**2.2. Dielectric behaviors**

**2.1. Crystal growth**

108 Ferroelectric Materials – Synthesis and Characterization

To obtain a single crystal of (Ba1-*x*Ca*x*)TiO3, we used the floating zone (FZ) technique (Fig. 3(a)) that allowed us to grow a single crystal with high purity. According to the reported phase equilibria in the system (1-*x*)BaTiO3-*x*CaTiO3 (Fig. 4), only the crystal with a congruent melting composition (*x*=0.27 report by DeVries and Roy[15] or *x*=0.227 report by Kuper et al.[16]) can be directly grown from the melt. Surprisingly, using the FZ technique, we could grow a single crystal of (Ba1-*x*Ca*x*)TiO3 with a perovskite structure for a wide composition range of 0.02 ≤ *x* ≤ 0.34 with a high growth rate of 20 mm/h.[6] Fig. 3(b) shows a rod of crystal obtained by this method. It was found that the crystal can be stably grown under an atmosphere of Ar or N2 gas. The crystal was yellowish but transparent. The Laue X-ray diffraction patterns clearly

**Figure 3.** (a)A schematic drawing of the floating zone (FZ) technique used to grow the (Ba1-*x*Ca*x*)TiO3 single crystal. (b) Photograph of the (Ba1-*x*Ca*x*)TiO3 single crystal grown by the FZ technique and its Laue X-ray back diffraction patterns

**Figure 4.** Phase equilibria in the system (1-*x*)BaTiO3-*x*CaTiO3 reported by (a) DeVries and Roy[15] and (b) Kuper et al.[16]

along the [001]c direction of the perovskite structure.

indicated that the obtained (Ba1-*x*Ca*x*)TiO3 crystal had a perovskite structure.

Figure 5 shows the temperature dependence of the dielectric permittivity of a (Ba1-*x*Ca*x*)TiO3 single crystals in the temperature range from 2K to 400K. Compared with the polycrystalline ceramics, the (Ba1-*x*Ca*x*)TiO3 single crystal showed a very sharp change of dielectric response at the phase transition, allowing easy determination of the transition temperatures. Similar to the polycrystalline ceramics[12], the Curie point was nearly independent of the Ca concentration. However, the *T*–*O* and *O*-rhombohedral (*R*) ferroelectric transitions shifted to lower tempera‐ tures as the Ca concentration increased. For compositions of *x*>0.23, these two transitions completely disappeared, and the *T*-phase was the only stable ferroelectric phase in the crystal. This situation is very similar to that of PbTiO3, in which the *T*-phase is the only stable ferroelec‐ tric structure. Another interesting finding was that the dielectric permittivity was nearly unchanged for temperatures lower than the Curie point for *x*>0.23. This unique behavior provides the possibility of designing electronic devices that operate stably in a wide tempera‐ ture range from the boiling point of water all the way down to absolute zero Kelvin using (Ba1 *<sup>x</sup>*Ca*x*)TiO3.

As is well known, the dielectric response in displacive-type ferroelectrics is dominated by phonon dynamics, particularly the soft-mode behavior. Lyddane–Sachs–Teller (LST) relation‐ ship predicts that the dielectric permittivity is inversely proportional to the soft-mode frequen‐ cy. As a step toward understanding the temperature independence of the dielectric response of (Ba1-*x*Ca*x*)TiO3 (*x* >0.23), we performed confocal micro-Raman scattering measurements for the (Ba1-*x*Ca*x*)TiO3 (*x* =0.233) single crystals to clarify its soft-mode dynamics.[14] In contrast to BaTiO3, a well-defined soft phonon mode was observed for temperatures lower than the Curie point in the (Ba1-*x*Ca*x*)TiO3 (*x* =0.233) single crystals. The temperature dependence of the softmode frequency agreed qualitatively with the dielectric permittivity through the Lyddane– Sachs–Tellerrelationship(Fig.6).This result clearlyindicates thattheuniquedielectric response of (Ba1-*x*Ca*x*)TiO3 (*x*=0.233) is directly derived from its soft-mode dynamics.

**Figure 5.** Dielectric behaviors of (Ba1-*x*Ca*x*)TiO3 single crystals.

**Figure 6.** Comparison of (a) dielectric permittivity of (Ba1-*x*Ca*x*)TiO3 single crystals and (b) the phonon frequency of the Slater soft-mode.

### **2.3. Phase diagram and quantum phase transitions**

From the temperature change of dielectric permittivity of a (Ba1-*x*Ca*x*)TiO3 crystal, we have established its phase diagram in a composition range of *x* ≤ 0.34 for temperature down to 2 K (Fig. 7). Compared with the phase diagram proposed by Mitsui and Westphal for ceramics,[12] our phase diagram has been expanded to a composition up to *x*=0.34 mole and to temperatures as low as 2 K. These expansions of composition and temperature allow us to reveal some unexpected phenomena in this system: (1) ferroelectric *R-* and *O-*phases become unstable as the Ca concentration increased, and they are predicted to disappear at *x*>*x*<sup>c</sup> O-R =0.18 and *x*>*x*<sup>c</sup> T-<sup>O</sup>=0.233, respectively; and (2) the ferroelectric *T*-phase is a ground state for *x*> *x*<sup>c</sup> T-O.

**Figure 7.** Top panel: phase diagram of (Ba1-*x*Ca*x*)TiO3 crystals. Left of bottom panel: Change of critical exponent *γ* for the dielectric susceptibility in the *T-O* phase transition with the composition. Right of bottom panel: Variation of the dielectric susceptibility measured at 2.5 K with the composition.

One important finding is that ferroelectric–ferroelectric quantum phase transitions occur in (Ba1-*x*Ca*x*)TiO3 crystals. The occurrence of ferroelectric–ferroelectric quantum phase transitions is supported by two experimental facts: the compositional dependence of *T*O-R and *T*T-O transition temperatures and the temperature dependence of the dielectric susceptibility in the crystals at compositions close to *x*<sup>c</sup> T-O. The theoretical and experimental studies[17-19] on quantum phase transitions indicate that (a) for a quantum ferroelectric, the transition temper‐ ature depends on the substitution concentration (i.e., on an effective order parameter) as

$$T\_c \propto \left(\mathbf{x} - \mathbf{x}\_c\right)^{1/2},\tag{1}$$

as opposed to the classical relationship,

**0 100 200 300 400 500**

*T* **(K)**

**Figure 6.** Comparison of (a) dielectric permittivity of (Ba1-*x*Ca*x*)TiO3 single crystals and (b) the phonon frequency of the

From the temperature change of dielectric permittivity of a (Ba1-*x*Ca*x*)TiO3 crystal, we have established its phase diagram in a composition range of *x* ≤ 0.34 for temperature down to 2 K (Fig. 7). Compared with the phase diagram proposed by Mitsui and Westphal for ceramics,[12] our phase diagram has been expanded to a composition up to *x*=0.34 mole and to temperatures as low as 2 K. These expansions of composition and temperature allow us to reveal some unexpected phenomena in this system: (1) ferroelectric *R-* and *O-*phases become unstable as

**Figure 7.** Top panel: phase diagram of (Ba1-*x*Ca*x*)TiO3 crystals. Left of bottom panel: Change of critical exponent *γ* for the dielectric susceptibility in the *T-O* phase transition with the composition. Right of bottom panel: Variation of the

*x***=0.233**

O-R =0.18 and *x*>*x*<sup>c</sup>

T-O.

T-

*x***=0.233**

**Ba1-***<sup>x</sup>* **Ca***<sup>x</sup>* **TiO3**

> **2 S**

*x***=0**

**103**

the Ca concentration increased, and they are predicted to disappear at *x*>*x*<sup>c</sup>

<sup>O</sup>=0.233, respectively; and (2) the ferroelectric *T*-phase is a ground state for *x*> *x*<sup>c</sup>

**106**

**2.3. Phase diagram and quantum phase transitions**

dielectric susceptibility measured at 2.5 K with the composition.

Slater soft-mode.

**/**w**2**

**)**

**2**

**s (1/cm-1**

**104 0.1**

**(b)** e*~***1/**w

**1**

e

**/103**

110 Ferroelectric Materials – Synthesis and Characterization

**10 (a)**

$$T\_c \propto \left(\mathbf{x} - \mathbf{x}\_c\right). \tag{2}$$

(b) The inverse dielectric susceptibility varies with temperature as

$$\mathcal{X}^{-1} \propto \left(T - T\_c\right)^2 \tag{3}$$

for the quantum mechanical limit instead of the classical Curie law

$$
\mathbb{Z}^{-1} \propto \left( T - T\_c \right). \tag{4}
$$

Our phase diagram clearly shows that the *T-O* and *O-R* phase transitions deviate from the classical relationship (equation (2)) for *x*≤0.06, and exactly follow equation (1) for *x*>0.06 with *x*c equal to *x*<sup>c</sup> T-O=0.233 and *x*<sup>c</sup> O-R=0.18, respectively.

To examine point (b), we have analyzed the temperature variation of the inverse dielectric susceptibility of *T-O* phase transition in the (Ba1-*x*Ca*x*)TiO3 crystals with the following equation,

$$\mathcal{X}^{-1} \propto \left(T - T\_c\right)^\vee. \tag{5}$$

Figure 8 shows two typical examples: one for *x*=0 and another for *x*=0.23 close to *x*<sup>c</sup> T-O. For pure BaTiO3 with *x*=0, the classical Curie law with *γ*=1 was observed to be operative. In contrast, for *x*=0.23, the critical exponent *γ* for the susceptibility was found to have a value of 2, which is predicted for the quantum phase transition (equation (3)). The left of the bottom panel in Fig. 7 shows the variation of *γ* with *x.* It is clear that the *γ* value changes from the value of the classical limit to that of the quantum limit as *x* increases from 0 to *x*<sup>c</sup> T-O. This fact again indicates that a quantum phase transition indeed occurs at zero Kelvin in the system when the Ca concentration increases. Interestingly, a dielectric anomaly was observed for the *T-O* quantum phase transition at *x*= *x*<sup>c</sup> T-O as shown at the right of the bottom panel of Fig. 7. At a temperature of 2.5 K, close to zero Kelvin, the crystal with *x*=0.23 close to *x*<sup>c</sup> T-O shows a maximum value of dielectric susceptibility in the system.

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