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The temperature dependence of the soft mode frequency and the damping constant, which are determined by the analyses, are plotted by solid circles and open squares in Fig.

Raman Scattering Study on the Phase Transition Dynamics of Ferroelectric Oxides

Synchronizing with the softening of the soft mode, the damping constant is increased as approaching T c. Temperature Dependencies of a the soft mode frequency and b the damping constant of CdTiO3. The displacement pattern of the soft mode is obtained by first principle calculations. The calculations were conducted with a pseudopotential method based on a density functional perturbation theory with norm-conserving pseudopotentials, which was implemented in the CASTEP code. It has been previously proposed that the hybridization between O-2p and empty Ti-3d orbitals triggers the non-centrosymmetric displacement of Ti thorough a second-order Jahn-Teller SOJT effect to induce the ferroelectricity as exemplified by BaTiO 3.

Therefore, the ferroelectricity in CdTiO 3 can not be explained only by the B -O orbital hybridization. Ti and O ions that do not participate in the O-Cd-O bonds are omitted for simplicity. In a prototypical Pm -3 m structure of the perovskite-type oxide, an A -site ion is surrounded by neighboring twelve O ions.

In the CdTiO 3 of paraelectric Pnma structure, however, the number of closest oxygen ions for Cd ion is reduced to four due to the octahedral rotations. Note that the octahedral rotations in the Pnma structure are different from each other according to the direction; the rotation is in-phase for a [] direction whereas those for [] and [] are anti-phase. As a result, CdTiO 3 has three different O-Cd-O bonds along [], [], and [] directions, whose equilibrium structures are respectively denoted by 1 , 2 , and 3 in the Fig.

The result of first-principles calculations indicates that the soft mode displacement is inherently dominated by the asymmetric stretching of the O-Cd-O bond along the [] direction denoted by 3. By comparing three equilibrium structures for O-Cd-O bonds presented in the right-hand side of Fig. Therefore, there is a remaining degree of freedom for further O-Cd-O off-center ordering due to the covalent bonding along the [] direction.

This remaining degree of freedom would be the origin of the ferroelectric soft mode.

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As indicated above, the covalency of the A -site ion plays an important role in the ferroelectricity in addition to that of B -site ion. The A -site substitution with an isovalent ion is instructive for the better understanding on the mechanism of the ferroelectricity in the perovskite-type oxides.

This section is thus devoted to the effect of the isovalent Ca-substitution on the ferroelectricity of CdTiO 3. Dielectric measurements were performed by a LCZ meter. Temperature dependencies of the dielectric constants in CCT- x are presented in Fig. The Ca concentration dependence of T c , which is estimated from the dielectric peaks, is shown in Fig.

The temperature dependence of the soft mode frequency is presented in Fig.

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The soft modes in CCT- x increase in frequency and intensity with distance from T c on cooling, exhibiting typical behavior of the soft-mode-driven phase transition. According to the Lyddane—Sachs—Teller relationship Eq. Therefore, if the dielectric properties of the CCT- x system are governed by the lattice dynamics, the soft mode frequency at the lowest temperature should decrease with increasing x because the dielectric constant near 0 K increases with x see Fig. The Raman spectra at 4 K presented in the bottom panels in Fig.

This result suggests that the phase transition of CCT- x can be discussed in terms of lattice dynamics. Note that, the soft mode become observable at low temperature in CCT This is a typical characteristic of precursory softening of the soft mode in the quantum paraelectric state.

Contour plots for the temperature dependence of soft mode spectra in CCT-x upper panels.

Phase Transition Dynamics

Figure 7 presents the partial electronic density of states p-DOS for CdTiO 3 left and CaTiO 3 right obtained by the first-principles calculations, where the p -orbital of oxygen, the s -orbital of Cd and Ca, and the total DOS are denoted by oppositely hatched and blank areas, respectively. As shown in the figure, the s -orbital of Cd ion has strong hybridization with the p -orbital of oxygen.


This characteristic substantially agrees with the difference in electronegativity of Cd and Ca ions, where those in Cd and Ca are 1. A calculated charge density distributions around the O-Cd Ca -O bonds along the [] direction are indicated in Fig.

As shown in the panels, larger charge density is observed between Cd and O ions in CdTiO 3 , confirming its strong covalency. In contrast, Ca-O bond is nearly ionic. Therefore, Ca-substitution suppresses the freezing of the asymmetric stretching vibration with the off-centering of Cd Ca by decreasing the covalency. Since the asymmetric stretching of the O-Cd-O bond along [] direction is suggested to be the origin of the soft mode, Ca-substitution results in the suppression of the softening of the soft mode. The present result confirms that the ferroelectric instability of CCT- x stems from A -site covalency.

Blue, red, and black curves represent p orbital of oxygen ions, s orbital of Cd and Ca ions, and total DOS. Cross sections of charge density around 4-coordinated Cd left and Ca right ions obtained by first principle calculations.

Submission history

Note that the upper and lower cross sections include O—Cd Ca —O bonds, which are involved in O—Cd Ca —O chains along the b and a direction, respectively. It has been known that the quantum fluctuation plays a non-negligible role in the phase transition dynamics when the T c goes down near 0 K, where the transition is suppressed though the dielectric permittivity reaches to several tens thousand.

This effect is known as "quantum paraelectricity", which was first proposed as an origin of the giant dielectric plateau of SrTiO 3 at the low-temperature. In the classical case as mentioned before, the soft mode frequency can be described as Eq. Therefore, Eq. The soft mode frequency can thus be expressed as. In this treatment, the temperature dependence of the squared soft mode frequency, which is inversely proportional to the dielectric permittivity, in no longer linear with respect to temperature, but saturated with the constant value near 0 K.

The schematic illustration describing the variation of the temperature dependence of the soft mode frequency as a function of T1. See text for the detail. The classical limit is also presented for comparison. Note that the value of T 1 is varied with constant intervals. As presented in the figure, the phase transition is completely suppressed when T 1 is sufficiently large, whereas it recovers as decreasing T 1.

Interestingly, the T 1 dependence of the transition temperature becomes extremely sensitive when the transition temperature is close to 0K, suggesting that the quantum paraelectric-ferroelectric transition can be induced by subtle perturbation such as the isotope substitutions. Temperature dependencies of the squared frequencies of the soft mode in STOx with various x. Solid lines in the figures denote temperature dependencies calculated by Eq. The isotopically induced phase transition of the quantum paraelectric SrTiO 3 is a good example for the quantum paraelectric-ferroelectric phase transition driven by the soft mode.

Here we show the Raman scattering study on the soft mode in SrTi 16 O 1- x 18 O x 3 STO x as functions of temperature and the isotope substitution rate x. Figure 10 presents the temperature dependence of the squared soft mode frequency in STO x for 0.

As indicated in the figure, the softening of the soft mode in STO saturates at the low-temperature region near 0 K, showing excellent agreement with the theory. Since the square of the soft mode frequency is inversely proportional to the dielectric permittivity as indicated by the LST-relation as mentioned before, it is clear that the dielectric plateau in SrTiO 3 stems from the soft mode dynamics.

Note that the soft mode is nominally Raman inactive in the centrosymmetric structure as for the paraelectric phase of SrTiO 3 due to the selection rule. The local non-centrosymmetric regions grow with 18 O-substitution to activate the soft mode spectrum even in the paraelectric phase as indicated in Fig. A mechanism of such defect-induced Raman process and an expected spectral profile are discussed in detail in Ref.

With the isotope substitution, softening of the soft mode is enhanced and the phase transition takes place above the critical concentration x c as manifested by the hardening of the soft mode on cooling in the low temperature region.

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