**2.5. Polarization and strain responses**

For many technical applications, understanding the physical properties of a ceramics sample is of great importance. Figure 11 shows the variation of polarization, bipolar-, and unipolarfield-induced strains with the Ca substitution in the (Ba1-*x*Ca*x*)TiO3 ceramics, which were measured at room temperature. One interesting finding is that the saturation polarization is nearly insensitive to the Ca substitution within the limit of solid solution[8] as shown in Fig. 11(a). This finding is predictable when considering the composition dependence of the tetragonality in the (Ba1-*x*Ca*x*)TiO3 system. As shown in Fig. 9(b), the tetragonality has an approximate value of 1.01 within the solid solution limit. As mentioned above, the tetragon‐ ality of the ferroelectric perovskite oxides is predicted to be proportional to its spontaneous polarization by the theoretical calculations.[20, 21]

For the strain response under an electric field, with the exception of BaTiO3 (*x*=0), (Ba1-*x*Ca*x*)TiO3 ceramics show nearly the same level of strain response under the same electric field. For examples, the strain was observed to be approximately 0.08% for a unipolar field of 40 kV/cm, which corresponds to a level of piezoelectric response of 200 pm/V. The observation of the large strain response in BaTiO3 is not surprising because its *T-O* phase transition is located at a temperature close to room temperature. Around the phase transition, a large response of physical properties generally occurs.

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)

For many technical applications, understanding the physical properties of a ceramics sample is of great importance. Figure 11 shows the variation of polarization, bipolar-, and unipolarfield-induced strains with the Ca substitution in the (Ba1-*x*Ca*x*)TiO3 ceramics, which were measured at room temperature. One interesting finding is that the saturation polarization is nearly insensitive to the Ca substitution within the limit of solid solution[8] as shown in Fig. 11(a). This finding is predictable when considering the composition dependence of the tetragonality in the (Ba1-*x*Ca*x*)TiO3 system. As shown in Fig. 9(b), the tetragonality has an approximate value of 1.01 within the solid solution limit. As mentioned above, the tetragon‐ ality of the ferroelectric perovskite oxides is predicted to be proportional to its spontaneous

For the strain response under an electric field, with the exception of BaTiO3 (*x*=0), (Ba1-*x*Ca*x*)TiO3 ceramics show nearly the same level of strain response under the same electric field. For examples, the strain was observed to be approximately 0.08% for a unipolar field of 40 kV/cm, which corresponds to a level of piezoelectric response of 200 pm/V. The observation of the large strain response in BaTiO3 is not surprising because its *T-O* phase transition is located at a temperature close to room temperature. Around the phase transition, a large response of

Change of potential energy of Ba7/8Ca1/8TiO3 along the path shown in (c).

polarization by the theoretical calculations.[20, 21]

physical properties generally occurs.

**2.5. Polarization and strain responses**

114 Ferroelectric Materials – Synthesis and Characterization

**Figure 11.** (a) Polarization, (b) bipolar-field and (c) unipolar-field strains under electric field in the (Ba1-*x*Ca*x*)TiO3 ce‐ ramics.
