*5.2.1. Propagation dependence of the soft mode in STO18*

STO18 undergoes the structural phase transition from cubic to tetragonal at about *T*<sup>0</sup> =108 K, which is slightly higher than *T*<sup>0</sup> =105 K of STO16. On further cooling at *Tc* =24 K, however, STO18 transforms to the ferroelectric phase with orthorhombic structure (C2v). [25] Ferroelec‐ tric properties of highly substituted STO18 (higher than 33%) were confirmed by the enormous increase of the dielectric constant and the appearance of the hysteresis curve.

All the samples used in many laboratories were provided from Itoh's laboratory. The samples are thin plates of size 0.3 x 2 (or 3) x 7 mm3 with the widest plane [110]c, and the longest and the shortest edges are parallel to [001]c and [110]c, respectively. In the early stage, most samples used were not a single tetragonal domain but included many domains with different Zdirections. Thus, the reports measured with multi-domain samples were controversial and could not give consistent results. Fig.9

4

Figure 9 is the summary of the temperature dependence of the soft mode frequency in STO18 measured by various experiments. The data are from samples with a single tetragonal domain except for the neutron diffraction.

Fig.9 Temperature dependence of the ferroelectric soft mode of SrTiO3. *Tc* of STO18 are slightly different for various experiments but here it is assumed to be about 24K. Lines are guide for eyes. ○and □are neutron diffraction of STO18.[34] ×are hyper Raman of STO16 [35] and ▲(red) are hyper Raman of STO18 above *Tc* **Figure 9.** Temperature dependence of the ferroelectric soft mode of SrTiO3. *Tc* of STO18 is slightly different for various experiments but here it is assumed to be about 24 K. The solid lines are guide for eyes. ○ and □ are neutron diffraction of STO18 [34]. **<sup>×</sup>** are hyper Raman of STO16 [35] and ▲ are hyper Raman of STO18 above *Tc* measured under the elec‐ tric field [27]. ● are Raman scattering below *Tc* in STO18-99 [36]. Below *Tc*, only the Raman data clearly show the soft mode behavior in STO18.

measured under the electric field.[27] ● are Raman scattering below *Tc* in STO18-99.[36] Below *Tc* , Only the Raman data clearly show the soft mode behavior in STO18. Hyper Raman data under DC-electric field from STO18-96 show the softening similar to STO16 above *Tc*, but they do not give reliable data below *Tc*.[27] Neutron diffraction data hardly detect the influence of the phase transition [34]. Thus, the space group of STO18 below *Tc* has not yet been determined definitely. As we shall show later, it is probably because of the inhomogeneity of STO18 due to the existence of the paraelectric phase as a matrix of ferroelectric domains. Below *Tc*, only the Raman data succeeded in getting reliable soft mode frequencies [36]. Note that all these data show that the freezing of the soft mode (perfect softening) does *not* occur.

Later, highly substituted single-domain samples, STO18-99, became available. The results discussed below are the data obtained from such samples. For the correct assignment and interpretation, we found that the following features should be carefully taken into account [36]:

All the samples used in many laboratories were provided from Itoh's laboratory. The samples are thin plates of size 0.3 x 2 (or 3) x 7 mm3 with the widest plane [110]c, and the longest and the shortest edges are parallel to [001]c and [110]c, respectively. In the early stage, most samples used were not a single tetragonal domain but included many domains with different Zdirections. Thus, the reports measured with multi-domain samples were controversial and

4

Figure 9 is the summary of the temperature dependence of the soft mode frequency in STO18 measured by various experiments. The data are from samples with a single tetragonal domain

Fig.9 Temperature dependence of the ferroelectric soft mode of SrTiO3. *Tc* of STO18 are slightly different for various experiments but here it is assumed to be about 24K. Lines are guide for eyes. ○and □are neutron diffraction of STO18.[34] ×are hyper Raman of STO16 [35] and ▲(red) are hyper Raman of STO18 above *Tc* measured under the electric field.[27] ● are Raman scattering below *Tc* in STO18-99.[36] Below *Tc* , Only the Raman data clearly show the soft mode behavior in

**Figure 9.** Temperature dependence of the ferroelectric soft mode of SrTiO3. *Tc* of STO18 is slightly different for various experiments but here it is assumed to be about 24 K. The solid lines are guide for eyes. ○ and □ are neutron diffraction of STO18 [34]. **<sup>×</sup>** are hyper Raman of STO16 [35] and ▲ are hyper Raman of STO18 above *Tc* measured under the elec‐ tric field [27]. ● are Raman scattering below *Tc* in STO18-99 [36]. Below *Tc*, only the Raman data clearly show the soft

Hyper Raman data under DC-electric field from STO18-96 show the softening similar to STO16 above *Tc*, but they do not give reliable data below *Tc*.[27] Neutron diffraction data hardly detect the influence of the phase transition [34]. Thus, the space group of STO18 below *Tc* has not yet been determined definitely. As we shall show later, it is probably because of the inhomogeneity of STO18 due to the existence of the paraelectric phase as a matrix of ferroelectric domains. Below *Tc*, only the Raman data succeeded in getting reliable soft mode frequencies [36]. Note that all these data show that the freezing of the soft mode (perfect softening) does *not* occur.

could not give consistent results.

16 Ferroelectric Materials – Synthesis and Characterization

Fig.9

STO18.

mode behavior in STO18.

except for the neutron diffraction.


In the ferroelectric phase of STO18, five modes are observed in the low-frequency part below 50 cm-1 as one can see in Fig.10 and 11. Three of them are from the structural soft modes, as A1 (from A1g) at 44 cm-1 and A2 and B1 (from Eg) at 11 and 17.5 cm-1, respectively. Since these modes are nonpolar, the line width is narrow and does not depend on *K* → *<sup>p</sup>* (Fig.11(b) and (c)). In contrast, the behavior of the ferroelectric soft modes, TO(A1) and TO(B2), very broad and sensitive to *K* → *<sup>p</sup>* (Fig.11). It is also sensitive to the scattering geometries. For example, the polarization dependence of the spectra with *K* → *<sup>p</sup>* // X at a temperature well below *Tc* (at about 7K) is shown for three different geometries in Fig.10. Note that in the geometry (c) of Fig.10, the VV(Z, Z) spectrum is quite different from other geometries, since even for the same *K* → *p*, the selection rules are different for different geometries.

Spectra for *K* → *<sup>p</sup>* // Y (not shown in Fig.11) were found to be exactly the same as *K* → *<sup>p</sup>* // X, sug‐ gesting that there are two kinds of small domains with *P* <sup>→</sup> // X and *<sup>P</sup>* <sup>→</sup> // Y equally distributed in the sample, as we will discuss later.

In Fig. 11, the spectra for *K* → *<sup>p</sup>* // Z and *K* → *<sup>p</sup>* // X look similar except that there are two broad peaks in *K* → *<sup>p</sup>* // Z spectra while in *K* → *<sup>p</sup>* // X there is only one. Moreover, their temperature dependences are essentially different as we shall show in the next section.

It should be emphasized that the *K* → *<sup>p</sup>* // X + Y spectra (Fig.11(c)) are qualitatively different from other directions in the following two points: (1) The strong and broad peak at 20 cm-1 in the VV spectra of *K* → *<sup>p</sup>* // Z and *K* → *<sup>p</sup>* // X appears in *K* → *<sup>p</sup>* // X + Y at a lower frequency 17 cm-1. This means that the depolarization field in *K* → *<sup>p</sup>* // X + Y is weaker than that in *K* → *<sup>p</sup>* // Z or *K* → *<sup>p</sup>* // X, so that the mode at 17 cm-1 is not a pure TO(A1) mode. (2) In *K* → *<sup>p</sup>* // X + Y, there appear spurious modes (arrows in Fig. 11(c)) that are unable to assign to any mode in STO18. As we shall discuss later, these modes are originated from the matrix of the paraelectric phase.

Our final assignment of displacement and the symmetry of the ferroelectric soft modes are illustrated in Fig. 12(a) for temperature above *Tc* and in Fig. 12(b) for well below *Tc*. All the features of the observed spectra, including the subtle differences in the polarized spectra and the expected selection rules, were consistently explained *only* if the spontaneous polarization *P* <sup>→</sup> is either parallel to the tetragonal axes X or Y. *<sup>P</sup>* <sup>→</sup> cannot be along one of the cubic directions

**Figure 10.** Three different scattering geometries for the same propagation direction, *K* → *<sup>p</sup>* // X. Broad mode at 20 cm-1 is TO(A1) and the shoulder in HV and VH of (b) (shown by arrows) is TO(B2) [36] In the geometry (c), the VV(Z, Z) spec‐ trum is quite different from other geometries.

as one might expect for the Slater mode [26], since it is neither parallel to X ± Y (=[100]c or [010]c ) nor *Z* (=[001]c). If *P* <sup>→</sup> were parallel to one of the cubic axes, observed spectra never satisfy any selection rules. Reason for this unexpected result will be given later.

As one can see in Fig. 12(b), the doubly degenerate TO(Eu) splits into TO(A1) + TO(B2) when it propagates along Z, but when it propagates in the X-Y plane, TO(A1) changes to TO(B1) , of which displacement is perpendicular to *P* <sup>→</sup> . Since the displacement of TO(A1) is parallel to *<sup>P</sup>* → , it is reasonable that the frequency of TO(A1) (20 cm-1) is higher than TO(B1) (17.5 cm-1) and its intensity is stronger. Similar to the very strong external DC-field effect observed in STO16 (Fig. 8(c)), these differences are due to the effect of the depolarization field produced by *P* → .

### *5.2.2. Temperature dependence of soft mode in STO18*

Figure 13 is the temperature dependence of Raman spectra for different *K* → *<sup>p</sup>* directions.

In Fig. 13(a) only the broad and underdamped mode TO(A1) softens and becomes weaker on approaching *Tc* as it should be. In contrast, Fig. 13(b) shows that two modes, TO(A1) and

**Figure 11.** Polarized spectra at the lowest temperature for different propagation vector *K* → *<sup>p</sup>*. (a) *K* → *<sup>p</sup>* // Z, (b) *K* → *<sup>p</sup>* // X, and (c) *K* → *<sup>p</sup>* // X + Y. Assignment of the peaks are given in red letters. Arrows in (c) are the spurious modes due to the paraelectric structure existing below *Tc* as the matrix of ferroelectric domains.

TO(B2) clearly soften. It is clear that frequencies of the soft modes never go down to zero. Instead in the *K* → *<sup>p</sup>* // Z spectra, in addition to the soft modes, a quasi-elastic mode appears near *Tc* (arrows in Fig. 13(b)). It is observed not only below *Tc* but also above *Tc*. Moreover, it is observed only when *K* → *<sup>p</sup>* is out of the X-Y plane such as for *K* → *<sup>p</sup>* // Z and *K* → *<sup>p</sup>* // Z + Y [36, 37].

The absence of the quasi-elastic mode in *K* → *<sup>p</sup>* // X (Fig. 13(b)) implies that it is *not* an overdamped soft mode as in the KDP case (section 4.1) but is a relaxational mode originated from the large fluctuations of *P* <sup>→</sup> near *Tc* in the X-Y plane.

as one might expect for the Slater mode [26], since it is neither parallel to

TO(A1) and the shoulder in HV and VH of (b) (shown by arrows) is TO(B2) [36] In the geometry (c), the VV(Z, Z) spec‐

spectra never satisfy any selection rules. Reason for this unexpected result will be given later.

As one can see in Fig. 12(b), the doubly degenerate TO(Eu) splits into TO(A1) + TO(B2) when it propagates along Z, but when it propagates in the X-Y plane, TO(A1) changes to TO(B1) , of

it is reasonable that the frequency of TO(A1) (20 cm-1) is higher than TO(B1) (17.5 cm-1) and its intensity is stronger. Similar to the very strong external DC-field effect observed in STO16 (Fig.

In Fig. 13(a) only the broad and underdamped mode TO(A1) softens and becomes weaker on approaching *Tc* as it should be. In contrast, Fig. 13(b) shows that two modes, TO(A1) and

8(c)), these differences are due to the effect of the depolarization field produced by *P*

Figure 13 is the temperature dependence of Raman spectra for different *K*

**Figure 10.** Three different scattering geometries for the same propagation direction, *K*

<sup>→</sup> were parallel to one of the cubic axes, observed

→

<sup>→</sup> . Since the displacement of TO(A1) is parallel to *<sup>P</sup>*

→

→ ,

→ .

*<sup>p</sup>* directions.

*<sup>p</sup>* // X. Broad mode at 20 cm-1 is

X ± Y (=[100]c or [010]c ) nor *Z* (=[001]c). If *P*

trum is quite different from other geometries.

18 Ferroelectric Materials – Synthesis and Characterization

which displacement is perpendicular to *P*

*5.2.2. Temperature dependence of soft mode in STO18*

Another peculiar nature of the ferroelecticity of STO18 is the fact that even for a sample with 99% substitution, STO18 is *not* homogeneous contrary to the report by Taniguchi et al. [39], because the ferroelectric domains with *P* <sup>→</sup> //*X* and *<sup>P</sup>* <sup>→</sup> //*<sup>Y</sup>* coexist in the matrix of the paraelectric structure. The appearance of the spurious peaks in the *K* → *<sup>p</sup>* // X + Y spectra (Fig. 13(c)), which is FMR (Ferroelectric Micro-Domain) intrinsic to STO16, verifies the inhomogeneity of STO18.

The presence of the quasi-elastic mode in *K* → *<sup>p</sup>* // Z and its absence in *K* → *<sup>p</sup>* // X in STO18 were also reported by the independent two reports by Takesada et al. [17, 38] and Taniguchi et al. [39], respectively. As mentioned above, however, their conclusions are inconsistent with our results. Therefore, let us discuss briefly their results from our point of view.

**a.** Perfect softening of the Slater mode in STO18 was claimed in ref. [17] using a highresolution spectrometer. Their results, however, are based on *only* the spectra with *K* → *<sup>p</sup>* // Z. Above *Tc*, they assign a *very weak bump* below 5 cm-1 sitting on the strong quasielastic (our relaxational) mode as the underdamped soft mode TO(Eu). But below *Tc*, no such a narrow mode is observed and the strong quasi-elastic (our relaxational) mode is

**Figure 12.** Ferroelectric soft-mode frequency surfaces for various directions of phonon propagation vector *K* → *p*. <sup>X</sup> =[110] <sup>c</sup> and Y = 11 ¯0 <sup>c</sup> are the crystal axes of the tetragonal structure. (a) In the paraelectric phase near *Tc*. (b) In the ferroelectric phase well below *Tc*. Figure 12(b) is drawn for the case of *P* <sup>→</sup> // Y. Since in a sample of STO18, do‐ mains with *P* <sup>→</sup> // X and *<sup>P</sup>* <sup>→</sup> // Y coexist (in the matrix of paraelectric phase), observed spectra are the sum of the similar figure with the X and Y axes exchanged. Red arrows are the displacements in the X-Y plane and blue arrows are those out of the X-Y plane.

suddenly assigned as the overdamped soft modes TO(A1) and TO(B2) . (We note that the lowest mode is *not* A1 *but* B2 contrary to the assignment by Takesada et al. [17].) From the computer fitting of the overdamped spectra below *Tc*, they obtain the extremely steep dropping of the frequency of TO(B2) down to zero. However, as mentioned in the case of KDP, such analysis is very ambiguous and the sudden qualitative change of the soft mode at *Tc* is unnatural. From these results they concluded that STO18 is an ideal displacivetype ferroelectrics induced by the Slater-type soft mode. If such a perfect softening took place at *Tc* it would be much more clearly observed in the *K* → *<sup>p</sup>* // X spectra, since as shown

**Figure 13.** Temperature dependence of Raman spectra for different propagation directions: (a) *K* → *<sup>p</sup>* // X, (b) *K* → *<sup>p</sup>* //*Z*, (c) *K* → *<sup>p</sup>* // X + Y. The spectra for *K* → *<sup>p</sup>* // Y was identical to those for *K* → *<sup>p</sup>* // X. Note that in (b), a relaxational mode (blue lines) appears near *Tc* and in (c), a spurious mode (the arrows) is seen at about 5 cm-1 and no *pure* TO(A1) is observed.

suddenly assigned as the overdamped soft modes TO(A1) and TO(B2) . (We note that the lowest mode is *not* A1 *but* B2 contrary to the assignment by Takesada et al. [17].) From the computer fitting of the overdamped spectra below *Tc*, they obtain the extremely steep dropping of the frequency of TO(B2) down to zero. However, as mentioned in the case of KDP, such analysis is very ambiguous and the sudden qualitative change of the soft mode at *Tc* is unnatural. From these results they concluded that STO18 is an ideal displacivetype ferroelectrics induced by the Slater-type soft mode. If such a perfect softening took

¯0 <sup>c</sup> are the crystal axes of the tetragonal structure. (a) In the paraelectric phase near *Tc*. (b) In

<sup>→</sup> // Y coexist (in the matrix of paraelectric phase), observed spectra are the sum of the similar

**Figure 12.** Ferroelectric soft-mode frequency surfaces for various directions of phonon propagation vector *K*

figure with the X and Y axes exchanged. Red arrows are the displacements in the X-Y plane and blue arrows are those

→

*<sup>p</sup>* // X spectra, since as shown

<sup>→</sup> // Y. Since in a sample of STO18, do‐

→ *p*.

place at *Tc* it would be much more clearly observed in the *K*

the ferroelectric phase well below *Tc*. Figure 12(b) is drawn for the case of *P*

<sup>X</sup> =[110] <sup>c</sup>

mains with *P*

out of the X-Y plane.

and Y = 11

20 Ferroelectric Materials – Synthesis and Characterization

<sup>→</sup> // X and *<sup>P</sup>*

in Fig. 13(a), in the *K* → *<sup>p</sup>* // X spectra no quasi-elastic mode appears near *Tc*. Furthermore, if the perfect softening were related to the Slater-type mode, the direction of *P* <sup>→</sup> should be parallel to cubic axis, which contradicts our results. Therefore, the essential difference between their interpretation and ours cannot be attributed to the difference of the resolution of the spectra.

**b.** Another essentially incorrect result was reported by Taniguchi et al. related to the homogeneity of STO18 [39]. They measured O<sup>18</sup> concentration dependence of Raman spectra for SrTi(O18x O161-x)3. In this paper, they measured the spectra again *only* with a single *K* → *<sup>p</sup>*, not *K* → *<sup>p</sup>* // Z but *K* → *<sup>p</sup>* // X, in which the quasi-elastic mode is absent. Then it was concluded that the homogeneous ferroelectric phase changes into theferroelectricparaelectric phase coexistence state as thesystem approaches quantum critical point *x* =0.33 and that highly substituted STO18 undergoes the ideal soft-mode-type quantum phase transition. As the evidence of the criticality, they claim that the mode at 15 cm-1 which is the Eg mode intrinsic to the paraelectric phase appears only in the low substitution samples below *x* =0.33. However, as we have shown in the *K* → *<sup>p</sup>* // X + Y spectra (Fig. 13(c)), in addition to the FMR at 5 cm-1, the Eg mode at 15 cm-1 (in our case 14.5 cm-1) is certainly observed even in a very highly substituted sample with *x* =0.99. This is a clear evidence of the existence of the paraelectric phase as the matrix of the ferroelectric domains. In other words, the *inhomogeneity is the intrinsic property* of STO18. Therefore, the transition of STO18 with high x is *not* the ideal (homogeneous) soft-mode-type quantum phase transition. In this case, they should have measured the *K* → *<sup>p</sup>* dependence more carefully, especially the *K* → *<sup>p</sup>* // X + Y spectra, since they have noticed from our results [36] that *P* <sup>→</sup> is not along the cubic axes which suggests that the soft mode cannot be the Slater mode.
