**5.1. Soft mode in STO16**

SrTiO3 undergoes a structural (anti-ferro-distortive) phase transition at *T*<sup>0</sup> =105*K* from cubic O*h* <sup>1</sup> (*Pm*3 ¯*m*) to tetragonal D4*<sup>h</sup>* <sup>18</sup> (4 / *mcm*) structure. This transition is induced by the freezing of a zone boundary nonpolar soft mode *R*<sup>25</sup> ' (structural soft mode), which is an alternative rotational vibration of TO6 octahedra around one of the cubic axes (Z-axis [001]c) [26]. As shown in Fig. 8(b), the crystal axes in the tetragonal phase are rotated by 45° from the cubic axes and the unit cell is a rectangular parallelepiped elongated along one of the cubic Z-axes. It should be noted, therefore, that the multi-domain effect must be carefully taken into account since a sample below *T*0 is usually consisted from a number of tetragonal domains with different Z-directions. It is especially important for the case of STO18 as we shall show in the next section. Below *T*<sup>0</sup> in the tetragonal phase, the triply degenerate structural soft mode splits into two Raman-active A1g + Eg modes and their frequencies increase on further cooling reaching 44 and 11 cm-1, respectively.

Besides the "structural soft mode" there is another soft mode, the so-called ferroelectric soft mode in SrTiO3. It is the lowest transverse mode (TO1) among the 4 zone-center Γ15 modes in the cubic symmetry, which is known as the Slater mode (Fig. 8(a)). Frequency of the corre‐ sponding LO mode is much higher (170 cm-1) than TO1. The large LO/TO splitting indicates the very strong polar nature of this mode. Temperature dependence of the ferroelectric mode has been extensively studied. Above *T*0 in the cubic phase, it is a doubly degenerate Eu mode regardless of the propagation directions *K* → *<sup>p</sup>*. Below *T*0 in the tetragonal phase, however, the symmetry of this mode depends on the *K* → *<sup>p</sup>*. If it propagates along Z-axis, the degeneracy remains but when it propagates in the X-Y plane, TO1 splits into A2u + Eu [27] (also see Fig. 12(a) in the next section).

The softening of A2u and Eu was first observed by IR spectroscopy [28] and later confirmed by neutron scattering [29] and hyper-Raman scattering [27, 30]. Its frequency (90cm−1) at room temperature decreases to about 15 cm−1 in low temperature. In spite of the significant softening, it rounds off at about 30 K and never freezes down to zero K. Thus, STO16 is known as quantum paraelectric crystal since quantum fluctuation at low temperatures hinders the freezing of the soft mode.

**Figure 8.** (a) Perovskite structure and the ferroelectric soft mode (Slater mode). (b) Blue and red squares are the unit cell in the cubic and the tetragonal phase, respectively. (c) Electric field effect on Raman spectra of ferroelectric soft mode in STO16 at *T*=101 K [32].

Although the ferroelectric soft mode is Raman-inactive, surprisingly it was found that if a DC electric field is applied its intensity and frequency drastically increase [31]. The field effect was observed even at higher temperature near *T*0 as shown in Fig. 8(c) [32]. This is another evidence of the extremely strong polar nature of the ferroelectric soft mode in SrTiO3. In other similar crystals, for example in KNbO3, the effect is much weaker than in SrTiO3 [33].

#### **5.2. Soft mode in SrTiO3 18 (STO18)**

**5. Ferroelectric SrTiO3**

14 Ferroelectric Materials – Synthesis and Characterization

**5.1. Soft mode in STO16**

¯*m*) to tetragonal D4*<sup>h</sup>*

zone boundary nonpolar soft mode *R*<sup>25</sup>

regardless of the propagation directions *K*

symmetry of this mode depends on the *K*

12(a) in the next section).

soft mode.

O*h* <sup>1</sup> (*Pm*3

respectively.

SrTiO3 is a typical crystal with perovskite structure (Fig. 8(a)) and it is easy to get a pure transparent single crystal. As a well-known incipient ferroelectric, it is one of the most widely studied dielectric crystals [24]. In 1999, Itoh et al. found the ferroelectricity in SrTiO3 by the isotope substitution of O16 by O18 [25]. Since then, there has been considerable renewed interest

STO16 and STO18, respectively. First, we will review the soft mode behavior in the normal

SrTiO3 undergoes a structural (anti-ferro-distortive) phase transition at *T*<sup>0</sup> =105*K* from cubic

vibration of TO6 octahedra around one of the cubic axes (Z-axis [001]c) [26]. As shown in Fig. 8(b), the crystal axes in the tetragonal phase are rotated by 45° from the cubic axes and the unit cell is a rectangular parallelepiped elongated along one of the cubic Z-axes. It should be noted, therefore, that the multi-domain effect must be carefully taken into account since a sample below *T*0 is usually consisted from a number of tetragonal domains with different Z-directions. It is especially important for the case of STO18 as we shall show in the next section. Below *T*<sup>0</sup> in the tetragonal phase, the triply degenerate structural soft mode splits into two Raman-active A1g + Eg modes and their frequencies increase on further cooling reaching 44 and 11 cm-1,

Besides the "structural soft mode" there is another soft mode, the so-called ferroelectric soft mode in SrTiO3. It is the lowest transverse mode (TO1) among the 4 zone-center Γ15 modes in the cubic symmetry, which is known as the Slater mode (Fig. 8(a)). Frequency of the corre‐ sponding LO mode is much higher (170 cm-1) than TO1. The large LO/TO splitting indicates the very strong polar nature of this mode. Temperature dependence of the ferroelectric mode has been extensively studied. Above *T*0 in the cubic phase, it is a doubly degenerate Eu mode

→

→

remains but when it propagates in the X-Y plane, TO1 splits into A2u + Eu [27] (also see Fig.

The softening of A2u and Eu was first observed by IR spectroscopy [28] and later confirmed by neutron scattering [29] and hyper-Raman scattering [27, 30]. Its frequency (90cm−1) at room temperature decreases to about 15 cm−1 in low temperature. In spite of the significant softening, it rounds off at about 30 K and never freezes down to zero K. Thus, STO16 is known as quantum paraelectric crystal since quantum fluctuation at low temperatures hinders the freezing of the

<sup>18</sup> (4 / *mcm*) structure. This transition is induced by the freezing of a

' (structural soft mode), which is an alternative rotational

*<sup>p</sup>*. Below *T*0 in the tetragonal phase, however, the

*<sup>p</sup>*. If it propagates along Z-axis, the degeneracy

16 and SrTiO3

<sup>18</sup> as

in the low-temperature properties of SrTiO3. Hereafter, we refer to SrTiO3

STO16 and in the next subsection the soft mode in STO18 will be discussed.
