1. Introduction
Ferroelectric phase transitions are conventionally divided into two types: an order-disorder and a displacive-type.[1] In the former one, which is frequently seen in hydrogen-bond-type ferroelectrics such as KH2PO4 (KDP), local dipole moments
2. Ferroelectric soft mode in the classical scheme
Since a concept of the soft mode was proposed, many theoretical and experimental studies have been conducted to clarify its dynamics.[2-4] In this section, we discuss the soft mode behavior in the classical scheme.
2.1. Theoretical background of the ferroelectric soft mode in the classical scheme
The potential of the soft mode is approximately described by accounting only a biquadratic phonon-phonon interaction as follows:
where
where the inside of the brackets is redefined by the soft mode frequency
Since
leading to
We thus obtain
Defining
it finally comes to Cochran’s law
where
As seen in the above equation, the soft mode decreases its frequency with approaching
2.2. Experimental observation of the ferroelectric soft mode in the classical scheme
The experimental observation of the typical soft mode behavior in the perovskite-type ferroelectric oxide, CdTiO3, is presented in this section.
CdTiO3 possesses an orthorhombic
The confocal micro-Raman scattering is the one of useful techniques to investigate the soft mode dynamics in the displacive-type ferroelectric phase transition due to the following reason; In general, the complicated domain structure forms across the ferroelectric phase transition, where the principle axis of the crystal orients to various directions, which are allowed by the symmetry relation between the paraelectric and the ferroelectric phases. The size of the individual domain generally ranges over several nanometers to microns. When the domains are smaller than the radiated area of the incident laser, the observed Raman spectrum is composed of signals from differently oriented domains, leading to difficulty in the precise spectral analyses. With application of the confocal micro-Raman scattering, on the other hand, we can selectively observe the spectrum from the single domain region, because its spatial resolution reaches sub-microns not only for lateral but also depth directions.[8] Note that the Raman scattering can observe the soft mode only in the non-centrosymmetric phase due to the selection rule, therefore the critical dynamics of the soft mode is in principle investigated in the ferroelectric phase below
Flux-grown colorless single crystals of CdTiO3 with a rectangular solid shape were used for this study. The two samples have dimensions of approximately 0.3×0.2×0.1 mm3 and 0.2×0.1×0.1 mm3. Since the present samples were twinned, we carefully determined the directions of the axes in the observed area by checking the angular dependence of the Raman spectra with the confocal micro-Raman system, whose spatial resolution is around 1
Figure 2 shows the temperature dependence of the Raman spectrum of CdTiO3, which is observed in Y(ZX,ZX)-Y scattering geometry with several temperatures from 85.0 K to 40.0 K. In the vicinity of
where
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. 3. Synchronizing with the softening of the soft mode, the damping constant is increased as approaching
The temperature dependence of the soft mode frequency obeys the Cochran’s low (Eq. 1.8) with
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.[9] Figure 4a presents phonon dispersion curves of CdTiO3 in the paraelectric
3. A role of covalency in the ferroelectric soft mode
As indicated above, the covalency of the
Ceramics form of Ca-doped CdTiO3, Cd1-
Temperature dependencies of the dielectric constants in CCT-
The temperature dependence of the soft mode frequency is presented in Fig. 6 as a function of
Figure 7 presents the partial electronic density of states (p-DOS) for CdTiO3 (left) and CaTiO3 (right) obtained by the first-principles calculations, where the
4. Ferroelectric Soft Mode in the Quantum Scheme
It has been known that the quantum fluctuation plays a non-negligible role in the phase transition dynamics when the
4.1. Theoretical background of the ferroelectric soft mode in the quantum scheme
In the classical case as mentioned before, the soft mode frequency can be described as Eq. 1.3. At the low-temperature, where the influence of the quantum fluctuation can not be ignored, the approximation <
At the sufficiently low-temperature, where only the Γ-point soft mode that is the optical phonon with lowest energy can be thermally excited, any other optical phonons are not effective in the second term except for λ =
As in the classical case,
gives
We thus obtain the soft mode frequency at the low-temperature,
with
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 qualitative behavior of the soft mode in the quantum para/ferroelectrics is schematically illustrated as a function of
The classical limit is also presented for comparison. Note that the value of
4.2. Experimental observation of the ferroelectric soft mode in the quantum scheme
The isotopically induced phase transition of the quantum paraelectric SrTiO3 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(16O1-
Figure 10 presents the temperature dependence of the squared soft mode frequency in STO18-100
With the isotope substitution, softening of the soft mode is enhanced and the phase transition takes place above the critical concentration
The
In STO18-32, the strongly rounded soft mode behavior keeps us from fitting with Eq. 3.7. However, we can determine γ
5. Summary
In the present review, we overviewed the soft mode behavior in both classical and quantum schemes through Raman scattering experiment in the CdTiO3 and the 18O-substituted SrTiO3. The analyses show excellent agreement qualitatively between the fundamental theory and experimental observations. The systematic studies with Raman scattering experiments and the first-principles calculations on Ca-substituted CdTiO3 clarified that the covalency between constituent cation and oxygen plays an essential role in the origin of the soft mode. We hope the present review serves better understanding on the mechanism of displacive-type ferroelectric phase transition.
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