**2.1. General properties of Polar phonons**

If the displacement of a phonon *Q* produces an electric dipole moment *μ* <sup>→</sup> =*e* \* *Q* ( *e* \*: effective charge), the mode is called a polar mode or polar phonon. A soft mode is not always a polar mode, for example, the soft mode in the *α* −*β* transition of quartz (SiO2) is nonpolar. However, for ferroelectric transitions it is always a polar mode.

Due to its interaction with any kind of electric field, the mechanism of the scattering of light from polar modes is much more complicated than nonpolar modes [4]. The Raman spectrum of polar phonons has several characteristics as follows (Fig.1):


### **2.2. General properties of soft modes**

**i.** Above *Tc*, the soft mode is the lowest frequency TO phonon. Below *Tc*, the soft mode becomes a totally symmetric mode (*A*1). Therefore, it is Raman-active. In typical cases, its frequency follows the equation

$$
\rho \alpha\_{TO}^2(q=0) = a \left| T - T\_c \right| \tag{1}
$$

Raman Spectra of Soft Modes in Ferroelectric Crystals http://dx.doi.org/10.5772/60613 5

freezing of the mode pattern changes the symmetry of the crystal. The interpretation of a soft mode, however, is not easy because its Raman spectrum is often observed without any definite peak and becomes a broad quasi-elastic peak near *Tc*. Therefore, interpretations of Raman spectra of polar modes, such as soft modes in ferroelectric crystals must be carefully done. Otherwise, incorrect conclusions might be obtained. In the following sections, several exam‐ ples related to this problem are shown for the cases of KDP, ferroelectric SrTiO3, and proton-

charge), the mode is called a polar mode or polar phonon. A soft mode is not always a polar mode, for example, the soft mode in the *α* −*β* transition of quartz (SiO2) is nonpolar. However,

Due to its interaction with any kind of electric field, the mechanism of the scattering of light from polar modes is much more complicated than nonpolar modes [4]. The Raman spectrum

**i.** Frequency of a polar phonon *Q* depends on the direction of its propagation vector

↼ *<sup>d</sup>* ∝ −*μ*

**ii.** As shown in Fig. 1(b), TO interacts with infrared (IR) photon and propagates as the

**iii.** In the IR spectra, reflectivity between LO and TO frequency is high and almost flat, while in Raman spectra they are observed as separate peaks (Fig. 1(d, e)).

**i.** Above *Tc*, the soft mode is the lowest frequency TO phonon. Below *Tc*, the soft mode

<sup>2</sup> ( 0) *TO <sup>c</sup>*

becomes a totally symmetric mode (*A*1). Therefore, it is Raman-active. In typical cases,

*q aT T* = = - (1)

→

phonon (LO). Generally speaking, the frequency of LO is higher than TO because of the depolarization field. If the polar nature of *Q* is strong (and much stronger than the anisotropy of the crystal), large LO/TO splitting is expected to take place (Fig.1).

coupled mode "polaritons". In Raman spectra, polaritons can be observed only with

<sup>→</sup> / *ε*. If *K* →

*<sup>p</sup>* is parallel to *μ*

<sup>→</sup> =*e* \* *Q* ( *e* \*: effective

<sup>→</sup> , it is a

*<sup>p</sup>* is perpendicular to *μ*

<sup>→</sup> , it is a longitudinal optic

ordered Ice crystals.

*K* →

**2. Characteristics of polar phonons**

4 Ferroelectric Materials – Synthesis and Characterization

**2.1. General properties of Polar phonons**

**2.2. General properties of soft modes**

its frequency follows the equation

for ferroelectric transitions it is always a polar mode.

of polar phonons has several characteristics as follows (Fig.1):

*<sup>p</sup>* due to the depolarization field *E*

transverse optic phonon (TO) and if *K*

a very small angle forward-scattering geometry [5].

w

If the displacement of a phonon *Q* produces an electric dipole moment *μ*

**Figure 1.** Various properties of polar LO and TO phonon in a cubic crystal. (a) LO/TO splitting in space, (b) dispersion of polariton, (c) dielectric constant, (d) IR reflection spectra, and (e) Raman spectrum.


$$(\omega\_{\rm TO} \not\!\omega\_{\rm LO})^2 = \mathfrak{e}\_{\rm o} \not\!\mathfrak{e}\_{\rm O} \tag{2}$$

**iv.** Integrated intensity of the soft mode Raman spectra is related to the *real part* of the low frequency dielectric constant *ε*(*ω*) via Kramers–Krong relation and Eq. (4) in the next section,

$$\operatorname{Re}\varepsilon(\boldsymbol{\alpha}) - \varepsilon\_{\boldsymbol{\alpha}} = \frac{2}{\pi} P \Big| \frac{\boldsymbol{\alpha}\prime}{\boldsymbol{\alpha}\prime^2 - \boldsymbol{\alpha}^2} \Big( \frac{I\left(\boldsymbol{\alpha}\prime\right)}{n\left(\boldsymbol{\alpha}\prime\right) + 1} \Big) d\boldsymbol{\alpha}\prime \tag{3}$$

An example of this relation was demonstrated in the case of KDP [7].
