If θ = π and α₁ = α₂, the above expression is the same as that for antiferroelectrics proposed by Kittel [41] . Because of the last Dzialoshinski-Moriya term, the above free energy favors the canted arrangement of dipoles rather than antiferroelectric one as shown in Figure 10(a).

From the condition, ¶F/¶θ = 0, we get the following relation

As the observed value of θ is 154˚ at room temperature, the coefficient δ of the antisymmetrical term is about half of the symmetrical term γ in this compound. From the relations

we have an expression for the dielectric susceptibility as

When we assume, c is given as

It is evident that the dielectric susceptibility shows a small cusp just like antiferroelectrics. As θ may vary from π to 154˚ with decreasing temperature, the coefficient γ' changes gradually. Dielectric constant may show a broad anomaly around T_c, which is smeared by two-dimensional surface strain as pointed out by Tagantsev et al. [42] . The weak dielectric behavior may be attributed to the two-dimensional effect, which favors for dipole moment to cant from the tetragonal axis in SrBi₂Ta₂O₉.

The phase transitions in Bi-layered perovskites with m = 5 have not been known in comparison to SrBi₂Ta₂O₉. The crystal structure is orthorhombic with space group B2cb at room temperature (Figure 11) [43] . The high-temperature phase is believed to be tetragonal (I4/mmm). The dielectric measurement of Sr₂Bi₄Ti₅O₁₈ at frequencies of 1 kHz, 10 kHz and 100 kHz was shown in Figure 12 [44] . Two dielectric anomalies of Sr₂Bi₄Ti₅O₁₈ were found at 551 K (T_c) and 730 K (T_H) although Mouri et al. reported dielectric anomalies at 428 K and 558 K [45] . The anomaly at T_c is almost the same as reported by Subbarao [25] . The successive phase transitions were reported also in Pb₂Bi₄Ti₅O₁₈ with m = 5 [45] [46] . Very little is known about dielectric behavior in Bi-layered perovskites, in particular, at high temperatures.

The peak value of dielectric constant is about 1700 (100 kHz) which is one order larger than SrBi₂Ta₂O₉ (ε ~ 260) but smaller than that of BaTiO₃ (ε ~ 14,000). The rather sharp dielectric anomaly was observed in the case of Bi-layered perovskites with m = 5. This evidence is consistent with the Curie-Weiss behavior in soft mode of Sr₂Bi₄Ti₅O₁₈ discussed later.

Recently, Onodera et al. pointed out close similarities between Bi-layered perovskites and ferroelectric thin films, and discussed the ferroelectric instability in thin films with two-dimensionality [9] [47] . Figure 1 and Figure 13 show a comparison of dielectric behavior in bulk BaTiO₃ crystal, BaTiO₃ thin film (1000 Å) and Bi-layered perovskite SrBi₂Ta₂O₉. The small and broad dielectric anomalies are commonly observed in BaTiO₃ thin films and bulk SrBi₂Ta₂O₉ crystal.

The decrease and non-clear behavior of dielectric constant were also observed around T_c 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 [48] .

As discussed previously, the two-dimensional crystal structures are clear in both crystals. The crystal structure of Bi-layered perovskites is highly anisotropic along the pseudo-tetragonal c-axis. The single crystal of SrBi₂Ta₂O₉ is easily cleaved along the pseudo-tetragonal c-axis, because the semiconducting Bi₂O₂ layer interacts weakly with upper and lower perovskite-like groups by van der Waals interaction. These dielectric and structural evidences may suggest the common mechanism for dielectric properties and the appearance of ferroelectricity for two crystals.

Experiments of Bi-layered perovskites are much easier than those of ultra-thin films because Bi-layered perovskites are bulk crystals. In addition, Bi-layered perovskites are free from misfit strain between ultra-thin film and substrate. The thickness was controlled by the stacking number of perovskite layers, m, of Bi-layered perovskites. The analogy in crystal structure and dielectric behavior between BaTiO₃ ultra-thin films and Bi-layered perovskites may give us a perspective for the size effect and the ferroelectric nature of ultra-thin films.

The soft modes of Bi-layered perovskites were studied extensively for Sr-compounds with m = 2, 4, 5, because Sr-compounds have a clear soft mode, while the exact crystal structures and phase transitions of this series of compounds have not been clarified yet.

Ferroelectricity in Bi-layered perovskites SrBi₂Ta₂O₉, Sr₂Bi₄Ti₅O₁₈ does not appear along the pseudo-tetragonal c-axis but along the a-axis normal to the pseudo-tetragonal axis, while along the tetragonal c-axis in BaTiO₃. Crystal structure analysis of SrBi₂Ta₂O₉ shows that TaO₆ octahedra is still distorted along the pseudo-tetragonal c-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 c-axis is interrupted by the existence of semiconducting Bi₂O₂ layers in SrBi₂Ta₂O₉, the strongly correlated -O-Ta-O-Ta-O- chain should play an essential role for the appearance of ferroelectricity as those in BaTiO₃ [63] .

The shape of octahedron is not a regular octahedron even in the high-temperature paraelectric phase; this indicates that Bi-layered perovskites are antiferroelectric in the high-temperature tetragonal phase. In the ferroelectric phase, this crystal favors canted octahedral arrangements below T_c, which results in the net spontaneous polarization along the a-axis. This situation is just the same as that reported in the case of weak ferromagnetic materials.

On the other hand, this pseudo-two-dimensional character of crystal structure is just suitable for fabrication of thin films. The spontaneous polarization (P_s) is relatively large (5.8 ~ 10 μC/cm² along the a-axis at room temperature), while P_s = 26 μC/cm² in a representative perovskite ferroelectric BaTiO₃ along the tetragonal c-axis.

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 L_c ~ 10 - 50 nm along the polar axis and L_a ~ 1 - 2 nm normal to the polar axis [64] . The needle-shaped correlation region is sketched in Figure 18, where La is several unit-cells, of the same order as the thickness of a 180˚ domain wall, and L_c is about 25 ~ 125 unit-cells in the case of BaTiO₃. Therefore it is considered that the stability of the ferroelectricity may be affected by the thickness of the thin films.

Measurements of thickness dependence of ferroelectricity are generally not so easy, because of the preparation of good quality of thin films, additional surface effects such as depolarization fields and space-charge effects. These effects generally influence the ferroelectric behavior in thin films.

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(Zr_0.2Ti_0.8)₃ films down to a thickness of 10 unit cells (40 Å) [65] . 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 [66] . 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 [67] [68] .

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 19) [69] .

Recent Raman scattering studies showed that ultra-thin BaTiO₃ films grown commensurately on SrTiO₃ substrate have a spontaneous polarization as thin as 4 unit cells (16 Å) [70] . The ferroelectric phase transition temperature T_c is shown as a function of BaTiO₃ film thickness in Figure 20. Recent experimental and theoretical works showed the critical thickness is much smaller than those previously reported.

It should be pointed out that the close analogy of ferroelectric behavior between Bi-layered perovskite SrBi₂Ta₂O₉ and thin film of so-called typical ferroelectric perovskite BaTiO₃. The layered ferroelectric SrBi₂Ta₂O₉ is a bulk crystal itself, but is considered to be a good example of ultra-thin ferroelectric model with

two monolayers of perovskite TaO₆ units, free from any misfit lattice strain and interface charge layer with electrodes. We summarized recent dielectric properties of Bi-layered perovskites and discuss the analogy of these two types of ferroelectrics.

As Sr₂Bi₄Ti₅O₁₈ is a compound with five perovskite layers sandwiched by Bi₂O₂ semiconducting layers, it is considered as a model of thin film with five perovskite unit cells. While the ferroelectricity of thin films may be suppressed below six unit cells after Junquera and Ghosez [67] , Sr₂Bi₄Ti₅O₁₈ shows ferroelectricity along the a-axis, but not along the tetragonal c-axis. Similar dielectric properties have been observed in SrBi₂Ta₂O₉ which have two perovskites. The spontaneous polarization appears along the a-axis and the dielectric constant shows rather weak temperature dependence. Even in the case of SrBi₂Ta₂O₉ (m = 2) and Sr₂Bi₄Ti₅O₁₈ (m = 5), the soft mode still exists, although it becomes to be highly overdamped near T_c and additional relaxation modes appear.

According to the calculations of local field by Luttinger and Tisza [71] , the strong ferroelectric interaction between Ti (or Ta) ions and O ions appears along the tetragonal c-axis. Although the strongly correlated -O-Ti-O-Ti-O- chain plays an important role for the appearance of ferroelectricity in bulk BaTiO₃ [63] , the -O-Ta-O-Ta-O- chain along the pseudo-tetragonal c axis is interrupted by the existence of semiconductor Bi₂O₂ layers in SrBi₂Ta₂O₉. The strength of interaction between Ti ion and adjacent O ions is half, but antiferroelectric in the a-b plane after Luttinger and Tisza. However, the configuration of distortion parameter p in Figure 10 suggests us that the interaction is strong but antiferroelectric along the pseudo tetragonal c-axis, and ferroelectric along the a-axis in SrBi₂Ta₂O₉. It is expected that a crossover of ferroelectric to antiferroelectric interaction will be realized in thin films of BaTiO₃, considering the similarity of dielectric properties between SrBi₂Ta₂O₉ bulk crystal and thin BaTiO₃ films mentioned above. It might be expected that the canted arrangement of dipole moments may be induced even in thin BaTiO₃ films by the two-dimensionality.

The suppression of ferroelectricity has been reported for nana-particles of BaTiO₃ and PbTiO₃, where the critical size is 20 ~ 30 unit cells [72] . 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 Bi-layered perovskites and ferroelectric thin films suggests that the bulk Bi-layered ferroelectrics are a good model of ferroelectric ultra-thin films with a few layers of perovskite units, free from any misfit lattice strain with substrate and surface charges at the interface with electrodes. In ultra-thin perovskite films, ferroelectric interactions are still prominent and the octahedra prefer an antiferroelectric arrangement rather than ferroelectric one along the tetragonal or pseudo-tetragonal axis (normal to the plane). In the case of a few layers less than m = 6, octahedra have a dipole moment, though first principles calculation suggests the soft-mode distortion ξ = 0, i.e. a non-polar structure [68] . The soft mode may exist even in ultra-thin films but changes to be highly overdamped near the ferroelectric phase transition temperature T_c. Moreover an additional relaxation mode is induced, which means an occurrence of coupling with displacive (soft mode) and order-disorder (relaxation mode) nature in ultra-thin films. As Sr₂Bi₄Ti₅O₁₈ has perovskite-like groups less than six unit cells, it is a critical material whether the ferroelectricity persists or not. The mechanism mentioned above might be applicable for SrBi₂Ta₂O₉ with two perovskite units (m = 2) and Sr₂Bi₄Ti₅O₁₈ with five perovskite groups (m = 5), because of the close similarity in dielectric behavior. Based on experiments on Bi-layered perovskites, it may be possible that there is no critical thickness for the appearance of ferroelectricity in ferroelectric thin films in principle.

Masanori Fukunaga,Akira Onodera,Masaki Takesada, (2016) Ferroelectricity in Layered Perovskites as a Model of Ultra-Thin Films. World Journal of Condensed Matter Physics,06,224-243. doi: 10.4236/wjcmp.2016.63022