**6. LO/TO splitting in the ferroelectric Ice-XI**

### **6.1. Structure of Ice-XI**

single *K* →

especially the *K*

The reason why *P*

one.

→

*5.2.3. Why the soft mode is not the slater mode ?*

rhombohedral polar clusters generates *P*

be the increase of the orientational fluctuations of *P*

axes. Spontaneous polarization *P*

macroscopic ferroelectric domains.

*<sup>p</sup>*, not *K* →

22 Ferroelectric Materials – Synthesis and Characterization

*<sup>p</sup>* // Z but *K*

→

samples below *x* =0.33. However, as we have shown in the *K*

transition. In this case, they should have measured the *K*

*<sup>p</sup>* // X, in which the quasi-elastic mode is absent. Then it was

→

→

<sup>→</sup> parallel *not* to the cubic axes but to the tetragonal

<sup>→</sup> in the X-Y plane. Its coupling with the soft

<sup>→</sup> generated by this process would percolate and grow to the

*<sup>p</sup>* // X + Y spectra, since they have noticed from our results [36] that *P*

<sup>→</sup> is not along the cubic axes (as expected from the softening of the Slater

*<sup>p</sup>* // X + Y spectra (Fig. 13(c)),

*<sup>p</sup>* dependence more carefully,

<sup>→</sup> is

<sup>→</sup> // <sup>110</sup> *<sup>c</sup>*. The

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

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

not along the cubic axes which suggests that the soft mode cannot be the Slater mode.

mode) but along the tetragonal axes would be closely related to the existence of the rhombo‐ hedral (parallel to [111]c) nano-scale clusters in STO18 ,which were found in the NMR experi‐ ment by Blinc et al. [40]. They revealed that below 70 K rhombohedral polar clusters are formed in the tetragonal matrix. These clusters subsequently grow in concentration, freeze out, and percolate, leading to an inhomogeneous ferroelectric state below *Tc*. This shows that the elusive ferroelectric transition in STO-18 is indeed connected with local symmetry lowering and implies the existence of an order–disorder component in addition to the displacive soft mode

Referring to this result we have shown in a recent paper [37] that the dipole interaction between

calculation shows that a pair of dipoles located in the same X-Y plane gives the strongest attractive interaction than any other possible pair interaction. The unification of such a pair of

Thus, the dipole interaction model is consistent with various peculiarities observed in the Raman spectra [36], such as the inhomogeneity, the imperfect softening, and the appearance of the relaxational mode near *Tc*. The origin of the relaxational mode observed near *Tc* would

Finally, we admit that the question "why the isotope substitution of O16 by O18 makes the

the rhombohedral polar clusters is the very probable reason for the formation of *P*

mode propagating along the Z axis would suppress the perfect softening.

ferroelectric transition possible in SrTiO3" has not yet been fully solved.

The most common solid phase of water at normal pressure is the hexagonal ice (ice-Ih, D6h 4 , P63 / mmc) which freezes at 273 K. In the Ih phase, however, due to the so-called ice-rules (Bernal–Fowler rules), protons on the hydrogen bonds are randomly distributed and below 100 K protons essentially freeze in place, leaving them disordered [41]. The residual entropy *Sres* =*R*ln(2 / 3)*N* [42] predicts the existence of the proton ordered state phase (known as ice-XI), but it would take a geological scale of time to transform into the ordered phase. In 1972, Kawada succeeded in obtaining the ice XI crystal at normal pressure by doping a small amount of KOH [43]. The calorimetric study [44] confirmed that ice-XI is stable for H2O below *Tc* = 72 K. As shown in Fig. 14, in both ice-Ih and ice-XI, four water molecules are in a unit cell. In ice-XI, the c-component of the dipole moment of each H2O molecule aligns ferroelectrically and the b-component aligns anti-ferroelectrically.

**Figure 14.** (a) Structures of ice-Ih and ice-XI. Four water molecules A, B, C, and D are in a unit cell [45]. (b) Projection to a–b plane. A unit cell is shown by the rhombus (thick green). The dotted hexagon shows the ice-Ih structure. Arrows represent the dipole moment of each molecule.

In contrast to neutron and theoretical studies, spectroscopic studies on ice-XI phase are very few. This is primarily because of the difficulty in growing a single crystal suitable for optical measurements. Recently, we have succeeded in growing a single crystal of ice-XI, which enabled us to measure the polarized Raman spectra [46]. Since the transition from Ih to XI is strongly of first order, no soft mode is expected for this transition. To observe whether the ferroelectric order was realized in our samples or not, the effect of the external electric field was tested on the sharp peak at 610 cm−1 which appears only in ice-XI. Although the effect was not significant compared to the ferroelectric soft mode in SrTiO3 (Fig. 8(c)), the electric field (max 4 kV/cm) applied along the c-axis certainly increased its Raman intensity. 5

Figure 15(a) is the comparison of the wide frequency range spectra between ice-Ih and ice-XI measured with low resolution. Changes in the spectra were clearly recognized in the transla‐ tional (below 350 cm-1), librational (350–1200 cm-1) and the stretching (above 2800 cm-1) mode range. The polarization dependences of these spectra were analyzed satisfactorily. The bending mode range (1300–2700cm-1) was too complicated to assign them properly [46]. Fig.15

Fig.15 (a) Raman spectra in a wide frequency range. Scattering geometry is a(c,∗)b with no polarization analyzer. Ice XI is in red and ice Ih (heated above *Tc* ) is in black. (b) High **Figure 15.** (a) Raman spectra in a wide frequency range. Scattering geometry is a(c, ∗)b with no polarization analyzer. Ice-XI is in red and ice-Ih (heated above *Tc*) is in black. (b) High-resolution Raman spectra in the translational mode range. Scattering geometry is a(cc)b Very strong peak at 237 cm-1 which corresponds to the A1 component of LO is ob‐ served [46].

resolution Raman spectra in the translational mode range. Scattering geometry is a(cc)b. **Very** 

#### **strong peak at 237 cm-1 which corresponds to the A1 component of LO is observed.** [46] **6.2. LO/TO splitting of the translational mode**

LO/TO splitting of polar modes in ice-Ih has been a topic of long-standing discussion and particularly the translational mode spectra below 350 cm−1 were the controversial subjects. In 1991, Klug et al. [47] suggested from the analysis of IR reflection spectra of ice-Ih that a peak near 230 cm−1 is TO mode and the corresponding LO mode would be about 4 cm−1 higher than the TO. However, no direct evidence of the LO/TO splitting has yet been provided. Therefore, measurement of the Raman spectra of ice-XI and comparison with that of ice-Ih are important to solve the long-standing questions.

In contrast to neutron and theoretical studies, spectroscopic studies on ice-XI phase are very few. This is primarily because of the difficulty in growing a single crystal suitable for optical measurements. Recently, we have succeeded in growing a single crystal of ice-XI, which enabled us to measure the polarized Raman spectra [46]. Since the transition from Ih to XI is strongly of first order, no soft mode is expected for this transition. To observe whether the ferroelectric order was realized in our samples or not, the effect of the external electric field was tested on the sharp peak at 610 cm−1 which appears only in ice-XI. Although the effect was not significant compared to the ferroelectric soft mode in SrTiO3 (Fig. 8(c)), the electric field

Figure 15(a) is the comparison of the wide frequency range spectra between ice-Ih and ice-XI measured with low resolution. Changes in the spectra were clearly recognized in the transla‐ tional (below 350 cm-1), librational (350–1200 cm-1) and the stretching (above 2800 cm-1) mode range. The polarization dependences of these spectra were analyzed satisfactorily. The bending mode range (1300–2700cm-1) was too complicated to assign them properly [46].

5

(max 4 kV/cm) applied along the c-axis certainly increased its Raman intensity.

Fig.15 (a) Raman spectra in a wide frequency range. Scattering geometry is a(c,∗)b with no polarization analyzer. Ice XI is in red and ice Ih (heated above *Tc* ) is in black. (b) High resolution Raman spectra in the translational mode range. Scattering geometry is a(cc)b. **Very strong peak at 237 cm-1 which corresponds to the A1 component of LO is observed.** [46]

**6.2. LO/TO splitting of the translational mode**

**Figure 15.** (a) Raman spectra in a wide frequency range. Scattering geometry is a(c, ∗)b with no polarization analyzer. Ice-XI is in red and ice-Ih (heated above *Tc*) is in black. (b) High-resolution Raman spectra in the translational mode range. Scattering geometry is a(cc)b Very strong peak at 237 cm-1 which corresponds to the A1 component of LO is ob‐

LO/TO splitting of polar modes in ice-Ih has been a topic of long-standing discussion and particularly the translational mode spectra below 350 cm−1 were the controversial subjects. In 1991, Klug et al. [47] suggested from the analysis of IR reflection spectra of ice-Ih that a peak near 230 cm−1 is TO mode and the corresponding LO mode would be about 4 cm−1 higher than the TO. However, no direct evidence of the LO/TO splitting has yet been provided. Therefore,

Fig.15

24 Ferroelectric Materials – Synthesis and Characterization

served [46].

In the higher-resolution spectra in ice-XI, clear polarization dependences were observed and successfully assigned most of the translational modes by taking into account the depolarization effect based on the simplified point mass model for each water molecule [46]. 6

Displacements of the water molecules of the related translational modes are shown in Fig. 16. Among the 9 translational modes in ice-Ih, the degenerate E1g and E2g are lifted as E2g → A1 + A2 and E1g → B1 + B2, respectively. Although the displacement patterns are similar for these modes, the frequencies of the modes parallel to ±*b* axes are expected to become higher than those parallel to the *a* axes due to the depolarization field. Therefore, for the case of *K* → *<sup>p</sup>*//*c* , LO/ TO splitting is expected for the modes shown in the squares in Fig. 16. **Fig.16( New)**  

**Figure 16.** Dsplacement patterns and the symmetry of translational modes of ice-XI. The modes parallel to ±*b* axes are LO (237 cm-1) and those parallel to *a* axes are TO mode (231 cm-1).

Fig.16 Dsplacement patterns and the symmetry of translational modes of ice XI. The modes parallel to *b* axes are LO (237cm-1) and those parallel to *a* axes are TO mode (231cm-1). The assignment of LO/TO splitting was confirmed by the *K* → *<sup>p</sup>* dependence of Raman spectra shown in Fig. 17. The most important feature is the fact that the very strong mode at 237 cm−1 is seen only in the geometry b(*c*, *c*)*b* ¯ (Fig. 17(a)) and in a(c, c)b in Fig. 15(b). In all other

geometries and polarizations, no frequency shift is observed. This means that the shift of the peak from 231 cm−1 to 237 cm−1 takes place only when *K* → *p* has a component parallel to the *b*axis. In other words, the modes at 237 cm−1 should be assigned as the LO modes while the mode at 231 cm−1, of which displacements are perpendicular to *b* is the corresponding TO mode. When the effect of the local depolarization electric field is larger than the crystal anisotropy, the LO/TO character dominates the symmetry of modes. Therefore, it is more appropriate to assign the peak at 237 cm−1 as the LO mode with mixed symmetry of A1/B2 and the peak at 231 cm−1 as the TO mode with mixed symmetry of A2/B1. Contribution of B2 component to LO was confirmed in the a(b,c)c spectra shown in Fig.7 of ref.[46]. Fig.17

7

Fig.17 Dependence of the Raman spectra on the phonon propagation direction *K <sup>p</sup>* . (a) **Figure 17.** Dependence of the Raman spectra on the phonon propagation direction *K* → *<sup>p</sup>* . (a) *K* → *<sup>p</sup>*//*b* and (b) *K* → *<sup>p</sup>*//*c* . Very strong peak at 237 cm-1 is observed only in (cc) spectrum in (a) corresponding to the A1 component of LO.

*K* // *b p* and (b) *<sup>K</sup>* // *<sup>c</sup> <sup>p</sup>* **Very strong peak at 237 cm-1 is observed only in (cc) spectrum in (a) corresponding to the A1 component of LO.** The present result is the first experimental confirmation of the LO/TO splitting in ice. The 6±0.5 cm-1 of the LO/TO splitting agrees with 4cm−1 obtained in the IR reflection spectra of ice-Ih by Klug et al*.* [47] and also close to 10 cm−1 of the calculation by Marchi et al*.* [48]. Another feature of the LO (A1/B2) mode at 237 cm−1 is the fact that its intensity is strong only in the spectra with polarization (cc). From the selection rule, the A1 mode is active

also in (aa) and (bb). However, the intensity in (aa) and (bb) is much weaker than that in (cc). This indicates that the Raman polarizability tensor *R*cc =∂*α*cc / ∂*Q*A1 is much larger than *R*bb or *R*aa . Large *R*cc may be attributed to the depolarization field parallel to the c-axis caused by the partial ferroelectric order in ice-XI.

The behavior of the highest translational mode near 326 cm−1 seems to be more complicated. It does not depend on *K* → *<sup>p</sup>* and polarization. Its intensity does not significantly increase by the transformation from ice-Ih to ice-XI. The differences from other translational modes cannot be explained by the simple point mass model. Maybe the modes with the displacement parallel to the c-axis (upper two modes in Fig. 16), are more complicated due to the long-range Coulomb force along the c-axis, which induces the interaction with other degrees of freedom such as the librational motion of water molecules.
