Ordering of polar state and critical phenomena in the quantum relaxor potassium lithium tantalate K 1 x Li x TaO 3. Hiroko YOKOTA

Transcription

1 Ordering of polar state and critical phenomena in the quantum relaxor potassium lithium tantalate K 1 x Li x TaO 3 K 1 x Li x TaO 3 Hiroko YOKOTA February 2008

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3 Contents 1 BACKGROUND AND PURPOSE OF THE RESEARCH Introduction Two types of phase transition mechanism Quantum paraelectricity Relaxor Quantum paraelectric KTaO Effects of Li ion doping into quantum paraelectric KTO Purpose of the present research HISTORY DEPENDENCE OF THE ORDER PARAMETER IN K 1 x Li x TaO Introduction Experimental History dependence of SH intensity Point group at low temperature Experimental results History dependence of SH intensity Point group at low temperature Analyses Conclusions EXTREMELY SLOW KINETICS OBSERVED IN K 1 x Li x TaO Introduction Experimental Experimental results Analyses Bi-exponential model Avrami theory Conclusions

4 4 CONTENTS 4 PRE-TRANTISITIONAL REGION ABOVE THE POLAR PHASE OF K 1 x Li x TaO Introduction Experimental X-ray diffraction measurements Neutron elastic scattering experiments Neutron diffuse scattering experiments Experimental Results X-ray diffraction measurements Neutron elastic scattering Neutron diffuse scattering Analyses The intermediate state of KLT Diffuse scattering from PNRs Polar state of KLT below T p Conclusions Li CONCENTRATION DEPENDENCE OF DIELECTRIC RESPONSES IN K 1 x Li x TaO Introduction Experimental Single crystal growth and determination of Li concentration Dielectric measurements Experimental Results Analyses Cole-Cole plot analysis Vogel-Fulcher analysis Conclusions QUANTUM PARAELECTRIC / RELAXOR-FERROELECTRIC CROSSOVER REGION OF K 1 x Li x TaO Introduction Experimental Neutron inelastic scatterings Dielectric measurements Experimental results Temperature dependences of TO and TA phonons Temperature dependence of dielectric constant Analyses The Barrett formula for dielectric constant and TO phonon 87

5 CONTENTS Experimental determination of the Barrett parameters Two-state model Conclusions CRITICAL PHENOMENON OF K 1 x Li x TaO Introduction Experimental Experimental results & Analyses Determination of critical Li concentration by SHG microscope Electric field dependence of dielectric constant Conclusions SUMMARY 113 Acknowledgments 119 Publication & Presentation Lists 121

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7 Chapter 1 BACKGROUND AND PURPOSE OF THE RESEARCH This chapter first describes the key concepts in ferroelectricity which they relate to the present study; in particular, nature of quantum paraelectrics and relaxor is discussed. Next, previous research on KTaO 3 doped with Li is summarized. Lastly, the purpose of the present research is clarified, thus emphasizing what is still lacking in the understanding of this system. 1.1 Introduction Materials are defined as ferroelectrics when they possess a spontaneous polarization P s in the absence of an external electric field and the direction of P s can be reversed by an electric field. Some ferroelectrics show a huge dielectric constant of several tens of thousands with a low dielectric loss of less than 10 %, a high spontaneous polarization of nearly 100 µ C/cm 2, a large piezo and electro-mechanical coupling coefficient and pyroelectricity, and remarkable electro-optical and nonlinear optical effects. Because of these properties, ferroelectrics are used in many kinds of electronic and optoelectronic devices, e. g., high capacitor ceramic condensers, surface wave filters, ultrasonic probes in medical diagnostics, position and driving controllers of scanning tunneling microscopes and of micro machines, temperature sensors, optical phase modulators, and laser wavelength conversion devices. In addition to the above applications, recent research has been extensively directed toward non-volatile memory devices such as 1 and 0 digital units that exploit up and down spontaneous polarizations, to realize an FeRAM with very low electric consumption and high processing speed. (see Fig. 1.1) 7

8 The study of ferroelectrics started at the discovery of NaK(C 4 H 4 O 6 )4H 2 O (Rochelle salt) in J. Valasek found that a Rochelle salt shows an electric hysteresis. After 1935 when the ferroelectricity of potassium dihydrogen phosphate KH 2 PO 4 was found, hydrogen bond ferroelectrics have been one of the main topics in ferroelectrics researches. 2 Around 1942, the ferroelectricity of barium titanate BaTiO 3 (BTO) with a perovskite structure was discovered independently in Japan, the United States, and the Soviet Union. 3 5 At present, the number of ferroelectrics is increasing and is currently approximately 300 (excluding miscellaneous solid solutions and isomorphous compounds). 6 In particular, perovskite oxide ferroelectrics ABO 3 are most thoroughly studied because of their simple structure, and stability to chemical and physical stimuli. Additionally, substitutions of various kinds of ions with different ionic radii into A and B sites induce various physical properties, e. g., ferroelectricity, anti-ferroelectricity, relaxor, quantum paraelectricity, and multi ferroic natures. Understanding these properties of full varieties is indeed an essential problem of condensed matter physics and extensive research has been done so far. In the followings sections, a key concept of ferroelectric phase transitions is briefly described, then characteristics of quantum paraelectricity and relaxor nature will be summarized. Figure 1.1: Applications of ferroelectrics 8

9 1.1.1 Two types of phase transition mechanism Ferroelectric phase transitions are classified into two categories. One is a displacive type and the other is an order-disorder type. In the displacive ferroelectric phase transition, a macroscopic polarization occurs as the result of relative displacements of cations and anions below T C. The type of phase transition is characterized by the soft phonon. With decreasing temperature, the frequency of a low lying transverse optic mode (TO mode) decreases toward zero at the Brillouin zone center (Γ point). At T C where a dielectric constant takes a maximum value, it becomes equal to zero and the structure phase transition to a polar state is induced as a result of the lattice instability. This is caused by the competition between a long range Coulomb force which favors a ferroelectric phase and a restoring short range force driving to a paraelectric state. On the contrary, in the order-disorder type, permanent dipole moments of molecules exist even above T C. Above T C, the directions of dipole moments are randomly distributed and a net polarization does not occur. Below T C, they are aligned in a specific direction and a spontaneous polarization appears. The critical slowing down of the relaxation time which becomes infinite towards T C, is a characteristic of this transition type. In reality, both types of ferroelectric phase transitions coexist and it is rather difficult to distinguish which is dominant, particularly around the phase transition temperature Quantum paraelectricity Recently, new research directions of fundamental physics in ferroelectrics have been recognized. These are quantum paraelectrics and relaxors. In quantum paraelectrics, the ferroelectric phase transition does not take place although the dielectric constant increases toward a low temperature. In 1952, J. H. Barrett explained this phenomenon treating the Slater theory of BaTiO 3 (BTO) quantum-statistically. 7,8 He pointed out implicitly that the zero point fluctuations prevent the appearance of a ferroelectric phase. Later, this phenomenon was Figure 1.2: Temperature dependences of dielectric constant in STO16 and STO18 termed quantum paraelectricity. A typical example is SrTiO 3 (STO), which was studied by K. A. Muller and his collaborators. 9 They analyzed experimental results of the dielectric constant using the Barrett formula and pointed out the necessity of taking 9

10 into account the interaction between different soft modes (anharmonic terms appearing in the Hamiltonian expanded with the normalized coordinates of phonons) to explain the experimental results. Prototypical instances of quantum paraelectrics are STO, KTO, and calcium titanate CaTiO 3. Because the quantum paraelectricity is an effect created by a subtle balance between a TO soft phonon and quantum fluctuations, external stimuli (e. g., external electric field, doping of impurity, or applying pressure) easily induce the ferroelectricity. Therefore quantum paraelectrics are sometimes called incipient ferroelectrics. In 1999, M. Itoh et al. reported that a substitution of oxygen isotope (O-18) induces ferroelectricity in STO. 10 They measured the dielectric constant and found that a peak appears around 23 K in the case of 90 % O-18 replaced STO, as shown in Fig This is the first report that ferroelectricity is induced in a quantum paraelectric with isotope substitution Relaxor The discovery of relaxor was made by Smolenskii and his co-workers early in They observed a diffused nature of a ferroelectric phase transition in Ba(Ti, Sn)O 3. In the 1960s, the prototype relaxor Pb(Mg 1/3 Nb 2/3 )O 3 (PMN) was discovered by the same group in the Ioffe Physico-Technical Institute in the former Soviet Union. Numerous research projects were started in the 1990s, mainly because the largest electro-mechanical coupling constant was quite attractive for piezoelectric applications. Another characteristic of a relaxor is a remarkable dielectric dispersion at the low frequency region. This phenomenon is thought to be originated from the existence of localized polar region in nano-scale, i.e., polar nano region (PNR), which appears at a characteristic temperature (Burns temperature) higher than the peak temperature of the dielectric constant. The size distribution of PNR can explain at least qualitatively the characteristic dielectric dispersion. Researches of relaxors are still very active and more than half of the reports presented in international conferences on ferroelectrics are concerned with relaxors. In spite of these efforts, the origin of relaxor is not fully understood at present Because of its intrinsic heterogeneity originated from random bonds and random forces in relaxors, it shows a quite complex response to the external stimulus and no model can explain the individual behavior of each relaxor. Furthermore, the nature of the random force which permits PNRs in high temperature cannot be well established yet. It is interesting to compare the PNR temperature region in relaxors with the coexistence temperature region of para and ferroelectric states in the first order phase transition. The former is several hundred Kelvin, while the latter is only around a few K at most. 10

11 1.2 Quantum paraelectric KTaO 3 In the present research, we choose KTO as a basal quantum paraelectric crystal because a pure KTO shows no structural phase transition down to near 0 K, and no elaborate pre-cautions on ferroelastic domain formation are needed in comparison with another typical quantum paraelectric STO. STO undergoes a structural phase transition from cubic m 3m to tetragonal 4/mmm at 105 K. 22 This phase transition is induced by a softening of the zone boundary R 25 mode, caused by a rotation of oxygen octahedral, and creates three types of ferroelastic domains. These domains sometimes make Figure 1.3: Crystal structure of KTO the analysis of experimental results difficult because the domain structure is not easily controllable. KTO has a centro-symmetric cubic structure as shown in Fig Precise dielectric and optical properties are reported by S.H.Wemple. 23 Measurements of temperature dependences of the soft mode and the dielectric constant 23,30,31 were also investigated. These results indicate that the TO phonon becomes soft towards a negative Curie temperature of -10 K, and the dielectric constant increases and is saturated to the value of 4000 without any peaks. Therefore, it keeps the cubic structure of m 3m down to nearly 0 K. The fact is a typical characteristic of the quantum paraelectricity. In the m 3m phase, it is anticipated that no second harmonic generation (SHG) occurs because of its inversion symmetry. However, a weak non linear optical effect was observed by the hyper-raman scattering. 28,32,33 To explain it, theoretical models which consider localized point defects and the dynamical frustration are proposed. 32,34 R. J. Migoni et al. insisted that a strong anharmonic coupling between an electron shell of oxygen ion and Ta ion ligand breaks a lattice stability in KTO. 35 This kind of anhamonic coupling is also pointed out by G. Shirane, J. D. Axe, and A. S. Chaves et al. From the temperature dependence of TO phonon, they concluded that the behavior of TO phonon cannot be described without anhamonic coupling between TO mode and other mode like longitudinal acoustic (LA) or transverse acoustic (TA) mode. 24,27,36 As mentioned above, lower symmetry regions exist locally even in a pure KTO. So the present author considered that these regions would correspond to PNRs in relaxors, and this structural heterogeneity could be the common feature of the quantum paraelectricity and relaxors, and the quantum paraelectricity may be explained on this standpoint. With this motivation, I started the present research. 11

12 1.3 Effects of Li ion doping into quantum paraelectric KTO Various kinds of studies have been performed on the external stimulus induced effect in KTO, in particular, the impurity effect. Among them, substitutions of Nb into Ta sites, and Li into K sites are mostly investigated. The former induces a ferroelectric phase transition, 37 while the latter effect has not been clearly understood yet. In Li doped KTO (KLT), Li ion replaces K ion to maintain the charge neutrality. Since the ionic radius of Li is so small (0.76 Å) comparing with that of K (1.38 Å), Li ion rattles in the KTO lattice cage. On the Li position, many experimental and theoretical studies have been made. Y. Yacoby et al. carried Figure 1.4: Temperature dependences of dielectric constant in KLT with different x out a Raman scattering experiment under hydrostatic pressure and found that the impurity induces a peak which does not appear in KTO. 38,39 Because this peak exists even at room temperature, they concluded that Li ion does not occupy the inversion center position. NMR, 40 pyroelectricity, 41 ultrasound, 42 and X-ray diffraction measurements 43 confirmed that Li ion occupies the off-center position which is shifted by Å along one of the six equivalent [001] directions. S. A. Prosandeev et al. estimated the amount of Li shift and its direction using the first principle calculation. 47 They assumed a supercell whose Li concentration is 12.5 % and computed the energy reduction in both cases of full ionic relaxation and of fixed ideal positions except Li ion, and concluded that the system takes the lowest energy when Li ion shifts by 0.8 Å along the [001] direction with the fixed position case. In the full relaxation case, the calculated equilibrium position of Li off-center is Å along the [001] direction. They also pointed out that the energy of full relaxation is five times lower than that of fixed situation. From these results, they concluded that Li ions shift Å along the [001] direction. Thus these studies revealed that the Li off-centering induces a large electric dipole moment. The substitution of Li ions induces a remarkable effect on the dielectric constant: A peak appears in dielectric constant as shown in Fig. 1.4, and the peak temperature shifts higher with increasing Li concentration. The appearance of a peak in dielectric constant would signify the ferroelectricity below the peak temperature. 44,45 However, the understanding of the polar state of KLT in low 12

13 temperature region has not been settled for long time. In 1980s, intense discussions had been made whether the state below the dielectric peak is a ferroelectric state or a polar glass state. E. J. Courtens et al. insisted that KLT shows the first order ferroelectric phase transition with 100 µ m order domain from electric-field-induced birefringence. 48 W. Kleemann et al. measured the temperature dependence of birefringence and derived a conclusion that the critical Li concentration is around 2.2 %. Above the critical concentration, KLT undergoes a ferroelectric phase transition with first order. And below 2.2 %, it becomes polar glass state The result of Raman scattering carried out by Boatner and Yacoby verified that a ferroelectric phase transition takes place above 2.2 %. 38,39,41 Because the soft phonon splits into A and E modes around a transition temperature but remains finite value, they insisted that KLT shows an order-disorder phase transition. Andrews performed X-ray diffraction study on different Li concentration specimens and concluded that KLT undergoes a ferroelectric phase transition above 5 % of Li concentration. 52 On the other hand, U. T. Höchli et al. claimed that KLT is a polar glass in whole Li concentration range from the results of NMR, ultrasonic, and dielectric measurements Because the spin lattice relaxation time estimated from the NMR showed no critical behavior, they concluded that KLT does not undergo a ferroelectric phase transition. They also insisted that the Li concentration dependence of freezing temperature of Li hopping supports their conclusion. In 1990s, J. Toulouse et al. reported that KLT is a relaxor. 60,61 They performed dielectric and neutron diffuse scattering measurements and observed the diffuse scattering caused by PNR for KLT whose concentration is above 6 %. However, the further experimental evidences are necessary to verify that KLT is a relaxor, and this is the starting point of my research. 1.4 Purpose of the present research The present study has three main purposes. The first purpose is to confirm that KLT is a real relaxor. The characteristic behaviors of relaxors are remarkable frequency dependence of dielectric constant, history and time dependence of the order parameter and the existence of PNR. To verify these behaviors in KLT, we perform dielectric measurements, SHG microscope observations, X-ray diffraction, and neutron diffuse scattering experiments. These experimental details and results are described precisely in chapters 2, 3, 4, and 5. The second purpose is to clarify the phase transition mechanism, i.e., the origin of the dielectric peak of KLT. As I mentioned above, doping of Li ion induces a 13

14 peak in dielectric constant. At this peak temperature, the softening of TO phonon is expected, however, TO phonon does not show any softening. On the contrary, Li substitution raises the frequency of TO phonon. This phenomenon eliminates the naive explanation that the Li doping weakens the effect of the quantum fluctuation and the ferroelectric phase transition occurs as a result of the inherent TO phonon softening at a finite temperature. The experimental results focusing on this phase transition mechanism are described in chapter 6. The third purpose is to observe the critical phenomenon in KLT. This is expected to explain the characteristic dielectric behavior observed at the Li critical concentration, as in the prototype relaxor/ferroelectric solid solutions at the morphotropic phase boundary. The results are described in chapter 7. Table 1.1: Chronology of main researches on KTO, KLT, and relaxor year KTO and KLT references relaxor references 1950 Dielectric measurement of KTO Discovery of relaxor Prototype relaxor PMN was found. 71, Optical properties and band structure of KTO; 23 dielectric constant, optical properties, Seebeck effect, hall effect and conductivity 1967 Inelastic scattering measurement of KTO Raman scattering measurement of KTO and KLT 38, Dielectric measurement of KLT 63 The term of relaxor began to be used s Strong discussions on the polar state of KLT; polar glass state or ferroelectric state? 1982 Giant electro-mechanical effect in PZN Existence of polar nano region (PNR) 75, Time dependence of Bragg intensity 64 Superparaelectric model History dependence of dielectric constant Dipole glass model SHG measurements of KLT 65 Time dependence of birefringence induced by strains History dependence & random field model Relaxor nature of KLT Time evolution Time dependence of dielectric constant Monoclinic phase in PZT at MPB 81 First principle calculation in PMN-PbTiO History dependence of dielectric constant 68 Polarization rotation model with first principle calculation 83 Diffuse scattering measurment of KLT Soft mode above Burns temperature First principle calculation for KLT 46 Soft mode condensation with a uniform phase shift First principle calculation for KLT Diffuse scattering measurement of KLT 69 Critical phenomenon 86 14

15 References 1 J. Valasek, Piezo-Electric and Allied Phenomena in Rochelle Salt, Phys. Rev. 17, (1921) 2 G. Busch, and P. Scherrer, Naturwissenschaften 23, 737 (1935) 3 B. Wul, and I. M. Goldman, Compt. Rend. Acad. Sci. URSS 46, 139 (1945) 4 B. Wul, and I. M. Goldman, Compt. Rend. Acad. Sci. URSS 49, 177 (1945) 5 B. Wul, and I. M. Goldman, Compt. Rend. Acad. Sci. URSS 51, 21 (1946) 6 T. Mitsui, Springer Handbook of Condensed Matter and Materials Data, ferroelectric oxide and non oxide including liquid & polymer 7 J. C. Slater, The Lorentz Correction in Barium Titanate, Phys. Rev. 78, (1950) 8 J. H. Barrett, Dielectric Constant in Perovskite Type Crystals, Phys. Rev. 86, (1952) 9 K. A. Muller, and H. Burkard, SrTiO 3 : An intrinsic quantum paraelectric below 4 K, Phys. Rev. B 19, (1979) 10 M. Itoh, R. Wang, Y. Inaguma, T. Yamaguchi, Y-J. Shan, and T. Nakamura, Ferroelectricity Induced by Oxygen Isotope Exchange in Strontium Titanate Perovskite, Phys. Rev. Lett. 82, G. A. Smolenskii, A. I. Agranovskaya, Soviet Physics Solid State 1, 1429 (1959) 12 L. E. Cross, Relaxor Ferroelectrics, Ferroelectrics 76, (1987) 13 G. A. Smolenskii, J. Phys. Soc. Jpn. 28 Suppl., 26 (1970) 14 A. J. Bell, Calculations of dielectric properties from the superparaelectric model of relaxors, J. Phys.: Condens. Matter 5, 8773 (1993) 15 S. B. Vakhrushev, B. E. Kvyatkovsky, A. A. Naberezhnov, N. M. Okuneva, and B. P. Toperverg, Glassy phenomena in disordered perovskite-like crystals, Ferroelectrics 90, (1989) 16 S. N. Dorogovtsev, and N. K. Yushin, Acoustical properties of disordered ferroelectrics, Ferroelectrics 112, (1990) 15

16 17 D. Viehland, S. J. Jang, L. E. Cross, and M. Wutting, Local polar configurations in lead magnesium niobate relaxors, J. Appl. Phys. 69, (1991) 18 H. Quian, L. A. Bursill, Int. J. Mod. Phys. B 10, 2027 (1996) 19 B. E. Vugmeister, and H. Rabitz, A phenomenology of relaxor ferroelectrics, Ferroelectrics 201, (1997) 20 V. Westphal, W. Kleemann, and M. D. Glinchuk, Diffuse phase transitions and random-field-induced domain states of the relaxor ferroelectric PbMg 1/3 Nb 2/3 O 3, Phys. Rev. Lett. 68, (1992) 21 A. E. Glazounov, A. K. Tagantsev, and A. J. Bell, Evidence for domain-type dynamics in the ergodic phase of the PbMg 1/3 Nb 2/3 O 3 relaxor ferroelectric, Phys. Rev. B 53, (1996) 22 R. A. Cowley, Lattice Dynamics and Phase Transition of Strontium Titanate, Phys. Rev. 134, A981-A997 (1964) 23 S. H. Wemple, Some Transport Properties of Oxygen-Deficient Single-Crystal Potassium Tantalate (KTaO 3 ), Phys. Rev. 137, A1575-A1582 (1964) 24 G. Shirane, R. Nathans and V. J. Minkiewicz, Temperature Dependence of the Soft Ferroelectric Mode in KTaO 3, Phys. Rev. 157, (1967) 25 C. H. Perry, and T. F. McNelly, Soft Mode in KTaO 3, Phys. Rev. 154, 456 (1967) 26 P. A. Fleury, and J. M. Worlock, Electric-Field-Induced Raman Scattering in SrTiO 3 and KTaO 3, Phys. Rev. 174, (1968) 27 A. S. Chaves, F. C. S. Barreto and L. A. A. Ribeiro, Model for the Low- Temperature Lattice Anomaly in SrTiO 3 and KTaO 3, Phys. Rev. Lett. 37, (1976) 28 H. Vogt, and H. Uwe, Hyper-Raman scattering form the incipient ferroelectric KTaO 3, Phys. Rev. B 29, 1030 (1984) 29 H. Vogt, The influence of defect-induced polarization clusters on frequency and width of the soft mode in nominally pure KTO 3, Ferroelectrics 125, 313 (1992) 30 W. R. Abel, Effect of Pressure on the Static Dielectric Constant of KTO 3, Phys. Rev. B 4, (1971) 16

17 31 G. A. Samara, and B. Morosin, Anharmonic Effects in KTO 3 : Ferroelectric Mode, Thermal Expansion, and Compressibility, Phys. Rev. B 8, (1973) 32 H. Vogt, Hyper-Rayleigh scattering from the bulk of nominally pure KTaO 3, Phys. Rev. B 41, (1990) 33 H. Vogt, Hyper-Rayleigh Scattering as a Probe of Quasistatic Lattice Distortions in Nominally Pure KTaO 3, Ferroelectrics 107, (1990) 34 W. Prusseit-Elffroth, and F. Schwabl, Second harmonic light scattering in paraelectric perovskites, Appl. Phys. A 51, (1990) 35 R. J. Migoni, H. Bilz, and D. Bauerle, Origin of Raman Scattering and Ferroelectricity in Oxidic Perovskites, Phys. Rev. Lett. 37, 1155 (1976) 36 J. D. Axe, J. Harada and G. Shirane, Anomalous Acoustic Dispersion in Centrosymmetric Crystals with Soft Optic Phonons, Phys. Rev. B 1, (1970) 37 S. Triebwasser, Study of Ferroelectric Transitions of Solid-Solution Single Crystals of KNbO 3 -KTaO 3, Phys. Rev. 114, (1959) 38 Y. Yacoby, and A. Linz, Vibrational properties of KTaO 3 at critical points in the Brillouin zone, Phys. Rev. B 9, (1974) 39 Y. Yacoby, S. Just, Differential Raman scattering from Impurity soft modes in mixed crystals of K 1 x Na x TaO 3 and K 1 x Li x TaO 3, Solid State Commun. 15, (1974) 40 F. Borsa, U. T. Höchli, J. J. Van der Klink, and D. Rytz, Condensation of Random-Site Elecric Dipoles: Li in KTaO 3, Phys. Rev. Lett. 45, (1980) 41 U. T. Höchli, H. E. Weibel, and L. A. Boatner, Stabilisation of polarised clusters in KTaO 3 by Li defects: formation of a polar glass, J. Phys. C: Solid State Phys. 12, L563-L567 (1979) 42 U. T. Höchli, H. E. Weibel, and W. Rehwald, Elastic and dielectric-dispersion in the dipole glass K 1 x Li x TaO 3, J. Phys. C 15, (1982) 43 H. Yokota, Y. Uesu, C. Malibert, and J. M. Kiat, Second harmonic generation and X-ray diffraction studies of the pretransitional region and polar phase in relaxor K (1 x) Li x TaO 3, Phys. Rev. B 75, (2007) 17

18 44 J. J. van der Klink, and S. N. Khanna, Off-center lithium ions in KTaO 3, Phys. Rev. B 29, (1984) 45 M. G. Stachiotti, R. L. Migoni, Lattice polarization around off-center Li in Li x K 1 x TaO 3, J. Phys.: Condensed. Matt. 2, (1990) 46 R. I. Eglitis, V. A. Trepakov, S. E. Kapphan, and G. Borstel, Large-scale computer modeling of Li impurities in KTaO 3 and K 1 x Li x Ta 1 y Nb y O 3 perovskite solid solutions, Crystal Engineering 5, (2002) 47 S. A. Prosandeev, E. Cockayne, and B. P. Burton, Energetics of Li atom displacements in K 1 x Li x TaO 3 : First-principles calculations, Phys. Rev. B 68, (2003) 48 E. Courtens, Birefringence measurements on KTaO 3 :Li, J. Phys. C: Solid State Phys. 14, L37-L42 (1981) 49 W. Kleemann, S. Kutz, and D. Rytz, Cluster Glass and Domain State Properties of KTaO 3 : Li, Europhys. Lett. 4, (1987) 50 W. Kleemann, S. Kutz, F. J. Schafer, and D. Rytz, Strain-induced quadrupolar ordering of dipole-glass-like K 1 x Li x TaO 3, Phys. Rev. B 37, 5856 (1988) 51 H. Schremmer, W. Kleemann, and D. Rytz, Field-induced sharp ferroelectric phase-transition in K Li TaO 3, Phys. Rev. Lett. 62, (1989) 52 S. R. Andrews, X-ray-scattering study of the random electric-dipole system KTaO 3 -Li, J. Phys. C: Solid State Phys. 18, (1985) 53 U. T. Höchili, H. E. Weibel, and L. A. Boatner, Extrinsic Peak in the Susceptibility of Incipient Ferroelectric KTaO 3 :Li, Phys. Rev. Lett. 41, (1978) 54 U. T. Höchli, Dynamics of Freezing Electric Dipoles, Phys. Rev. Lett. 48, (1982) 55 J. J. van der Klink, D. Rytz, F. Borsa, and U. T. Höchli, Collective effects in a random-site electric dipole system: KTaO 3 :Li, Phys. Rev. B 27, (1983) 56 J. J. Van der Klink, and F. Borsa, NMR study of the quasi-reorientational dynamics of Li ions in KTaO 3 :Li, Phys. Rev. B 30, (1984) 57 U. T. Höchli and D. Baeriswy, Coexisting polar configurations in potassiumlithium tantalite, J. Phys. C: Solid State Phys. 17, (1984) 18

19 58 S. Rod, F. Borsa, and J. J. van der Klink, Order and disorder in pure and doped KTaO 3 :A181 Ta NMR study, Phys. Rev. B 38, (1988) 59 U. T. Höchli, and M. Maglione, Dielectric relazation spectroscopy and the ground state of K 1 x Li x TaO 3, J. Phys.: Condens. Matter. 1, (1989) 60 J. Toulouse, B. E. Vugmeister, and R. Pattnaik, Collective Dynamics of Off- Center Ions in K 1 x Li x TaO 3 : A Model of Relaxor Behavior, Phys. Rev. Lett. 73, (1994) 61 G. Yong, J. Toulouse, R. Erwin, S.M. Shapiro and B. Hennion, Pretransitional diffuse neutron scattering in the mixed perovskite relaxor K 1 x Li x TaO 3, Phys.Rev. B 62, (2000) 62 J. K. Hulm, B. T. Matthias, and E. A. Lond, A Ferromagnetic Curie Point in KTaO 3 at Very Low Temperatures, Phys. Rev. 79, (1950) 63 U. T. Höchli, H. E. Weibel, and L. A. Boatner, Extrinsic Peak in the Susceptibility of incipient Ferroelectric KTaO 3 :Li, Phys. Rev. Lett. 41, (1978) 64 W. A. Kamitakahara, C. -K. Loong, G. E. Ostrowski, and L. A. Boatner, Timedependent phase transformation in KTaO 3 :Li, Phys. Rev. B 35, (1987) 65 G. A. Azzini, G. P. Banfi, E. Giulotto, and U. T. Höchli, Second-harmonic generation and origin of polar configuration in KTaO 3 :Li, Phys. Rev. B 43, W. Kleemann, V. Schonknecht, D. Sommer, and D. Rytz, Dissipative quantum tunneling and absence of quadrupolar freezing in glassy K Li TaO 3, Phys. Rev. Lett. 66, (1991) 67 F. Alberici-Kious, J. P. Bouchaud, L. F. Cugliandolo, P. Doussineau, and A. Levelut, Aging in K 1 x Li x TaO 3 : A Domain Growth Interpretation, Phys. Rev. Lett. 81, (1998) 68 P. Doussineau, T. de Lacerda-Aroso, and A. Levelut, Search for memory and return to disorder in potassium-lithium tantalate crystal, J. Phys.: Condens. Matter 12, (2000) 69 S. Wakimoto, G. A. Samara, R. K. Grubbs, E. L. Veturini, L. A. Boatner, G. Su, G. Shirane, and S. -H. Lee, Dielectric properties and lattice dynamics of Ca-doped K 0.95 Li 0.05 TaO 3, Phys. Rev. B 74, (2006) 19

20 70 G. A. Smolenskii, and V. A. Isupov, Dokl. Akad. Nauk SSSR 9, 653 (1954) 71 G. A. Smolenskii, and A. I. Agranovskaya, Zh. Thek. Fiz. 28, 1491 (1959) 72 G. A. Smolenskii, and A. I. Agranovskaya, Sov. Phys. Sol. State 1, 1429 (1959) 73 W. A. Shulz, J. V. Biggers, and L. E. Cross, J. Amer. Ceram. Soc. 61, (1978) 74 J. Kuwata, K. Uchino, and S. Nomura, Dielectric and Piezoelectric Properties of 0.91Pb(Zn 1/3 Nb 2/3 )O PbTiO 3 Single Crystals, Jpn. J. Appl. Phys. 21, (1982) 75 G. Burns, and F. H. Dacol, Glassy polarization behavior in ferroelectric compounds Pb(Mg 1/3 Nb 2/3 )O 3 and Pb(Zn 1/3 Nb 2/3 )O 3, Solid State Commun. 48, (1983) 76 G. Burns, and F. H. Dacol, Crystaline ferroelectric with glassy polarization behavior, Phys. Rev. B 28, (1983) 77 L. E. Cross, Relaxor Ferroelectrics, Ferroelectrics 76, (1987) 78 D. Viehland, S. J. Jang, L. E. Cross, and M. Wuttig, Freezing of the polarization fluctuations in lead magnesium niobate relaxors, J. Appl. Phys. 68, (1990) 79 V. Westphal, W. Kleemann, and M. D. Glinchuk, Diffuse phase transitions and random-field-induced domain sattes of the relaxor ferroelectric PbMg 1/3 Nb 2/3 O 3, Phys. Rev. Lett. 68, (1992) 80 E. V. Colla, E. Y. Koroleva, N. M. Okuneva, and S. B. Vakhrushev, Long-Time Relaxation of the Dielectric Response in Lead Magnoniobate, Phsy. Rev. Lett. 74, (1995) 81 B. Noheda, D. E. Cos, G. Shirane, J. A. Gonzalo, L. E. Cross, and S. -E. Park, A monoclinic ferroelectric phase in the Pb(Zr 1 x Ti x )O 3 solid solution, Appl. Phys. Lett. 74, (1999) 82 L. Bellaiche, and D. Vanderbilt, Intrinsic Peezoelectric Response in Perovskite Alloys: PMN-PT versus PZT, Phys. Rev. Lett. 83, (1999) 83 H. Fu, and R. E. Cohen, Polarization rotation mechanism for ultrahigh electromechanical response in single-crystal piezoelectrics, Nature 403, (2000) 20

21 84 P. M. Gehring, S. Wakimoto, Z. -G. Ye, and G. Shirane, Soft Mode Dyanmics above and below the Burns Temperature in the Relaxor Pb(Mg 1/3 Nb 2/3 )O 3, Phys. Rev. Lett. 87, (2001) 85 K. Hirota, Z. -G. Ye, S. Wakimoto, P. M. Gehring, and G. Shirane, Neutron diffuse scattering from polar nanoregions in the relaxor Pb(Mg 1/3 Nb 2/3 )O 3, Phys. Rev. B 65, (2002) 86 Z. Kutnjak, J. Petzelt, and R. Blinc, The giant electromechanical response in ferroelectric relaxors as a critical phenomenon, Nature 441, (2006) 21

22

23 Chapter 2 HISTORY DEPENDENCE OF THE ORDER PARAMETER IN K 1 x Li x TaO 3 In this chapter, experimental results on the marked history dependence of the order parameter in K 1 x Li x TaO 3 are described. We find that the behaviors of the optical second harmonic (SH) intensities on several paths, i.e., zero field cooling (ZFC), zero field heating after zero field cooling (ZFH/ZFC), field heating after zero field cooling (FH/ZFC), field cooling (FC), and field heating after field cooling processes (FH/FC), are different. In ZFC and ZFH/ZFC processes, the SH intensities do not appear in the whole temperature range. In the case of FH/ZFC, the SH intensity, which vanishes at low temperatures, begins to increase at a higher temperature upon heating. This phenomenon is one of the characteristic behaviors observed in the canonical relaxor Pb(Mg 1/3 Nb 2/3 )O 3. Therefore, this result strongly indicates that K (1 x) Li x TaO 3 is a relaxor. H. Yokota, T. Oyama, and Y. Uesu Second-harmonic-generation microscopic observations of polar state in Li-doped KTaO 3 Phys. Rev. B 72, (2005) 2.1 Introduction In 1980 s, there have been heated discussions whether the long-range order develops at low temperature in KLT or not. U. T. Höchli et al. insisted that the 23

24 ferroelectric phase transition with the long-range order does not take place due to the lack of an anomaly in the elastic compliance and the vanishing spontaneous bulk polarization P s after removing an electric field at all temperatures In addition to these experimental facts, they found that the observed magnitude of P s of KLT is 10 times larger than the calculated value from a Li off-centering dipole moment, and they noticed that the P s should be induced by the formation of clusters and KLT is in a polar glass state at low temperature. On the contrary, R. L. Prater et al. reported that a ferroelectric phase transition takes place from Raman scatterings and the polarization dependence of a transmittance light. 11 In particular, with the transmittance experiment, they obtained the temperature dependence of the depolarization of light and concluded that KLT with Li concentration above 4 % shows a clear ferroelectric phase transition. In the case of KLT with higher Li concentration, e.g., 4 %, the depolarization ratio is one above a certain temperature and it decreases rapidly with decreasing temperature. If the low temperature phase is a polar glass, the ratio should remain one at low temperature because the anisotropic index of refraction is almost averaged out to be isotropic in the whole region of a sample. Thus they disputed a polar glass model. Though these discussions were substantial, no decisive conclusion has been obtained up to now. Recently, J. Toulouse et al. reported that KLT is a relaxor like Pb(Mg 1/3 Nb 2/3 )O 3 which is originated from an intrinsic structural inhomogeneity. 12,13 They carried out dielectric and neutron diffuse scattering experiments. Because of relaxor-like dielectric dispersions and diffuse scatterings related to the polar nano regions, they claimed that KLT is a relaxor. It should be noted that relaxors are characterized by following 4 points; 14 (i) remarkable dielectric dispersion in a low frequency region, 15 (ii) existence of polar nano regions above the dielectric peak temperature, (iii) history dependence of the order parameter, and (iv) a slow kinetics of the order parameter under an electric field J. Toulouse et al. confirmed only a part of these criteria and further experimental evidences are necessary to judge KLT as a real relaxor. Therefore we perform the experiments to clarify that KLT satisfies all the above-mentioned criteria. In this chapter, we describe experimental results on a history dependence of the order parameter. Concerning this phenomenon, Höchli et al. measured the temperature dependence of P s and found its history dependence in KLT-1.6%. 34,35 However, they did not interpret the peculiar behavior. Furthermore, the measurement of a net polarization cannot make clear whether the polar state is localized or the long-range order develops. We perform the experiments using the second harmonic generation (SHG) microscope which enables to observe the 2-D polarization distributions in a sample from the SH wave intensity distribution. The SHG is also a powerful tool for investigating ferroelectric phase transitions, because it is so sensitive to the change in polarization that it can detect a quite small polarization with the order of tens nc/cm 2. SHG 24

25 observations has been already reported on KLT, however, they did not touch with the history dependence Additionally, by exploiting the fact that an SHG tensor component is the polar third rank, we determine the possible point group of the low temperature phase of KLT from the polarization dependence of the SH intensity. 2.2 Experimental History dependence of SH intensity The measurement is performed with using the SHG microscope. Because SH intensity is proportional to the square of the second order non-linear optical coefficient d, the possibility of occurrence of SH waves is determined from the crystal symmetry. In the case of centro-symmetric crystals, the whole components of third order tensor become zero and SH waves do not generate from the crystals. Thus, SHGM is a very effective equipment to detect the breakdown of centro-symmetry in other words the occurrence and change of polarization. The optical system of SHG microscope is illustrated in Fig.2.1. We use a pulsed wave from Q switched Nd 3+ :YAG (yttrium aluminum garnet) laser with wavelength of 1064 nm, repetition frequency of 20 Hz, fluence of 15 mw/pulse, and the radius of 2.5 mm. Therefore, the light intensity per pulse is 76 mw/cm 2. It passes through a halfwave plate to change the polarization of the fundamental wave and illuminates a specimen, and SH waves with wavelength of 532 nm generated in a specimen are collected by an objective followed by an infra-red absorption filter, a multi-layer interference filter for 532 nm and an analyzer. SH intensity inside a specimen is obtained by a charge coupled device (CCD) camera with an image-intensifier, which is connected to a computer. The special feature of present work is that the distributions of SH waves from samples can be obtained in two dimensional images. This makes us to understand the inhomogeneity in the system. For the experiments, we use several specimens with different Li concentrations (x = 1.5, 2.6, 2.9, 5.1, 5.3, 6.8, 6.9, 9.5 % ). The average size of specimen is 5x3x0.5 mm 3 with a (100) plate. For the optical measurement, both surfaces of samples are optically polished with oxide aluminum. In KLT, a remarkable photo current effect at low temperature was reported by Klein et al. 43,44 Taking into account it, transparent electrodes like ITO are not suitable to put on the top and bottom surfaces. Thus, two narrow rectangular electrodes apart by 3 mm are put on the top surface by Au evaporation. We confirm the electric field distribution by using a finite elements method and the almost homogeneous electric field is applied in the illuminated region. Specimens are set into a cryostat with silver paste. The temperature of the specimen is cooled down to 24 K with a speed of 3 K/min, while 25

26 Figure 2.1: Optical system of second harmonic generation microscope. 1 K/min on heating. The measurements are performed in the following five processes; (i) zero field cooling (ZFC), (ii) zero field heating after ZFC (ZFH/ZFC), (iii) field heating after ZFC (FH/ZFC), (iv) field cooling (FC), and (v) FH after FC (FH/FC). The electric field of 80 V/mm is applied along the [001] direction Point group at low temperature In order to determine the average symmetry of the field-induced polar state of KLT- 2.6%, polarization dependences of SH intensity are measured at 24 K after FC process. The procedure of observing the SH intensity is same as in the measurement of the history dependence. We measure the SH intensity within the following four different conditions; (i) as a function of the rotation angle of the polarizer with the analyzer fixed parallel to the direction of the electric field (//[001]), (ii) as a function of the rotation angle of the polarizer with the analyzer fixed perpendicular to the [001] direction, (iii) as a function of the rotation angle of the analyzer with the polarizer fixed parallel to [001], and (iv) as a function of the rotation angle of the analyzer with the polarizer fixed perpendicular to the [001] direction. 26

27 2.3 Experimental results History dependence of SH intensity Temperature dependences of SH intensity are shown in Fig.2.2. Here, the results of KLT-2.6% and KLT-6.8% are shown in Fig. 2.2 (a) and (b), respectively. The other concentrations show almost the same behaviors as these specimens. In ZFC and ZFH/ZFC processes, no marked SH intensity is observed in the measurement temperature region. In FC, SH intensity starts to increase dramatically at a certain temperature. Below this temperature, it increases gradually with decreasing temperature. We find that the temperature where SH intensity begins to increase shifts to higher temperature with increasing Li concentration. In FH/FC process, SH intensity generated from the specimen at low temperature decreases with raising the temperature and vanishes around a temperature of T p. In KLT-2.6%, T p corresponds with the temperature where SH intensity appears, while there is a disagreement between these two temperatures in KLT-6.8%. In FH/ZFC, SH intensity dose not appear at low temperature even applying an electric field. With heating the sample, weak SH intensity observed at low temperature starts to increase abruptly at certain temperature T 1 (42 K for KLT-2.6% and 83 K for KLT-6.8%), shows a peak and decreases and finally vanishes at a temperature T 2. T 2 is consistent with T p where SH intensity disappears in FH/FC process. From these results, we confirm the marked history dependence of the order parameter in KLT. Figure 2.2: Temperature dependences of SH intensity for KLT-2.6% (a) and KLT-6.8% (b), respectively. In Fig. 2.3, the images of SH waves from the samples are shown. In the figures, a bright region indicates that a strong SH intensity occurs from a sample and a dark area means a weak SH intensity generates. It should be also noted that the SHG intensity is not homogeneously distributed in the sample as shown in SH images, where bright spots and dark regions coexist. We estimate the average size of the polar region to be around 1 µm from Fig.2.3. The origin of the inhomogeneity 27

28 Figure 2.3: SHG images of KLT-2.6%. (a) indicates those observed in the FH after ZFC process, (b) in the FH after FC. Brighter parts produce stronger SH waves. In (a), SHG appears only in a limited temperature range below 48 K of T p, while in (b), SHG is observed in whole temperature range below T p. is attributed to the coexistence of paraelectric and ferroelectric phases. The latter is found to be accompanied with tiny micro domains, which will be discussed in chapter 4. It should be noted that this heterogeneity is not caused by the difference in Li concentration from site to site. Because T p of each site in the sample is observed to take the same value. To confirm that the inhomogeneity of SH waves from KLT specimens comes from the above-mentioned nature, we measure the SH intensity of non linear optical crystal Mg-doped LiNbO 3 (LN) which produces homogeneous SH waves. As a consequence, almost homogeneous SH waves occur from LN. Therefore, the distribution of SH waves seen in KLT reflects the intrinsic heterogeneity in KLT Point group at low temperature In Fig. 2.4, polarization dependences of SH waves under different conditions are shown. Fig.2.4 (a) indicates the result with the analyzer fixed parallel to [001]. A sinusoidal curve of the periodicity of 180 degree is observed. The maximum is 28

29 obtained when the polarization direction of the fundamental wave is parallel to the electric field. In the case of the analyzer fixed perpendicular to the field direction, a sinusoidal curve of the periodicity of 90 degree is obtained as shown in Fig. 2.4 (b). Minima are obtained when the polarization direction of the fundamental wave is parallel and perpendicular to the field direction. In Fig. 2.4 (c) and (d), the results measured as a function of the polarization direction of the SH wave with the polarizer fixed parallel and perpendicular to the field direction are shown respectively. Sinusoidal curves of the periodicity of 180 degree are obtained. These results are best fitted with theoretical curves assuming the point group of tetragonal 4mm as discussed in the session Analyses To explain the results of the polarization dependences shown in Fig. 2.4, we first assume the point group tetragonal 4mm with the polar axis parallel to the cubic [001]. Non-zero SHG tensor components d i j (the Voigt notation is adopted here. i=1 3, j=1 6) are described as follows. d 15 d 15 d 15 d 15 d 33 (2.1) Here Kleinman s law on a transparent nonlinear optical crystal is taken into account. There exist two independent components d 15 and d 33. SHG intensities I 1, I 2, I 3 and I 4 corresponding to cases (i), (ii), (iii) and (iv) are expressed as follows, respectively. I 1 = I 2 10 [sin2 (θ θ 10 ) + p 1 cos 2 (θ θ 10 )] 2 (2.2) I 2 = I 2 20 [p 2 cos (θ θ 20 ) sin (θ θ 20 )] 2 (2.3) I 3 = I 2 30 [p 3 cos (θ θ 30 )] 2 (2.4) I 4 = I 2 40 [p 4 cos (θ θ 40 )] 2 (2.5) Here I i0 (i = 1 4) means the intensity of a fundamental laser wave, θ i0 the direction of the electric field (//[001]), and p i is expressed using SHG tensor components d 15 and d 33 as p 1 = d 33 /d 32 = d 33 /d 15, p 2 = 2d 15, p 3 = d 33, p 4 = d 32 = d 15. (2.6) Values determined by fitting experimental results in Fig.2.4 are p 1 = 2.82, p 2 = 1.87, p 3 = 2.63, p 4 = (2.7) 29

30 From these values, the following ratio is consistently obtained. d 33 /d 15 = 2.86 ± (2.8) These values are almost same as the previous reports as shown in Table 2.1. We assume that the point group is tetragonal 4mm, but our results can be also explained by tetragonal 4, or lower symmetry point group, e.g., orthorhombic mm2. However the result of X-ray diffraction experiments shows that the most plausible crystal system is tetragonal. 46 Then the present experiment concludes that the average symmetry of the low temperature phase below T p of KLT is 4mm or 4. Table 2.1: Experimentally determined SHG tensor component ratio d 33 /d 15 d 33 /d 15 Li concentration x [% ] E [V/mm] Reference 2.86± ± ± , ± , 2.6, 3.4, ± 4% ± 4% Conclusions Under the SHG microscope, we observe a marked history dependence of the polar phase induced by the electric field in KLT specimens. This remarkable history dependence is one of the relaxor criteria as noted above. ZFH/ZFC, FH/ZFC, and FH/FC processes follow different paths. The result indicates that the ground state of KLT is inhomogeneous where the polar phase is coexisted with non-polar cubic phase. The history dependence of the order parameter is quite similar to canonical relaxor PMN. 47 For understanding the phenomenon more quantitatively, further experiments are necessary, especially that for disclosing nature of the polar nanocluster. The experiments concerning this problem are described in chapter 4. From the polarization dependences of the SHG intensity together with the results of X-ray diffraction, we disclose that a point group below T p is 4mm. 30

31 Figure 2.4: Polarization dependences of SH intensity of KLT-2.6% at 24 K in FC process. (a) indicates the case (i) of rotating the polarizer with the analyzer fixed parallel to the electric field E (//[001]), (b) the case (ii) of rotating the polarizer with the analyzer fixed perpendicular to E, (c) the case (iii) of rotating the analyzer with polarizer fixed parallel to E, and (d) the case (iv) of rotating the analyzer with polarizer fixed perpendicular to E. Solid lines indicate fitted curves calculated using eqs.(2.1) (2.4). 31

32 References 1 U. T. Höchli, H. E. Weibel, and L. A. Boatner, Stabilisation of polarised clusters in KTaO 3 by Li defects: formation of a polar glass, J. Phys. C: Solid State Phys. 12, L563-L567 (1979) 2 F. Borsa, U. T. Höchli, J. J. Van der Klink, and D. Rytz, Condensation of Random-Site Elecric Dipoles: Li in KTaO 3, Phys. Rev. Lett. 45, (1980) 3 U. T. Höchli, H. E. Weibel, and W. Rehwald, Elastic and dielectric-dispersion in the dipole glass K 1 x Li x TaO 3, J. Phys. C 15, (1982) 4 W. Kleemann, S. Kutz, and D. Rytz, Cluster Glass and Domain State Properties of KTaO 3 : Li, Europhys. Lett. 4, (1987) 5 U. T. Höchili, H. E. Weibel, and L. A. Boatner, Extrinsic Peak in the Susceptibility of Incipient Ferroelectric KTaO 3 :Li, Phys. Rev. Lett. 48, (1978) 6 U. T. Höchli, Dynamics of Freezing Electric Dipoles, Phys. Rev. Lett. 48, (1982) 7 J. J. van der Klink, D. Rytz, F. Borsa, and U. T. Höchli, Collective effects in a random-site electric dipole system: KTaO 3 :Li, Phys. Rev. B 27, (1983) 8 J. J. Van der Klink, and F. Borsa, NMR study of the quasi-reorientational dynamics of Li ions in KTaO 3 :Li, Phys. Rev. B 30, (1984) 9 U. T. Höchli and D. Baeriswy, Coexisting polar configurations in potassiumlithium tantalite,. Phys. C: Solid State Phys. 17, (1984) 10 U. T. Höchli, and M. Maglione, Dielectric relazation spectroscopy and the ground state of K 1 x Li x TaO 3, J. Phys.: Condens. Matter. 1, (1989) 11 R. L. Prater, L. L. Chase, and L. A. Boatner, Raman scattering studies of the impurity-induced ferroelectric phase transition in KTaO 3 :Li, Phys. Rev. B 23, (1981) 12 J. Toulouse, B. E. Vugmeister, and R. Pattnaik, Collective Dynamics of Off- Center Ions in K 1 x Li x TaO 3 : A Model of Relaxor Behavior, Phys. Rev. Lett. 73, (1994) 32

33 13 G. Yong, J. Toulouse, R. Erwin, S.M. Shapiro and B. Hennion, Pretransitional diffuse neutron scattering in the mixed perovskite relaxor K 1 x Li x TaO 3, Phys. Rev. B 62, (2000) 14 L. E. Cross, Relaxor Ferroelectrics, Ferroelectrics 76, (1987) 15 A. J. Bell, Calculations of dielectric properties from the super-paraelectric model of relaors, J. Phys.: Condens. Matter 5, (1993) 16 G. Burns, and F. H. Dacol, Glassy polarization behavior in ferroelectric compounds Pb(Mg 1/3 Nb 2/3 )O 3 and Pb(Zn 1/3 Nb 2/3 )O 3, Solid State Commun. 48, (1983) 17 G. Burns, and F. H. Dacol, Crystaline ferroelectric with glassy polarization behavior, Phys. Rev. B 28, (1983) 18 S. B. Vakhrushev, B. E. Kvyatkovsky, A. A. Nabereznov, N. M. Okuneva, and B. P. Toperverg, Ferroelectrics 90, 173 (1989) 19 S. B. Vakhrushev, A. Nabereznov, S. K. Sinha, Y. P. Feng, and T. Egami, J. Phys. Chem. Solids 57, 1517 (1996) 20 A. Nabereznov, S. Vakhrushev, B. Dorner, D. Strauch, and H. Moudden, Eur. Phys. J. 11, 13 (1999) 21 D. La-Orauttapong, J. Toulouse, J. L. Robertson, and Z. -G. Ye, ibid. 64, (2001) 22 K. Hirota, Z. -G. Ye, S. Wakimoto, P. M. Gehring, and G. Shirane, Neutron diffuse scattering from polar nanoregions in the relaxor Pb(Mg 1/3 Nb 2/3 )O 3, Phys. Rev. B 65, (2002) 23 S. B. Vakchrushev, B. E. Kvyatkovsky, A. A. Nabereznov, N. M. Okuneva, and B. P. Toperverg, Neutron scattering from disordered perovskite-like crystals and glassy phenomena, Physica B 156 & 157, (1989) 24 N. de Marhan, E. Husson, G. Calvarin, and A. Morell, Structural Study of a Poled PbMg 1/3 Nb 2/3 O 3 Ceramic at Low Temperature, Mat. Res. Bull. 26, (1991) 25 V. Westphal, W. Kleemann, and M. D. Glinchuk, Diffuse Phase Transitions and Random-Field-Induced Domain States of the Rlaxor Ferroelectric PbMg 1/3 Nb 2/3 O 3, Phys. Rev. Lett. 68, (1992) 33

34 26 R. Sommer, N. K. Yushin, and J. J. van der Klink, Polar metastability and an electric-field-induced phase transition in the disordered perovskite Pb(Mg 1/3 Nb 2/3 )O 3, Phys. Rev. B 48, (1993) 27 O Bidault, M Licheron, E Husson and A Morell, The onset of an electric fieldinduced ferroelectric-like phase in the perovskite, J. Phys.: Condens. Matter 8, (1996) 28 E. V. Colla, N. K. Yushin, and D. Viehland, Dielectric properties of (PMN) (1 x) (PT) x single crystals for various electrical and thermal histories, J. Appl. Phys. 83, (1998) 29 C. Ang, Z. Yu, P. Lunkenheimer, J. Hemberger, and A. Loidl, Dielectric relaxation modes in bismuth-doped SrTiO 3 : The relaxor behavior, Phys. Rev. B 59, (1999) 30 K. Fujishiro, T. Iwase, Y. Uesu, Y Yamada, B. Dkhil, J. M. Kiat, S. Mori, and N. Yamamoto, Optical and Structural Studies of Long-Range Order Development in Relaxor Ferroelectrics, J. Phys. Soc. Jpn. 69, (2000) 31 E. V. Colla, E. Yu. Koroleva, N. M. Okuneva, and S. B. Vakhrushev, Long- Time Relaxation of the Dielectric Response in Lead Magnoniobate, Phys. Rev. Lett. 74, (1995) 32 E. V. Colla, E. Yu. Koroleva, N. M. Okuneva, and S. B. Vakhrushev, Field Induced Kinetic Ferroelectric Phase Transition in Lead Magnoniobate, Ferroelectrics 184, (1996) 33 B. Dkhil, and J. M. Kiat, Electric-field-induced polarization in the ergodic and nonergodic states of PbMg 1/3 Nb 2/3 O 3 relaxor, J. Appl. Phys. 90, (2001) 34 U. T. Höchli, P. Kofel, and M. Maglione, Dipolar relaxation and limit of ergodicity in K 1 x Li x TaO 3, Phys. Rev. B 32, (1985) 35 P. Doussineau, T de Lacerda-Aroso, and A. Levelut, Search for memory and return to disorder in potassium-lithium tantalate crystals, J. Phys. Condens. Matter 12, (2000) 36 G. A. Azzini, G. P. Banfi, and E. Giulotto, and U. T. Höchili, Second-harmonic generation and origin of polar configuration in KTaO 3 :Li, Phys. Rev. B 43, (1991) 34

35 37 P. Voigt, and S. Kapphan, Variations of the Second Harmonic Intensity at the Low Temperature Phase Transition in K 1 x Li x TaO 3 under Electric Field, Ferroelectrics 124, (1991) 38 P. Voigt, and S. Kapphan, Experimental study of second harmonic generation by dipolar configurations in pure and Li-doped KTaO 3 and its variation under electric field, J. Phys. Chem. Solids 55, (1994) 39 P. Voigt, S. Kapphan, L. Oliveira, and M. S. Li, Second harmonic generation and thermally stimulated depolarization current investigation of K 1 x Li x TaO 3, Radiation Effects and Defects in Solids 134, (1995) 40 S. A. Prosandeev, V. S. Vikhnin, and S. Kapphan, Percolation with constraints in the highly polarizable oxide KTaO 3 :Li, Eur. Phys. J. B 15, (2000) 41 S. A. Prosandeev, V. S. Vikhnin, and S. Kapphan, Percolative Clusters in KTaO 3 :Li, Integrated Ferroelectrics 32, (2001) 42 C. auf der Horst, J. Licher, S. Kapphan, V. Vikhnin, and S. Prosandeyev, SHG Properties of Pure and Doped Incipient Ferroelectrics KTaO 3 and SrTiO 3 under Applied Electric Fields, Ferroelectrics 264, (2001) 43 R. S. Klein, and G. E. Kugel, Photoconductivity in KTaO 3 :Li single crystals, Phys. Rev. B 50, (1994) 44 R. S. Klein, G. E. Kugel, M. D. Glinchuk, R. O. Kuzian, and I. V. Kondakova, Observation and interpretation of photocurrents in KTaO 3 :Li single crystals, Optical Materials 4, (1995) 45 Y. Fujii, and T. Sakudo, Electric-field-induced optical second-harmonic generation in KTaO 3 and SrTiO 3, Phys. Rev. B 13, 1161 (1976) 46 S. R. Andrews, X-ray-scattering study of the random electric-dipole system KTaO 3 -Li, J. Phys. C: Solid State Phys. 18, (1985) 47 A. A. Bokov, and Z. -G. Ye, Recent progress in relaxor ferroelectrics with perovskite structure, J. Mater. Science 41, (2006) 35

36

37 Chapter 3 EXTREMELY SLOW KINETICS OBSERVED IN K 1 x Li x TaO 3 In this chapter, experimental results are first described for the extremely slow kinetics of the order parameter under an electric field, which is one of the characteristic behaviors of relaxors. After cooling down to the lowest temperature without an electric field, a sample is heated up to a certain temperature and an electric field is applied at this temperature. When the sample is fixed at a low temperature, the SH intensity does not appear for more than 2 hours. On the other hand, the SH intensity increases with time at higher temperatures. The time evolution strongly depends on temperature. Below the critical temperature of 40 K, the SH intensity increases dramatically after an incubation time of 30 minutes. Above the critical temperature, it shows a relatively quick response to the electric field. We analyze the results with the Kolmogorov- Avrami-Chandra theory and explain the results through the difference in the growth mechanism of polar regions. H. Yokota, and Y. Uesu Extremely slow time evolution of the order parameter under an electric field in relaxor K 0.97 Li 0.03 TaO 3 J. Phys.: Condens. Matter 19, (2007) 37

38 3.1 Introduction Nucleation and growth processes in various kinds of the first order phase transition have been extensively investigated as a fundamental subject of non-equilibrium thermodynamics and statistical physics. 1 4 Domain reversal phenomena in ferroelectrics belong to the same category of physics when a domain state is presumed to be a thermodynamic phase. There exist a lot of researches on this stand point since the phenomenon is also important in practical applications. 5 7 The domain switching time of usual ferroelectrics is known to be several nano seconds. Exceptionally, it has been observed that a transformation time from cubic to tetragonal in classic ferroelectrics BaTiO 3 (BTO) takes several msec around the phase transition temperature, which is six orders of magnitude greater than that of ordinal ferroelectrics. However, an incubation time of KLT is around 10 3 seconds order as described later in section 3.3. This is still six orders longer than BTO as shown in Fig. 3.1 and this kind of quite long relaxation time is one of the characteristic behaviors in relaxor systems. In relaxor ferroelectrics, the time approaching to the equilibrium state under an electric field is several ten minutes The origin of the slow kinetics in relaxors is still open at present. 11 The present paper reports that the order parameter (OP) of K Li TaO 3 (KLT-2.6% ) with nearly critical concentration of Li 1 also shows extremely slow time evolution under an electric field E. This phenomenon resembles to the time evolution observed in a prototype relaxor Pb(Mg 1/3 Nb 2/3 )O 3 (PMN) The time dependences of some physical quantities have been already investigated in KLT, i.e, the spontaneous polarization P s determined by the piezoelectric effect, 12 neutron Bragg intensities, 13 strain-induced and stress-induced birefringences, 14,15 and dielectric constants. 16 However measurements with more sensitive method in wider temperature range are needed to elucidate the origin of complicate behavior of domain nucleation and growth processes in KLT. We have recently investigated the temperature dependence of the OP of KLT-2.6% on different paths of the (E,T) space. 17 Based upon these experimental results, we measure precisely the time evolution of the OP of KLT-2.6% at several temperatures, using the optical second harmonic generation (SHG) microscope, because the SHG is a quite sensitive tool for detecting P s. The results are analyzed with the Kolmogorov-Avrami-Chandra theory by fitting the experimental data to obtain characteristic parameters which specify the growth dimensionality and possibility. It is also well established that SHG tensor components are proportional to P s, i.e., the ferroelectric OP, by experiments 18 and by the group-theoretical consider- 1 The Li concentration of 2.6% is revealed to be critical because of its location near the tri-critical point and the depletion point of quantum paraelectricity. 38

39 Figure 3.1: Switching times in ferroelectrics and relaxors. ation. 19 Although several researches have been performed in KLT using SHG in order to discuss the behavior of the OP, these were concerned with phenomena in the equilibrium state. 20, Experimental KLT-2.6% is chosen as a specimen. The sample is a (100) rectangular plate with edges parallel to the 100, the area being 9.4x5.0 mm 2 and the thickness 0.85 mm. Both surfaces are polished for optical measurements. A pair of narrow gold electrodes separated by 3 mm are evaporated on the top surface to apply an electric field along the [001] direction. A special care is paid to avoid the photocurrent effect observed below 70 K. For this purpose, the position of incident laser beam is adjusted to be between the two electrodes. A detail of optical system and sample condition are described in chapter 2. All measurements are performed in the following thermal cycles. First the sample is cooled down to 24 K without an electric field E, and heated without E to a certain temperature at which a time evolution measurement is performed. When the temperature becomes stable, the electric field of 80 V/mm is applied and the field-induced SH intensity is observed as a function of time. After finishing a measurement, the sample is heated to room temperature without E and kept during several hours to erase the memory effect. We select the field magnitude of 80 V/mm owing to the fact that previous measurements of the history dependence of the SH intensity were performed under the same field, 17 which enables us to examine the relation between the saturated SH intensity in time evolution experiments and the field-cooling (FC) SH intensities. Another reason is that stronger fields produce free charges in a sample to cause remarkable photo-currents. Seven temperatures below T p of 50 K are chosen to observe the temporal evolution: T = 30, 31.1, 33.4, 35.3, 39.7, 42 and 43 K. 39

40 3.3 Experimental results Figure 3.2 shows the time evolution of the SH intensity at each temperature. Below 35 K, the SH intensity is almost null and it does not increase within the experimental time of 2 hours in spite of applying E. At 35.3 K, the SH intensity is found to increase slightly with time. At 39.7 K, it begins to increase dramatically after a long incubation time of 40 minutes. This interesting phenomenon seems to be overlooked in the previous researches. At 42 and 43 K, the SH intensity seems to increase in two steps: It increases rapidly for a first few minute after applying E, and then slowly toward the saturated value. Figure 3.2: Time evolutions of SH intensities under an electric field of 80 V/mm. The SH intensity at each temperature is normalized to the saturated value. It is interesting to compare the present results with our previous measurements of SH intensities in different cooling and heating processes with or without E (Fig. 3.3). 17 In the figure, the results of the time evolution are also indicated with arrows. It is disclosed that the SH intensities finally reach the values of FH after FC processes above 35 K. These results show that the FH after FC process is an equilibrium state under the applied field, but the FH after ZFC is a transitional process depending on the heating speed. However, below 35 K, the FH after ZFC path cannot attain the corresponding point in the FH after FC process. This means that the temperature region below 35 K is a non-ergodic region in KLT-2.6% with the applied field of 80 V/mm. It should be pointed out that the time evolution of the OP in KLT-2.6% resembles to that of PMN. This fact also supports that KLT is a relaxor

41 Figure 3.3: The relation between the time evolutions and the history dependences of SH intensities. The arrows show the temperatures where the time evolutions are measured. 3.4 Analyses The present results reveal that the time evolution of the SH intensity strongly depends on the temperature where E is applied. In particular, the behavior of SH intensity, i.e., the square of the OP, varies drastically within a few Kelvin around 40 K. In this sense, 40 K is a critical temperature T*. Below T*, the SH intensity develops abruptly after a long incubation time. Above T*, SH intensity apparently grows in two-steps as shown in Fig. 3.2: The fast evolution is followed by the slow one. Therefore, the temporal behavior above T* is first analyzed by the bi-exponential model Bi-exponential model This model is simply phenomenological and is used to determine two relaxation times. The time evolution of the OP is expressed as P = 1 exp( t/τ 1) + r exp( t/τ 2 ) 1 + r (3.1) with three fitting parameters, fast and slow relaxation times τ 1 and τ 2, respectively, and the portion of weight r. I (2ω) is fitted using the square of the OP. The experimental results at 42 and 43 K can be well described by eq.(3.1) as shown in Fig these fittings, 3 fitting parameters are obtained:τ 1 = 155 sec, τ 2 = 5030sec and r = at 42 K, and τ 1 = 58.8 sec, τ 2 = 6367 sec and r = at 43 K. τ 1 at 41

42 Figure 3.4: Results of the fitting of time evolutions of SH intensities using the bi-exponential model at 42 K (a) and 43 K (b), respectively. Solid lines indicate the calculated values. 43 K is found to be one third of τ 1 at 42 K. This means that the OP response to E becomes faster at higher temperature above T*. Since all measurements are performed below the transition temperature of 50 K, the temporal development of SH intensities is strongly coupled with the growth of domains with the same polarity of applied electric field. In order to explain the slow kinetics of the OP of KLT-2.6%, we examine 3 theories concerning the domain growth, i.e., the Avrami theory, 1 3 the thermally activated domain growth theory 16 and the Chandra theory, 5,6 and find that the Avrami-Chandra theory can explain the whole present results as follows Avrami theory The Avrami theory is an equation to describe the nucleation and growth process in crystallization. In this theory, it is assumed that the new phase is nucleated by germ nuclei which are embeded in the old phase. In the crystallization process, some germ nuclei become to growth nuclei overcoming an activation energy. To estimate the volume of transformed new phase, he summed up the volume which is transformed from each germ nuclei considering statistically the overlapping area. In the Avrami theory, the total volume v of nuclei is described with fitting parameters k and n as v = 1 exp( kt n ) (3.2) Assuming that the SH intensity is proportional to v, we fit the time evolution of SH intensity I (2ω) using eq.(3.2), where I (2ω) is normalized to the saturated FC value. The results at 42 K and 39.7 K are shown in Fig. 3.5, from which fitting parameters k and n are determined. The same fitting procedure is also performed 42

43 Figure 3.5: Results of the fitting of time evolutions of SH intensities using the Avrami theory at 43 K (a) and 39.7K (b). Solid lines indicate the calculated values. with the data obtained at 43 K, and the results are summarized in Table 3.1. The determined exponent n is about 0.5 above 40 K and between 4 and 5 at 39.7 K. In order to examine the effect of n and k on the relaxation time to the equilibrium state, we calculate v for two cases; (i) changing n with fixed k, (ii) changing k with fixed n. In Fig. 3.6 (a), (b), and (c), the results with fixing k at 0.001, 0.01, and 0.1 are shown, respectively. When n is smaller than 1, v increases rapidly. When n is larger than 1, a long relaxation time appears. In Fig. 3.6 (d) and (e), the results with fixing n at 0.5 and 4 are shown. When n is 0.5, the relaxation time is relatively short, while v increases gradually with reducing k at n = 4. Therefore, n is more effective to the existence of the long relaxation time. Table 3.1: Fitting parameters determined by the Avrami theory Temperature [K] n k k* [sec 1 ] * k* is defined as (k*) n = k to compare the cases with different n. According to Avrami, n changes depending on nucleation and growth velocities. If we presume that nucleation and growth velocities are constant with respect to time, n takes 2, 3 or 4 depending on the domain shape, linear, plate-like and polyhedron, respectively. When the number of nuclei is constant and domains grow with the diffusion-controlled velocity, n takes 0.5. The present analysis dis- 43

44 Figure 3.6: The calculated results using eq. (3.2) with changing n and k. (a) indicates the case of k = 0.001, (b) k = 0.01, and (c) k = 0.1. (d) indicates the case of n = 0.5, and (e) n = 4. closes that the exponent n changes from 0.5 to about 4 at 40 K (T*). This indicates that the temporal behavior above T* is governed by the diffusion-controlled process with a constant nuclei number, while domains grow in the three-dimensional directions with the nucleation number proportional to t below T*. However, when we make close examination on Fig. 3.5, some discrepancies are found between experimental and fitting results, in particular at 39.7 K. The most important point of the present result is the existence of a long incubation time. Thus we take a special notice for the time region around the incubation time to reanalyze the data. The fitting result using the data at times within an hour is shown in Fig When n takes 6.8, the calculated result around the incubation time is better fitted than those of n = 4.8. But this value is not explained by the Avrami theory. In ordinal ferroelectrics, it is known that the switching property is explained by the Kolmogorov-Avrami theory. In this theory, the required time for phase transition is expressed as ( π ) 1 τ = 3 I 4 0 τ 0 (3.3) Here, I 0 is the Maxwell-Boltzmann factor, τ 0 = (Γ 0 ν 3 0 ) 1 4 with the microscopic nucleation rate Γ 0 and the microscopic growth rate ν 0. Eq. (3.3) cannot explain the extremely long switching time and the drastic acceleration of the volume of transformed region. To reproduce the long switching time observed in BTO, Chandra 44

45 Figure 3.7: The fitting result (solid line) at 39.7 K with n = 6.8. adopted a long-range strain effect into the Gibbs free energy as a form of 1 2 K(s i s) 2 to extend the Avrami theory. The strain mismatch produced by the coexistence of a paraelectric and a ferroelectric state is the origin of the long-range force and she succeeded to explain the quite long switching time. She pointed out also that the most striking result is the existence of a critical droplet radius R, which does not exist in the Kolmogorov-Avrami theory. In her theory, the competition between a short-range attractive force and a long-range strain force determines R. When a radius of R is smaller than R, a droplet does not grow with thermal fluctuations. On the contrary, when R is larger than R, the droplet shows an enormous growth. The discrepancy between the Avrami theory and the present experimental results, in particular at 39.7 K, indicates that the extra long incubation time cannot be explained without considering the effect of the long-range strain field. In the Chandra model, the time evolution of the average order parameter φ is determined by the self-consistent equations as ( φ(t) = 1 2 exp 4π 3 T 0 ) I(T t)r 3 (t)dt (3.4) dr dt = (1 + η φ) (3.5) ( 1 1+η φ ) 2 I(t) = I0 (3.6) 45

46 with I 0 exp{ F[φ(R 0 )]/k BT}. (3.7) k B T Here, η = γ/e, and γ expresses the coefficient related with the long-range strain field in the Helmholtz free energy as expressed by eq. (3.8), and e the effective field strength. γ = 1 cp 2 0 ( 2Kq 2 3c e (3c e + K) ) (3.8) where c is the coefficient of the sixth power of the order parameter in the free energy, c e is the shear elasticmodulus,q the electrostrictive coefficient, and P 0 the satulated spontaneous polarization. It should be noted that the Kolmogolov-Avrami theory is the special case of the Chandra theory with η = 0, which means there is no long range force in the system. From eqs. (3.4) (3.7), Chandra described the incubation time τ inc as 1 τ inc /τ I 1 4 (1 η) (3.9) Using reasonable values of coefficients appearing in the above equations, a quite long incubation time whose order is 6 times longer than usual ferroelectric switching time was obtained with η = 0.6. In KLT, its order is still 6 times larger in comparison with BTO. To yield this value, η should be To summarize the above discussion, the Kolmogorov-Avrami theory is not enough to explain the existence of a quite long incubation time. To interpret this observed phenomenon, the long-range strain force to suppress the growth of nuclei should be considered. In relaxor systems, this kind of long-range force is originated from the random force which is produced by the intrinsic heterogeneity in the nature. Here, we would point out that the present results of SHG are different phenomena from the photoluminescence which were reported in KTO and KLT. 23,24 The SHG spectrum spread over only less than 1nm, while that of the photoluminescence spectra excited by the two-photon process spread over 100 nm. Moreover the intensity of the SHG is much stronger than that of photoluminescence induced by the two-photon process, and the latter is hidden in the background of the SHG in the present experiment. Although the SHG and photoluminescence have different origins, i.e, the former being structural and the latter electronic, it is interesting to notice that the cross-over time changes at 40 K in the photoluminescence experiment, 24 which coincides with T in our experiments. The origin of the coincidence is not clear and awaits the further experiments. 46

47 Lastly we would compare the present results with time evolutions in PMN. Although both behaviors of long time evolution of the OP are quite similar, the phenomena observed in KLT-2.6% are much more complicated as described above. In the case of PMN, no critical temperature exists and all time evolutions at various temperatures can be scaled using a normalized time, 11 while such a treatment cannot be applied to KLT-2.6%. New microscopic approaches are expected to explain the whole phenomena in KLT, which would also shed light to the origins of relaxor, in particular the concrete origin of the random force Conclusions In summary, the present experiments reveal that the OP of KLT-2.6% develops with quite long relaxation times under an electric field. This process depends sensitively on temperature, and within a narrow temperature range around 40 K (T ), the behavior drastically changes. Above T, the OP develops soon after the application of the field and then increases gradually to the equilibrium FC value. In 35 K T T, the OP develops rapidly with a long incubation time of several ten minutes. Below 35 K, the OP does not exhibit an increase under an electric filed and the region is non-ergodic in the sense that no path exists from the ZFC to the FC path in the (E, T) diagram. The time evolution of the OP cannot be well fitted by the Avrami theory in particular below T. This phenomenon is explained by introducing the long-range strain field produced by substituted Li ions. 47

48 References 1 M. Avrami, Kinetics of Phase Change. I General Theory, J.Chem.Phys. 7, (1939) 2 M. Avrami, Kinetics of Phase Change. II Transformation-Time Relations for Random Distribution of Nuclei, J.Chem.Phys. 8, (1940) 3 M. Avrami, Granulation, Phase Change, and Microstructure Kinetics of Phase Change. III, J.Chem.Phys. 9, (1941) 4 A. N. Kolmogorov, Bull.Acad.Sci.USSR Phys.Ser. 3, 355 (1937) 5 P. B. Littlewood, and P. Chandra, Delayed Nucleation at a Weakly First-Order Transition, Phys. Rev. Lett. 57, (1986) 6 P. Chandra, Nucleation in the presence of long-range interactions, Phys. Rev. A 39, (1989) 7 Y. Ishibashi, and Y. Takagi, Note on Ferroelectric Domain Switching, J. Phys. Soc. Jpn. 31, (1971) 8 W. Kleemann, and R. Lindner, Dynamic behavior of polar nanodomains in PbMg 1/3 Nb 2/3 O 3, Ferroelectrics 199, 1-8 (1997) 9 K. Fujishiro, T. Iwase T, Y. Uesu, Y. Yamada, B. Dklil, J. M. Kiat, S. Mori, and N. Yamamoto, Optical and Structural Studies of Long-Range Order Development in Relaxor Ferroelectrics, J. Phys. Soc. Jpn. 69, (2000) 10 B. Dkhil, and J. M. Kiat, Electric-field-induced polarization in the ergodic and nonergodic states of PbMg 1/3 Nb 2/3 O 3 relaxor, J. Appl. Phys. 90, (2001) 11 Y. Uesu, K. Ishikawa, and Y. Yamada, Colossal Response Induced by Hetero- Structure Fluctuation, Trans. Mater. Res. Soc. Jpn 28, (2003) 12 U. T. Höchli, P. Kofel, and M. Maglione, Dipolar relaxation and limit of ergodicity in K 1 x Li x TaO 3, Phys. Rev. B 32, (1985) 13 W. A. Kamitakahara, C. K. Loong, G. E. Ostrowski, and L. A. Boatner, Timedependent phase transformation in KTaO 3 :Li, Phys. Rev. B 35, (1987) 48

49 14 W. Kleemann, S. Kutz, F. J. Schafer, and D. Rytz, Strain-induced quadrupolar ordering of dipole-glass-like K 1 x Li x TaO 3, Phys. Rev. B 37, (1988) 15 W. Kleemann, V. Schonknecht, D. Sommer, and D. Rytz, Dissipative quantum tunneling and absence of quadrupolar freezing in glassy K Li TaO 3, Phys. Rev. Lett. 66, (1991) 16 F. Alberici-Kious, J. P. Bouchaud, L. F. Cugliandolo, P. Doussineau, and A. Levelut, Aging in K 1 x Li x TaO 3 : A Domain Growth Interpretation Phys. Rev. Lett (1998) 17 H. Yokota, T. Oyama, and Y. Uesu, Second-harmonic-generation microscope observations of polar state in Li-doped KTaO 3 under an electric field, Phys. Rev. B 72, (2005) 18 R. C. Miller, Optical Harmonic Generation in Sigle Crystal BaTiO 3, Phys.Rev. 134, A1313-A1319 (1964) 19 J. Jerphangon, Invariants of the Third-Rank Cartesian Tensor: Optical Nonlinear Susceptibilities, Phys.Rev. B 2, (1970) 20 G. A. Azzini, G. P. Banfi, E. Giulotto, and U. T. Höchli, Second-harmonic generation and origin of polar configuration in KTaO 3 :Li, Phys. Rev. B 43, (1991) 21 P. Voigt, and S. Kapphan, Experimental study of second harmonic generation by dipolar configurations in pure and Li-doped KTaO 3 and its variation under electric field, J. Phys. Chem. Solids 55, (1994) 22 J. Toulouse, B. E. Vugmeister, and R. Pattnaik, Collective Dynamics of Off- Center Ions in K 1 x Li x TaO 3 : A Model of Relaxor Behavior, Phys. Rev. Lett. 73, (1994) 23 E. Yamaichi, K. Watanabe, and K. Ohi, Photoluminescence in KTaO 3 Single Crystal, J. Phys. Soc. Jpn. 56, (1987) 24 I. Katayama, and K. Tanaka, Nature and Dynamics of Photoexcited States in KTaO 3, J. Phys. Soc. Jpn 75, (2006) 25 V. Westphal, W. Kleemann, and M. D. Glinchuk, Diffuse phase transitions and randomfield-induced domain states of the relaxor ferroelectric PbMg 1/3 Nb 2/3 O 3, Phys. Rev. Lett. 68, (1992) 49

50

51 Chapter 4 PRE-TRANTISITIONAL REGION ABOVE THE POLAR PHASE OF K 1 x Li x TaO 3 It is well established that the relaxor behavior is related to the dynamics of polar nano regions. Here, our finding of the intermediate state just above the polar phase is described. In this state, a net polarization does not appear but a small lattice strain is generated. This peculiar phenomenon is observed through measurement with the second harmonic generation microscope combined with X-ray and neutron scattering experiments. We determine two characteristic temperatures T p and T d for KLT-2.6% and KLT-6.8%. T p is the ferroelectric transition temperature where polar nano regions are transformed to ferroelectric micro-domains, whilet d is the Burns temperature where polar nano regions start to appear. Additional neutron diffraction studies below T p, and neutron diffuse scattering experiments above T p on KLT-6.8% confirm the above idea. Based upon these results, we propose a new model of the polarization state at low temperature in KLT. H. Yokota, Y. Uesu, C. Malibert, and J. M. Kiat Second harmonic generation and X-ray diffraction studies of the pretransitional region and polar phase in relaxor K (1 x) Li x TaO 3 Phys. Rev. B75, (2007) 51

52 4.1 Introduction It is known that the relaxor behavior is strongly connected with the structural inhomogeneity, i.e., the existence of polar nano regions (PNRs) in the cubic matrix. The characteristic dielectric dispersion of relaxors observed in a low frequency region can be explained at least qualitatively by the size distribution of PNRs. Burns et al. proposed that PNRs nucleate at much higher temperature than the dielectric peak temperature. They measured the temperature dependence of the refractive index of the prototype relaxor Pb(Mg 1/3 Nb 2/3 )O 3 (PMN) and found that the refractive index begins to deviate from a linear behavior at the characteristic temperature which is now termed Burns temperature T d. 1,2 They denoted that this deviation is caused by the appearance of PNRs in the non-polar region. Based on their idea, many experiments, in particular neutron diffuse scatterings, have been carried out to confirm PNRs. In chapters 2 and 3, the history dependence and the slow kinetics of the order parameter in KLT were described. 3,4 Because these behaviors are characteristic natures in the prototype relaxors, we definitely concluded that KLT is in the category of relaxor. As for the neutron inelastic scattering, J. Toulouse et al. and S. Wakimoto et al. have already obtained the basic results. 5,6 J. Toulouse et al. measured the diffuse scattering of KLT-6% and 13% and pointed out that the diffuse scattering is only observed around the reciprocal lattice points with at least two or three odd Miller indices. However they only showed a schematic picture of the diffuse scattering distribution. Furthermore they did not mention the Burns temperature. To clarify these points, we perform SHG, X-ray diffraction and neutron scattering experiments on KLT-2.6% and KLT-6.8%. 4.2 Experimental X-ray diffraction measurements X-ray diffraction measurements are performed on a highly accurate two-axis diffractometer with the Bragg-Brentano geometry using Cu-K β wavelength (λ= nm) from an 18 kw rotating anode generator, equipped with a liquid He cryostat. The lattice constants are determined from profiles of (400) reflections using the pseudo Voigt analysis. KLT-2.6% and KLT-6.8% are chosen as specimens. We use (001) plate specimens with an area of 5x3 mm 2 and thickness of 0.65 mm. 52

53 4.2.2 Neutron elastic scattering experiments Neutron diffraction experiments are performed using a triple-axis spectrometer T1-1HQR installed in JRR-3M in Japan Atomic Energy Agency with the incident energy of neutron beams of mev (λ = nm). The horizontal collimator sequence is a monochromator-20 -sample-40 -detector. Sample dimensions are 6.0x5.5x3 mm 3 for KLT-2.6% and 4.5x4.5x3 mm 3 for KLT-6.8%. Temperature dependences of Bragg intensity are measured for each sample. To carry out the neutron diffuse scatterings, it is necessary to determine the exact phase transition temperature of each specimen. For this purpose, we measure temperature dependences of Bragg intensities without an electric field. (220) reflection for KLT-2.6% and (200), (110), and (220) reflections for KLT-6.8% are used. To confirm the existence of thermal hysteresis, the (220) for KLT-2.6% and the (200) for KLT-6.8% are measured on cooling and heating processes Neutron diffuse scattering experiments Neutron diffuse scattering experiments are also performed using the T1-1HQR installed in JRR-3M in JAEA with the incident energy of neutron beams of mev (λ = nm). The horizontal collimator sequence is a monochromator- 40 -sample-20 -analyze- open- detector. The incident slit is 2 x 3 cm 2. Because the horizontal receiving area at a specimen is 4.5 mm, a dispersion angle is lower than For the measurement, a single crystal of KLT-6.8% with 4.5x4.5x3 mm 3 is used. The measurement is performed between 10 and 300 K. After cooling to the lowest temperature, the sample is heated up to 90 K and a temperature is fixed there. The two dimensional distribution of diffuse scattering intensities is obtained in the reciprocal space of h, k with 0.08 step. The exposure time is 600 seconds for each point. To observe the temperature dependence of the diffuse scattering intensity, we choose a reciprocal lattice point of (1.032, 1, 0) where the effect of Bragg scattering is not dominant. In this case, the exposure time is 1200 seconds. Finally, the q dependence of diffuse scattering is measured at 160 K and 84.2 K along the h direction with k = Experimental Results X-ray diffraction measurements Absolute values of lattice constants of pure KTO, KLT-2.6% and KLT-6.8% at room temperature are determined using several (h00) reflections. The result is shown in Fig.4.1 with the previously reported values for comparison. 7 A small dif- 53

54 Figure 4.1: Dependence of the absolute lattice constant a of KLT on Li concentration x at r. t. The solid squares indicate the present results and the open triangles indicate those reported. 7 ference of 0.01% is observed in the absolute values, but the relative variations with the Li concentration are consistent with the reported values. The lattice constant a(x) of KLT-x decreases almost linearly with x. Assuming that the lattice contraction is caused by an effective ionic radius R 0 (Li + ) of Li +, we obtain the following relation, { a(x) = a(pure KTO) 1 + R } R(K + ) x (4.1) where R(K + ) is an ionic radius of K + and From the experiment, we obtain R = R 0 (Li + ) R(K + ). (4.2) R R(K (4.3) ) When we take into account the fact that Li ions occupy off-center positions with deviations δ from the ideal A sites, the effective radius of Li ion is expressed as δ+r(li + )/2. From the present experiment and using the ionic radii of K + (0.164 nm for the coordination number of 12) and Li + ( nm depending on the coordination number), 8 δ is estimated to be nm, which coincides well with the value of 0.12 nm obtained by the NMR measurement. 9 This implies that the deviation δ is equally oriented along the <100> direction in KTO rigid lattices of KLT-2.6% and KLT-6.8%. 54

55 Figure 4.2: X-ray profiles of KLT-2.6% at different temperatures. (a), (b), (c), (d), and (e) are the results at 12, 40, 65, 90, and 120K, respectively. In Fig. 4.2, the spectra of (4 0 0) of KLT-2.6% are shown. It is seen that spectra begin to split as a result of tetragonal deformations at around 100 K and 140 K in KLT-2.6% and KLT-6.8%, respectively. With increasing Li concentration, a split becomes more significant. The temperature dependences of lattice constants of KTO, KLT-2.6% and KLT- 6.8% are plotted in Fig.4.3. These are measured in the ZFC process. We also make the measurements in the FC and FH/FC processes. However, as the remarkable photoconductivity rises with X-ray illumination below 70 K, an effective electric field cannot be applied. 10,11 With cooling the sample, one parameter increases and the other decreases. In the case of KLT-2.6%, the temperature dependence of the lattice volume is shown in Fig.4.4. With cooling the sample, it decreases monotonically. Around 100 K, the lattice volume begins to deviate from the high temperature linear approximation and shows the anomaly at around 50 K. These temperatures coincide well with T d and T p, respectively. The fact, in particular the existence of T d, is also verified by the temperature dependence of the full width at half maximum (FWHM) of (400) spectra as shown in Fig As shown in Fig.4.3, the temperature dependence of the lattice constant of pure KTO is well-fit by the Debye formula, ( ) 3 T θd /T a = a 0 + A 9RT θ D 0 t 3 dt, (4.4) exp t 1 with parameters a 0 = (3) nm, A = 10 7 nm mol/j and θ D = 312(16) K. 55

56 Figure 4.3: Temperature dependences of lattice constants of KTO (open squares), KLT-2.6% (gray triangles) and KLT-6.8% (solid circles). The Debye fitting for KTO is plotted with the dotted line. Although the result agrees qualitatively with the reported values, 3,12 a substantial difference exists: The present experiment using the high-precision diffractometer reveals that tetragonal deformation develops in 2 steps. With decreasing temperature, a small deformation appears at T d ( 100 K for KLT-2.6%, and 140 K for KLT-6.8%) prior to a larger deformation at T p (48 K for KLT-2.6%, and 90 K for KLT-6.8%). The tetragonality t defined by 2(c - a)/(c + a) is plotted together with SH intensities in the FH/FC process in Fig.4.6. It should be noted that the electric-field induced SHG appears at T p and not at T d. This fact strongly suggests that a pretransitional region exists between T d and T p. This region is accompanied with a small tetragonal deformation that is non-polar on average because of a lack of SHG. Below T p, the temperature dependence of tetragonality coincides well with that of SH intensity under an electric field Neutron elastic scattering In Fig. 4.7, temperature dependences of the neutron Bragg reflection intensity are shown. Fig. 4.7 (a) shows the result of the (220) reciprocal point in KLT-2.6%, and (b), (c), and (d) show the results of (200), (110), and (220) in KLT-6.8%, respectively. The Bragg intensity exhibits a remarkable increase at 48 K in KLT-2.6% 56

57 Figure 4.4: Temperature dependence of the lattice volume in KLT-2.6%. Figure 4.5: Temperature dependence of the FWHM in KLT-2.6%. 57

58 Figure 4.6: Temperature dependences of tetragonalities of KLT-2.6% (solid squares) and KLT-6.8% (solid triangles) in the ZFC process and SH intensities of KLT-2.6% (open squares) and KLT-6.8% (open triangles) in the FH/FC process. and 87 K in KLT-6.8%. The result indicates that the phase transition temperatures of KLT-2.6% and KLT-6.8% are 48 K and 87 K, respectively. These temperatures are consistent well with T p where SH intensity disappears as described in chapter 2. From Fig. 4.5 (a) and (b), a thermal hysteresis is not clearly seen in the case of KLT-2.6%, while there exists around 5.7 K of thermal hysteresis in KLT-6.8%. These thermal hysteresis widths agree with those of SHG observations. The large increase of Bragg intensity results from the change in crystal mosaicity between the high and low temperature phases: If the state below T p is a ferroelectric phase accompanied by tiny micro domains, the appearance of the mosicity due to micro-domains could break the secondary extinction rule and Bragg intensity begins to increase Neutron diffuse scattering Because the phase transition temperature of KLT-6.8% is 87 K, the distribution of the diffuse scattering is measured at 90 K. It is known in relaxors that polar nano regions (PNRs) starts to appear at the Burns temperature T d which is higher than the dielectric maximum temperature. Therefore, measurements to observe diffuse scatterings from PNRs should be performed above the phase transition temperature. In Fig. 4.8 (a) and (b), the diffuse scattering distributions around (110) at room temperature and 90 K are shown, respectively. A streak elongating along the [110] direction is observed at both temperatures. The origin is not clear at present. 58

59 Figure 4.7: Temperature dependences of Bragg intensity. (a) (220) of KLT-2.6%, (b), (c), (d) (200), (110), and (220) of KLT-6.8%, respectively. 59

60 However it exists even at room temperature, therefore the existence is not related to the change in the polar state in KLT. So we neglect the effect when we analyze the diffuse scattering due to PNRs. At 90 K, the intensity along [ 110] direction becomes stronger, which signifies the occurrence of the mosicity. From Fig. 4.8 (b), it is observed that the tail of the diffuse scattering intensity expands along the [100] and [010] directions to form a crossed rod. In Fig.4.8(c) and (d), the results of the diffuse scatterings around the (220) reciprocal point measured at room temperature and 90 K are shown, respectively. The diffuse scattering from PNRs does not appear at the (220) reciprocal point. This phenomenon is coincident with the result of Toulouse et al. The temperature dependence of the diffuse scattering at (1.032, 1, 0) reciprocal point is shown in Fig Below 80 K, the diffuse scattering intensity begins to increase with decreasing temperature probably because of the influence of Bragg intensity. With heating the specimen, the diffuse scattering increases abruptly and shows a peak around 85 K, which indicates the first order nature of the ferroelectric phase transition at T p. In Fig. 4.10, the q dependence of the diffuse scattering is shown. Although a clear disappearance of the diffuse scattering intensity which is expected at T d is not observed, the intensity decreases toward 160 K. This fact does not contradict our conclusion that T d is located around 140 K by our X-ray diffraction measurement. 4.4 Analyses The intermediate state of KLT The most important point of this chapter is the finding of existence of pre-transitional state between the non-polar cubic and the polar tetragonal state. This becomes possible as a result of the complementary measurements with different kinds of equipments like a polarity-sensitive SHG microscope and a strain-sensitive X-ray diffractometry. The temperature interval of the intermediate state is determined to be about 50 K for both KLT-2.6% and 6.8%. Dipole moments generated by the off-center of Li ions align along one of the symmetry equivalent < 100> directions in the non-polar cubic matrix. Below T d, each dipole moment begins to correlate and form PNRs as a result of intra-correlations of Li dipoles. This interaction is thought to cause the deviation from the Curie Weiss law for dielectric constants and generate the diffuse scatterings. Further cooling down to T p, PNR starts to interact with each other and the size distribution of PNR appears. This is the origin of the characteristic frequency dispersion of the dielectric response in KLT. Above T p, the polarization does not develop with a long range order, and is restricted in local areas with a random orientation in the non-polar cubic matrix. Thus, very 60

61 Figure 4.8: Two dimensional images of the diffuse scattering of KLT-6.8%. (a), (b) show the results at (110) reciprocal point at room temperature and 90K, (c), (d) correspond to the results at (220) point at room temperature and 90K, respectively. 61

62 Figure 4.9: The temperature dependence of diffuse scattering intensity at ( ) in KLT-6.8% Figure 4.10: The q dependence of the diffuse scattering in KLT-6.8%. The inset figure shows the pure contribution of PNRs to the diffuse scattering, which is obtained by subtracting the background measured at 160 K. 62

63 Figure 4.11: The intensity difference between 160 K and 84.2 K. Solid line indicates the fitting result with using Lorentz function. weak SH intensity is generated. On the other hand, the lattice strains are induced by the average of polarization fluctuation < ( P) 2 > through the electrostrictive effect. Lattice strains x 1, x 3 and tetragonality t are expressed as x 1 = Q 13 < ( P) 2 >, x 3 = Q 33 < ( P) 2 >, and t = x 3 x 1 = (Q 33 Q 31 ) P 2. (4.5) Below T p, the coefficients of electro-strictive Q 13 and Q 33 take the same value as in the paraelectric and intermediate phases, while P is replaced by P s. Because the magnitude of P s is much larger than P, large strains appear and the tetragonality becomes also large below T p Diffuse scattering from PNRs As mentioned above, the interval between two characteristic temperature T d where the tetragonality appears and T p where SHG is generated is about 50 K. In this temperature region, it is expected that PNRs with a small local strain produce the diffuse scatterings. We perform the neutron diffuse scattering in this temperature region and observe the crossed rod-like pattern elongating along the [100] and [010] directions as shown in Fig. 4.8 (b). This pattern is in good agreement with the results of J. Toulouse et al. 5 From the temperature dependence of a diffuse scattering intensity, it shows a peak around 87 K. The increase of the diffuse scattering intensity could be related to the increase of number of PNR. Its sudden drop at T p indicates the change in PNRs to the micro domains with much larger volume. The peak intensity also contains the contribution from the fluctuation of Li dipole moment. From Fig. 4.10, the diffuse scattering intensity at 84.2 K is stronger than that 63

64 of 160 K. Subtracting the result of 160 K as a background, we obtain the q dependence of diffuse scattering intensity as shown in the inset figure. To estimate the correlation length of PNR, we fit the data with the Lorentz function and calculate the FWHM value from Fig The correlation length is calculated with the following equation. ζ = a 1 π FWHM = π = 71.8 [Å]. (4.6) J. Toulouse et al. obtained the correlation length of KLT-6% as 46 Å. The result we obtained is almost the same order that they reported. Further experiments are needed to estimate the exact value of correlation length Polar state of KLT below T p The polar state of KLT at low temperature has not been clearly understood. Here, we propose a new model based on the experimental results of SHG microscope, X-ray diffraction, and neutron elastic and diffuse scatterings. As we described in chapter 2, the order parameter shows a remarkable history dependence. In ZFC process, no significant SH intensity occurs. From the X-ray diffraction measurement, the tetragonality begins to occur below T d in KLT-2.6% and KLT-6.8%. This suggests that a structural transition from cubic to tetragonal would occur with keeping the centro-symmetry. In this case, the point group should be 4/mmm. It is experimentally clarified that no evidences, which show a phase transition, have been observed so far. Therefore, it is concluded that this intermediate state is the heterogeneous region with PNRs. Concerning the phase below T p, the centro-symmetry is broken in ZFH after FC process because strong SH waves are observed. If the ground state is centro-symmetric, the electric-field induced SHG would be vanished as soon as removing an electric field. Since this kind of decline is not observed in our present measurement, the point group below T p should be tetragonal 4m, and the phase is ferroelelectric. In addition to this fact, a quite remarkable increase is observed below T p in the neutron elastic scatterings in KLT. This is originated from the break of the secondary extinction law due to the appearance of micro ferroelectric domains. This explains the fact that no net SHG and no net piezoelectric effect appear in KLT without an electric field, because the size of a polar region is smaller than that of the transverse coherence length of the laser. Based upon these considerations, the polarization states in 3 different processes in KLT are illustrated as shown in Fig In the ZFC process, the high temperature phase is non-polar cubic Pm 3m with uncorrelated individual shifts of Li 64

65 Figure 4.12: Landscape of the polarization states in 3 different processes in KLT. More precise image is shown in Fig.8.2. and below T d, PNRs appear in the cubic matrix. In the vicinity of T d, the density of PNR is not large and each polar region fluctuates independently to form a super-paraelectric state. The macroscopic symmetry of the intermediate phase is tetragonal as a result of the anisotropy of electro-strictive constants (eq.4.5). On approaching to T p, PNRs start to interact with each other to increase the area with size-distribution, which provides characteristic dielectric dispersions. Below T p, a ferroelectric phase transition takes place accompanied by the micro-domain state. In the FC process, the polarization direction is easily aligned below T p and a macroscopic polarization can be produced and maintained down to the low temperature region. In the ZFC process, at a sufficiently low temperature, a random force originating from the random distribution of Li dipoles freezes a polar state, and the electric field cannot change the polarization direction. Thus, in the FH/ZFC process, a macroscopic polarization cannot be produced under an electric field up to a temperature near 40 K, where the thermal excitation overcomes the random force. It should be noted that 40 K is the temperature T which separates the rapid and slow time evolutions of the order parameter described in chapter Conclusions X-ray diffraction studies and SHG observations are performed in KLT-2.6% and KLT-6.8%. As a consequence, we make it clear that strains appear around T d which is about 50 K higher than T p where SH waves are generated. This phenomenon 65

66 suggests that there exists a pre-transitional region between T p and T d. In this region, a macroscopic polarization does not appear but small strains appear. This intermediate state can be disclosed due to the combination of SHGM and X-ray diffraction measurements. Additionally, the existence of polar nano regions in this intermediate state is confirmed by the neutron diffuse scattering experiment. Lastly, we would stress that KLT-2.6% and KLT-6.8% belong to a class of perovskite relaxors where the A-sites are occupied by different kinds of isovalent atoms. Because almost all relaxors have a structure with B-sites occupied by 2 kinds of heterovalent ions, KLT will be an important prototype relaxor for understanding the nature of relaxors. 66

67 References 1 G. Burns, and F. H. Dacol, Glassy polarization behavior in ferroelectric compounds Pb(Mg 1/3 Nb 2/3 )O 3 and Pb(Zn 1/3 Nb 2/3 )O 3, Solid State Commun. 48, (1983) 2 G. Burns, and F. H. Dacol, Crystaline ferroelectric with glassy polarization behavior, Phys. Rev. B 28, (1983) 3 H. Yokota, T. Oyama, and Y. Uesu, Second-harmonic-generation microscopic observations of polar state in Li-doped KTaO 3, Phys. Rev. B 72, (2005) 4 H. Yokota, and Y. Uesu, Extremely slow time evolution of the order parameter under an electric field in relaxor K 0.97 Li 0.03 TaO 3, J. Phys.: Condens. Matter 19, (2007) 5 G. Yong, J. Toulouse, R. Erwin, S.M. Shapiro and B. Hennion, Pretransitional diffuse neutron scattering in the mixed perovskite relaxor K 1 x Li x TaO 3, Phys.Rev. B 62, (2000) 6 S. Wakimoto, G. A. Samara, R. K. Grubbs, E. L. Venturini, L. A. Boatner, G. Xu, G. Shirane, and S. -H. Lee, Dielectric properties and lattice dynamics of Ca-doped K 0.95 Li 0.05 TaO 3, Phys. Rev. B 74, (2006) 7 E. A. Zhurova, V. E. Zavodnik and V. G. Tsirel son, Crystallography Reports. 40, 753(1995) 8 R. D. Shannon, and C. T. Prewitt, Effective ionic radii in oxides and fluorides, Act. Cryst. B 25, (1969) 9 J. J. van der Klink, D. Rytz, F. Borsa, and U. T. Höchli, Collective effects in a random-site electric dipole system: KTaO 3 :Li, Phys. Rev. B 27, (1983) 10 R. S. Klein, G. E. Kugel, M. D. Glinchuk, R. O. Kuzian, and I. V. Kondakova, Photoconductivity in KTaO 3 :Li single crystal, Phys. Rev. B 50, (1994) 11 R. S. Klein, G. E. Kugel, M. D. Glinchuk, R. O. Kuzian, and I. V. Kondakova, Observation and interpretation of photocurrents in KTaO 3 :Li single crystals, Optical Materials 4, (1995) 67

68 12 S. R. Andrews, X-ray-scattering study of the random electric-dipole system KTaO 3 -Li, J. Phys. C: Solid State Phys. 18, (1985) 68

69 Chapter 5 Li CONCENTRATION DEPENDENCE OF DIELECTRIC RESPONSES IN K 1 x Li x TaO 3 In this chapter, results and analyses of dielectric measurements on KLT with different Li concentrations are described. The experimental results are analyzed with the Debye relaxation model and the Vogel-Fulcher law, which have been commonly used to explain the relaxor behavior. From the Cole-Cole plot, we clarify that the dielectric behavior of KLT is quite complicated: at lower Li concentrations, it is mono dispersive in the whole temperature region. On the other hand, three dispersions occur for high Li concentrations. The dispersion at the highest temperature region corresponds to the occurrence of relaxor state that is related to polar nano regions. The dispersion in the intermediate temperature region originates from a ferroelectric phase transition. The dispersion appearing in the lowest temperature region could be related to ferroelectric domain fluctuations. Two semi-circles are observed in the Cole-Cole plots at high temperatures with Li concentrations above 4.4%. These could be associated with the hopping of Li dipoles with 180 and 90 degrees. H. Yokota, A. Okada, I. Ishida, and Y. Uesu Li Concentration Dependence of Dielectric Responses of Quantum Relaxor K 1 x Li x TaO 3 69

70 J. Jpn. Appl. Phys B, (2007) 5.1 Introduction Relaxors are characterized by large and broad dielectric constants with strong frequency dispersion, 1 17 and large piezoelectricity. Because of these features, relaxors attract much attention and have been extensively studied. Nevertheless, the origin of the relaxor s behavior is not clear at present. Most of the investigated relaxors are categorized as lead-oxides with B-sites occupied by two heterovalent cations with different ionic radii. On the other hand, KLT is a relaxor characterized by isovalent cation replacement in A-sites. Thus KLT has a simpler structure than lead-oxide relaxors and is a suitable subject for elucidating the nature of relaxors. In the present chapter, we report the result of dielectric measurements of KLT with different Li concentrations and their analyses using the Debye-type relaxation model and the Vogel-Fulcher law to clarify the effect of Li concentration. 5.2 Experimental Single crystal growth and determination of Li concentration The KLT single crystals with eight different Li concentrations (starting composition ratio x s = 2.5, 5, 7.5, 10, 12.5, 15, 17.5, 25 mol%) used in the present experiments are grown by the self-flux method with a slow-cooling technique of Ta 2 O 5, Li 2 CO 3 and an excess of K 2 CO 3 as a flux. To obtain large single crystals, cooling and heating cycles are repeated above the melting point. This process suppresses the creation of crystalline germs. 32 We obtain plane habit, the maximum size being around 1 cm 3. Li concentration x in in a single-crystal KLT-x (x s = 5, 10, 15%) is determined using the SHG microscope. 33 Some researchers claim that x in can be determined from the temperature T m at which the dielectric constant exhibits maximum. However the dielectric constant of KLT shows frequency dispersion and T m depends strongly on the frequency. Therefore this method cannot be applied to define the phase transition temperature T p. On the other hand, the SHG is effective, because it is quite sensitive to the appearance of macroscopic polarization. In our case, the SHG in zero field heating after the field cooling process can be used to determine T p accurately. The values of our samples are determined by the empirical relation between x in and T p :T p = 535x 2/3 in.34 Judging from the SHG image, T p is distributed around 3 or 4 K in the whole specimen. This determines the accuracy of Li concentration to be ± 0.3 %. The concentration x in of other samples which are not large enough to measure the SH intensity are estimated using the relation- 70

71 ship between x s and x in : x in = 0.35x s. 32 The results are shown in Table 5.1, which shows that this relationship holds well in our results. In the following, we use x in as Li concentration x unless specified otherwise. Table 5.1: Li concentration x s in starting materials for crystal growth, and Li concentration x in in crystals and transition temperature T p. x s [%] x in [%] T p [K] x in was estimated using the empirical relation between x s and x in Dielectric measurements Samples used for dielectric measurements are rectangular (100) plates with the area of about 5x5 mm 2 and the thickness of 0.3 mm. Gold electrodes are evaporated on top and bottom surfaces. Samples are fixed with a holder plate in a cryostat by silver paste. Dielectric measurements are performed using a Solartron SI-1255B frequency response analyzer with the SI-1296 dielectric interface (from 5 Hz to 5 khz) and an HP4192A LCR bridge (from 1 khz to 1 MHz). To check the equivalence of results obtained from these two equipments, measurements are carried out with same frequency of 1 khz for each sample. The discrepancy was found to be below 3 % under the same conditions. The measurements are made during cooling and heating processes to examine thermal hysteresis. No thermal hysteresis is observed in all dielectric measurements. 5.3 Experimental Results The temperature dependences of real (ɛ ) and imaginary (ɛ ) parts of the dielectric constant at 10 khz are shown as a parameter of x in Fig. 5.1(a) and 5.1(b), respectively. Similar behaviors were observed at other frequencies. For x<2.6 %, 71

72 Figure 5.1: Temperature dependences of dielectric constant of K 1 x Li x TaO 3 with different Li concentrations (x = 0.9, 1.6, 2.6, 2.7, 4.4, 5.6, 6.1, 8.8%) at 10 khz. ɛ shows a round peak at T m. Below T m, it decreases sharply, then increases with further decrease of temperature. This phenomenon indicates that the soft-phonon behavior related to quantum paraelectricity is superposed upon the Li dipole ordering with low Li concentrations. A similar phenomenon was also reported for Bi-doped SrTiO With increasing x, Li dipole ordering depresses the effect of zero point fluctuations and the increase of ɛ stops. It is also observed that T m shifts to a higher temperature with x. For x>2.7 %, shoulders appear below and above T m. The shoulder observed above T m shifts to higher temperature with frequency in the same manner as the dielectric peak. The shoulder on the low-temperature side is more clearly observed at a lower frequency. The behavior of ɛ is much more complicated than that of ɛ. When x is low, only two peaks are visible. With increasing x, more than two shoulders and peaks appear at around T m. Figure 5.2 shows the temperature dependence of the dielectric constant of KLT- 0.9% as a parameter of frequency. The characteristic frequency dispersion of relaxors is observed in both ɛ and ɛ around T m. The peak value of ɛ decreases and T m shifts to higher temperature with increasing frequency. It is also found, in the present experiment, that another dielectric dispersion exists at around 90 K which is about 40 K above T m, as shown in the inset of Fig. 5.2(b). This new dispersion is related to the onset of polar nano regions (PNRs) in the cubic matrix. 35 In Fig. 5.3, the temperature dependence of dielectric constant of KLT-4.4% is shown as a parameter of frequency. In ɛ, a small shoulder, which is more striking in higher frequency region, appears above T m. From Fig. 5.3(b), an additional dispersion which is not clearly seen in lower Li concentration becomes obvious. Temperature where a shoulder appears in ɛ is slightly different from the temperature of an additional peak in ɛ. The shoulder of ɛ shifts to higher temperature with the increase of x, while a dispersion observed in ɛ does not strongly depends on x. 72

73 Figure 5.2: Temperature dependence of dielectric constant of KLT-0.9% with the parameter of frequency: (a) ɛ and (b) ɛ. Figure 5.3: Temperature dependence of dielectric constant of KLT-4.4% with the parameter of frequency: (a) ɛ and (b) ɛ. The temperature dependence of the dielectric constant of KLT-6.1% is shown in Fig The difference between KLT-6.1% and KLT-4.4% is that three frequency dispersions appear throughout the entire frequency region in KLT-6.1% but such a phenomenon is not observed in KLT-4.4%. The highest frequency dispersion exhibits the relaxor-like dispersion where peak values of ɛ and ɛ decrease and T m increases with frequency. The second dispersion, which is less marked, corresponds to the ferroelectric phase transition and ɛ shows a sudden drop at the ferroelectric phase transition T p of 75 K in the low-frequency region. The third dispersion observed below T p could be attributed to ferroelectric domain fluctuations. The above experimental results are analyzed using the Cole-Cole plot based on the Debye relaxation model and the Vogel-Fulcher law, which is often used as a criterion of a relaxor. 73

74 Figure 5.4: Temperature dependence of dielectric constant of KLT-6.1% with the parameter of frequency: (a) ɛ and (b) ɛ. 5.4 Analyses Cole-Cole plot analysis The Debye relaxation model gives the following equations for ɛ and ɛ. ɛ = ɛ( ) + ɛ = ɛ(0) ɛ( ) 1 + (ωτ) 2 (5.1) {ɛ(0) ɛ( )}ωτ} 1 + (ωτ) 2 (5.2) Here, ω is the angular frequency, τ is the relaxation time, and ɛ(0) and ɛ( ) are dielectric constants at zero frequency and at infinite frequency, respectively. From eqs. (5.1) and (5.2), we obtain the following Cole-Cole relation. { ɛ (ω) } 2 ɛ(0) + ɛ( ) + ε (ω) 2 = 2 { } 2 ɛ(0) ɛ( ). (5.3) Here, the single dispersion with one relaxation time is assumed. As examples of the Cole-Cole plot analyses, the results for KLT-0.9%, 4.4%, and 6.1% are shown as a parameter of temperature in Fig. 5.5, 5.6, and 5.7, respectively. In KLT-0.9%, only one semicircle (SC-I) is obtained in the whole temperature region. Below 50K and above 70K, it becomes difficult to get the Cole-Cole plot because of a lack of frequency dispersion. With raising the temperature, the radius of semicircle becomes small. At lower temperatures, the shape of semicircle is little bit distorted. In the case of KLT-4.4%, the results are shown for the temperature region between 60 K and 90 K, where the dispersion is clearly observed. Below 65 K, only 74 2

75 Figure 5.5: Cole-Cole plots of KLT-0.9% at several temperatures. Solid square at 50 K, solid circle at 55 K, at 60 K, solid diamond at 65 K,and solid triangle at 70K. Figure 5.6: Cole-Cole plots of KLT-4.4% at several temperatures. at 60 K, solid diamond at 65 K, solid triangle at 70 K, solid circle at 75 K, open square at 80 K, + at 85 K and open diamond at 90 K. Figure 5.7: Cole-Cole plot of KLT-6.1% at several temperatures. at 60 K, solid triangle at 70 K, open square at 80 K, open circle at 90 K, solid square at 100 K, + at 110 K, * at 120 K, and solid diamond at 130 K. 75

76 one semicircle (SC-I) is clearly observed. With increasing temperature, another semicircle (SC-II) appears at 75 K and SC-I becomes smaller and finally disappears at 105 K. This phenomenon is not observed in KLT when x is less than 4.4%. In KLT-6.1%, both SC-I and SC-II are observed. Above 110 K, a small semicircle (SC-I) is obtained. With decreasing temperature, the radius becomes larger and the position of the center of the semi-circle shifts to higher value of ɛ. At 100 K, a part of SC-II becomes visible on the left side of the SC-I. With cooling the sample, SC-II becomes clear and its radius gets larger. Below 70 K, SC-I almost disappears. We summarize here the results of dielectric dispersions of KLT with different x: In KLT with a lower Li concentration, only SC-I is observed in the whole temperature region. Above 4.4% of Li concentration, SC-II appears in low temperature region. Because the SC-I is observed in all samples, it corresponds to the appearance of PNRs. The temperature where SC-I appears shows a good agreement with T d. With decreasing the temperature, the size of PNRs distributes and makes the SC-I distorted. It is interesting to note that the SC-I is smoothly changed into the characteristic relaxor behavior. On the other hand, the origin of SC-II would be related to the appearance of ferroelectricity. The reason why the SC-II does not appear below KLT-4.4% is not clear at present. The possible reason is that at lower Li concentration the effect of quantum paraelectricity becomes stronger and the SC-II is hidden in the sea of the quantum paraelectricity. Another reason would be that KLT with low Li concentration is canonical relaxor which does not exhibit ferroelectric transition. However, this assumption contradicts with the neutron diffraction experiments as described in chapter 4. In the single-dispersion system, it is known that the center of the semicircle is on the lateral axis. In KLT, this kind of semicircle does not appear in any temperature region. This indicates that the behavior of the dielectric constant in KLT cannot be explained by the simple single-dispersion state. Thus we fit the data taking into account the multi-dispersion system expressed by ɛ = ɛ(0) + ɛ(0) ɛ( ) 1 + (iωτ) β. (5.4) Here, β is a parameter ranging from 0 and 1. In KLT, β takes between 0.45 and 0.90 and it does not clearly depend on the Li concentration. With raising the temperature, the value of β shows a slight increase. This fact means that the dielectric response of KLT becomes mono-dispersive at higher temperatures. From the temperature dependence of relaxation time τ, activation energy E a and characteristic frequency ν 0 are deduced using the Arrhenius law expressed by ν = ν 0 exp[ E a /k B T]. (5.5) 76

77 The result is shown in Table 5.2, which shows that there exist two relaxation processes of Li dipole moments with different E a. This phenomenon has already been discussed by Pattnaik et al. 36 and is attributed to Li dipole hopping of 90 degrees (π/2 relaxation) and of 180 degrees (π relaxation) among 6 equivalent positions. Judging from the fact that the activation energy of SC-II is three times as large as that of SC-I, SC-I corresponds to π/2 relaxation and SC-II to π relaxation. It is also found that E a does not show a clear dependence on Li concentration. This result is consistent with those in ref. 28. On the contrary, ν 0 scatters with respect to x. This is probably due to the fact that ν 0 is sensitive to experimental errors in the Cole-Cole plot fittings. Table 5.2: Activation energy and characteristic frequency of KLT with different x determined using the Cole-Cole plot and the Vogel-Fulcher law. Cole-Cole plot Vogel-Fulcher law x [% ] E a [K] ν 0 [10 11 Hz] E a [K] ν 0 [10 11 Hz] T VF [K] , , , , E a of x>4.4 % exhibits two values corresponding to two semicircles on the Cole-Cole plots Vogel-Fulcher analysis The Vogel-Fulcher law expresses the relationship between T m and the corresponding frequency ν as expressed by ν = ν 0 exp[ E a /k B (Tmax T VF )]. (5.6) Here, T VF is the Vogel-Fulcher temperature at which the dynamics of Li dipoles is frozen. The experimental results well fit this equation; the fitting parameters are tabulated in Table 5.2. It is shown that T VF increases monotonically with x, while ν 0 changes dramatically at the critical concentration of 3 %. On the contrary, E a does not show any dependence on x. 77

78 5.5 Conclusions Single crystals of KLT with different Li concentrations were grown by the self-flux method. Li concentrations in the specimens were determined using the empirical relation between x and T p, and T p was determined by SHG microscopic observation. With increasing x, a clear dielectric dispersion which collates with the occurrence of PNRs appears above T m. At high Li concentration, the sudden drop of ɛ occurs as a result of a ferroelectric phase transition. In addition to this, the third dispersion corresponds to the domain fluctuation appears. The results are analyzed using the Cole-Cole plot and the Vogel-Fulcher law. From the Cole-Cole plot, two semicircles which relate to dynamics of PNR and the domain fluctuation are obtained at high Li concentration. The analyses using the Vogel-Fulcher law disclose that ν 0 and T VF change at x of about 3 %. 78

79 References 1 Z. G. Lu, and G. Calvarin, Frequency dependence of the complex dielectric permittivity of ferroelectric relaxors, Phys. Rev. B51, (1995) 2 C. S. Hong, W. C. Su, R. C. Chang, H. H. Nien, and Y. D. Juang, Dielectric behavior of Pb(Fe 2/3 W 1/3 )-PbTiO 3 relaxors : Models comparison and numerical calculations, J. Appl. Phys. 101, (2007) 3 D. U. Spinola, I. A. Santos, L. A. Bassora, J. A. Erias, and D. Garcia, Dielectric Properties of Rare Earth Doped (Sr, Ba)Nb 2 O 6, Ceramics, J. Euro. Ceram. Soc. 19, (1999) 4 X. Wei, and X. Yao, Analysis on dielectric response of polar nanoregions in paraelectric phase of relaxor ferroelectrics, J. Appl. Phys. 100, (2006) 5 A. A. Bokov, and Z. -G. Ye, Phenomenological description of dielectric permittivity peak in relaxor ferroelectrics, Solid State Commu. 116, (2000) 6 S. Said, J. P. Mercurio, Relaxor behaviour of low lead and lead free ferroelectricceramics of the Na 0.5 Bi 0.5 TiO 3 -PbTiO 3 and Na 0.5 Bi 0.5 TiO 3 -K 0.5 Bi 0.5 TiO 3 systems, J. Euro. Ceram. Soc. 21, (2001) 7 Z. Yu, A. Z. Jing, P. M. Vilarinho, and J. L. Baptista, Dielectric properties of Ba(Ti, Ce)O 3 from 10 2 to 10 5 Hz in the temperature range K, J. Phys.: Condens. Matter 9, (1997) 8 S. J. Butcher, and N. W. Thomas, Ferroelectricity in the system Pb 1 x Ba x (Mg 1/3 Nb 2/3 )O 3, J. Phys. Chem. Solids 52, (1991) 9 M. wolters, and A. J. Burggraaf, Relaxational polarization and diffuse phase transitions of La-substituted Pb(Zr, Ti)O 3 -ceramics, Mat. Res. Bull. 10, (1975) 10 B. Jimenez, C. Alemany, J. Mendiola, and E. Maurer, Phase transitions in ferroelectric ceramics of the type Sr 0.5 Ba 0.5 Nb 2 O 6, J. Phys. Chem. Solids 46, (1985) 11 T. W. Cline, L. E. Cross, and S. T. Liu, Dielectric behavior of strontium barium niobate (Sr 0.5 Ba 0.5 Nb 2 O 6 ) crystals, J. Appl. Phys. 49, (1978) 12 K. Keizer, G. J. Lansink, and A. J. Burggraaf, Anomalous dielectric behaviour of La(III) substituted lead titanate ceramics, J. Phys. Chem. Solids. 39, (1979) 79

80 13 F. D. Morrison, D. C. Sinclair, and A. R. West, Electrical and structural characteristics of lanthanum-doped barium titanate ceramics, J. Appl. Phys. 86, A. A. Bokov, Y. H. Bing, W. Chen, Z. -G. Ye, S. A. Bogatina, I. P. Raevski, S. I. Raevskaya, and E. V. Sahkar, Empirical scaling of the dielectric permittivity peak in relaxor ferrelectrics, Phys. Rev. B 68, (2003) 15 A. A. Bolov, and Z. -G. Ye, Universal relaxor polarization in Pb(Mg 1/3 Nb 2/3 )O 3 and related materials, Phys. Rev. B 66, (2002) 16 A. A. Bolov, and Z. -G. Ye, Freezing of dipole dynamics in relaxor ferroelectric Pb(Mg 1/3 Nb 2/3 )O 3 -PbTiO 3 as evidenced by dielectric spectroscopy, J. Phys.: Condens. Matter 12, L541-L548 (2000) 17 V. P. Bovtoun, an M. A. Leshchenko, Two dielectric contributions due to domain/cluster structure in the ferroelectrics with diffused phase transitions, Ferroelectric 190, (1997) 18 D. Vielahnd, S. J. Jang, L. E. Cross, and M. Wutting, Freezing of the polarization fluctuations in lead magnesuium niobate relaxors, J. Appl. Phys. 68, (1990) 19 D. Vieland, S. J. Jang, and L. E. Cross, Local polar configurations in lead magnesium niobate relaxors, J. Appl. Phys. 69, (1991) 20 D. Vielhand, S. J. Jang, L. E. Cross, and M. Wutting, Deviation from Curie- Weiss behavior in relaxor ferroelectrics, Phys. Rev. B 46, (1992) 21 W. B. Zhang, X. W. Zou, Z. Z. Jin, and D. C. Tian, Dynamic heterogeneous structure relaxation of supercooled liquids, Phys. Rev. E 61, (2000) 22 Y. Moriya, H. Kawaji, T. Tojo, and T. Atake, Specific-Heat Anomaly Caused by Ferroelectric Nanoregions in Pb(Mg 1/3 Nb 2/3 )O 3 and Pb(Mg 1/3 Ta 2/3 )O 3, Phys. Rev. Lett. 90, (2003) 23 L. E. Cross, Relaxor Ferroelectrics, Ferroelectrics 76, (1987) 24 A. J. Bell, Calculation of dielectric properties from the super-paraelectric model of relaxors, J. Phys.: Condens. Matter 5, (1993) 25 X. Long, and Z. -G. Ye, Relaxor behavior in the solid solution between dielectric Ba(Mg 1/3 Nb 2/3 )O 3 and ferroelectric PbTiO 3, Appl. Phys. Lett. 90, (2007) 80

81 26 C. Ang, and Z. Yu, Dielectric relaxor and ferroelectric relaxor: Bi-doped paraelecric SrTiO 3, J. Appl. Phys. 91, (2002) 27 C. Ang, and Z. Yu, Phonon-coupled impurity dielectric modes in Sr 1 1.5x Bi x TiO 3, Phys. Rev. B 61, (2000) 28 C. Ang, Z. Yu, J. Hemberger, P. Lunkenheimer, and A. Loidl, Dielectric anomalies in bismuth-do ped SrTiO 3 : Defect modes at low impurity concentrations, Phys. Rev. B 59, (1999) 29 Z. Y. Cheng, R. S. Katiyar, X. Yao, and A. S. Bhalla, Temperature dependence of the dielectric constant of relaxor ferroelectrics, Phys. Rev. B 57, (1998) 30 Z. Y. Cheng, R. S. Katiyar, X. Yao, and A. Guo, Dielectric behavior of lead magnesium niobate relaxors, Phys. Rev. B 55, (1997) 31 Z. Y. Cheng, L. Y. Zhang, and X. Yao, Investigation of glassy behavior of lead magnesium niobate relaxors, J. Appl. Phys. 79, (1996) 32 J. J. Van der Klink, and D. Rytz, Growth of K 1 x Li x TaO 3 crystals by a slowcooling method, J. Cryst. Growth 56, (1982) 33 Y. Uesu, S. Kurimura, and Y. Yamamoto, Optical second harmonic images of 90 domain structure in BaTiO 3 and periodically inverted antiparallel domains in LiTaO 3, Appl. Phys. Lett. 66, (1995) 34 J. J. van der Klink, D. Rytz, F. Borsa, and U. T. Höchli, Collective effects in a random-site electric dipole system: KTaO 3 :Li, Phys. Rev. B 27, (1983) 35 H. Yokota, Y. Uesu, C. Malibert, and J. M. Kiat, Second harmonic generation and X-ray diffraction studies of the pretransitional regionand polar phase in relaxor K (1 x) Li x TaO 3, Phys. Rev. B 75, (2007) 36 R. K. Pattnaik, J. Toulouse, and B. George, Relazation of Li-dipole pairs in the disordered perovskite K 1 x Li x TaO 3 and the effect of external electric field, Phys. Rev. B 62, (2000) 81

82

83 Chapter 6 QUANTUM PARAELECTRIC / RELAXOR-FERROELECTRIC CROSSOVER REGION OF K 1 x Li x TaO 3 In order to clarify the polarization state in the quantum paraelectric and relaxor crossover region, we perform dielectric measurements, neutron inelastic scattering experiments and SHG microscope observations on KLT with different Li concentrations. This chapter describes the results and analyses of these experiments. With an increasing temperature, the quantum paraelectricity that dominates at low temperatures begins to disappear: A dielectric constant of KLT with low Li concentrations decreases monotonically with increasing temperature. We analyze these results with the Barrett formula and determine the parameters T 1 and T C which characterize the quantum paraelectricity. These parameters change the tendencies at a specific Li concentration. This means that the effect of zero point vibration becomes weaker above this concentration. We also measure temperature dependences of transverse optic (TO) and transverse acoustic (TA) phonons. TO and TA phonons show no softening around the dielectric peak temperatures. Therefore, the ferroelectric phase transition of KLT is not displacive but relaxational. To interpret the appearance of the dielectric peak, we propose a two state model where the local field is taken into account. The peak shift of the dielectric constant with increasing Li concentration can be qualitatively explained by this model. 83

84 6.1 Introduction The temperature dependence of the dielectric constant of quantum paraelectrics can be described by the Barrett formula, 1 which is obtained by extending the Slater potential energy written for BaTiO 3 (BTO) by a quantum-mechanical treatment. 2 Here, we should notice that the treatment by Barrett is different from an unharmonic phonon approach which is sometimes used in the textbook of ferroelectrics. Thus, in this chapter we take fully into account this difference when we apply the Barrett formula to analyze the dielectric constant and transverse optic (TO) phonon of KTO and KLT. KLT-x% is believed to be a quantum paraelectric for 0<x<2 %, while a dipolar glass 3 9 or a ferroelectric state for 2 <x<6 % A small but distinct tetragonal lattice distortion was observed in KLT with Li concentration larger than 2.6 % by X-ray diffraction experiments. 13,14 Furthermore, with higher Li concentration and at higher temperature regions, KLT displays relaxor behaviors characterized by remarkable frequency dispersion, 15 the appearance of polar nano-regions, 16,17 the history dependence of the order parameter 18 and its slow time-evolution. 19,20 Although extensive research has attempted to clarify the complicated dielectric behaviors in KLT, the cross-over region from the quantum paraelectric to ferroelectric state is not fully understood. The investigation in this region is important for elucidating the origin of quantum paraelectricity and its impurity effect. As a result, we precisely measured the temperature dependence of KLT dielectric constant, paying attention to KLT with low Li concentration, and temperature dependences of low-lying TO and transverse acoustic (TA) phonons by neutron inelastic scattering experiments. From these experiments, characteristic parameters in the Barrett formula are determined from dielectric and phonon experiments as a function of the Li concentration and compared with the results of SrTiO 3 (STO). TA and TO phonons show no softening at the ferroelectric phase transition temperature T p in KLT. This fact confirms the relaxational nature of KLT at T p. Based upon this experimental fact, a simplified two-state model is proposed to describe the temperature dependence of dielectric constant near T p. 6.2 Experimental Neutron inelastic scatterings Neutron inelastic scattering experiments are performed using triple-axis spectrometers T1-1HQR and 4G-GPTAS installed in JRR-3M in Japan Atomic Energy Research Agency with the incident energy of neutron beams of mev (λ = Å) and mev (λ = Å), respectively. For these measurements, the hori- 84

85 zontal collimator sequence is monochromater (fixed) sample analyzer detector. The constant Q scanning is adopted. For KLT-6.8%, TA branch at (2, Z, 0) with Z = -0.05, -0.10, -0.15, at 10, 30, 60, 74, 79, 86, 91, 95, 101 K are measured. The dimensions of samples are 17x17x4 mm 3 for KTO, and 15x10x5 mm 3 for KLT-2.6%, and 4.5x4.5x3 mm 3 for KLT-6.8%. T p of KLT-2.6% and KLT-6.8% are determined by abrupt changes of (200) and (220) neutron Bragg reflection intensities originating from the break of the secondary extinction law due to the appearance of micro ferroelectric domains below T p. The determined T p, 48 K for KLT-2.6% and 90 K for KLT-6.8% coincide well with those determined by SHG measurements. For the measurement of TO branch of KTO, the reciprocal point of (2, Z, 0) with Z = -0.05, , is adopted. Here, Z is a relative reciprocal point in a Brillouin zone measured by a reciprocal unit vector b, where the phonon wave vector q is expressed as q = Z b. The measurements are performed at 10, 20, 30, 40, 50, 75, 100, and 125 K. The TO phonon spectrum is also measured at (2, 2, 0) reciprocal point at 10, 20, 30, 40, 50, 75, 100 and 125 K. In the case of KLT-2.6%, both TO and TA phonons are measured; TA phonon of (2, -0.1, 0) at 9, 20, 35, 42, 46, 50, 55, 60, 70, 80, 100 K, and TO phonon of (2, 2, 0) at 20, 25, 35, 42, 45, 46, 50, 55, 60, 70, 80, 100 K, and (2, -2, 0) at 11, 43, 55, 60, 72, 77, 101, 150, 176 K Dielectric measurements For dielectric measurements, (100) plate samples with edges parallel to the <100> are used. The Li concentrations of the samples are x = 0, 1.5, 2.4, 2.6, 3.3, 4.4, 5.6, 6.1, and 8.8%. The experimental details are described in chapter 5. Dielectric measurements are performed using two equipments: a Solartron SI-1255B frequency response analyzer with SI-1296 dielectric interface (from 5 Hz to 1 khz) and an HP 4192A LCR bridge (from 1 khz to 1 MHz). Temperature dependence of dielectric constants is measured in the range of 18 K to 290 K on cooling and heating processes. 6.3 Experimental results Temperature dependences of TO and TA phonons Well defined peaks of TO phonons of KTO and KLT-2.6%, and TA phonons of KTO, KLT-2.6% and KLT-6.8% are observed in the measured temperature range. Examples of TA and TO phonon dispersions are shown in Fig FWHM values of TO and TA phonons increase with Li concentration. This is due probably to the lowering of the single crystal quality with Li substitution. It should be noted 85

86 Figure 6.1: Profiles of neutron inelastic scattering of KTO, KLT-2.6% and KLT-6.8%. (a) indicates the TO branch of KTO around (2, 0, 0) at 30 K, (b) the TA phonon of KTO around (2, -0.1, 0) at 30 K, (c) TO phonon of KLT- 2.6% around (2, 2, 0) at 30 K, (d) TA phonon of KLT-2.6% around (2, -0.1, 0) at 30 K, and (e) TA phonon of KLT-6.8% around (2, -0.1, 0) at 30K. 86

87 that energies of TO and TA phonons increase with the Li concentration, and TO phonons of KLT-6.8% could not be detected because the phonon energy is beyond the energy limit of neutron spectrometers used in the present experiments. TO phonon temperature dependences of KTO, KLT-2.6%, and TA phonons of KTO, KLT-2.6% and KLT-6.8% are shown in Fig. 6.2 (a) and (b), respectively. The result of KTO coincides well with previous reports and shows the typical quantum paraelectric behavior: 21,22 The phonon energy decreases with decreasing temperature but remains at finite values down to 0 K. Interestingly, the curve of low-lying TO phonon of KLT-2.6% lies constantly higher by about 1 mev, as compared to KTO throughout the observed temperature range. This is in complete disagreement with the expectation that the critical softening of the optical mode is assumed to be the driving mechanism of the phase transition. We could not observe the split to A 1 and E branches below T p, mainly because of the weak intensity of inelastically scattered neutrons due to the coexistence of high temperature and ferroelectric phases at low temperatures. TA phonon frequencies of 3 crystals also decrease monotonically with decreasing temperatures and no anomaly is observed in the entire temperature region. However the FWHM of TA phonon of KLT-2.6% increases below T p as shown in Fig.6.2(c), which signifies the splitting of two branches, A 1 and E modes, and a ferroelectric phase transition from cubic m 3m to tetragonal 4mm takes place Temperature dependence of dielectric constant Figure 6.3 shows temperature dependences of real part of the relative dielectric constant of KLT at 1 MHz as s parameter of Li concentration. It is shown that the substitution of Li ions induces a peak and the peak temperature shifts higher with increasing Li concentration x. This phenomenon is also observed for all other frequencies. In lower temperature regions, the dielectric constant of KLT decreases monotonically with increasing temperature from 0 K, then begins to increase and has a peak as shown in an inset of Fig.6.3. This tendency is observed in KLT particularly with lower x and when x exceeds 4.4%, the tendency disappears. Thus we define tentatively the cross-over region as 0< x <4.4 % and intensive studies are made in this region. 6.4 Analyses The Barrett formula for dielectric constant and TO phonon Barrett extended the Slater s theory of BTO and leaded an expression for the dielectric constant of quantum paraelectrics. (He himself did not use the term quantum 87

88 Figure 6.2: Temperature dependences of TO phonon energies of KTO and KLT-2.6% observed at (2, 2, 0) (a), TA phonon energies of KTO, KLT-2.6% and KLT-6.8% at (2,-0.1,0) (b) and full widths of half maximum of TA phonon profiles of KLT-2.6% (c). Solid square KTO, open diamond KLT-2.6% and open triangle KLT-6.8%. The arrows indicate ferroelectric phase transition temperatures for each sample. Figure 6.3: The temperature dependence of real part ɛ of the complex dielectric constant of KTO and KLT-x% at 1MHz as a parameter of Li concentration. Solid lines are drawn for guide to eyes. The inset figure is an enlarged figure of KLT. 88

89 paraelectricity ). 1 In Slater s theory, 2 Ba and O ions are fixed at the ideal positions and only Ti ions are treated as isolated harmonic oscillators. The potential energy of the Ti ion in an external electric field is expressed as φ = A(x 2 + y 2 + z 2 ) + B 1 (x 4 + y 4 + z 4 ) + 2B 2 (x 2 y 2 + y 2 z 2 + z 2 x 2 ) q(xe x + ye y + ze z ). (6.1) Barrett treated the potential energy using quantum mechanics and calculated the energy perturbation from which the partition function was obtained. 1 Following the usual process of statistical physics and taking the Lorentz correction into account, he obtained the temperature dependence of dielectric constant known as the Barrett formula: M ɛ = T 1 2 coth ( T 1 2T ) TC. (6.2) Here T 1 = ω/k B with phonon angular frequency ω. T 1 is the temperature where the quantum effect becomes significant. T C is the hypothetical Curie temperature. Concerning the temperature dependence of a low-lying TO phonon, a similar equation to eq.(6.2) is obtained for the square of phonon angular frequency ω q from the Hamiltonian expanded by phonon normal coordinates Q with the mean field approximation. 23,24 Under the assumption of displacive transition, the potential energy V(Q) is expressed as V(Q) = 1 2 mω2 0 Q γq4. (6.3) In the eq. (6.3), the first term denotes the elastic energy of the short range force with the nearest neighbor atoms, and the second term means an unharmonic oscillation. Using the mean field approximation, following relationships are obtained. < Q > 0 = Q 0 < Q 2 > 0 = Q δ < Q 3 > 0 = Q δQ 0 < Q 4 > 0 = Q δQ δ2 (6.4) where Q 0 = < Q > 0, and δ =< Q 2 > 0. Substituting eq.(6.4) into eq.(6.3), we obtain V(Q) = 1 2 m(ω γδ)Q γq mω2 0 δ γδ2. (6.5) The stabilization condition gives (mω γδ + γq2 0 )Q 0 = 0. (6.6) 89

90 From the equipartition law, the following relationship is obtained. < Q 2 > cl = k B T/mΩ 2 (T). (6.7) Since the TO phonon of the Brillouin zone center approaches to zero near the ferroelectric phase transition temperature of T C, a soft phonon frequency ω q (T) can be written as mω 2 q=0 = 3γ(δ δ C) = 3γk B T T C mω(t). (6.8) Using the mean field theory, the dynamic susceptibility is expressed as χ q (ω, T) = 1 m ( (6.9) ω 2 q ω2). At ω = 0, eq.(6.8) and eq.(6.9) leads the Curie-Weiss law of χ q=0 (0, T) = 1 A(T T C ) (6.10) where A = 3γk B /mω 2. At low temperature region, we have to treat the system quantum mechanically and < Q 2 > in eq.(6.4) is described as δ(t) =< Q 2 > qu = { } 2 2mΩ exp( Ω/k B T) (6.11) Compare the case of the equi-partition law (eq.(6.7)). From eq.(6.9), eq. (6.8), and eq.(6.11), we obtain finally mω 2 (T) = 1 A ( T1 2 coth T ) 1 2T T C. (6.12) This relation is illustrated in Fig. 6.4: The softening of TO phonon toward T C is upheaved by the zero point vibration and prevents the ferroelectric phase transition. Eq.(6.2) and eq.(6.12) apparently satisfy the Lyddane-Sachs-Teller relationship. However, because these expressions stand on different deduction processes with different potential energies, special care should be taken, when we compare the parameters determined by the two equations. We will discuss this point later Experimental determination of the Barrett parameters Below T m, the Barrett formula is applied to experimental results of dielectric behavior of KLT with low x. Since the frequency dispersion is found to be less than 4 % for all samples, the fittings are performed for the results measured at 1 MHz. 90

91 Figure 6.4: Illustration of the effect of quantum fluctuation in quantum paraelectrics. The results are shown in Fig.6.5 for 5 samples with different x. Obtained parameters are summarized in Table 6.1, where previous results of KTO and STO are shown for comparison As shown in Fig.6.5, the coincidence between experimental and calculation results are excellent in low temperature regions. However, disagreement is found at higher temperatures in KLT due to the fluctuation of Li dipole moment. In Fig.6.6(a) and (b), Li concentration dependence of fitting parameters T 1 (a) and T C (b) are shown. Positive T C of KTO becomes negative and decreases further with the increases in x. This result coincides with the observation of TO phonon hardening with Li substitution as shown in Fig.6.2(a). Fig.6.6(a) shows that the x dependence of T 1 changes around 2.5 %. T 1 increases monotonically with x, but decreases with x above 2.6 %. This fact suggests that above 2.6 % of the Li concentration, cooperative movements of Li dipole moments become dominant over the quantum fluctuations. This tendency appears also in the x -dependence of T C as shown in Fig.6.6(b). Therefore, application of the Barrett formula is limited below the specific concentration of 2.5 %. Interestingly, around this concentration the order of ferroelectric phase transition changes from 2nd to 1st which is obtained by SHG measurements. 14 (chapter 7) The same analysis is performed for the temperature dependence of TO phonon using eq.(6.12). In this fitting procedure, the data number of KLT-2.6% is so limited to perform accurate analyses, and for KLT-6%, the concentration is too high to be beyond the limit of the Barrett formula. Therefore, we cannot apply the Barrett formula to these crystals. Fitting result of KTO is shown in Fig.6.7 and 91

92 Figure 6.5: Temperature dependences of ɛ of KTO (a), KLT-1.5% (b), KLT-2.4% (c), KLT-2.6% (d) and KLT- 3.3% (e). Solids lines are the fitting curves with the Barrett formula. Figure 6.6: Li concentration dependence of the Barrett parameters T 1 (a) and T C (b). 92

93 Table 6.1: Barrett parameters determined from the dielectric measurements. M T 1 T C ω [x10 12 Hz] KTO KLT-1.5% KLT-2.6% KLT-2.8% KLT-3.3% KTO ref. 26 KTO ref. 27 KTO ref. 28 STO ref.29 STO ref. 30 STO ref. 24 obtained parameters are summarized in Table 6.2 together with that of STO. It is noticeable that values of the Barrett parameters of KTO determined by two experiments, i.e., dielectric and phonon measurements are quite different. Muller and Burkard suggested that the single-mode approximation in the Barrett formula is not sufficient and the coupling of other modes is necessary to formulate the exact expression of quantum paraelectricity. 24 Their idea motivated to derive modified expressions which take into account the mode coupling, although these approaches are phenomenological As for the disagreement of the Barrett parameters in dielectric and TO phonon measurements of KTO, the single-mode treatment in the Barrett formula could be the most probable origin of disagreement. In particular, the analysis of a low-lying TO phonon could be suffered much by the incomplete treatment, while the analysis of dielectric constant is not significantly affected because the experimentally obtained dielectric constants include mode coupling to other phonons, as expressed by Kurosawa 35 and Fleury et al. 36 As a consequence, the mode coupling effect would be squeezed into the Barrett parameters without modifying the Barrett formula. In Table 6.1, the frequency of zero-point vibration determined from T 1 is tabulated. The average frequency is 1.6x10 12 Hz which contributes the upheaval of TO phonon (see Fig. 6.4). It is interesting to compare results of KTO with that of STO. In STO, the Barrett parameters determined by the dielectric constant nearly coincides with those determined by phonon measurements as shown in Table 6.1 and 6.2. It indicates that the mode coupling effect of KLT is larger than STO. This is supported by 93

94 Figure 6.7: Temperature dependence of TO phonons of KTO, KLT-2.6% and KLT-6%. Results of the THz spectroscopy (ref.31) are also shown for the sake of comparison. The solid line is the fitting result of KTO using eq. (6.12). neutron inelastic measurements performed by Axe et al. 22 They suggested that the mode coupling between TO and TA phonons are important to explain the phonon dispersions. Table 6.2: Barrett parameters determined from TO phonon measurements. ma T 1 T C KTO STO Two-state model As mentioned above, the ferroelectricity induced by Li substitution in KLT is not associated with the lattice instability due to a soft TO phonon, but the Li dipole relaxation mode could be a most probable origin. As a result, we propose a classical two-state model to describe the order-disorder state, and explain qualitatively the appearance of dielectric peak and the shift of peak temperature with Li concentration. 94

95 Let us assume the numbers of Li dipole µ directed toward +z and -z directions to be x + and x respectively, and the total number of dipole moment be x = x + + x. Under an external electric field E ex, the polarization P is given by P = x + µ x µ = xµ tanh { µ k B T ( E ex + γ 3ɛ 0 P )} (6.13) where γ means the Lorentz correction factor. With α = µ/k B T and Γ = γ/3ɛ 0, the susceptibility is expressed by ( ) P ɛ 0 χ = = xµαsech2 {α(e ex + ΓP)} E 1 xµαsech 2 (6.14) {α(e ex + ΓP)} For T>T C, eq.(6.14) becomes where ɛ 0 χ = xµ2 /k B T T C (6.15) T C = xγµ2 k B (6.16) This is the Curie-Weiss law. Below T C, spontaneous polarization in the vicinity of T C is given by P 2 s = 3k3 B xγ 3 µ 4 T 2 C (T C T). (6.17) Substituting eq.(6.17) into eq.(6.14), we obtain the susceptibility as ɛ 0 χ = T C 2Γ(T C T) (6.18) The shift of T C to higher temperatures with increasing Li concentration can be explained by eq.(6.16). In Fig.6.8, calculated results of the temperature dependence of dielectric constant are shown as a parameter of x. This figure grasps substantial features of experimental results shown in Fig Using eq.(6.16), the Lorentz correction factor Γ is estimated to be between 0.15 and 0.19 from T C determined by the SHG measurement and a Li dipole moment of the point charge model. 18 The first principle calculation determined also Γ by taking into account the interaction between TO phonon and Li dipole moment. 37 The value is which is almost half the value of our estimation based upon the two-state model, which treats the Li dipole motion independently and neglects the coupling with other modes. In addition, the point 95

96 Figure 6.8: Simulation results of the dielectric constant of KLT with different Li concentration calculated with the two-state model. The curves 1, 2, 3 and 4 correspond to x = 1.3, 2.0, 2.9 and 3.8% respectively. charge approximation is not enough to estimate the Li dipole moment because the shift of Li ions promotes the displacement of oxygen ions to produce large dipole moments. These factors contribute to the lack of concordance between our results and the first principle calculation. 37 However, we stress that the simplified two-state model can explain the substantial features of Li substitution effects on quantum paraelectric KTO. This means that KLT can be classified into a class of ferroelectrics of classical order-disorder type. This is still a naive approach and cannot explain the frequency dispersion and the first-order nature of KLT. 6.5 Conclusions Dielectric and phonon measurements are performed on KLT with different concentrations of Li. The dielectric constant shows a peak with Li substitution and peak temperature shifts to higher temperature. Phonon measurements confirm that both TO and TA phonon energies decreases toward 0 K, but no instability is observed at the peak temperatures of the dielectric constant. Particular attention is paid to the cross-over region, where the quantum paraelectric state changes to a ferroelectric phase. In this region, the temperature dependence of the dielectric constant and TO phonon is fitted by the Barrett formula and the parameters in the theory are 96

97 determined as a function of Li concentration. It is found that these parameters, T 1 and T C, change their tendencies around the specific Li concentration of 2.5%. Besides, in the case of KTO, significant differences in sign and magnitude are observed between the parameters determined from the dielectric constant and TO phonon behaviors. This is attributed to mode couplings, which were not appropriately considered in the Barrett formula. The behavior of the dielectric constant in the cross-over region can be qualitatively explained by the simplified two-state model. The Lorentz correction factor deduced from the experiments using the twostate model is compared with the first principle theory. It would be interesting to compare the present results with those of oxygenisotrope replaced STO (STO18). STO18 induces ferroelectricity because of soft phonon condensation, 38,39 while the Li substitution induces ferroelectric transition of the order-disorder type. Yamada indicated that the essential point of these differences is influenced by the existence of quantum tunneling between the symmetric excited states: The oxygen isotope replaced STO exhibits the quantum tunneling, while KLT exhibits thermally-excited hopping. This would justify our simplified two-state model for KLT

98 References 1 J. H. Barrett, Dielectric Constant in Perovskite Type Crystals, Phys. Rev. 86, (1952) 2 J. C. Slater, The Lorentz Correction in Barium Titanate, Phys. Rev. 78, (1950) 3 F. Borsa, U. T. Höchli, J. J. Van der Klink, and D. Rytz, Condensation of Random-Site Elecric Dipoles: Li in KTaO 3, Phys. Rev. Lett. 45, (1980) 4 U. T. Höchli, H. E. Weibel, and L. A. Boatner, Stabilisation of polarised clusters in KTaO 3 by Li defects: formation of a polar glass, J. Phys. C: Solid State Phys. 12, L563-L567 (1979) 5 U. T. Höchli, H. E. Weibel and L. A. Boatner, Extrinsic Peak in the Susceptibility of Incipient Ferroelectric KTaO 3 :Li, Phys. Rev. Lett. 41, (1978) 6 I. N. Geifman, A. A. Sytikov, V. I. Kolomytsev, and K. Krulikovskil, Sov. Phys. JETP 53 (6), 1212 (1981) 7 U. T. Höchli, Dynamics of Freezing Electric Dipoles, Phys. Rev. Lett. 48, (1982) 8 J. J. van der Klink, D. Rytz, F. Borsa, and U. T. Höchli, Collective effects in a random-site electric dipole system: KTaO 3 :Li, Phys. Rev. B 27, (1983) 9 P. Voigt, and S. Kapphan, Experimental study of second harmonic generation by dipolar configurations in pure and Li-doped KTaO 3 and its variation under electric field, J. Phys.: Chem. Solids. 55 (9), (1994) 10 B. E. Vugmeister, and M. Glinchuk, Sov. Phys. JETP 52, 482 (1980) 11 E. Courtens, Birefringence measurements on KTaO 3 :Li, J. Phys. C: Solid State Phys. 14, L37-L42 (1981) 12 L. L. Chase, E. Lee, R. L. Prater, and L. A. Boatner, Brillouin spectra of K 1 x Li x TaO 3 under poled and zero-field-cooled conditions, Phys. Rev. B 26, (1982) 13 S. R. Andrews, X-ray-scattering study of the random electric-dipole system KTaO 3 -Li, J. Phys. C: Solid State Phys. 18, (1985) 98

99 14 H. Yokota, Y. Uesu, C. Malibert, and J. M. Kiat, Second-harmonic generation and x-ray diffraction studies of the pretransitional region and polar phase in relaxor K (1 x) Li x TaO 3, Phys. Rev. B 75, (2007) 15 J. Toulouse, B. E. Vugmeister, and R. Pattnaik, Collective Dynamics of Off- Center Ions in K 1 x Li x TaO 3 : A Model of Relaxor Behavior, Phys. Rev. Lett. 73, (1994) 16 G. Yong, J. Toulouse, R. Erwin, S.M. Shapiro, and B. Hennion, Pretransitional diffuse neutron scattering in the mixed perovskite relaxor K 1 x Li x TaO 3, Phys. Rev. B 62, (2000) 17 S. Wakimoto, G. A. Samara, R. K. Grubbs, E. L. Venturini, L. A. Boatner, G. Xu, G. Shirane, and S. -H. Lee, Dielectric properties and lattice dynamics of Ca-doped K 0.95 Li 0.05 TaO 3, Phys. Rev. B 74, (2006) 18 H. Yokota, T. Oyama and Y. Uesu, Second-harmonic-generation microscopic observations of polar state in Li-doped KTaO 3, Phys. Rev. B 72, (2005) 19 H. Yokota, and Y Uesu, Extremely slow time evolution of the order parameter under an electric field in relaxor K 0.97 Li 0.03 TaO 3, J. Phys.: Condens. Matter 19, (2007) 20 W. A. Kamitakahara, C. K. Loong, G. E. Ostrowski, and L. A. Boatner, Timedependent phase transformation in KTaO 3 :Li, Phys. Rev. B 35, (1987) 21 G. Shirane, R. Nathans, and V. J. Minkiewicz, Temperature Dependence of the Soft Ferroelectric Mode in KTaO 3, Phys. Rev. 157, (1967) 22 J. D. Axe, J. Harada, and G. Shirane, Anomalous Acoustic Dispersion in Centrosymmetric Crystals with Soft Optic Phonons, Phys. Rev. B 1, (1970) 23 H. Thomas, in Structural Phase Transitions and Soft Modes, edited by E. J. Samuelsen, E. Andersen, and J. Feder (Universitetsforlaget, Oslo, 1971) 24 K. A. Muller, and H. Burkard, SrTiO 3 : An intrinsic quantum paraelectric below 4 K, Phys. Rev. B 19, (1979) 25 H. Vogt, and H. Uwe, Hyper-Raman scattering form the incipient ferroelectric KTaO 3, Phys. Rev. B 29, 1030 (1984) 99

100 26 W. R. Abel, Effect of Pressure on the Static Dielectric Constant of KTaO 3, Phys. Rev. B 4, (1971) 27 G. A. Samara, and B. Morosin, Anharmonic Effects in KTaO 3 : Ferroelectric Mode, Thermal Expansion, and Compressibility, Phys. Rev. B 8, (1973) 28 R. P. Lowndes, and A. Rastogi, J. Phys. C 6, 932 (1973) 29 H. E. Weaver, Dielectric properties of Single Crystals of Single crystals of SrTiO 3 at Low temperatures, J. Phys. Chem. Solids 11, (1959) 30 E. Sawaguchi, A. Kikuchi, and Y. Kodera, Dielectric Constant of Strontium Titanate at Low Temperatures, J. Phys. Soc. Jpn. 17, (1962) 31 V. Zelezny, A. Pashkin, J. Petzelt, M. Savinov, V. Trepakov, and S. Kapphan, Soft-Mode Study in Li-Doped KTaO 3, Ferroelectrics 302, (2004) 32 J. Dec, and D. Kleemann, From Barrett to generalized quantum Curie-Weiss law, Solid State Commun. 106, (1998) 33 M. Yuan, C. L. Wang, Y. X. Wang, R. Ali, and J. L. Zhang, Effect of zero-point energy on the dielectric behavior of strontium titanate, Solid State Commun. 127, (2003) 34 M. Marques, C. Arago, and J. A. Gonzalo, Quantum paraelectric behavior of SrTiO 3 : Relevance of the structural phase transition temperature, Phys. Rev. B 72, (2005) 35 T. Kurosawa, Polarization Waves in Solids, J. Phys. Soc. Jpn. 16, (1961) 36 P. A. Fleury, and J. M. Worlock, Electric-Field-Induced Raman Scattering in SrTiO 3 and KTaO 3, Phys. Rev. 174, (1968) 37 S. A. Prosandeev, E. Cockayne, and B. P. Burton, R. I. Eglitics, Energetics of Li atom displacements in K 1 x Li x TaO 3 : First-principles calculations, Phys. Rev. B 68, (2003) 38 T. Shigenari, K. Abe, T. Takemoto, O. Sanaka, T. Akaike, Y. Sakai, R. Wang, and M. Itoh, Raman spectra of the ferroelectric phase of SrTi 18 O 3 : Symmetry and domains below T C and the origin of the phase transition, Phys. Rev. B 74, (2006) 100

101 39 M. Takesada, M. Itoh, and T. Yagi, Perfect softening of the Ferroelectric Mode in the Isotope-Exchanged Strontium Titanate of SrTi 18 O 3 Studied by Light Scattering, Phys. Rev. Lett. 96, (2006) 40 Y. Yamada, N. Todoroki, and S. Miyashita, Theory of ferroelectric phase transition in SrTiO 3 induced by isotope replacement, Phys. Rev. B 69, (2004) 101

102

103 Chapter 7 CRITICAL PHENOMENON OF K 1 x Li x TaO 3 This chapter first describes the analysis using the Landau-Devonshire phenomenological theory applied to the temperature dependence of the second harmonic generation. Because the sign of the forth-order power term of the order parameter in the free energy formula changes from positive to negative at 2±0.8 %, we conclude that the tri-critical point lies at this Li concentration x c. This value coincides with the specific Li concentration where the quantum paraelectricity becomes dominant below it (chapter 6). Second, the results of the dielectric measurements under a DC electric field that are carried out for KLT around this specific Li concentration are described. For KLT specimens with higher Li concentration which show the first order phase transition without an electric field, the peak of dielectric constant ɛ max becomes sharper and enlarges upon increasing the electric field. Under the critical electric field, ɛ max is twice as large as that of the zero field measurement. Above the critical electric field, the dielectric constant decreases and becomes broad, and the super-critical state appears. On the other hand, ɛ max decreases monotonically with increasing the electric field for KLT with a Li concentration lower than x c. This is a typical electric field effect which appears in ferroelectrics which show the second order phase transition. The critical phenomenon is important for figuring out the origin of the giant response observed in relaxors, and it signifies that the phase diagram with three parameters, i.e., T, x, E, is necessary to explain the peculiar behavior of relaxors. The (T, x, E) phase diagram for KLT is determined by the above experiments. 103

104 H. Yokota and Y. Uesu Critical Behavior of K 1 x Li x TaO 3 with the Tri-Critical Concentration of Li under an Electric Field J. Phys. Soc. Jpn. Letters 77, No.2 (2008) (in print) 7.1 Introduction Perovskite oxides exhibit various kinds of physical properties. In particular, solid solutions and impurity-doped perovskites show novel characteristics that cannot be explained by a simple melange of materials. These properties are not only important in the fundamental physics of condensed matters but are also attractive in practical applications. It is known that these pluralistic materials could manifest huge responses at a specific phase boundary of constituent elements. The colossal piezoelectric effect at the morphotropic phase boundary (MPB) of relaxor/ferroelectric (R-F) solid solutions, 1 the Invar effect of magnetic alloys, 2 and the colossal magnetic resistance effect of some complex manganese oxides 3 are examples. Recently Kutnjak et al. pointed out that the colossal piezoelectric response at the MPB can be consistently explained by the critical behaviour in a tertiary phase diagram of temperature (T), composition ratio (x) and electric field (E) instead of the bidimensional phase diagram (T, x) as had been discussed thus far. 4 The important point of their claim is that R-F materials with the MPB composition are close to the line of critical end points (LCEP). They also stressed that this phenomenon is ubiquitous and is expected to be observed in other kinds of mixed materials. Here we show experimentally that KTaO 3 (KTO) doped with a Li critical concentration is located in the vicinity of the LCEP in the quantum paraelectric-relaxor ferroelectric crossover region, which is the origin of the large dielectric constant. 7.2 Experimental Single crystals with different Li concentrations were grown by the self-flux method with Ta 2 O 5, Li 2 CO 3 and K 2 CO 3 as starting materials, where K 2 CO 3 was used as a flux (See chapter 2.5 in detail). 4 The Li concentration of KLT was identified from the T p at which a polar phase appears, using the empirical relation T p = 535x 2/3. 5 T p was determined by the temperature dependence of optical second harmonic (SH) intensities in zero field heating after field cooling process, because a conspicuous history dependence of the order parameter is observed. 6 Accuracy of the Li concentration x is estimated from the distribution T p of T p in a sample. 104

105 T p is ±2 K which is determined from the SHG images, and the empirical relation between T p and x determines x to be ±0.2 %. Five KLT-x% samples with different x (=1.5, 2.6, 5.1, 6.8, 6.9 %) are used for the SHG measurements. Typical dimensions of these samples are 7x4 mm 2 in area with the thickness of 0.5 mm. Two gold electrodes are evaporated on the top surface of the (100) plane separated by a distance of 3 mm along the [001] direction, and the laser beam position is adjusted so as to be located just between the two electrodes. The optical system is described in detail in chapter 2. For the measurement of the bias field effect on the dielectric constant ɛ, three kinds of KLT with x = 1.5, 3.3 and 5.3 % are selected because these Li concentrations are located in the neighborhood of the tri-critical point. Dielectric measurements are performed using a Solartron SI-1255B frequency response analyzer and an HP4192A LCR bridge which are used in the dielectric dispersion measurements (chapter 5). An AC field amplitude of 1 V/mm and DC bias fields of 0, 100, 200, 300 V/mm for the SI-1225B and 0, 100, 200, 300, 400 V/mm for the HP4192A are applied to samples. Measurements are made in the heating process, but a bias field is applied at room temperature before cooling a sample. The average sample size is 4x4 mm 2 in area and 0.1 mm in thickness along the [001] direction. Gold electrodes are evaporated on the both top and bottom surfaces. Thus the bias electric field is applied along the [001] direction. 7.3 Experimental results & Analyses Determination of critical Li concentration by SHG microscope Figure 7.1 shows the temperature dependence of SH intensities of KLT-x%. SH intensities of all samples decrease monotonically with increasing temperature and vanish at T p. This result confirms that KLT undergoes a ferroelectric phase transition from polar tetragonal 4mm to centro-symmetric cubic m 3m. 7 The phenomenological Landau-Devonshire theory is applied to analyze the results of the SH intensities. By expanding the free energy up to the 6th power of the polarization, the free energy of a ferroelectric crystal with a single domain is expressed as F = 1 2 α(t T 0)P βp γp6. (7.1) Considering the equilibrium and stability conditions, we obtain the temperature dependence of spontaneous polarization P s as; P 2 s = β + β 2 4αγ(T T 0 ). (7.2) 2γ 105

106 Figure 7.1: Temperature dependences of SH intensities of KLT-x. The temperature dependence of an SH intensity determines the coefficients in the free energy, exploiting the fact that the SH intensity is proportional to the square of polarization. For this fitting procedure, the coefficient α is fixed at the value obtained from the temperature dependence of the dielectric constant of each sample using the Curie-Weiss law. The results are shown in Fig.7.1, where the calculated results using fitting parameters are indicated with solid lines. The coincidence between the experiment and the calculation is satisfactory in the vicinity of T p. The fluctuation of the SH intensities in low temperature region could be attributed to domain fluctuations and does not affect substantially the present fitting. The determined parameters are summarized in Table 7.1. It should be noted that the sign of the coefficient β changes from negative to positive between x = 1.5±0.2 % and 2.6±0.2 %. This indicates that the order of the ferroelectric phase transition changes from 2nd to 1st at the tri-critical point x c of 2±0.8 %. The maximum width of thermal hysteresis is expressed as T max = β2 4αγ. (7.3) With the obtained values of the coefficients in the free energy, T max is calculated and shown in Table 7.1. The values of K for KLT-2.6% and 3.1 K for KLT- 6.8% are not contradictory with the experimental results of 0 K and 5.7 K, when the experimental ambiguity of T p is taken into account. In Table 7.1, the value of γ of KLT-1.5% is larger than those of other KLTs. This is a natural consequence of 106

107 Table 7.1: Fitting parameters and thermal hysteresis width T max. x [% ] α [x10 5 ] β γ T max the second-order phase transition which gives no thermal hysteresis (see eq.(7.3)) Electric field dependence of dielectric constant Figure 7.2 shows temperature dependences of ɛ at 10 Hz under different DC bias fields (E b ) for KLT-3.3%. Without a bias field, the dielectric constant shows a steep increase near T = 50.2 K with a dielectric peak of 6170 as shown in Fig. 7.2(a). E b enhances an increase in the dielectric constant and shifts the peak temperature to a higher temperature. Under E b of 200V/mm, a sharp peak of near appears (Fig. 7.2(c)). With further increases of E b, the peak is broader with a smaller peak (Fig.7.2(d) and 2(e)). This is the typical phenomenon of the super-critical state. Therefore, E b = 200 V/mm is the critical field. In these experiments, we could not observe a clear jump of ɛ at the transition temperature, because ɛ was measured under continuous temperature changes of 2 K/min for heating and 3 K/min for cooling. Instead the temperature derivative and the peak value of ɛ of KLT-3.3% are plotted against the bias electric field. The result is shown in Fig.7.3(a) and (b). Both quantities are found to exhibit maxima at 200 V/mm. These are characteristic behaviors in the first-order transition. As an example of the second order phase transition, the results of KLT-1.5% are shown in Fig No critical phenomena are observed under an electric field; The temperature derivative and peak value have maxima at E = 0 in KLT-1.5% as shown in Fig.7.5. It is interesting to note that the bias field effect depends on the frequency and is more remarkable in lower frequency regions. The dielectric constant does not show a strong dependence on E b at frequencies larger than 5 khz. This would be a typical phenomenon in canonical relaxors, 8,9 because the spontaneous polarization develops quite slowly under a bias field. This phenomenon is related to the random field originated from the long-range strain field that prevents the development of polarization. The sample with x = 5.3% shows almost similar behavior to the sample with x = 3.3%. 107

108 Figure 7.2: Temperature dependences of dielectric constant ɛ of KLT-3.3% measured at 10 Hz. (a) indicates the result of E b = 0 V/mm, (b) 100 V/mm, (c) 200 V/mm (critical field), (d) 300V/mm (super critical field) and (e) 400V/mm (super critical field). Figure 7.3: Electric field dependences of the temperature derivative (a) and the peak value ɛ max of dielectric constant (b) in KLT-3.3%. Here (a) is plotted as a parameter of T m T, where T m is the peak temperature. The electric field is applied along the [001] direction. 108

109 Figure 7.4: Temperature dependences of dielectric constant ɛ of KLT-1.5% measured at 10 Hz. (a) indicates the result of E b = 0 V/mm, (b) 100 V/mm, (c) 200 V/mm, (d) 300V/mm and (e) 400V/mm. Figure 7.5: Electric field dependences of the temperature derivative (a) and the peak value of ɛ max dielectric constant (b) in KLT-1.5%. Here (a) is plotted as a parameter of T m T, where T m is the peak temperature. 109

110 This fact indicates that the LCEP of KLT does not strongly depend on x under the examined electric fields. Further experiments are necessary to obtain the tertiary phase diagram in wider phase space. From these SHG and dielectric measurements, the (T, x, E) phase diagram is obtained and is shown in Fig.7.6. Comparing the case of Pb(Mg 1/3 Nb 2/3 )O 3 - PbTiO 3 (PMN-PT), 4 one end point of x is not pure relaxor but quantum paraelectrics. However, this feature quite resembles to that observed in PMN-PT. Therefore it is expected in KLT with the critical concentration where a large piezoelectric effect would appear in this region. 7.4 Conclusions Temperature dependences of the dielectric constant under a bias electric field along the [001] direction together with SH intensities of KLT-x% with different Li concentrations reveals the 3-dimensional (T, x, E) phase diagram of KLT. In this diagram, the tri-critical point with respect to x is determined to be 2±0.8 % and the electric field induced critical points of KLTs with x = 3.3 % and 5.3 % are located at 200V/mm. It should be pointed out that characteristic x c coincides with the Li concentration where the influence of the quantum paraelectricity starts to decline. 110

111 Figure 7.6: The (T-x-E) phase diagram of KLT-x. The red line indicates the line of critical end points. Experimental results of KLT-3.3% and KLT-5.3% under different DC bias fields are indicated by open circles and solid circles, respectively. 111

112 References 1 B. Jaffe, W. R. Cook, and H. Jaffe, Piezoelectric Ceramics (Academic, London, 1971) 2 R. J. Weiss, Proc. Phys. Soc. 82, 281 (1963) 3 T. Kimura, Y. Tomioka, R. Kumai, Y. Okimoto and Y. Tokura, Diffuse Phase Transition and Phase Separation in Cr-Doped Nd 1/2 Ca 1/2 MnO 3 : A Relaxor Ferromagnet, Phys. Rev. Lett. 83, 3940 (1999) 4 Z. Kutnjak, J. Petzelt, and R. Blinc, The giant electromechanical response in ferroelectric relaxors as a critical phenomenon, Nature 441, (2006) 5 J. J. Van der Klink, and D. Rytz, Growth of K 1 x Li x TaO 3 crystals by a slowcooling method, J. Cryst. Growth 56, (1982) 6 J. J. van der Klink, D. Rytz, F. Borsa, and U. T. Höchli, Collective effects in a random-site electric dipole system: KTaO 3 :Li, Phys. Rev. B 27, (1983) 7 H. Yokota, T. Oyama, and Y. Uesu, Second-harmonic-generation microscopic observations of polar state in Li-doped KTaO 3, Phys. Rev. B 72, (2005) 8 The canonical relaxor is defined as a relaxor where the polar nano regions appear in the nonpolar matrix down to low temperatures, exchanging kinetic energy with the surrounding matrix. Typical example is Pb(Mg 1/3 Nb 2/3 )O 3. See ref A. A. Bokov, and Z. -G. Ye, Recent progress in relaxor ferroelectrics with perovskite structure, J. Mater. Sci. 41, (2006) 112

113 Chapter 8 SUMMARY In this dissertation, I aim at elucidating the complicated polar states in KTaO 3 (KTO) doped with Li, which has been long discussed but no established concept has been obtained. For this purpose, I point out the importance of a noticeable relationship between the quantum paraelectric state and the relaxor/ferroelectric phase in this system, and perform comprehensive experiments using the second harmonic generation microscope, X-ray diffractions, neutron elastic and inelastic scatterings, neutron diffuse scatterings, and dielectric spectroscopy. The obtained results are described in chapters 2 to 7, and are summarized as follows. In chapter 2, we describe the result of a marked history dependence of the SH intensity. With the SHG microscope, we find that the behaviors of the order parameter on several paths, i.e., ZFC, ZFH/ZFC, FH/ZFC, FC, FH/FC, and FH/ZFC, are different. This remarkable history dependence is caused by polar nano regions (PNRs) generated locally in the cubic matrix. The phenomenon is interpreted by the random field which restrains the development of the long-range order. In this sense, a history dependence is one of the characteristic behaviors observed in the prototype relaxor Pb(Mg 1/3 Nb 2/3 )O 3. Therefore the present result strongly indicates that KLT is a true relaxor. From the polarization dependence of the order parameter, we disclose that a probable point group at low temperature is 4mm or 4. In chapter 3, our finding of an extremely slow kinetics of SH intensity under an electric field is described. After cooling down to the lowest temperature without an electric field, a specimen is heated up to a certain temperature without an electric field. The time dependence of the SH intensity is measured after applying a DC bias field. When a fixed temperature is low, the long-range order does not develop. 113

114 With raising a fixed temperature, the increase of SH waves is observed. This kind of time evolution of the order parameter has already been observed in prototype relaxors. Thus this result also supports our conclusion that KLT is a relaxor. Here, we analyze this behavior with the Avrami-Chandra theory and find the existence of a specific temperature which separates the character of slow kinetic behaviors, and discuss the origin of this phenomenon. It is pointed out that the long-range strain field is responsible for this slow kinetics. In chapter 4, the results of X-ray and neutron scattering experiments are described. From these experiments, we find the intermediate state just above the polar phase. This pre-transitional region exists between T d and T p of the ferroelectric phase transition temperature where SH intensity appears under an electric field. Without the electric field, a net polarization does not appear but a small lattice strain is generated. This pre-transitional region is successfully observed by the measurements of the second-harmonic generation microscope combined with X-ray and neutron scattering experiments. At T p, PNRs are transformed into ferroelectric micro-domains, while T d is the Burns temperature where PNRs start to appear. With these experimental evidences, we propose a model of polarization state at low temperature of KLT. In chapter 5, we describe the results and analyses of dielectric measurements. We find that the dielectric constant shows only one dispersion around a peak temperature in lower Li concentration specimens, while three dispersions appear in higher Li concentration. We discuss the origin of these dispersions. Using the Debye relaxation model and the Vogel-Fulcher law, we estimate the relaxation time and activation energy for each Li concentration, and we find that these physical parameters do not show a remarkable Li concentration dependence. In chapter 6, the results of temperature dependences of TO, TA phonons and dielectric constants are described. We disclose that the dielectric constant of KLT with low Li concentrations increases monotonically with decreasing temperature below a peak temperature. We analyze these results with the Barrett formula, assuming that the phenomenon of the quantum paraelectricity could be described for KLT with lower Li concentration in low temperature. This analysis reveals that the quantum paraelectricity still remains below 3.3 % of Li concentration. The same analysis is also applied for the TO phonon behavior of KTO and we find that the Barrett parameters determined from these two experiments are different. 114

115 This discrepancy indicates that a contribution from other phonons to the Lyddane- Sachs-Tellar relationship which connects phonons and dielectric constants should be considered. To interpret the appearance of dielectric peak, we propose a twostate model where the local field is taken into account. The peak shift of dielectric constant with Li concentration is described qualitatively by this model. In chapter 7, our new finding of the critical behavior in KLT is discussed. By applying the Landau-Devonshire phenomenological theory to the temperature dependence of SH intensity, the sign of the coefficient of the 4th power expansion of the polarization in the free energy changes from positive to negative at 2±0.8 % of the Li concentration. The fact indicates that this Li concentration is at the tricritical point. It is interesting that this concentration agrees well with the specific concentration at which the Barrett parameters change the tendencies. We carry out the dielectric measurement under an applying electric field around the tri-critical Li concentration and find a critical behavior under a specific electric field where a dielectric constant takes the largest value. Above this electric field, the width of dielectric constant becomes wide and its value decreases, and the state changes into a super-critical state. From these results, we conclude that a critical behavior similar to that of PMN also exists in KLT. The present result conforms that the location of the system at the critical end point is the origin of the large susceptibility of the system. We propose the tertiary phase diagram (T, x, E) of KLT. In the following, the polar state of KLT disclosed by the present research is summarized. (i) Without an electric field Conclusion 1: The phase below T p is a ferroelectric phase with tiny micro-domains, but the structure is inhomogeneous and paraelectric phase still exists. This is supported by the X-ray diffraction measurement, which reveals the marked grow of the tetragonality below T p, and SHG microscopic observations which finds no SHG below T p. By the dielectric measurement, an increase of dielectric constant below a peak temperature is observed in KLT with Li concentration below 3.3 %. This dielectric behavior can be fitted by the Barrett formula and the quantum paraelectric state exists in this temperature and Li composition regions. All above-mentioned experimental results indicate that the whole region of the sample does not transform to the ferroelectric phase but paraelectric state remains partially. Additionally, an enormous increase of neutron Bragg intensities at T p suggests that the micro domain state develops below T p. The domain size is estimated to be small enough in comparison with the transverse coherence length 115

116 of the laser because the SH waves are not observed by SHG microscope. The above presumption concerning the polar state below T p is also sustained by the dielectric measurements: A broad dielectric peak with marked frequency dispersion around the dielectric peak falls steeply under a DC bias electric field. This is a typical ferroelectric phase transition of the first order. The neutron inelastic scatterings reveal that the FWHMs of both TO and TA phonons increase with Li concentration. This is due to the split of T 1 mode to A 1 and E modes at the ferroelectric phase transition temperature. Weak TA phonon intensity supports our conclusion of the coexistence of paraelectric and ferroelectric phases. Conclusion 2: The pre-transitional region exists between T p and T d. This is the region where PNRs appear at T d (Burns temperature) and are transformed into ferroelectric micro domains below T p. Neutron diffuse scatterings disclose the ridge shape patterns elongated along the [100] and [010] directions in the pre-transitional region. It is the evidence of the existence of PNRs. The slight change of tetragonality observed by the X-ray diffraction is induced by the appearance of PNRs via the electrostrictive effect. In the temperature dependence of the dielectric constants, three kinds of dispersion are clearly observed. Among them, the dispersion appearing at the highest temperature corresponds to the occurrence of PNRs commonly seen in relaxor system and is caused by the size distribution of PNRs in the cubic matrix. This result supports our result that T d denotes the Burns temperature. (ii) Under an electric field in FC process Conclusion 3: Under an electric field of FC process, a ferroelectric phase appears accompanied by macroscopic domains below T p. The development of the long-range order is confirmed by SHG microscope observation. Judging from the inhomogeneity of SH image, the complete ferroelectric phase transition is not realized and the paraelectric state still exits under an electric field applied in the present experiments. This heterogeneity is the origin of the characteristic behaviors of relaxors. (iii) In FC after ZFC process Conclusion 4: Non-ergodic behavior is observed at Low temperatures In FH/ZFC process, SH intensity does not appear at low temperatures. This result suggests that the development of the long-range order is restrained by the random field. With raising the temperature to T, a steep increase of SH intensity is observed. It indicates that a potential barrier becomes low and the effect of thermal fluctuation overcomes the random force. To obtain further knowledge of the polar state, the observations by a polarization microscope are performed. In ZFH/ZFC process, the domain structure does not appear even at low tem- 116

117 peratures below T p. On the other hand, the change of the interference color is observed. This indicates that a macroscopic strain corresponding to the results of X-ray diffraction experiments appears below T p. Figure 8.1: Temperature dependence of SHG (a) and polarization microscope images on FH/ZFC (b) and those in FC (c) processes. In SHG measurements, an electric field is applied along the [001] direction. In the case of polarization observations, an electric field is parallel to the [100] direction. In FH/ZFC process, the interference color pattern is almost same with that of ZFH/ZFC process. With heating the sample up to T, complicated domain structures appear with boundaries elongated along the [100], [010], [011] and [0-11] directions.(fig.8.1(b)-2,3) This result is consistent with the SHG microscope observations: microscopic domains change to macroscopic domains around T.(Fig.8.1(a)-2,3) Above T p, the interference color disappears in the whole region, which means that the sample is in the isotropic cubic phase. (In fact, this phase is tetragonal with quite small strain. But the interference observations cannot detect it.) In FH/FC process, the domain structure whose boundary is parallel to the [010] direction is clearly observed.(fig.8.1(c)-1,2) With raising the temperature, the boundary becomes obscure and vanishes around T p.(fig.8.1(c)-4) 117

118 Figure 8.2: Polar state model without E (a) and without E (b) below T p. A hatched regions indicate paraelectric phase, and other regions ferroelectric phase with different directions of P s. In Fig.8.2(b), arrows indicate the direction of P s. Applying a DC electric field at low temperature and raising the temperature, a domain pattern with edges parallel to the [011] and [01-1] directions is obtained by SHG microscope as shown in Fig. 8.1(a). In this case, the DC field is applied along the [001] direction in a sample plate. Therefore, two kinds of 90 degree domain are observed as shown in Fig.8.1(a)-6. The 90 degree domain structure is also distinguished by polarization microscope observations shown in Fig. 8.1(c) where an electric field is applied perpendicular to the sample plate. The polarization direction of each domain is illustrated in Fig.8.1(c)-5. Additionally, paraelectric states can be seen in the polarization images. Fig.8.2 illustrates the polar state in KLT without (Fig.8.2(a)) and with an electric field (Fig.8.2(b)). Without an electric field, a macroscopic domain structure does not appear and each domain size is estimated to be in the micron order. The paraelectric regions still exist in part. Fig.8.2(b) shows the macroscopic domain structure under an electric field below T p. Also in this state, a paraelectric state still remains in part as shown in the hatched regions. 118