**4. Phase transition of KDP (KH2PO4)**

### **4.1. Raman spectrum of soft mode in KDP**

( )

 w

( )

4

. The "intensity" of the two poles,

=0) as an ideal softening in displacive-

increases, while the other pole gradually vanishes.

<sup>1</sup> ( )2 b

=0, GVWF corresponds to DHO and two poles move parallel to *ω* ' axis and reach

≠0 (GVWF), two poles behave similar to the case (a) but even for *ω*0→0 neither of

0

w gg

the imaginary axis when *ω*<sup>0</sup> =*γ* / 2 and they separately move on the *ω* '' axis, one toward the origin and another toward −*iγ*. In other words, if *ω*<sup>0</sup> <*γ* / 2, DHO cannot be discrimi‐

**d.** If *γ* <*γ* ′ , behaviors of two poles are similar to the case (b) except that the "stronger" pole

Behavior of various *χ*(*ω*) can be understood in terms of the three parameters (*ω*<sup>0</sup> , *γ*, *<sup>γ</sup>* ′

GVWF. Figure 4 represents schematically the paths of the pole on approaching *Tc*. There are many paths leading to the divergence of *χ*(*ω*). The stability limit leading to the phase transition takes place when one of the poles moves toward the origin of complex **ω** plane

type transitions. Ideal relaxation in order–disorder-type transitions (DB) corresponds to

The behavior of the poles in the complex **ω** plane is shown in Fig. 3 for the case of the softening

4, a <sup>2</sup>

+ -¢ ¢ <sup>=</sup> +-+ - ¢ ¢ (8)

(9)

Fig.3 (New)

fixed

(10)

) of

2 2

 g g w w

*i*

In general, this form can be described by the two poles Ω1,2 in the complex **ω** plane as follows:

( ) 1 2 1 2

2 2

ë û - - ¢

w

ì ü ï ï P P = + í ý ï ï W- W- î þ

2 0

w gg gw

0 ( ) (0) *<sup>i</sup>*

0

( ) <sup>2</sup> c

( ) ( )

w gg

1,2 <sup>0</sup> <sup>2</sup> <sup>2</sup>

c w

1,2 0

*i*

g g

*i*

(*ω*0→0) with constant *<sup>γ</sup>* and *<sup>γ</sup>* ′

nated from DB.

**a.** If *γ* > >*γ* ′

**b.** If *γ* >*γ* ′

−*iγ* ′

(*γ* axis in Fig.4).

g w gg g

2

P =- ± + - ¢ ¢¢ é ù

.

them can reach the origin. They stop at −*i γ* and −*iγ* ′

**c.** If *γ* =*γ* ' (VWF), two poles reach *ω* '' axis only when *ω*<sup>0</sup> =0 .

The pole of DHO moves in the DHO-plane (*ω*<sup>0</sup> , *γ*, *<sup>γ</sup>* ′

lies further from the origin than the weaker pole −*iγ*.

*<sup>Π</sup>*1,2, also changes; a mode toward −*i<sup>γ</sup>* ′

<sup>+</sup> ¢ W =- ± - - ¢

where

w gg

cw

8 Ferroelectric Materials – Synthesis and Characterization

 c

> The first experimental study of the soft mode was done by Kaminov and Damen in 1968 on the ferroelectric phase transition of KDP (KH2PO4) [10]. The structure changes from tetragonal D2d (*I* 4 ¯2*d*) to orhorhombic C2v (*Fdd*2). KDP is a unique ferroelectric crystal because of its large isotope effect on *Tc*, being 122K and 212K for KDP and DKDP (KD2PO4), respectively. Its origin was first attributed by Blinc et al. from the IR spectra to the tunneling of a proton between two PO4 ions [11]. The soft mode has B2 symmetry, which is Raman-active in the (xy) polarized spectrum. Contrary to most high-frequency modes, the lowest strong mode is a quasi-elastic mode that shows no peak down to zero frequency. Kaminov et al. analyzed the spectra by DHO and concluded that the characteristic frequency *ω*<sup>0</sup> <sup>2</sup> goes to zero at *Tc* and suggested that

> the results are consistent with either the soft mode model [12] or the collective-tunneling-mode model [13, 14]. Their conclusions, however, were obtained simply assuming that the damping

Fig.4

)const.,0( 0 

, VWF )0,( <sup>0</sup>

3

Fig.4 Behavior of the generalized susceptibility GVWF in ),,( <sup>0</sup> space. Various paths of the pole to the origin of the complex -plane are shown for DHO **Figure 4.** Behavior of the generalized susceptibility GVWF in (*ω*<sup>0</sup> , *<sup>γ</sup>*, *<sup>γ</sup>* ′ ) space. Various paths of the pole to the ori‐ gin of the complex *ω* plane are shown for DHO (*ω*0→0, *<sup>γ</sup>* =const.), VWF (*ω*<sup>0</sup> , *<sup>γ</sup>* <sup>=</sup>*<sup>γ</sup>* ′ →0), GVWF (*ω*<sup>0</sup> , *<sup>γ</sup>* ′ →0), and DB (*γ* '→0') cases.

, GVWF )0,( <sup>0</sup>

constant *γ* is a constant. Quasi-elastic spectra are often observed in other crystals. (We shall call it simply as an overdamped mode unless it is necessary to discriminate them explicitly.) As mentioned in the previous section, it is not easy to analyze such a spectrum. It might be an overdamped soft mode (DHO) or a Debye-type relaxational mode (DB). Small white circles mean the position of the pole for 0 <sup>0</sup> and the big black circles denote the DB relaxation mode.

and DB )0( ''

cases.

Temperature dependence of Raman spectra of overdamped soft modes is often shown in the *log scale* instead of the *linear scale* because of the significant increase of intensity as shown in Fig. 5(a) in linear scale. In the pioneering data by Kaminov et al. [10] and also in the papers by Tominaga et al. [16] for KDP and by Takesada et al. [17] for SrTiO3 (see section 5), spectra are shown in log scale. We note, however, that this might mislead its interpretation. In the log scale spectra, the product of the Bose factor *n*(*ω*) and Im*χQ* is replaced by the sum of them. Then the low intensity part (i.e., the tail of the overdamped spectra) is apparently strengthened since the Bose factor is a monotonically increasing function toward infinity (*n* (*ω* =0)=*∞* ), while Im *χ<sup>Q</sup>* is zero at *ω* =0 (Im*χQ*(*ω* =0)=0). As a consequence, the line shapes look quite different from that in the linear scale. For example, as in Fig. 5(b), the curvature of spectra changes from convex (*T* < <*Tc*) to concave (*T* ≈*Tc*). It looks as if there exist two components in the overdamped spectra. Figure 5(c) is the Im*χQ*(*ω*, *T* ) spectra corresponding to the linear scale spectra by removing the Bose factor contribution [15]. Its temperature dependence is very smooth toward

**Figure 5.** Temperature dependence of Raman spectra of KDP. (a) *I*(*ω*) in linear scale [15]. (b) Similar spectra *I*(*ω*) in log scale [16]. (c) Spectra of Im *χ<sup>Q</sup>* = *I*(*ω*) / *n* + 1 in linear scale. (See the text for the difference between these spectra.)

*Tc* without any qualitative change of the line shape. (The apparent peak in Fig. 5(c) is simply due to the fact Im*χQ*(*ω* =0)=0.)

### **4.2. Displacive or order–disorder ?**

constant *γ* is a constant. Quasi-elastic spectra are often observed in other crystals. (We shall call it simply as an overdamped mode unless it is necessary to discriminate them explicitly.) As mentioned in the previous section, it is not easy to analyze such a spectrum. It might be an

, GVWF )0,( <sup>0</sup>

and DB )0( ''

space. Various

) space. Various paths of the pole to the ori‐

cases.

→0), GVWF


<sup>0</sup> and the big black circles

3

Temperature dependence of Raman spectra of overdamped soft modes is often shown in the *log scale* instead of the *linear scale* because of the significant increase of intensity as shown in Fig. 5(a) in linear scale. In the pioneering data by Kaminov et al. [10] and also in the papers by Tominaga et al. [16] for KDP and by Takesada et al. [17] for SrTiO3 (see section 5), spectra are shown in log scale. We note, however, that this might mislead its interpretation. In the log scale spectra, the product of the Bose factor *n*(*ω*) and Im*χQ* is replaced by the sum of them. Then the low intensity part (i.e., the tail of the overdamped spectra) is apparently strengthened since the Bose factor is a monotonically increasing function toward infinity (*n* (*ω* =0)=*∞* ), while Im *χ<sup>Q</sup>* is zero at *ω* =0 (Im*χQ*(*ω* =0)=0). As a consequence, the line shapes look quite different from that in the linear scale. For example, as in Fig. 5(b), the curvature of spectra changes from convex (*T* < <*Tc*) to concave (*T* ≈*Tc*). It looks as if there exist two components in the overdamped spectra. Figure 5(c) is the Im*χQ*(*ω*, *T* ) spectra corresponding to the linear scale spectra by removing the Bose factor contribution [15]. Its temperature dependence is very smooth toward

overdamped soft mode (DHO) or a Debye-type relaxational mode (DB).

Fig.4 Behavior of the generalized susceptibility GVWF in ),,( <sup>0</sup>

gin of the complex *ω* plane are shown for DHO (*ω*0→0, *<sup>γ</sup>* =const.), VWF (*ω*<sup>0</sup> , *<sup>γ</sup>* <sup>=</sup>*<sup>γ</sup>* ′

paths of the pole to the origin of the complex

**Figure 4.** Behavior of the generalized susceptibility GVWF in (*ω*<sup>0</sup> , *<sup>γ</sup>*, *<sup>γ</sup>* ′

Small white circles mean the position of the pole for 0

, VWF )0,( <sup>0</sup>

denote the DB relaxation mode.

→0), and DB (*γ* '→0') cases.

)const.,0( 0 

(*ω*<sup>0</sup> , *<sup>γ</sup>* ′

Fig.4

10 Ferroelectric Materials – Synthesis and Characterization

Later studies by various groups however, revealed that the interpretation on the origin of the phase transition by Kaminov et al. [10] is not conclusive even from the Raman spectroscopic point of view. It still remains unclear whether it is a displacive type or an order–disorder type.

Difficulty of the interpretation of Raman spectra comes from the following reasons :


**iv.** Tominaga et al. insist that the phase transition of KDP is the order–disorder type [20]. The main evidence for their interpretation is based on the observation of extra Raman peaks above *Tc* in the 500–1000 cm-1 range, which should not be observed if the PO4 units are the regular tetrahedra with S4 symmetry. Thus, the existence of those peaks would indicate the PO4 tetragonal unit is *not* regular but already distorted to C2 symmetry above *Tc*. If it were true, the two different distortions of PO4 (Fig. 6(b)) would cause the order–disorder transition. However, the observation of the extra peaks does not necessarily mean that PO4 is statically distorted, because two protons vibrate coherently with PO4 tetrahedron so that vibrational modes of the H2-PO4 system have C2 symmetry and the extra peaks would be observed in Raman spectrum even if PO4 is regular tetrahedra [21].

6

**v.** The role of protons in this transition is not clear in the order–disorder model. The coupled proton–PO4 model proposed by Kobayashi [22] shown in Fig. 6(a) seems to be more realistic. As for the large difference of *Tc* between KDP and DKDP, no direct evidence for the proton tunneling as the origin of the isotope effect has so far been obtained. It was also revealed that there exists some difference of structure between KDP and DKDP. Details were discussed in [21]. Thus, it still remains unclear whether it is a displacive type or an order–disorder type. Fig. 6

**Figure 6.** Two models for the ferroelectric transition in KDP. (a) Soft mode pattern of a proton–PO4 coupled system proposed by Kobayashi [22]. (b) The order–disorder model of a deformed PO4 [20].

order–disorder model of a deformed PO4 [20].

Fig. 6 Two models for the ferroelectric transition in KDP. (a) Soft mode

pattern of a proton–PO4 coupled system proposed by Kobayashi [22]. (b) The

### **4.3. Influence of the ferroelectric domain**

**iv.** Tominaga et al. insist that the phase transition of KDP is the order–disorder type [20].

**v.** The role of protons in this transition is not clear in the order–disorder model. The

coupled proton–PO4 model proposed by Kobayashi [22] shown in Fig. 6(a) seems to be more realistic. As for the large difference of *Tc* between KDP and DKDP, no direct evidence for the proton tunneling as the origin of the isotope effect has so far been obtained. It was also revealed that there exists some difference of structure between KDP and DKDP. Details were discussed in [21]. Thus, it still remains unclear whether

Fig. 6 Two models for the ferroelectric transition in KDP. (a) Soft mode

pattern of a proton–PO4 coupled system proposed by Kobayashi [22]. (b) The

**Figure 6.** Two models for the ferroelectric transition in KDP. (a) Soft mode pattern of a proton–PO4 coupled system

order–disorder model of a deformed PO4 [20].

proposed by Kobayashi [22]. (b) The order–disorder model of a deformed PO4 [20].

6

even if PO4 is regular tetrahedra [21].

12 Ferroelectric Materials – Synthesis and Characterization

Fig. 6

it is a displacive type or an order–disorder type.

The main evidence for their interpretation is based on the observation of extra Raman peaks above *Tc* in the 500–1000 cm-1 range, which should not be observed if the PO4 units are the regular tetrahedra with S4 symmetry. Thus, the existence of those peaks would indicate the PO4 tetragonal unit is *not* regular but already distorted to C2 symmetry above *Tc*. If it were true, the two different distortions of PO4 (Fig. 6(b)) would cause the order–disorder transition. However, the observation of the extra peaks does not necessarily mean that PO4 is statically distorted, because two protons vibrate coherently with PO4 tetrahedron so that vibrational modes of the H2-PO4 system have C2 symmetry and the extra peaks would be observed in Raman spectrum

Raman spectrum measures the macroscopic region of the sample. Therefore, in the Raman spectra of ferroelectric crystals the appearance of the domains with various scales and orientations in the sample should not be ignored. For example, a strange oscillatory behavior of the Raman intensity was observed in KDP. Raman intensity of A1 mode, measured with the scattering geometry shown in Fig. 7(a), change oscillatory with temperature in the range 0<*Tc* −*T* <15*K* [23]. This behavior can be explained in terms of the interference of light reflected by the domain walls. Spontaneous polarizations ±*P* <sup>→</sup> in KDP are parallel to the c-axis and the thickness of 180 domain grows on cooling. The oscillatory behavior of Raman (Fig. 7(c)) is due to the periodical change of the polarization of light caused by the reflection from the domain walls and the birefringence between the neighbor domains as was confirmed by the similar oscillatory behavior the reflected light (Fig. 7(d)) [23]. 7

The influences of domains in Raman spectra are also important in the case of more complicated domain structures in the case of STO18, discussed in section 5.2. Fig. 7

**Figure 7.** Influence of domains on Raman spectrum. (a) Scattering geometry. (b) Laser path and the reflection from domain walls. (c) Oscillatory change of Raman intensity with temperatture. (e) Intensity of the reflected light [23].

Fig. 7 Influence of domains on Raman spectrum. (a) Scattering geometry. (b) Laser path and the reflection from domain walls. (c) Oscillatory change of Raman

intensity with temperatture. (e) Intensity of the reflected light [23].
