Ferroelectricity in Layered Perovskites as a Model of Ultra-Thin Films

The instability of thin ferroelectric films is discussed based on the close similarity of dielectric properties between bulk Bi-layered perovskites and thin BaTiO3 films. The dielectric properties of pseudo-two-dimensional layered perovskites suggest that the bulk layered ferroelectric is a good model of ultra-thin ferroelectric film with a few perovskite units, free from any misfit lattice strain. It seems plausible that the ferroelectric interaction is still prominent but shows a crossover from ferroelectric to antiferroelectric along the unique c-axis (perpendicular to the film plane); with decreasing thickness, the ferroelectricity appears within the plane, which results in so-called “canted ferroelectricity”. An extra relaxation mode induced by surface effect of thin films correlates with soft mode, which results in a new intermediate phase between the paraelectric and ferroelectric phases. These evidences may indicate no critical thickness even for ferroelectric ultrathin films.


Introduction
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 twodimensional 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 BaTiO 3 thin films and Bilayered perovskite compounds.This means that both materials have a common mechanism for ferroelectric activity.Among Bi-layered perovskites, SrBi 2 Ta 2 O 9 (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.

BaTiO 3 : Bulk Crystal and Thin Film
We will simply introduce fundamentals of barium titanate, BaTiO 3 , in the form of bulk crystal and thin films.Perovskite oxides have a basic structure of ABO 3 , 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 BaTiO 3 , 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/cm 2 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 ultrathin films of BaTiO 3 .

Figure 2.
Thickness dependence of dielectric constant of BaTiO 3 thin films on Pt/MgO substrate at room temperature [13]- [19].With decreasing thickness, the dielectric constant monotonically decreases from 700 (8000 Å thick) to 100.The solid line is a guideline.Detailed discussion is given in reference [17].The specific heat (C p ) shows a sharp and clear λ-type anomaly at T c in bulk BaTiO 3 [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] SrBi 2 Ta 2 O 9 (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 Ba-TiO 3 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.
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 Bi 2 O 2 layers.The perovskite BO 6 layers and Bi 2 O 2 layers are linked by weak van der Waals interaction.This structure is similar to that of thin perovskite films with   The spontaneous polarization (P s ) is relatively large and 5.8 ~ 10 μC/cm 2 along the a-axis (not along the pseudo-tetragonal c-axis) at room temperature [25], while P s = 26 μC/cm 2 along the tetragonal c-axis in BaTiO 3 [34].
The high-temperature paraelectric phase is tetragonal with space group I4/mmm (a = 3.927, and c = 25.142Å at 1000 K), where the TaO 6 octahedra take antiparallel arrangements along the tetragonal c-axis [30].The shape of TaO 6 octahedron is not perfect and is elongated along the c-axis even in the paraelectric tetragonal phase, which means the TaO 6 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 Bi 2 O 2 -layer are long.In the ferroelectric phase, this crystal favors canted octahedral TaO 6 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 TaO 6 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 TaO 6 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 TaO 6 [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 SrBi 2 Ta 2 O 9 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 TaO 6 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 P 1 , P 2 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 α 1 = α 2 , 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 tanθ 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 1 2 we have an expression for the dielectric susceptibility as ( ) ( ) 2 , cos sin .
When we assume 1 2 , χ 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 SrBi 2 Ta 2 O 9 .

Dielectric Behavior in Sr 2 Bi 4 Ti 5 O 18 with m = 5
The phase transitions in Bi-layered perovskites with m = 5 have not been known in comparison to SrBi 2 Ta 2 O 9 .
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 Sr 2 Bi 4 Ti 5 O 18 at frequencies of 1 kHz, 10 kHz and 100 kHz was shown in Figure 12 [44].Two dielectric anomalies of Sr 2 Bi 4 Ti 5 O 18 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 Pb 2 Bi 4 Ti 5 O 18 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 SrBi 2 Ta 2 O 9 (ε ~ 260) but smaller than that of BaTiO 3 (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 Sr 2 Bi 4 Ti 5 O 18 discussed later.The decrease and non-clear behavior of dielectric constant were also observed around T c with decreasing film thickness in (Ba, Sr)TiO 3 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 SrBi 2 Ta 2 O 9 is easily cleaved along the pseudo-tetragonal c-axis, because the semiconducting Bi 2 O 2 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.

Similarities of
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 BaTiO 3 ultra-thin films and Bi-layered perovskites may give us a perspective for the size effect and the ferroelectric nature of ultra-thin films.

Soft Mode and Relaxation Behavior in Bi-Layered Perovskites
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.

Soft Mode in SrBi 2 Ta 2 O 9 (m = 2)
The soft mode of SrBi 2 Ta 2 O 9 has been studied by Raman scattering by several researchers [38] [49]- [52].The square of soft mode frequency decreases toward to the highest phase transition temperature T H (850 K) and shows a clear anomaly at T c (610 K).The soft mode exists in the ferroelectric phase and shows highly overdamped behavior near T c , although dielectric constant does not show any clear anomaly at T c .An anomaly was detected in the share strain c 44 at 850 K [53], which corresponds to the high-temperature phase transition at T H .The temperature dependence of 2 s ω , the square of soft mode frequency, and the damping factor Γ of In low frequency region from 0.3 to 1.7 cm −1 , the relaxation mode of SrBi 2 Ta 2 O 9 was observed which is generally observed in order-disorder type ferroelectrics [53].The increase in intensity of relaxation mode was observed with increasing temperature towards T c .This means that the soft mode exists even in Bi-layered compound with two perovskite layers, although an order-disorder nature is induced additionally.

Soft Mode in SrBi 4 Ti 4 O 15 (m = 4)
The crystal structure of Bi-layered perovskites with m = 4 is orthorhombic (A2 1 am) at room temperature [54].The rotation of TiO 6 octahedra and displacement of Ti ions have been reported from their hypothetical tetragonal structure at high temperatures.The soft mode behavior of Bi-layered perovskites with m = 4 was studied systematically by Kojima [55].The ferroelectric-like phase transitions have been reported at 803 K for SrBi 4 Ti 4 O 15 and 1063 K for CaBi 4 Ti 4 O 15 respectively although the detailed series of phase transitions have not been confirmed yet.Figure 16 shows the temperature dependence of the square of the frequency 2 s ω of underdamped soft modes [55].It is found that this soft mode was observed up to about 580 K but showed softening towards T c = 803 K as observed in SrBi 2 Ta 2 O 9 .Similar soft mode behavior was reported also in CaBi 4 Ti 4 O 15 .
The existence of an additional intermediate phase may be possible also for this series of compounds.The lattice parameters show slight kinks around 600 K for SrBi 4 Ti 4 O 15 and 750 K for CaBi 4 Ti 4 O 15 [54], although no detailed works have been done for phase transitions.

Soft Mode in Sr 2 Bi 4 Ti 5 O 18 (m = 5)
The soft mode spectrum shows large temperature dependence at two temperatures, T c (551 K) and T H (730 K) [56]; the underdamped soft mode is clearly observed at room temperature, but changes to an overdamped one above 460 K.The extrapolation of soft mode frequency and line width shows an anomaly at T H .Moreover, an additional relaxation mode due to an order-disorder nature was observed which may be induced by the twodimensionality in Sr 2 Bi 4 Ti 5 O 18 .In an order-disorder phase transition, the relaxation time diverges at T c , and the HWHM (half width half maximum) becomes zero at T c .This typical order-disorder character observed in Sr 2 Bi 4 Ti 5 O 18 may be related to the layered structure, i.e. coupling of polarization fluctuation in intra-layer and inter-layer.
The similar behavior in soft modes is found in SrBi    [56].
been reported in the critical region near T c observed by diffuse scattering in BaTiO 3 [57]- [59].Recent NMR study also showed the coexistence of both order-disorder and displacive components in BaTiO 3 in the tetragonal ferroelectric to cubic paraelectric transition [60]- [62].The delicate balance between the soft mode behavior and relaxational order-disorder component will compete and modify dielectric properties and the sequence of phase transitions as observed in ultra-thin perovskite films and Bi-layered perovskites.

Ferroelectricity and Size Effect in Thin Films
Ferroelectricity in Bi-layered perovskites SrBi 2 Ta 2 O 9 , Sr 2 Bi 4 Ti 5 O 18 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 BaTiO 3 .Crystal structure analysis of SrBi 2 Ta 2 O 9 shows that TaO 6 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 Bi 2 O 2 layers in SrBi 2 Ta 2 O 9 , the strongly correlated -O-Ta-O-Ta-O-chain should play an essential role for the appearance of ferroelectricity as those in BaTiO 3 [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 ferroe-lectric 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/cm 2 along the a-axis at room temperature), while P s = 26 μC/cm 2 in a representative perovskite ferroelectric BaTiO 3 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 BaTiO 3 .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 0.2 Ti 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 BaTiO 3 films with SrRuO 3 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 BaTiO 3 films grown commensurately on SrTiO 3 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 BaTiO 3 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 SrBi 2 Ta 2 O 9 and thin film of so-called typical ferroelectric perovskite BaTiO 3 .The layered ferroelectric SrBi 2 Ta 2 O 9 is a bulk crystal itself, but is considered to be a good example of ultra-thin ferroelectric model with   two monolayers of perovskite TaO 6 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.

Critical Thickness
As Sr 2 Bi 4 Ti 5 O 18 is a compound with five perovskite layers sandwiched by Bi 2 O 2 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], Sr 2 Bi 4 Ti 5 O 18 shows ferroelectricity along the a-axis, but not along the tetragonal c-axis.Similar dielectric properties have been observed in SrBi 2 Ta 2 O 9 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 SrBi 2 Ta 2 O 9 (m = 2) and Sr 2 Bi 4 Ti 5 O 18 (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 BaTiO 3 [63], the -O-Ta-O-Ta-O-chain along the pseudo-tetragonal c axis is interrupted by the existence of semiconductor Bi 2 O 2 layers in SrBi 2 Ta 2 O 9 .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 SrBi 2 Ta 2 O 9 .It is expected that a crossover of ferroelectric to antiferroelectric interaction will be realized in thin films of BaTiO 3 , considering the similarity of dielectric properties between SrBi 2 Ta 2 O 9 bulk crystal and thin BaTiO 3 films mentioned above.It might be expected that the canted arrangement of dipole moments may be induced even in thin BaTiO 3 films by the two-dimensionality.
The suppression of ferroelectricity has been reported for nana-particles of BaTiO 3 and PbTiO 3 , 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.

Summary
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 Sr 2 Bi 4 Ti 5 O 18 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 SrBi 2 Ta 2 O 9 with two perovskite units (m = 2) and Sr 2 Bi 4 Ti 5 O 18 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.

Figure 1 .
Figure 1.Dielectric constant of BaTiO 3 bulk crystal associated with a paraelectric-ferroelectric phase transition [10].The recent T c has been reported as 409.5 K in high quality sample as shown in Figure 4.The red line shows dielectric constants of SrBi 2 TaO 9 bulk ceramics with T c = 608 K [9].

Figure 3 .
Figure 3. Temperature dependence of dielectric constant of BaTiO 3 thin films with thickness of 4000 Å and 1000 Å. Dielectric constant of SrBi 2 Ta 2 O 9 (SBT) bulk ceramics is also referred for comparison where its T c is shifted to that of BaTiO 3 bulk crystal [13]-[19].The dotted line is the T c of BaTiO 3 bulk crystal (T c = 409.5 K).More detailed comparison is given in Figure 13.

Figure 6 .
Figure 6.Crystal structures of Bi-layered perovskites depending on stacking number (m) of perovskite-like groups.The atoms in A m−1 Bi 2 B m O 3m+3 within the unit cell are shown as green ball (A), red ball (B), yellow ball (O) and blue ball (Bi).The long axis is the c-axis.The crystal structure is orthorhombic but nearly tetragonal at room temperature.The BO 6 are shown as octahedra.Bulk BaTiO 3 with simple perovskite structure has been recognized as a structure with expanded Bi-layered perovskite with infinity perovskite units (m = ∞).

Figure 7 .
Figure 7. Crystal structure Bi-layered perovskite SrBi 2 Ta 2 O 9 , which consists of Bi 2 O 2 semiconducting layers interleaved with TaO 6 perovskite groups.This orthorhombic structure is nearly tetragonal with a ~ b.The c-axis is perpendicular to these stacking layers.

Figure 8 .
Figure 8. Small and broad dielectric behavior in SrBi 2 Ta 2 O 9 around T c .The peak value is only 260 and the half width of this anomaly is over 400 K.

Figure 11 .
Figure 11.Crystal structure of Bi-layered perovskite Sr 2 Bi 4 Ti 5 O 18 with m = 5 (half of the unit cell).The c-axis is perpendicular to the stacking layers of Bi 2 O 2 semiconducting layer and perovskite-like groups.
Dielectric Properties of BaTiO 3 Ultra-Thin Film and SrBi 2 Ta 2 O 9 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 BaTiO 3 crystal, BaTiO 3 thin film (1000 Å) and Bi-layered perovskite SrBi 2 Ta 2 O 9 .The small and broad dielectric anomalies are commonly observed in BaTiO 3 thin films and bulk SrBi 2 Ta 2 O 9 crystal.

Figure 12 .
Figure 12.Dielectric constant of Sr 2 Bi 4 Ti 5 O 18 at 1 kHz, 10 kHz and 100 kHz.Two clear anomalies were found at 551 K (T c ) and 730 K (T H ).

SrBi 2
Ta 2 O 9 are shown in Figure 14.The soft mode frequency does not follow the Curie-Weiss law.This soft mode disappears at T c , which means that this mode becomes Raman inactive above T c .The extrapolated temperature where this soft mode vanishes is 860 K, just close to T H .

Figure 16 .
Figure 16.Temperature dependence of square frequency (red point) of soft mode and its line width (blue point) in Sr 2 Bi 4 Ti 5 O 18 [56].The solid line and red curve indicate the Curie-Weiss law and the universal scaling law, respectively.

Figure 19 .
Figure 19.First-principles calculations of the energy of BaTiO 3 ferroelectric films with m unit cells as a function of the soft-mode distortion ξ [68].

Figure 20 .
Figure 20.The plot of the ferroelectric phase transition temperature T c versus film thickness of BaTiO 3 films grown on SrTiO 3 substrates, observed by Raman scattering [69].

Table 2 .
Crystal system and the existence of soft mode of typical Bi-layered perovskites.