**2.4. Ca off-centering predicted from first principles calculations**

As mentioned above, the ionic radius of Ca is approximately 16% smaller than that of Ba. The substitution of Ca for Ba will absolutely result in the shrinkage of the perovskite unit cell. As shown in Fig. 9(a), both *a*- and *c*-axes of the tetragonal structure shrink with increasing Ca concentration, resulting in the reduction of the unit cell. The volume of the unit cell of *x*=0.34 is approximately 3.6% smaller than that of pure BaTiO3. This chemical-pressure-induced reduction of the unit cell does not have significant influence on the Curie point (*T*<sup>c</sup> C-T) of the system (Fig. 9(c)).

In contrast, for the case of hydrostatic pressure, at the same level of unit-cell reduction, the Curie point is reduced to ~180 K, which is greatly lower than the ~400 K of pure BaTiO3 (Fig. 9(c)). The hydrostatic pressure gradually reduces the Curie point, leading to the complete disappearance of ferroelectricity in BaTiO3 at a level of 5% reduction of the unit cell. Although the chemical substitution of the smaller Ca for the bulky Ba also leads to the reduction of the unit cell, the effects of chemical pressure on the ferroelectricity in the (Ba1-*x*Ca*x*)TiO3 system were very different from what we would expect with the application of hydrostatic pressure.

Apparently, the reduction of the unit cell by the chemical pressure shrinks the oxygen octahedron in the perovskite structure, resulting in the reduction of available space for a Ti off-centering shift in the oxygen octahedron. Since the ferroelectricity is derived from the Tishift in the oxygen octahedron in the perovskite structure of BaTiO3, it is naturally expected that chemical-pressure-induced reduction of the unit cell in the (Ba1-*x*Ca*x*)TiO3 system should weaken the ferroelectricity of the system and result in a decrease in its Curie point. However, as shown in the phase diagram in Fig. 7 and Fig. 9, the Curie point remains nearly unchanged with the Ca substitution. Also, as shown in Fig. 9(b), the tetragonality (*c/a*), which is generally used to estimate the atomic displacement and thus the ferroelectricity and polarization in the tetragonal ferroelectric,[20, 21] maintains a constant value in the whole composition range in the (Ba1-*x*Ca*x*)TiO3 system. In accordance with such unchanged tetragonality, we observed that the saturation polarization is also nearly independent on the Ca substitution within the limits of a solid solution (see Fig. 11).[8] These facts suggest that in addition to the Ti displacement in the oxygen octahedron, there should be an additional atomic displacement to contribute to the ferroelectricity of the (Ba1-*x*Ca*x*)TiO3 system. The ionic radii of Ca and Ba are 1.34 Å and 1.61 Å, respectively. There is a difference of 0.27 Å between them. This suggests that the smaller Ca ion may have an off-centering displacement in the bulky Ba sites (Fig. 10(b)). that chemical‐pressure‐induced reduction of the unit cell in the (Ba1‐*<sup>x</sup>*Ca*x*)TiO3 system should weaken the ferroelectricity of the system and result in a decrease in its Curie point. However, as shown in the phase diagram in Fig. 7 and Fig. 9, the Curie point remains nearly unchanged with the Ca substitution. Also, as shown in Fig. 9(b), the tetragonality (*c/a*), which is generally used to estimate the atomic displacement and thus the ferroelectricity and polarization in the tetragonal ferroelectric,20,21 maintains a constant value in the whole composition range in the (Ba1‐*<sup>x</sup>*Ca*x*)TiO3 system. In accordance with such unchanged tetragonality, we observed that the saturation polarization is also nearly independent on the Ca substitution within the limits of a solid solution (see Fig. 11).8 These facts suggest that in addition to the Ti displacement in the oxygen octahedron, there should be an additional atomic displacement to contribute to the ferroelectricity of the (Ba1‐*<sup>x</sup>*Ca*x*)TiO3 system. The ionic radii of Ca and Ba are 1.34 Å and 1.61 Å, respectively. There is a difference of 0.27 Å between them. This suggests that the smaller Ca ion may have an off‐centering

displacement in the bulky Ba sites (Fig. 10(b)).

Apparently, the reduction of the unit cell by the chemical pressure shrinks the oxygen

shift in the oxygen octahedron in the perovskite structure of BaTiO3, it is naturally expected

concentration increases. Interestingly, a dielectric anomaly was observed for the *T-O* quantum

**-1000 0 1000 2000**

**(***T***-***T***T-O) 2 (K2 )**

**0 50**

*T***-***T***T-O (K)**

**Figure 8.** Change of the inverse dielectric susceptibility (*χ*=*ε*-1) near the *T-O* phase transition in (Ba1-*x*Ca*x*)TiO3 crystals.

As mentioned above, the ionic radius of Ca is approximately 16% smaller than that of Ba. The substitution of Ca for Ba will absolutely result in the shrinkage of the perovskite unit cell. As shown in Fig. 9(a), both *a*- and *c*-axes of the tetragonal structure shrink with increasing Ca concentration, resulting in the reduction of the unit cell. The volume of the unit cell of *x*=0.34 is approximately 3.6% smaller than that of pure BaTiO3. This chemical-pressure-induced

In contrast, for the case of hydrostatic pressure, at the same level of unit-cell reduction, the Curie point is reduced to ~180 K, which is greatly lower than the ~400 K of pure BaTiO3 (Fig. 9(c)). The hydrostatic pressure gradually reduces the Curie point, leading to the complete disappearance of ferroelectricity in BaTiO3 at a level of 5% reduction of the unit cell. Although the chemical substitution of the smaller Ca for the bulky Ba also leads to the reduction of the unit cell, the effects of chemical pressure on the ferroelectricity in the (Ba1-*x*Ca*x*)TiO3 system were very different from what we would expect with the application of hydrostatic pressure. Apparently, the reduction of the unit cell by the chemical pressure shrinks the oxygen octahedron in the perovskite structure, resulting in the reduction of available space for a Ti off-centering shift in the oxygen octahedron. Since the ferroelectricity is derived from the Tishift in the oxygen octahedron in the perovskite structure of BaTiO3, it is naturally expected that chemical-pressure-induced reduction of the unit cell in the (Ba1-*x*Ca*x*)TiO3 system should

reduction of the unit cell does not have significant influence on the Curie point (*T*<sup>c</sup>

*x***=0** *T***T-O=278.3K**

**Ba1-xCax**

*x***=0.23** *T***T-O= 36K**

**0.44** *<sup>x</sup>***=0.23**

**TiO3**

**0.40**

1 <sup>c</sup>/

**0.42**

of 2.5 K, close to zero Kelvin, the crystal with *x*=0.23 close to *x*<sup>c</sup>

**0.2**

**2.4. Ca off-centering predicted from first principles calculations**

**0.3**

1 <sup>c</sup>/

**0.4**

**0.5**

T-O as shown at the right of the bottom panel of Fig. 7. At a temperature

T-O shows a maximum value of

C-T) of the

phase transition at *x*= *x*<sup>c</sup>

system (Fig. 9(c)).

dielectric susceptibility in the system.

112 Ferroelectric Materials – Synthesis and Characterization

Figure 9. Change of (a) lattice constants and (b) tetragonality (*c/a*) of (Ba1‐*<sup>x</sup>*Ca*x*)TiO3 crystal. (c) Change of phase transition temperature as a function of unit cell volume (determined at room temperature);the hydrostatic pressure effect for pure BaTiO3 is also shown for comparison.7,22,23 **Figure 9.** Change of (a) lattice constants and (b) tetragonality (*c/a*) of (Ba1-*x*Ca*x*)TiO3 crystal. (c) Change of phase transi‐ tion temperature as a function of unit cell volume (determined at room temperature);the hydrostatic pressure effect for pure BaTiO3 is also shown for comparison.[7, 22, 23]

To examine the idea of Ca off‐centering in the bulky Ba sites, we performed first principles calculations for this system.7,24 Since Ca substitution tends to stabilize the tetragonal structure, we focused on the calculations in this structure to get information about Ca displacement. The results are summarized in Fig. 10. As shown in Fig. 10(a), a Ti shift along the [001] direction leads to the stability of the tetragonal phase in BaTiO3 (*x*=0). In our calculations for the substitution of Ca for Ba (*x*=1/8), we calculated the relative change in To examine the idea of Ca off-centering in the bulky Ba sites, we performed first principles calculations for this system.[7, 24] Since Ca substitution tends to stabilize the tetragonal structure, we focused on the calculations in this structure to get information about Ca dis‐ placement. The results are summarized in Fig. 10. As shown in Fig. 10(a), a Ti shift along the [001] direction leads to the stability of the tetragonal phase in BaTiO3 (*x*=0). In our calculations for the substitution of Ca for Ba (*x*=1/8), we calculated the relative change in potential energy for various locations of the Ca ion, as shown schematically in Fig. 10(c). As shown in Fig. 11(d), a Ca off-centering shift results in the lowering of the potential energy of the system. This result clearly indicates that the Ca off-centering stabilizes the structure of the (Ba1-*x*Ca*x*)TiO3 system. After tracing the variation of potential energy for moving Ca along various paths, we found that the most likely direction for a Ca shift is [113] since the potential energy is the lowest when Ca is shifted along this direction (Fig. 10(e)). The Ca-shift along the [113] direction seems to be incompatible with the tetragonal structure. However, if we consider the eight-site model similar to that assumed for Ti displacement in BaTiO3, then it becomes clear that Ca can displace along the equivalent directions [113], [1-13], [-113], [-1-13], or [11-3], [1-1-3], [-11-3], [-1-1-3]. The activation barrier for Ca moving between these equivalent states has been evaluated to be less than 3 meV. Therefore, thermal and spatial averaging among these states allows the preservation of the overall tetragonal symmetry detected from X-ray diffractions. It should be noted that the estimated displacement of Ca is approximately 0.1 Å (Fig. 10(d)), which is larger than the 0.05 Å shift of Ti in the tetragonal structure of BaTiO3 (see Ref. 1 and Fig. 10(a)).

**Figure 10.** (a) Two-dimensional contour map of potential energy of BaTiO3 as a function of Ti and O1 displacement along the [001] direction of the polar *c*-axis. (b) Schematic of Ca off-centering in the bulky Ba sites of the perovskite structure, in which the atomic shifts are shown by the arrows. (c) Direction of Ca-shift for the first principles calcula‐ tions of Ba7/8Ca1/8TiO3. (d) Change of potential energy of Ba7/8Ca1/8TiO3 along the [001], [111], and [113] directions. (e) Change of potential energy of Ba7/8Ca1/8TiO3 along the path shown in (c).
