Copyright 2008 by the American Chemical Society
VOLUME 112, NUMBER 21, MAY 29, 2008
REVIEW ARTICLE Notes of the Recent Structural Studies on Lead Zirconate Titanate J. Frantti* Laboratory of Physics, Helsinki UniVersity of Technology, P.O. Box 4100, FIN-02015 HUT, Finland ReceiVed: December 17, 2007; ReVised Manuscript ReceiVed: March 07, 2008
Atomic scale structure has a central importance for the understanding of functional properties of ferroelectrics. The X-ray and neutron diffraction studies used for the average symmetry determination of lead zirconate titanate [Pb(ZrxTi1-x)O3, PZT] ceramics and powders are reviewed. The results obtained through two frequently used local probes, transmission electron microscopy (TEM) combined with electron diffraction (ED) and Raman scattering measurements, are summarized. On the basis of these studies, structural trends as a function of composition x and temperature are outlined. There are two distinguished intrinsic structural features, (i) lead-ion shifts and (ii) local structural distortions related to different B cations and the spatial composition variation of x, which have a pronounced effect on the functional properties of PZT. Particular attention is paid to the morphotropic phase boundary (MPB) compositions for which a large number of different structural models have been proposed. Earlier symmetry considerations show that the monoclinic phase cannot serve as a continuous bridge between tetragonal and rhombohedral phases. This suggests that the two-phase coexistence has an important role for the piezoelectric properties. Near the MPB, the extrinsic contribution to piezoelectricity includes pressure (or electric-field)-induced changes in phase fractions and domain wall motion. It was recently shown that the domain contribution is crucial for the electromechanical properties of PZT in the vicinity of the MPB. The dependence of domain widths on crystal size and shape should also be properly accounted for when TEM/ED measurements complement X-ray and/or neutron diffraction experiments. The structurepiezoelectric property relations are summarized. 1. Introduction Lead zirconate titanate [Pb(ZrxTi1-x)O3, PZT] solid solutions, first introduced in 1953,1 are among the most widely used ferroelectric ceramics. The importance of PZT ceramics within morphotropic phase boundary (MPB) compositions is due to the exceptionally good piezoelectric properties they exhibit,2 which in turn has motivated numerous studies seeking physical mechanisms responsible for this extraordinary behavior. Crystal structure studies and the changes in atomic positions versus * To whom correspondence should be addressed. E-mail: jfr@ fyslab.hut.fi.
composition, temperature, pressure, or electric field have a central role for the understanding of these materials: nuclei positions, together with the electron density, determine the electric polarization vector on which the numerous applications are based. It is essential to understand how the polarization vector changes as a function of electric field, composition, temperature, or pressure. The purpose of the present paper is to review structural studies on the PZT system and to establish a link between atomic scale structure and functional properties. Despite its nominally simple perovskite structure, the number of structural models proposed for PZT is vast. This is mainly due to the fact that disorder
10.1021/jp711829t CCC: $40.75 2008 American Chemical Society Published on Web 04/24/2008
6522 J. Phys. Chem. B, Vol. 112, No. 21, 2008 Dr. Tech. Johannes Frantti graduated from the University of Oulu in 1992 with an M.Sc. in Electrical Engineering and received his Dr. Tech. in Materials Physics in 1997 from the University of Oulu. Between 1997 and 1999 and between 2000 and 2005, he was a researcher at the Tokyo Institute of Technology, and in 1999, he was working at the Naval Research Institute in Finland. In 2005, he received an Academy Research Fellowship and joined the Helsinki University of Technology. Dr. Frantti concentrates on the experimental and computational studies on functional materials, with keen interest in structural phase transitions in ferroelectric and magnetic oxides. Main methods applied in his research are neutron powder diffraction, Raman scattering, and density functional theory modeling.
plays a significant role in this system and the results largely depend on the experimental method: Raman scattering and transmission electron microscopy and electron diffraction (TEM/ ED) techniques are essentially local probes, whereas neutron and X-ray powder diffraction (NPD and XPD, respectively) measurements essentially provide information on the average structure. One cannot neglect the impact of the sample preparation technique: for instance, crystal symmetry depends also on the crystal size, which is why so many studies are dedicated to the size effect studies. Obviously, these effects cannot be neglected in the case of TEM/ED samples. Sample preparation techniques, such as solid state reaction or sol-gel methods, and processing temperatures and times, also affect the phase purity and homogeneity. Also, NPD and XPD measurements reveal the disorder, though it is often rather difficult to unambiguously assign a detailed structural model for a system like PZT. The quality and information which can be extracted from the collected data are also instrument dependent. Thus, usually a combination of different methods, such as those sensitive to the local order and long-range order, is necessary. We start by reviewing the results obtained by two frequently used local probes, Raman scattering and TEM/ED, in section 2.1. The results are summarized by identifying the types of disorder characteristic to PZT ceramics. This disorder is also revealed by high-resolution powder diffraction instruments as an anisotropic line broadening and is discussed in the context of average structure determination in section 2.2, with a summary of the structural features of the different phases of PZT. Structural changes as a function of temperature and composition are considered. For powder diffraction data analysis, the Rietveld refinement3 is among the most commonly applied methods. The Rietveld refinement possesses some wellknown difficulties, which, if not properly taken into account, may result in wrong structural parameters. Thus, testing a structural model is of central importance. The structural models were tested by computing Madelung energies and bond-valence sums for PZT, section 3. The rest of the paper deals with the relationship between the atomic scale structure and electrical and electromechanical properties. Particular attention is paid to the Pb-ion shifts and the rotation of oxygen octahedra versus Zr content, temperature, and pressure. The connection between Pb-ion shifts and electrical conductivity is also pointed out. This is crucial, as most of the applications based on these materials require that the material is a good insulator. The currently prevailing idea of polarization rotation is summarized, the inadequacies of the theory are pointed out, and models based on the latest results are described. 2. Atomic Scale Structures 2.1. Local Structure. The aristotype structure of PZT is cubic (space group Pm3jm) perovskite. However, this symmetry is valid only in a sense that sufficiently large volumes are probed. Though information from the local deviations from the average symmetries are embedded in the powder diffraction
Frantti TABLE 1: Irreducible Representations for the Normal Modes at the Brillouin Zone Center for Various Symmetries Reported for PZT (IR and R Stand for the Infrared and Raman Active Modes, Respectively) irreducible representation space group
optical modes
acoustical modes
Pm3jm P4mm Cm R3m R3c Pbam
3T1u(IR) x T2u 3A1(IR, R) x B1(R) x 4E(IR, R) 7A′(IR, R) x 5A′′(IR, R) 3A1(IR, R) x A2 x 4E(IR, R) 4A1(IR, R) x 5A2 x 9E(IR, R) 16Ag(R) x 16B1g(R) x 14B2g(R) x 14B3g(R) x 12Au x 11B1u(IR) x 17B2u(IR) x 17B3u(IR)
T1u A1 x E 2A′ x A′′ A1 x E A1 x E B1u x B2u x B3u
data, it is often better to carry out independent measurements to probe the local symmetries. For the forthcoming discussion, we emphasize that Raman scattering is a nondestructive technique, which implies that the sample preparation or measurement does not change the atomic scale structure of the sample, in contrast to the transmission electron microscopy technique. Raman Scattering Studies. Raman scattering is a phenomenon where excitations (such as the phonons discussed below) with specific symmetries modulate the electronic polarizability tensor in the course of time. In Raman spectra, this modulation is revealed as the scattered light intensity maxima shifted from the excitation light energy by an amount corresponding to the phonon energy. If a phonon is created (destroyed), the intensity maximum occurs at lower (higher) energy and the phenomenon is termed Stokes (anti-Stokes) scattering. In practice, this modulation is studied by irradiating the sample by a laser light at specified geometries. It is a characteristic feature of Raman scattering that rather small changes in the equilibrium atomic positions, either local or periodical, are often revealed as separate peaks in the spectrum. The Raman active modes one expects to observe in the case of average symmetries discussed in section 2.2 are given in Table 1. In practice, the macroscopic electric field associated with the longitudinal optical (LO) mode increases its frequency when compared to the transversal optical (TO) mode. This is frequently referred to as the LO-TO splitting. Typically, the intensity of the LO modes is lower than the intensity of the TO mode. The changes related to the average crystal structure frequently reveal themselves as a softnening of mode(s), and the phase transition studies as a function of temperature, pressure, or composition are commonly studied by Raman spectroscopy. In addition, phase transition is signaled also in the case of higher frequency modes. The peak splitting related to average symmetry changes from tetragonal to monoclinic and to rhombohedral phases was studied in refs 4 and 5. Raman scattering can also be used to study very local scale deviations from the average symmetry, such as those due to cation substitution. The case of lead titanate, PbTiO3, PT, based perovskites has been intensively studied, and there are numerous experimental data available for different symmetry breaking cases. For experiments, it has been a great benefit that PT can readily be prepared in a single crystal form and the lattice dynamics of PT is well documented.6 In the case of PT, the number of Raman active modes obeys the symmetry predictions (Table 1), though one must take into account the LO-TO splitting and modes due to the anharmonicity of the lowest frequency optical A1 mode (it was rather recent that the lowest frequency A1(TO) mode was identified; see ref 7). These anharmonicity effects are also seen in tetragonal PZTs8 and A-site substituted lead
Review Article titanates.9–11 The origin of the anharmonicity is related to the ferroelectric distortion: in the A1(TO) mode, the oxygen octahedra and Ti ion move rigidly against Pb ions, parallel to the c axis. Since the ferroelectric polarization points along the c axis, the force felt by the octahedra depends on whether it moves upward or downward. Thus, the potential corresponding to this (and other A1) normal modes is asymmetric, implying that the spacing between energy eigenvalues n + 1 and n decreases with increasing n.7–9 This is in contrast to the E symmetry modes, which correspond to the vibration perpendicular to the c axis. Due to the Boltzmann factor, the subpeaks of the A1(1TO) peaks show distinctive temperature dependence; at the lowest temperatures, only the highest frequency peak (n ) 1 f n ) 0 transition) remains. Anharmonic interactions between normal modes has been used to calculate the infrared reflectivity in BaTiO3, SrTiO3, and KTaO3 perovskites.12 Such an approach, where the low-frequency mode was coupled to the high-frequency mode, was applied to fit the reflectivity.12 This approach was used, since the independent oscillator model could not fit the entire frequency ranges. In contrast, the simple damped-harmonic-oscillator form was found adequate to fit the Raman data well.13 The softening of the A1(1TO) and the lowest frequency E symmetry mode with increasing temperature (P4mm f Pm3jm phase transition) has been studied in ref 13. In PZT samples, the interpretation of the lowest frequency spectral region is obscured by the simultaneous appearance of the A1 and E symmetry modes together with the very strong Rayleigh peak. The A1 symmetry modes preserve the crystal symmetry, which in turn suggests that the symmetry breaking modes, with E or B1 symmetry in the P4mm phase, are more informative for the phase transition studies as a function of composition in PZT. For instance, the B1 symmetry mode breaks the 4-fold symmetry, and the symmetry lowering P4mm f Cm is seen as a frequency difference between the B1 (P4mm phase) and A′′ (Cm phase) modes.4,14 In pure PT, the Raman active modes disappear once a transition to the cubic phase occurs at elevated temperatures (Table 1) and all Raman active modes predicted for the P4mm symmetry could be identified.7 This situation is drastically changed if the A and/or B cation sites are alloyed by other elements, with examples being Pb1-xCaxTiO3,10,11 Pb1-xLax(Zr0.4Ti0.6)O3,15 and PZT.16 For small alloying concentrations, the effect of A- and/or B-cation substitution can qualitatively be understood by considering the (i) mass effect, (ii) bond strength, and (iii) bond length on the normal modes of pure PT. Bond strength is largely affected by the changes in the nature of the bonding. Pb tends to be covalently bonded to its surrounding oxygen due to the lone electron pair it possesses. Thus, replacing Pb by a closed-shell ion, such as La or Ca, adds ionic bonding between the A cations and oxygen. In addition to the frequency shifts corresponding to the average structure, one often observes completely new modes in the spectra. These can either be an indication of a phase transition or local distortions in a matrix which still has the average symmetry unchanged. It is well-known that broad peaks are observed in the Raman spectra collected on PZT samples at the high-temperature cubic phase, even though no first-order Raman scattering is allowed for the ideal symmetry. The modes appearing in the cubic phase were assigned to the first-order scattering, activated by the local disorder, by studying the temperature dependence of the intensities.17 The deviations from the average symmetry are not limited to the high-temperature phase, but also the Ti-rich PZT possesses somewhat unusual behavior at low temperatures: the
J. Phys. Chem. B, Vol. 112, No. 21, 2008 6523 TABLE 2: Classification of Superlattice Reflections Commonly Found in the Rhombohedral Phases of PZT by TEM/ED Studies (Tilts Refer to Glazer Notation;27 Table Is Adapted from ref 24) type R1 R2 M1 M2
reflection
conditions
tilt
1
h)k)l h*k*l k)l k*l
none anone a+
/2{hkl}p /2{hkl}p 1 /2{0kl}p 1 /2{0kl}p 1
2-fold degenerate E symmetry modes were split into two at low temperatures.14,18 This splitting was increasing with increasing x. However, the high-resolution neutron powder diffraction pattern of the Pb(Zr0.20Ti0.80)O3 at 4 K did not indicate any symmetry lowering from the P4mm symmetry.19 The same behavior was observed in a closely related Pb(HfxTi1-x)O3 (PHT) system with 0.10 e x e 0.40.20,21 A similar phenomenon was reported in the case of a relaxor ferroelectric Pb0.775La0.15(Zr0.4Ti0.6)O3.22 A combined infrared and X-ray diffraction study revealed that there were more IR active modes than the average P4mm symmetry allows. X-ray diffraction study indicated that Pb ions were locally shifted to 〈110〉 directions. If one places Pb ions to the site symmetry 4d, the number of Raman and IR active modes is correspondingly increased.15 Thus, one has a picture where Pb ions are hopping between different disordered positions. The IR reflectivity was calculated by using the factorized form of the damped-harmonicoscillator model to calculate the permittivities along and perpendicular to the local polarization.15 Both dynamic and static displacements affect the atomic displacement parameters (ADPs) observed by X-ray and neutron diffraction techniques. Leadion shifts as a function of average composition x are summarized in section 2.2.5. Electron Diffraction Studies on Zr-Rich PZT. Transmission electron microscopy (TEM) and electron diffraction (ED) studies revealed extra reflections (which should be absent in the case of ideal symmetry) in rhombohedral PZTs.23 The origin of these reflections was more recently studied by TEM and ED together with neutron powder diffraction (NPD) techniques.24 In this study, superlattice reflections of the type R1, R2, M1, and M2 (see Table 2) were observed by ED, consistently with ref 23. Among these reflections, only R2 is consistent with R3c symmetry.24,25 We summarize the observations and interpretations given in ref 24 as follows: (i) M-type reflections were accompanied by satellites; (ii) different areas in the same grain resulted in changes in the relative intensities of the superlattice reflections so that in some cases the extra spots completely disappeared; (iii) by collecting data on samples prepared from single crystals and ceramics, it was found that the satellites around M points are a characteristic feature of only ceramics; (iv) by studying the temperature dependence of the extra reflections, they could be assigned to the ferroelectric state (no extra reflections were observed in the cubic phase); (v) octahedral tilts and distortions alone are insufficient to explain R1 and M superlattice reflections with the intensities obserVed (it was demonstrated that very weak superlattice reflections could be obtained using unrealistically large oxygen octahedra tilts and distortions); (vi) NPD experiments revealed only R2 reflections, consistently with an average R3c symmetry. To explain these observations, models based on locally ordered regions presenting antiparallel cation displacements were proposed (structural models based on NPD experiments are given in ref 26) and compared with experiments. The tendency of Pb ions to form four short bonds with oxygen can be seen also in the orthorhombic, tetragonal, and rhombohedral phases of PZT.26
6524 J. Phys. Chem. B, Vol. 112, No. 21, 2008 In other words, this means that Pb ion is not located at the center of the oxygen cuboctahedra, not even in the cubic phase. In ref 26, local structures capable of producing the extra reflections of the type {1/2, 1/2, 0}p (Figure 7) and {1/2, 1/2, 1/2}p (Figure 8), seen in TEM and ED studies of rhombohedral PZTs, at approximately the correct intensities are given. This picture gives a qualitative idea concerning the Pb-ion displacements with respect to its nearest neighbors. Correspondingly, one must consider the whole oxygen and B-cation network in order to determine the overall Pb-displacement pattern. The finite size effect was proposed as an underlying reason for the phenomena observed by TEM and ED techniques. This suggests that these phenomena, obserVed by TEM and ED techniques, do not occur in the case of bulk ceramics. However, the tendency of Pb ions to form four short bonds with oxygen is a feature observed through NPD studies.26 This tendency is also seen from the models constructed to explain the observed ED patterns.24 Similar conclusions were obtained through Monte Carlo simulations: M-like superlattice reflections are not present in pure bulk crystals but might be locally induced by surfaces over some range of temperature.28 Domains. Since the static domain configuration is obtained by minimizing the total free energy of the crystal including the energy associated with the crystal including the energy associated with crystal surfaces and domain walls, sample size and shape effects should be taken into account when data collected on different techniques are compared. Thus, it is not legitimate to assume that the domain configuration, which minimizes the energy of micrometer scale crystals within spherical particles would minimize thin, wedge shaped TEM samples. The principles of domain formation, the dependence of the domain width on the thickness of the sample and temperature, are wellknown (see, e.g., ref 29). This means that the domain widths observed in very thin TEM samples are not similar to the widths observed in large particle size powders. For instance, by applying Kittel’s law,30 the width d of the 180° domains is [σt/(/P02)]1/2, where t is the thickness, / is a constant depending on the dielectric constant, and σ is the energy per unit area.31 This behavior was confirmed by studies on Rochelle salt and KH2PO4.32 In the case of free-standing BaTiO3 samples and PbTiO3 films on SrTiO3 substrate, the domain widths as a function of thickness were studied by scanning transmission electron microscopy and X-ray scattering in refs 33 and 34, respectively. Kittel’s law was found to be valid in a large thickness range, including those characteristic to TEM samples. This in turn suggests that the domain widths in typical X-ray and neutron powder diffraction specimens are an order of magnitude larger than those in TEM specimens. Thus, even though the domain configuration of TEM samples might not be resolved by X-ray and neutron powder diffraction studies using a typical wavelength of 1 Å, the larger domain widths of bulk crystals eliminates the ambiguity. Thus, before the phenomena observed in TEM samples are applied to bulk samples, size- and shape-dependent phenomena should be taken into account. We note that polarized Raman scattering was recently applied for determining domain distribution in tetragonal PZT thin films.35,36 The method is based on the different selection rules the A1, B1, and E modes obey and allows a determination of the volume fractions of a and c axis oriented domains in a nondestructive manner. 2.1.1. Spatial Composition Variation. It is obvious that TEM/ ED and Raman scattering data cannot solely be understood without giving up the assumption of translational symmetry. Since the difference between ionic radii of isovalent Zr and Ti
Frantti ions is not sufficient for the introduction of long-range order, one must estimate how much the composition of a given volume element can deviate from the average composition. The twophase coexistence is related to the first-order transition between tetragonal and rhombohedral phases, which becomes apparent in an inhomogeneous system. The possibility of a bridging monoclinic phase is discussed in section 2.2.3. We start by giving a statistical estimation for the composition variation after which a summary of the two thermodynamical approaches given for PZT with MPB compositions is given. Random Distribution. We assume that the distribution of Zr and Ti obeys the binomial density function with probabilities p ) x and q ) 1 - p that the B-cation site is occupied by Zr and Ti ion, respectively (the solubility gap discussed below would further enhance the spatial composition variation). In principle, ordering effects can be taken into account by introducing conditional probabilities. The present approximation provides a microscopic explanation for the two-phase coexistence observed in the vicinity of the MPB. We consider the case of PZT with x ) 0.50: if we divide a large domain into cubes containing N primitive cells, roughly two-thirds of the cubes contain NZr ) xN Zr ions with Np - (Npq)1/2 e NZr e Np + (Npq)1/2. Assuming the cube edge to be 10 nm (typical spot sizes used in TEM and ED studies are between 1 and 5 nm), almost one-third of the cubes have x < 0.49 or x > 0.51. The idea of an inhomogeneous distribution of Zr and Ti ions is also consistent with the TEM and ED observations according to which there is a large spatial variation in ED patterns as discussed above. Surface phenomena are sensitive to local distortions. It is the feature of the binomial distribution that the variance is largest at around MPB composition (p ≈ q) and is zero only for pure PbTiO3 and PbZrO3. This is seen from the line widths which increase with increasing x when 0 e x e 0.50. This is supported by an atomic pair distribution function analysis carried out for PZT powders with x ) 0.40, 0.52, and 0.60:37 the MPB is a crossover point with maximum disorder. It is also important to note that even if the average composition of different grains would be exactly the same, the spatial variation can result in domains with different symmetries. From the microscopic point of view, attention should be paid to how Zr and Ti ions are distributed over the B-cation site. This issue was addressed in ref 38 through a Monte Carlo type model where the mean of the distribution of cluster volume was analyzed as a function of x. This study revealed that the sum of the mean cluster size of the Zr and Ti clusters is a minimum at x ) 0.50, with the most likely minimum volume in which an equal number of Zr and Ti ions is found to be 1 nm3 (though there correspondingly were many 1 nm3 volumes with an inequal number of Zr and Ti ions). It was argued that the increase in coherence length of the nanodomains, proposed as the mechanism for the appearance of the macroscopic monoclinic phase close to x ) 0.50 in ref 39 may be a consequence of some ordering of the B cations. According to the model proposed in ref 39, the local structures of the R3m and P4mm phases were considered to be monoclinic, and the Cm phase was interpreted to correspond to those composition and temperature values for which the local regions had grown sufficiently that diffraction techniques see a distinct Cm phase. For the forthcoming discussion, it is important to notice that the spatial composition variation is larger the smaller the spatial length scale is, as is also apparent from Figure 13 in ref 38. Monte Carlo simulations carried out for PZT with a broad range of x and random distribution of Zr and Ti in the B-cation site yielded the phase transition sequence P4mm f Cm f R3m.40 Virtual crystal
Review Article approximation (VCA, where Zr and Ti ions are replaced by the same fictitious average atom) does not reveal a Cm phase.40 The given treatment estimates the composition variation in an ideally disordered case. However, the only way to achieve strictly homogeneous samples in the atomic scale is to arrange the B cations in an ordered way. For the x ) 0.50 sample, this is achieved by constructing double perovskite structured crystals. Experimentally, the two extrema cases, perfect disorder and order, can easily be distinguished by X-ray or neutron diffraction techniques. As the studies dedicated to Ba2MnWO6 double perovskite show, ordering is clearly seen from the superlattice reflections in X-ray41 and neutron powder diffraction patterns.42 Also, Raman spectra are consistent with the observed diffraction patterns.43,44 As the numerous studies show, the difference in ionic radii (and, to a lesser extent, charge) in PZT does not provide sufficient driving force for a formation of the double perovskite structure. As proposed in ref 38, partial ordering may occur, although it is not easy to determine the degree of ordering in PZTs. The spatial distribution of Zr and Ti atoms is also strongly affected by the sample preparation technique. The effect of calcination temperature on the phase fractions of the tetragonal and rhombohedral phases on powders synthesized through a peroxide based route is discussed in ref 45. Compositional homogeneity in sol-gel processed PZTs was reviewed in ref 46 with a conclusion that Pb is not uniformly distributed throughout a typical gel and Ti and Zr are not randomly distributed in the gel polymeric backbone. Though the thermodynamical variables, such as temperature and pressure, are usually used for the controlling reaction and crystallization kinetics, it was demonstrated in ref 47 that under hydrothermal conditions the PZT particle or thin film characteristics, including size, shape, and chemical homogeneity, can be controlled by nonthermodynamic processing variables, such as stirring speed. Phase Coexistence near MPB. The PZT system is frequently assumed to be a complete binary solid solution of PbTiO3 and PbZrO3 without solubility gaps.48 According to ref 48, once a solid solution is formed at high temperature (typically above 800 °C), the chemical composition cannot be changed at low temperatures, but the system can have temperature-induced diffusionless structural phase transitions. As numerous studies have shown, there is a coexistence region near the MPB composition. The phase fractions near the MPB were estimated through the lever rule.49 This approach was criticized, as it correspondingly assumes a solubility gap,48 in contradiction with the general assumption. Instead, the phase coexistence was considered to be due to the frozen-in second metastable phase from thermal fluctuations.48 The merit of the latter approach is its ability to explain the experimentally known fact that the width of the coexistence region is inversely proportional to the particle size in a powder system. The validity of the assumption that the PZT system has no solubility gaps has been questioned in refs 50 and 51, and it was reported that PZT solutions exhibit a positive enthalpy of mixing that suggests a tendency toward immiscibility and phase decomposition. Correspondingly, miscibility gaps were proposed to replace the MPB and the paraelectric to ferroelectric transition lines of the diffusionless phase diagram51 given in ref 2 (which would be valid only if the cooling rate significantly exceeded the diffusion rate of Ti and Zr atoms). The analysis was restricted to temperatures above 420 K,51 although very similar conclusions should be valid at lower temperatures. Usually, the solubility gap increases with decreasing temperature. In the present case, the solubility gap and finite atomic diffusion rate would imply that the system minimizes its free energy by
J. Phys. Chem. B, Vol. 112, No. 21, 2008 6525 segregating into two different phases possessing different compositions and crystal symmetries. 2.2. Average Structure: Neutron and X-ray Powder Diffraction Studies. 2.2.1. Anisotropic Line Broadening. As the Raman scattering and electron diffraction studies reveal, deviations from the average symmetry are significant and the standard models commonly used in powder diffraction studies may not be sufficient. In the high-resolution diffraction studies, this is revealed as an anisotropic (hkl-dependent) line broadening, which was ascribed to a “microstrain” in ref 52. The line shapes necessary to model this type of broadening are rather complex.53,54 There are several factors which together are responsible for the anisotropic line broadening: (i) Zr substitution for Ti creates local strains due to the size difference. (ii) Spatial composition variation: Zr-rich clusters take larger volume per formula unit than Ti-rich clusters. (iii) Pb-ion shifts depend on the B-cation environment (i.e., Pb shifts vary spatially as a response to the spatial variation of the B cations). Although the mechanisms are interweaved, this treatment serves as a way to estimate which one is dominating at different composition regimes. Thus, the changes in Bragg reflection intensities and displacement parameters in the case of small Zr/Ti substitutions in PbTiO3/ PbZrO3 can largely be understood by considering mechanism i. Thus, following the treatment given in ref 55, small Zr substitution for Ti in PbTiO3 might be treated by considering the concentration of Zr atoms x and the difference between matrix (Ti) and solute (Zr) atoms ∆R. Now, the degree of distortion is characterized by the mean square atomic displacement 〈U2〉 ) jx(∆R2), where j is a constant. Static atomic displacements cause a reduction in diffraction peak intensity and increase in diffuse scattering and can be treated in the same way as thermal vibrations. However, this treatment is no more adequate for intermediate values of x, since there is no way of selecting such a matrix that experimentally observed features could be explained. Although mechanism i can explain changes in intensities (matrix and solute atoms have different scattering lengths and cross sections) and diffuse scattering (seen as an increased intensity in the Bragg reflection tail regions: the peak width at half-maximum is practically not affected by this mechanism) for small values of x, it does not provide a model for changes in Bragg reflection widths. This is why the phenomenological model given in ref 54 was adopted: it also takes into account the hkl-dependent Bragg reflection widths by introducing a microstrain broadening ΓS2 ) ∑HKL SHKLhHkKlL, H + K + L ) 4. Laue symmetry imposes restrictions on the allowed SHKL terms. Expressions for ΓS2 for each Laue symmetry are given in ref 56. It is also worth mentioning that usually there are several line broadening mechanisms. Especially, powders consisting of fine crystallites, typically obtained through wet chemical methods allowing preparation of the perovskite phase at rather moderate temperatures, show large broadening due to the small crystallite size, in addition to the anisotropic line broadening.56 A Williamson-Hall plot was constructed to separate crystallite and anisotropic strain broadening effects57 in tetragonal PZT. This study revealed that the lattice strain is greater along the [001] directions than along the [100] directions. On the other hand, samples prepared through solid state reaction often have an order of magnitude larger crystallite size. In the case of PZT with compositions in the vicinity of the MPB, the line widths are strongly increasing with decreasing temperature,52 demonstrating that the crystallite size and shape alone cannot account for the observed line broadening behavior.
6526 J. Phys. Chem. B, Vol. 112, No. 21, 2008 2.2.2. PbZrO3 and PZT with 0.55 e x < 1. The room temperature symmetry of PbZrO3 is orthorhombic Pbam.58,59 With a small Ti addition, the orthorhombic phase transforms to R3c (by symmetry, through a first-order transition).26,52,60 The characteristic feature of the R3c phase is that it allows the oxygen octahedra to be rotated about the pseudocubic diagonal and cation displacements. The R3c phase transforms to the R3m phase with increasing temperature, through a first-order transition when x e 0.88 (critical point) and further to the cubic phase through the first-order phase transition when x e 0.90 (critical point).61 In the case of rhombohedral compositions, it has been found that oxygen octahedra tilts increase with increasing x (see ref 26) and decreasing temperature.62,63 Orthorhombic Phase. A rough but illustrative picture for the Pb-ion behavior can be obtained by considering the tolerance factor t ) (RA + RO)/(2(RB + RO)), where RA, RB, and RO are the ionic radii for the A and B cations and oxygen. In the case of PbZrO3, t < 1 implies that the Pb ions are not able to fill the cuboctahedra, whereas Zr fills octahedra so tightly that it takes a larger share from the total volume by tilting the octahedra. In this sense, one can “construct” PbZrO3 by (i) considering the large Zr ions, which results in a a-a-c0 tilt corresponding to distorted oxygen octahedra and then (ii) displacing Pb ions inside the cuboctahedra in such a way that four short, essentially covalent, bonds with oxygen (constituting a PbO4 “pyramid”) are formed. A very similar situation occurs in a closely related PbHfO3 which is isostructural and has the same space group symmetry as PbZrO3.64 For clarity, it must be said that the given treatment is meant to be illustrative and not to give any causality relationships (in ref 64, the type of octahedral tilting and deformation was seen as a way to accommodate the nature of the Pb-O bonding). Now, the PbO4 pyramids alternate in the ab plane in such a way that antiferroelectric ordering is formed. Also the Zr ions experience similar shifts, although the absolute variation in Zr-O bonds (at 100 K, the shortest and largest Zr-O bonds were 2.044 and 2.205 Å) is not as large as in the case of Pb-O bonds (at 100 K, the shortest and largest Pb-O bonds were 2.41 and 3.105 Å). In contrast to Pb ions, Zr ions exhibit antiparallel [001] shifts.59 Also, the behavior of the Pb(1) and Pb(2) ions (specified in Table 4) versus temperature was found to be different. Namely, at 100 K, the Pb(1)-ion shift was significantly larger that of the Pb(2). Thus, changes in Pb(1) position were larger than those of Pb(2) once the cubic phase was approached. It is interesting to note that in the case of Pb(1) three of the Pb-O bonds in the PbO4 pyramid were much shorter than the fourth one so that Pb(1) is almost 3-fold coordinated, though the Pbam and R3m phases do not have a group-subgroup relationship nor is a common subgroup reported to exist between them. Interestingly, there is a very narrow (with respect to temperature and x) region where PbZrO3 and PZT with a very small amount of Ti with Pbam symmetry transforms into a ferroelectric rhombohedral phase (assigned to the high-temperature rhombohedral65 phase, i.e., to the R3m phase) at around 500 K. However, the stability region of this phase is very sensitive to impurities.65 Both phase transitions Pbam f R3m and R3m f Pm3jm are first-order transitions (first necessarily by the given symmetry arguments).65 Since the Pbion displacements from their ideal cubic sites are dominant in PbZrO3 (so that the Pb ions are shifted with respect to the Zr and oxygen ions, which remain at their ideal sites), they essentially measure the sublattice polarization. Thus, it is possible to simplify the structural model which in turn allows one to write down a simple free energy expansion65 for PbZrO3. The structural changes occurring in PbZrO3 as a function of
Frantti temperature were more recently addressed in ref 66. Although the intermediate phase between Pbam and Pm3jm phases has been experimentally verifed, it is worth noting that the disorder in this phase was reported to be significant.66 This was demonstrated through a combined pulsed neutron powder diffraction (suitable for studying local distortions through atomic pair distribution function analysis) and Rietveld refinement study where locally correlated Pb displacements along the c axis were found. These displacements were increasing with increasing temperature and resulted in local ferroelectric polarization (at 473 K, the Pb-ion displacement correlation length was estimated to be approximately 20 Å). The origin of ferroelectricity in this phase was proposed to be due to the interaction between these locally polarized regions. Thus, even the structure of PbZrO3 seems to possess local distortions which have a crucial impact on the physical properties. From this viewpoint, it is hardly a surprise that the disorder in a solid solution is even larger. 2.2.3. MPB Region and the Two-Phase Models. The MPB region has been a rather challenging area to be modeled, largely due to the fact that it separates tetragonal and rhombohedral phases. By symmetry, the transition between the two phases must be of first order, or the transition should occur via an intermediate phase. The rather recently observed monoclinic phase with Cm symmetry, which is a common subgroup of the R3m and P4mm phases according to the group-subgroup chains P4mm f Cmm2 f Cm and R3m f Cm, was proposed to serve as a bridging phase.67 We note that similar ideas were suggested for Zr-richer compositions.68 Thus, considerable focus was turned to the Cm phase, as it suggested that polarization rotation theory, summarized in section 4.2, might be able to explain the electromechanical properties of PZT and similar systems in the vicinity of the MPB. However, it soon turned out that twophase coexistence was preserved and, to our knowledge, no single-phase samples have been prepared with MPB composition.69 This in turn implies that, in addition to the intrinsic contribution to the piezoelectric response, also the extrinsic contribution in this system is important (section 4.2). Thus, in what follows, we pay attention to the changes in phase fractions Versus temperature on the Zr-rich side of the MPB. Structural Changes as a Function of Temperature. The balance between coexisting phases near MPB compositions is very delicate for temperature and pressure changes. To understand which phase, R3c or Cm, is the preferred one at low temperature, a high-resolution neutron powder diffraction study as a function of temperature was conducted on a PZT sample with x ) 0.54.52 The NPD data suggest the following phase transition sequence with decreasing temperature in the vicinity of MPB: Pm3jm f P4mm f Cm f R3m f R3c, where adjacent groups have a group-subgroup relationship (R3m is a common supergroup of Cm and R3c phases). As is seen from Figure 1 in ref 52, once the crystals are compressed with decreasing temperature, the R3c phase was favored at the expense of the Cm phase. The temperature and composition dependence clearly show that when the phase fraction of Cm phase decreased, the intensity of the superlattice reflections (most notably the peak at around 1.06 Å) were increasing together with the phase fraction of the R3c phase, confirming that the superlattice reflections are due to the R3c phase. This behavior is consistent with the composition variation: the distortion (monoclinic or rhombohedral) depends on the average composition and temperature. The Cm phase remained pseudotetragonal down to a temperature of 4 K: only its phase fraction diminished rapidly. This was accompanied by a line width broadening of the Cm phase. Though the monoclinic β angle and lattice parameters
Review Article
J. Phys. Chem. B, Vol. 112, No. 21, 2008 6527
Figure 1. Pb-ion displacements in (a) tetragonal (space group P4mm), (b) monoclinic (space group Cm), and (c) antiferroelectric orthorhombic (space group Pbam) phases. Shaded areas indicate the unit cells.
deviated sufficiently to allow the identification of the monoclinic distortion through Bragg peak splitting, the spontaneous polarization was still very far from the rhombohedral direction.52,63 This implies a first-order transition between the Cm and R3c phases. According to the group theoretical consideration carried out for perovskites,70 the transition between rhombohedral and monoclinic phases must be of first order, whereas the transition between the P4mm and Cm phases can be continuous. Due to the composition variation, some domains undergo the same phase transition sequence at higher (Zr-richer) or lower (Tiricher) temperatures. Whereas it is clear that the spatial composition variation affects the phase transition, it is an open question if the variation is just due to the statistical variation (section 2.1.1) or if it is further enhanced by the aforementioned solubility gap. The coexistence of the rhombohedral and Cm phase is consistent with the idea that they are energetically almost as favorable (as is suggested by the Madelung energies in section 3.1) and rather small changes in temperature, pressure, or an applied electric field can transform the Cm phase into a rhombohedral phase. At room temperature, the structural parameters of the Cm phase are close to the P4mm phase so that it is not possible to say if the Cm phase coexists with the P4mm phase. This was proposed to be the case on the Ti-rich side of MPB.71 Finally, we note that there have been numerous different structural models for PZT within MPB, including a plethora of different and mutually exclusive single- and two-phase models based on combinations of Cm, Pc, and Cc symmetries. Correspondingly, the structural parameters are wildly scattered, though the differences between the diffraction patterns are much smaller. Some of these studies seem to be attempts to fit PZT into the framework of the polarization rotation theory, though the structural models were not really supported by the X-ray or neutron powder diffraction patterns. These models were frequently based on the use of subgroups (Cc for R3c, Cm for P4mm and/or R3m, etc.), instead of the higher symmetry groups (an issue which is addressed in a broader context in ref 72). The study of superlattice reflections is useful for the determination of the space group symmetry.24 For example, replacing the R3c symmetry by its subgroup Cc predicts a 110 reflection at around 4.7 Å. The absence of the reflection(s), other than those corresponding to the R3c symmetry, allows a rejection of the Pc and Cc symmetry based models, though this is often omitted in the papers reporting the Pc or Cc symmetries. On the other hand, the superlattice reflection between 2.4 and 2.5 Å in neutron powder diffraction patterns is the most clear indication that a cell doubling transition has occurred: the Cm phase with one formula unit per primitive cell cannot explain
the diffraction pattern. This is a point where neutrons are necessary, since X-rays do not reveal the oxygen octahedra rotations.63 To our experience, studies conducted as a function of temperature and composition are particularly useful as the evolution of structural parameters can be followed. Since the number of structural models is ever increasing, we have summarized some of them in Table 3. The purpose is to clarify the relationships between different models; for example, both models 5 and 8 used the same symmetries, Cm and Cc, but for different phases, thus making the models completely different. It was pointed out in ref 79 that the phase fractions obtained using model 5 are not realistic. We note that the essential structural parameters in models 1, 2, and 8 are nearly the same. The most essential differences are that (i) the P4mm symmetry was replaced by the Cm symmetry, in accordance with the observed Bragg peak splitting, and (ii) the lowtemperature superlattice reflections, which are not explained by the R3m structure, were taken into account by using the R3c symmetry (model 2) or the subgroup of R3c with constraints (model 8). It is our opinion that models 3-7 are not consistent with the experimental observations. Model 9 introduces nanodomain mixture of the rhombohedral or tetragonal phases, which cannot be resolved by X-rays or neutrons with a wavelength ≈1 Å, in contrast to the interpretation given for the Cm phase in ref 39. Within this model, the monoclinic structure was taken as a diffraction artifact. The line widths of the R3c phase, notably the widths of superlattice reflections, indicate large crystal size. However, the domain widths in very thin TEM specimens and larger particle size crystals are different, as was discussed in section 2.1, and in this sense, one can question if model 9 is valid for the bulk crystals. What Makes the R3c Phase Interesting? The dielectric and piezoelectric properties peak at the MPB.2 It is worth noting that they remain significantly higher on the rhombohedral side of the phase boundary. The R3c space group is a generic rhombohedral symmetry in the sense that it allows both octahedral tilting (tilt system a-a-a-) and cation displacements. By tilting the oxygen octahedra, the crystal can partially compensate for the effect of a larger average B-cation (Zr and Ti) size which increases with increasing Zr content x and decreasing temperature. This is seen from the behavior of Zrrich samples: once crystal adopts the R3c symmetry, instead of the R3m symmetry, the volume per primitive cell is smaller. This is consistent with the notion that in the case it would cost a lot of energy to contract the oxygen octahedra (due to the short-range interactions which cause the mere compression of octahedra to be energetically unfavorable) overall energy gain can still be obtained by tilting the oxygen octahedra. This can
6528 J. Phys. Chem. B, Vol. 112, No. 21, 2008
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TABLE 3: Various Models Used for the Modeling of the Structures in the Vicinity of the MPBa model model model model model model model model model
phase 1
phase 2
x
T (K)
R3m R3c
P4mm Cm Pc Cc Cc P4mm
0.53 0.52, 0.53, 0, 54 0.52 0.52 0.52 0.515 < x < 0.530 0.525 < x < 0.60 0.52
e300 10 10 10 RT RT 20
1 2 3 4 5 6 7 8 9
Cm Cm Cm Cc R3m
Cm P4mm
analysis method
sample preparation method
X-ray neutron neutron
SSR SSR SW
X-ray X-ray neutron
SW SW SSR
notes
ref 49, 73 52, 63 74 75 76 77 78 79 81
b c c c d e f
a Since the diffraction patterns in different studies were similar, it was possible to identify phase 1 and phase 2 in each report. The average composition and temperature are indicated by x and T . Abbreviations SSR and SR refer to solid state reaction and semiwet route, respectively. b To see the behaviour of the octahedral tilt, the R3c symmetry was used to model the room temperature pattern, though the superlattice reflections characteristic of the R3c symmetry were unambiguously revealed only at low temperatures. At room temperature, the structure was close to the R3m phase with a small octahedral tilt angle. c The same powder diffraction pattern was used in models 3, 4, and 5. d The sample contained three phases, as was revealed by the peaks at around 3.18 and 2.56 Å. These peaks can be assigned to the ZrO2 phase.80 The neutron powder diffraction pattern of the ZrO2 phase contains several strong peaks overlapping the superlattice reflections at around 1.06 Å originating from the perovskite phase. It is exactly this region which was claimed to provide evidence for Cc symmetry in ref 76. Thus, ignorance of this phase further deteriorates the reliability of the refinement. e Constraints were used to decrease refinable parameters, making the difference between models 2 and 8 insignificant. In order to compare the structural parameters of the R3c and Cc phases, the rhombohedral structural parameters, given with respect to the triple hexagonal cell (lattice vectors (a, b, c) and fractional coordinates (x, y, z)), can be expressed in terms of the C-centered monoclinic cell (unique axis b, cell choice 1, lattice vectors (a′, b′, c′), and fractional coordinates (x′, y′, z′)) by using x′ x 2/3 0 0 3/2 0 0 the relationships (a′, b′, c′) ) (a, b, c)P and y′ ) Q y , where P ) 1/3 1 0 and Q ) P-1 ) -1/2 1 0 .82 Another often used z′ z -2/3 0 1 1 0 1
() ()
(
)
(
C-centered monoclinic cell (unique axis b) convention replaces the c axis by a′ + c′, in which case P )
(
)
1/2 0 -1 -1/2 1 0 . f Both phases were taken to be nanoscale crystals. 1 0 1
neatly be expressed through the equation VA/VB ≈ 6 cos2〈ω〉 1, where VA and VB are the polyhedral volumes for A and B cations and 〈ω〉 is the mean octahedral tilt angle.62 This expression means that when there is no chance for oxygen octahedra to decrease its volume the crystal can still be compressed by decreasing the volume of the cuboctahedra surrounding the A cation. This is not possible in the case of the P4mm, Cm, and R3m symmetries. 2.2.4. PZT with 0 e x e 0.50. Below 493 °C, lead titanate undergoes a first-order phase transition from the Pm3jm phase to the P4mm phase. The phase transition temperature decreases with increasing x, though the phase transition is of first order up to x ) 0.28 (critical point61). The ionic displacements in tetragonal PbTiO383 are reminiscent to those found in tetragonal PZT. The effect of increasing Zr concentration is to expand the a-axis length, whereas the c-axis length is almost constant until the MPB. The second distinguishing feature is that the Pb-ion shift toward the 〈110〉 directions increases with increasing x. Also, the difference between Zr- and Ti-ion fractional coordinates increases with increasing x. The Pb-ion displacements in the ab plane average out in tetragonal PZT so that there is no net polarization perpendicular to the c axis. This is often modeled by assuming that the four (xx0) sites, equivalent in the case of P4mm symmetry, are statistically occupied by Pb ion (this is contrasted to the case where Pb ion is located at the 1a site and anisotropic ADPs are used).63,84,85 The structural models used in Rietveld refinements take this into account by assuming that each site is occupied by a quarter of a Pb ion. In the case of PZT, the four sites are not equivalent, but to a first approximation, the occupation probability of each site does depend on the local B-cation configuration (how the eight nearest B cation sites are occupied). Thus, the Pb-ion displacements are different in the case of Zr- and Ti-rich areas. In the case of
(
)
2/3 0 2/3 1/3 1 1/3 -2/3 0 1/3
)
and Q )
an ideal symmetry, this corresponds to coherent Pb displacements. However, due to the local distortions following spatial composition variation (and changes in bond lengths), also other Pb-ion displacements (local disorder) occur. It is not straightforward to say which displacement configuration is energetically the most favorable and is most reliably addressed through experiments. This is demonstrated by PbZrO3 and PbHfO3: due to the large space available for Pb, one can construct various hypotetical PbO4 configurations (different from the experimental ones, such as the one corresponding to a ferroelectric ordering) under the constraint that ions have their nominal valences. It is not obvious when a change to another structure is favorable. 2.2.5. Structural Trends. Table 4 summarizes the average structures of the PZT system with typical structural parameters. Figure 1 shows the commonly assumed Pb-ion displacements for tetragonal PZT, PZT with x in the vicinity of the MPB (Cm symmetry), and PbZrO3. Figure 1 illustrates different ways to displace Pb ions to form short Pb-O bonds. 3. Testing the Structural Models Though the Rietveld refinement can yield good estimations for atomic positions and temperature factors, the reliability of the structural parameters depends heavily on the quality of the data. One common problem is the correlation between different parameters, such as the correlation between atomic displacement parameters and atomic positions or background. Thus, it is essential to check if the structural parameters are reasonable. Below, we summarize the results from two types of tests. The first considers the energetical aspect, Madelung energies, whereas the second one is based on the computation of ionic valence through the bond-valence sum method. 3.1. Madelung Energies. The Madelung energy was computed by applying the Ewald method.87,88 The Madelung energy
at ) ap ct ) cp
am ) ap - bp bm ) ap + bp cm ) cp
ah ) ap - bp bh ) bp - cp ch ) ap + bp + cp
ao ) ap - bp bo ) 2(ap + bp) co ) 2cp
P4mm
Cm
R3c
Pbam
Pb(1) Pb(2) Zr O1 O2 O3 O4 O5
Pb Zr/Ti O
Pb Zr/Ti O(1) O(2)
Pb Zr/Ti O(1) O(2)
Pb Zr/Ti O
ion
4g 4h 8i 4g 4h 8i 4f 4e
6a 6a 18b
2a 2a 2a 4b
4d 1b 1b 2c
1a 1b 3c
multiplicity and Wyckoff letter
x x x x x x 0 0
0 0 1 /6 - 2e - 2d
0 x x x
x 1 /2 1 /2 1 /2
0 1 /2 1 /2
x
y y y y y y 1 /2 0
0 0 1 /3 - 4d
0 0 0 y
y 1 /2 1 /2 0
0 1 /2 1 /2
y
0 1 /2 z 0 1 /2 z z z
s + 1/4 t 1 /12
0 z z z
0 z z z
0 1 /2 0
z
a-a-c0
a-a-a-
a0a0a0
a0a0a0
a0a0a0
tilt
at ) 3.978 Å, ct ) 4.149 Å x ) y ) 0.015 zZr ) 0.560, zTi ) 0.547 zO(1) ) 0.101 zO(2) ) 0.617 am ) 5.722 Å, bm ) 5.710 Å, cm ) 4.137 Å, β ) 90.498° xZr/Ti ) 0.477, zZr/Ti ) 0.551 xO(1) ) 0.449, zO(1) ) 0.099 xO(2) ) 0.212, yO(2) ) 0.257, zO2 ) 0.627 ah ) 5.824 Å, ch ) 14.372 Å s ) 0.032 tZr ) 0.014, tTi ) 0.025 e ) 0.012, d ) -0.003 ao ) 5.884 Å, bo ) 11.787 Å, co ) 8.231 Å xPb(1) ) 0.699, yPb(1) ) 0.130 xPb(2) ) 0.707, yPb(2) ) 0.123 xZr ) 0.242, yZr ) 0.124, zZr ) 0.249 xO(1) ) 0.296, yO(1) ) 0.097 xO(2) ) 0.278, yO(2) ) 0.156 xO(3) ) 0.036, yO(3) ) 0.262, zO(3) ) 0.220 zO(4) ) 0.297 zO(5) ) 0.270
ac ) 4.080 Å
typical values for structural parameters
100
295
20
295
773
T/K
xZr
1
0.80
0.52
0.30
0.54
59
26
84
63
52
ref
a In the case of the rhombohedral phases, the structural parameters refer to the hexagonal setting. The parameters s and t give the fractional shifts of A and B cations along the hexagonal ch axis, respectively. The parameter d describes the distortion of the oxygen octahedron, and the parameter e measures the oxygen octahedra tilt about ch.86 Customarily, R3c symmetry has been used to describe the R3m phase by setting e ) 0 so that the octahedral tilt is zero (correspondingly, the c axis is halved). Thus, only the R3c phase is specified. Lattice vectors and oxygen octahedra tilts are given in terms of pseudocubic lattice vectors. xZr refers to Zr content. Deviations from these symmetries are discussed in the text.
ac ) ap
lattice vectors
Pm3jm
space group
TABLE 4: Average Space Group Symmetries and Representative Values for Structural Parameters for the PZT Systema
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6530 J. Phys. Chem. B, Vol. 112, No. 21, 2008
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Figure 2. The Madelung energies (in units of 0Å/e2) of PZT crystals per formula unit (a) versus composition x at room temperature and (b) versus temperature in the case of the x ) 0.54 sample. EM values corresponding to the P4mm, Cm, R3c, and Pm3jm phases are indicated by blue squares, red diamonds, green triangles, and a black square, respectively. For the x ) 0.40 sample, the structural data for the 2.7 GPa value was adapted from ref 87. In the case of the two-phase samples, the larger error bars correspond to the Cm phase.
is given by the equation EM ) ∑kk′ zkzk′Rkk′/(0r), where the summation runs over all ions k and k′ (with charges zk and zk′, respectively) in a primitive cell (assumed to be charge neutral) and 0 is the permittivity of free space. Charges were assumed to be integer multiples of the electric charge e. Following refs 87 and 88, the coefficients Rkk′ are written as
1 Rkk′ ) c∑′H(c|xl′k′ - x0k|/r) - δkk′c/√π + 2 l′ π/(2c2s)∑′ G(π2|b(hkl)|2r2/c2) exp[2πib(hkl)(x0k′ - x0k)r2/c2] hkl
(1) where H(x) ) (1/x) erfc(x) (erfc is the complementary error function), G(x) ) (1/x) exp(x), δkk′ is the Kronecker delta function, xlk is the position of ion k at the primitive cell l, b(hkl) is a primitive reciprocal lattice vector, and V is the volume of the primitive cell, V ) sr3. Primes above the summation signs indicate that in the first (real space) sum the term xl′k′ ) x0k is omitted, and in the case of the second (reciprocal lattice) sum, the term b(hkl) ) 0 is omitted. Number c controls the convergence of the two sums, and was selected so that both the real and reciprocal lattice sums were of the same order of magnitude. The coefficients Rkk′ do not depend on the value of c (dRkk′/dc vanishes for all c). We fixed c at π. Crystal structure data for PZT ceramics were adapted from refs 52, 63, and 89 (high-pressure data). To simplify the computational task, we assumed that the fractional z coordinates for both B cations are the same (though they differed in the case of the tetragonal perovskites; see Table 6 in ref 63). Thus, for this case, the z coordinate values for the B cations were computed using the equation z(B) ) x × z(Zr) + (1 - x) × z(Ti) in the case of x ) 0.20, 0.30, and 0.40. A similar approximation was also done for the fractional c coordinates of the x ) 0.52 PZT sample. In addition, we assumed that Pb ions were fixed at origin. During the Rietveld refinement, they were allowed to shift toward the 〈110〉 directions. Ionic charges were fixed at their nominal values (+2 for Pb, +4 for the B cations, and -2 for O). An error estimation for the Madelung energy was carried out using the standard deviation values of the structural parameters. The error estimates of the structural parameters were assumed to be independent. It is worth noting that the error estimates are specific to the structural model. For instance, for the cubic symmetry, there is only one structural parameter, the a axis value, while there are 11 structural parameters for the Cm phase. This resulted in significantly larger error estimates for the Cm phase.
Figure 2 shows the room temperature Madelung energies for PZT ceramics versus x and, for the x ) 0.54 sample, versus temperature. The Madelung energy increases almost linearly with increasing x (when x e 0.50), as is seen from Figure 2a. This is related to the fact that the average bond lengths increase with increasing x. However, (i) octahedral tilt (R3c phase) and (ii) collective Pb-ion shifts (Cm phase) correspond to a small energy gain. Among the phases in concern, only the R3c phase allows octahedral tilt. Both are features which were neglected in the present treatment of tetragonal PZT (first by symmetry and second by the present approximation). It was interesting to note that, in the vicinity of the MPB, the P4mm (x ) 0.50), Cm, and R3c phases (x ) 0.52) had almost the same Madelung energies: the Cm phase had the lowest energy, while the energy of the R3c phase was almost the same, and approached the energy of the Cm phase with increasing x (Figure 2a). This is consistent with the observed phase transition sequence against x. As the difference between Madelung energies of the Cm and R3c phases in the vicinity of the MPB is small, it is not a surprise that a preparation of single-phase samples within this composition range is very difficult. Also, the evolution of Madelung energies versus temperature indicates that the difference between the Madelung energies of the Cm and R3c phases is small up to room temperature; see Figure 2b. The Madelung energy of the R3c phase at 583 K should be treated with caution, since the R3c phase fraction at this temperature was very small (correspondingly, the structural data were not very accurate). It is clear from Figure 2b that the Madelung energy solely does not explain the observed temperature-dependent phase fractions seen in PZT with the x ) 0.54 sample:52 the Cm and R3c phases had almost the same Madelung energies, and it was only at around 583 K that the difference became significant. Although the given treatment is rather simple, it points out that both (i) local cation shifts and (ii) octahedral tilts are mechanisms to be considered once the energetically favorable structures are searched. There is no reason to expect that the ground state would strictly obey any space group symmetry. The first mechanism is significant through the whole composition range, whereas the latter cannot be neglected at higher Zr concentrations. 3.2. Bond-Valence Sums. Computations of bond-valence sums (BVS) provides a way to test whether the bond lengths are reasonable. Although it is not sensitive for minor changes in symmetry, unless bond lengths are altered, it has proven to be useful for confirming the effect of the proposed Pb-, Zr-, and Ti-ion displacements.63 Namely, if Pb ions are constrained to be located in their ideal positions in the case of the P4mm
Review Article
J. Phys. Chem. B, Vol. 112, No. 21, 2008 6531
Figure 3. Abinit density functional theory code94 was used to estimate the change in ion positions and spontaneous polarization under biaxial x1 ) x2 (red spheres) and uniaxial x3 stress (blue squares). Panels a and b show the fractional coordinates of PbTiO3 and BaTiO3, respectively, panels c and d show the lattice parameters (filled marks) and c/a axes ratios (open marks) of PbTiO3 and BaTiO3, respectively, and panel e shows the spontaneous polarization. The computations were carried out within the local-density approximation and a plane wave basis. Norm-conserving pseudopotentials were generated using the OPIUM pseudopotential generator package.95,96 Details of the computational method are given in ref 97.
phase, the Pb-ion valence would be ≈ +1.8, whereas it is quite close to the nominal valence +2 once Pb ions are allowed to be displaced toward 〈110〉 directions. This is actually a rather general tendency for Pb ion and occurs also in rhombohedral phases. This is quite plausible, since the cuboctahedra is so large that, if Pb ion would occupy its center, it would result in valence deficient Pb ions. Thus, by forming four short bonds with oxygen, more reasonable valence values for Pb are achieved. Similarly, the average B-cation valence V(Bav) ) xV(Zr) + (1 - x)V(Ti) was +4 when Zr and Ti ions were allowed to have different fractional coordinates. In the case of the tetragonal PZT, it was demonstrated that if one insists to constrain the fractional coordinates to be the same, no Wyckoff 1b position in the unit cell would correspond to the nominal valence +4 . It is also worth pointing out that if Pb ions are constrained to be at the 1a site symmetry position, Pb ADPs are very large. On the other hand, constraining the B cations to have the same fractional coordinates yielded negative ADP values. Allowing Pb ions to be displaced toward 〈110〉 directions and Zr and Ti ions to have different fractional z-coordinates resulted in physically reasonable ADP values. From this point of view, it was important to check the validity of the structural models by confirming that both anomalous valence and ADP values can be eliminated by the same structural model. Oxygen valences in PZT systems were close to their nominal valence value -2. Also, oxygen ADP values were reasonable. Thus, as far as NPD studies are considered, it is a decent approximation to assume oxygen octahedra to fullfil the requirements of space group symmetry, in contrast to the case of cation positions. This also implied that it was not necessary to invoke the plausible mechanism, the expansion of Zr octahedra and contraction of Ti octahedra, to correct the anomalously large Zr and small Ti valences in the structural models used to model the NPD data. BVS were found to be consistent with the observed phase transition sequences so that reasonable valence values for each ion were obtained also in the case of the Cm and R3c phases.
The relationship between the oxygen octahedra and cuboctahedra volumes and octahedra tilts were pointed out in ref 62. Namely, R3c symmetry allows oxygen octahedra tilt, which means that the volume of oxygen octahedra increases with increasing tilt angle. This in turn allows the valences of each ion to be close to their nominal values. The same is not possible in the case of R3m symmetry, and thus, the observed oxygen octahedra tilts versus temperature and x seen in the case of PZT samples with x ) 0.52 and x ) 0.53 are consistent with the “constraints” set by nominal valences.63 4. Structure-Piezoelectricity Relationship 4.1. Role of the Pb 6s2 Lone Electron Pair in Tetragonal PZT. The physical origin of the Pb-ion displacement in a local scale is often connected to the 6s2 lone electron pair (L) which makes the displaced position energetically more favorable (one must also consider the 6p states, as discussed below). An insight to the important role of L can be obtained by considering the formation of PbTiO3 by alloying the litharge phase of PbO and TiO2, as shown in ref 90. In the case of Pb1-x(TiO)xO solid solution, a TiO2 group is substitued for a PbO L group and the extra oxygen fills the volume of L. Thus, due to L, Pb prefers to form four short Pb-O bonds in a pyramidal configuration, a tendency which is also seen in different phases of PZT: to fulfill this criteria, Pb ions are displaced from their average positions (although the resulting PbO4 pyramid is no more symmetric). In PbTiO3, the hybridization between Pb 6s and O 2p states explains why large tetragonal distortion is favored over the rhombohedral ground state found in BaTiO3 (Ba is essentially a closed-shell ion).91 This hybridization is important for electromechanical properties and partially explains why Pb containing perovskites generally have much better piezoelectric properties than those perovskites where the A cation is a closedshell ion. This is demonstrated in Figure 3, where the shift of the oxygen octahedra and Ti ion in BaTiO3 and PbTiO3 under the biaxial x1 ) x2 and uniaxial x3 stress are shown. In
6532 J. Phys. Chem. B, Vol. 112, No. 21, 2008 PbTiO3, the oxygen octahedra and Ti ion are shifted against Pb ions with the corresponding change in the spontaneous polarization. The second thing worth noting is that it is the oxygen octahedra which responds most readily to stress, with the shift of Ti ion being far smaller. Though a very similar response is seen in BaTiO3, the absolute ionic shifts are smaller than in the case of PbTiO3. The relative changes in spontaneous polarization as a function of stress follows structural changes in fractional coordinates and lattice parameters. Due to the Pb-O hybridization, the zero stress distortion in PbTiO3 is larger than that in BaTiO3. The relative changes in atomic positions and spontaneous polarization values are slightly larger in BaTiO3 than in PbTiO3, consistent with the notion in ref 92. Though the stress range in Figure 3 is rather large and not easily accessible through experiments, studies on very thin PbTiO3 films confirmed that the change in spontaneous polarization against compressive x1 ) x2 stress was small.92 Very recently, huge strain values in PbTiO3, c/a ) 1.1 (comparable to the highest c/a axis ratios shown in Figure 3c) were achieved by using electric-field pulses with a duration between 35 and 50 ns.93 4.1.1. Electronic Energy Band Structure and ConductiWity. PZT and Ba(ZrxTi1-x)O3 (BZT) systems provide an interesting case study, since the changes in their electronic band structures versus x are well documented. As was summarized in ref 98, there are two main differences between Pb perovskites and those containing closed-shell A-cations (such as Ba in the BZT system): (i) the band gap in the PZT system is almost constant, being 3.45 eV in PbTiO3 and 3.72 eV in PbZrO3, whereas the band gaps in BaTiO3 and BaZrO3 are 3.0 and 5.0 eV, respectively, and (ii) the presence of shallow Pb3+ centers. To understand these observations, band structure computations and electron paramagnetic resonance (EPR) spectroscopy studies were carried out for the PZT system, reported in refs 98 and 99 and summarized below. In BZT, the valence band edge consists of O 2p states and the conduction band consists of pure Ti/Zr d states.100 The band gap increases in BZT with x because the Zr 4d states lie 2 eV above the Ti 3d states, and Ba has little effect on the electronic structure because its states lie well away from the band gap. In PZT, the valence band has essentially the same width and character for all x, because this is determined by the Pb 6s-O 2p interaction. Similar to the case of the BZT system, the Ti/Zr d states increase rapidly in energy with x. The tight-binding computations for cubic PbTiO3 and PbZrO3 revealed that the conduction band minimum is composed of the Ti/Zr d states only at low Zr compositions and switches to Pb 6p states with increasing x.98,99 Shallow Pb3+ centers correspond to a local state slightly above the valence band maximum. EPR experiments revealed that the Pb3+ center acquires more Pb 6p character with increasing x. This was interpreted to be due to the local off-center displacement of the Pb ion. Thus, by lowering the symmetry of the center, the p character is introduced into its wave function. The Pb3+ hole trap binding energy was found to be between 0.14 and 0.26 eV.99 These findings were found to be consistent with Raman scattering studies of PZT, Nd-modified PbTiO3 (PNT), and Ndmodified PZT (PNZT) bulk ceramics using ultraviolet (wavelength 363.79 nm which corresponds to 3.4 eV, close to the band gap energy) and visible light (wavelength 514.532 nm which corresponds to 2.4 eV).99 In the case of UV light, several high-frequency modes above 1000 cm-1 were observed. Most interestingly, the frequency of the mode at around 1170 cm-1 in PZT ceramics showed anomalous behavior at the well-known phase transitions, corresponding to the changes in Pb-ion environment. The use of different systems (PZT, PNT, and
Frantti PNZT) allowed the assignment of these modes to electronic processes in the Pb3+ hole traps,101 in contrast to Zr and Ti. Once this information is combined with the Pb-ion displacements observed through NPD studies, one gains a very consistent picture of these systems. Now, these observations are important not only for understanding the factors affecting switchable polarization in Pb perovskites, but they also provide a physical model explaining the activation energies EA of conductivity in PZT. Values between 0.15 and 0.18 eV were calculated for the EA assuming the Poole-Frenkel mechanism for the conductivity.102 4.2. Models for the Electromechanical Properties in the Vicinity of the MPB. The enhanced piezoelectric properties of the PZT system near the MPB phase boundary are closely related to the two-phase coexistence. An efficient poling is possible for these compositions. Assuming coexisting tetragonal and rhombohedral domains, the number of possible poling directions is 14. Electrically poled ceramics, including PZT, belong to the symmetry group ∞m . The response of PZT ceramics to applied stress or electric field includes intrinsic contributions, due to the distortion of the crystal structure, and extrinsic contributions, due to the domain wall motions and phase changes.103 4.2.1. Extrinsic Contributions. The domain switching in the Pb(Zr0.49Ti0.51)O3 sample, where tetragonal and rhombohedral (volumetric phase fractions 79 and 21%, respectively) phases were present, was studied through in situ uniaxial compression neutron diffraction experiments.104 Data were analyzed via Rietveld refinement which allowed texture and lattice strains to be simultaneously determined. It was found that the rhombohedral phase responds more readily both to electric fields and to applied stress. The importance of the coexistence of the tetragonal and rhombohedral phases for domain switching was demonstrated through these experiments: the tetragonal PZT (x ) 0.40) showed no 90° domain switching under uniaxial compressive stress, whereas the two-phase sample with x ) 0.49 showed significant 90° domain switching. This was explained by considering the constraints set to a domain in a grain by the neighboring grains.105 By taking the mechanical and electrical compatibility conditions into account, it was shown that they are responsible for the known facts that the ceramics near the MPB composition can easily be poled, whereas the poling of polycrystalline tetragonal PZT is very difficult. As was discussed in ref 105, these conclusions would remain unchanged if one includes other phases, such as the monoclinic phase instead of the tetragonal phase. We note that the Cm phase was interpreted to relieve the stress which otherwise would be generated due to the interacting rhombohedral and tetragonal domains in ref 106. First-principles computations, discussed below, suggest that the minor monoclinic distortion has a rather marginal effect on the piezoelectric properties of the PZT system. 4.2.2. Intrinsic Contributions. Polarization Rotation Theory. To understand the exceptionally good electromechanical properties observed in the Pb(Zn1/3Nb2/3)O3-PbTiO3(PZN-PT) and Pb(Mg1/3Nb2/3)O3-PbTiO3 relaxor systems, an explanation, which is not based on the reorientation of domains and the corresponding motion of domain walls but to the polarization rotation, was suggested in ref 107, slightly after the monoclinic phase was reported to occur in the PZT system.67 Computations were carried out for BaTiO3 single crystal, the main finding being that the energetically favorable path to rotate the polarization vector by an applied electric field is from the 〈111〉 to 〈001〉 directions. This polarization rotation was found to be accompanied by a large strain along this direction. When another
Review Article path, such as from the [111] direction via the [011] direction to [001] directions, was used for the computation, significantly lower strain levels (and correspondingly piezoelectric constants) were found. According to this study, the polarization rotation alone can result in the giant piezoelectric response so that one could ignore the extrinsic contribution and concentrate on the intrinsic part. After the powder diffraction studies revealed the presence of the Cm phase in the vicinity of the MPB, the extraordinary properties of PZT in this composition range were soon interpreted to be due to the polarization rotation and the idea was further extended to other systems.108 By symmetry, the polarization vector in the Cm phase is in the (110) mirror plane so that the continuous rotation from the [111] to [001] direction is possible without symmetry changing phase transitions. Though certain polarization rotation paths should be energetically most favorable, there are few open issues when these ideas are applied to poled PZT ceramics with composition near the MPB: (i) in contrast to the well-defined single-crystal case, the symmetry group is ∞m; (ii) material contains two phases; (iii) ab initio computations dedicated for PZT found the largest piezoelectric d33 coefficients, namely, in the rhombohedral side of the MPB (much smaller values were found in the tetragonal side of the MPB);109 (iv) though the transition P4mm f Cm can be continuous, the transition Cm f rhombohedral must be discontinuous;70 and (v) according to the first-principles computations,97 symmetry lowering to the Cm space group alone does not correspond to the increase in piezoelectric constants. Issue iii is consistent with the experimental observations on PZT110 and PZN-PT and PMN-PT111 (the two latter systems also possess a MPB separating the rhombohedral and tetragonal phases). Issues i and ii imply that extrinsic contributions, most notably the domain wall motion and changes in the phase fractions, should be taken into account. Besides, there is hardly any reliable way of isolating the electrical response from two phases; instead, it is the two-phase response that the experiments yield. The intrinsic contribution is closely related to the phase stabilities, which play a significant role in the vicinity of the phase boundary. Thus, if one is sufficiently close to the boundary, the changes in the free energy by applied stress or electric field can be sufficient to destabilize the phase. These issues are discussed next. Phase Instability and Octahedral Versus Cuboctahedral Volumes. According to the thermodynamical study, based on the Landau-Ginzburg-Devonshire theory, once the thermodynamic coercive compressive pressure is approached but not passed, the crystal is destabilized, the free energy becomes shallow, and the piezoelectricity is greatly enhanced.112 Similar conclusions were found through first-principles studies in ref 97, where huge piezoelectric constants were found in the P4mm and R3m phases under hydrostatic pressures at which these phases were not stable. Symmetry dictates that for these phases polarization rotation is not possible. Also, the monoclinic phases were found to be instable in the phase transition pressure range. Consistent with the thermodynamical study,112 the piezoelectric constants of the stable phases were increasing once the phase transition pressure was approached.97 Thus, it is worth studying E if the increase of the elastic compliance slmjk is responsible for the enhanced piezoelectric properties near the phase boundary E regions, through the relationship dijk ) eilmslmjk , where dijk and eilm are piezoelectric constants, defined in ref 113. This type of mechanism was proposed to be responsible for the giant piezoelectric response observed in PZN-PT in refs 114 and 115. As the analysis given in ref 69 indicates, the existence of the Cm phase does not allow a continuous polarization rotation.
J. Phys. Chem. B, Vol. 112, No. 21, 2008 6533 Thus, even in the case of the predicted pressure-induced phase transition P4mm f R3c,97 there would be a two-phase coexistence in the transition pressure region due to the hysteresis effects, characteristic to the first-order transition. In such a case, the piezoelectric properties would be determined by the intrinsic properties of the coexisting phases and the extrinsic properties discussed in section 4.2.1. 5. Conclusions Structure-property relationships in the lead zirconate titanate (PZT) system were reviewed. The first part concentrated on the structure determination through X-ray and neutron powder diffraction techniques. Results obtained through two frequently used local probes, Raman scattering and transmission electron microscopy combined with electron diffraction, were summarized. Particular attention was paid to the structural models proposed for PZT near the morphotropic phase boundary (MPB) composition. Experiments have shown that PZT at that composition range contains two phases, the rhombohedral and monoclinic phases. Monoclinic distortion from the tetragonal distortion is rather small and may not have a significant role for the piezoelectricity. The role of domain configuration was found to be important for (i) the correct structure determination and (ii) the piezoelectric properties. When data collected on different techniques, such as electron diffraction and neutron powder diffraction, are compared, the size and shape dependence of the domain widths should be taken into account. Notably, the piezoelectric response of PZT with composition near the MPB can be divided into two parts: (i) the extrinsic part, which includes changes in phase fractions of the two coexisting phases and domain configuration necessitating domain wall motion, and (ii) the intrinsic part, which is largely determined by the changes within a domain. Recent in situ uniaxial compression neutron diffraction experiments showed the importance of extrinsic contribution, whereas the hybridization between Pb 6s and O 2p states largely explains why the intrinsic piezoelectric response of Pb-based perovskites is much larger than in the case of closed-shell A cations, such as Ba-based perovskites. Near the phase boundaries, the intrinsic part is further enhanced by the phase instability: the increase of certain elastic compliance tensor components is accompanied by the increase of the corresponding piezoelectric tensor components. Acknowledgment. An anonymous reviewer is acknowledged for constructive suggestions. The author is grateful for the Academy of Finland for financial support (Project Nos. 207071 and 207501). References and Notes (1) Sawaguchi, E. J. Phys. Soc. Jpn. 1953, 8, 615. (2) Jaffe, B.; Cook, W. R.; Jaffe, H. Piezoelectric Ceramics; Academic Press: London, 1971. (3) Young, R. A. The RietVeld Method; Oxford University Press: New York, 1996. (4) Lima, K. C. V.; Souza Filho, A. G.; Ayala, A. P.; Mendes Filho, J.; Freire, P. T. C.; Melo, F. E. A.; Arau´jo, E. B.; Eiras, J. A. Phys. ReV. B 2001, 63, 184105. (5) Souza Filho, A. G.; Lima, K. C. V.; Ayala, A. P.; Guedes, I.; Freire, P. T. C.; Melo, F. E. A.; Mendes Filho, J.; Arau´jo, E. B.; Eiras, J. A. Phys. ReV. B 2002, 66, 132107. (6) Freire, J. D.; Katiyar, R. S. Phys. ReV. B 1988, 37, 2074. (7) Foster, C. M.; Li, Z.; Grimsditch, M.; Chan, S. K.; Lam, D. J. Phys. ReV. B 1993, 48, 10160. (8) Frantti, J.; Lantto, V. Phys. ReV. B 1997, 56, 221. (9) Frantti, J.; Lantto, V. Phys. ReV. B 1996, 54, 12139. (10) Singh, A.; Sreenivas, K.; Katiyar, R. S.; Gupta, V. J. Appl. Phys. 2007, 102, 074110.
6534 J. Phys. Chem. B, Vol. 112, No. 21, 2008 (11) Tsukada, S.; Kato, K.; Kano, J.; Yamamoto, T.; Kojima, S. Ferroelectrics 2007, 355, 66. (12) Barker, A. S., Jr.; Hopfield, J. J. Phys. ReV. B 1973, 7, 3088. (13) Burns, G.; Scott, B. A. Phys. ReV. 1964, A1732, 135. (14) Frantti, J.; Lappalainen, J.; Eriksson, S.; Lantto, V.; Nishio, S.; Kakihana, M.; Ivanov, S.; Rundlo¨f, H. Jpn. J. Appl. Phys. 1999, 38, 5679. (15) Buixaderas, E.; Nuzhnyy, D.; Veljko, S.; Kamba, S.; Savinov, M.; Petzelt, J.; Kosec, M. Appl. Phys. Lett. 2007, 91, 112909. (16) Burns, G.; Scott, B. A. Phys. ReV. Lett. 1970, 25, 167. (17) Ikeuchi, Y.; Kojima, S.; Yamamoto, T. Jpn. J. Appl. Phys. 1997, 36, 2985. (18) Frantti, J.; Lantto, V.; Nishio, S.; Kakihana, M. Phys. ReV. B 1999, 59, 12. (19) Frantti, J.; Eriksson, S.; Hull, S.; Ivanov, S.; Lantto, V.; Lappalainen, J.; Kakihana, M. J. Eur. Ceram. Soc. 2004, 24, 1141. (20) Frantti, J.; Fujioka, Y.; Eriksson, S.; Hull, S.; Kakihana, M. Inorg. Chem. 2005, 44, 9267. (21) Frantti, J.; Fujioka, Y.; Eriksson, S.; Lantto, V.; Kakihana, M. J. Electroceramics 2004, 13, 299. (22) Buixaderas, E.; Kamba, S.; Petzelt, J.; Drahokoupil, J.; Laufek, F.; Kosec, M. J. Appl. Phys. 2007, 101, 074106. (23) Viehland, D. Phys. ReV. B 1995, 52, 778. (24) Ricote, J.; Corker, D. L.; Whatmore, R. W.; Impey, S. A.; Glazer, A. M.; Dec, J.; Roleder, K. J. Phys.: Condens. Matter 1998, 10, 1767. (25) According to a study [ Huang, H.; Zhou, L. M.; Guo, J.; Hng, H. H.; Oh, J. T.; Hing, P. Appl. Phys. Lett. 2003, 83, 3692. ] carried out on the x ) 0.90 sample, the R2 spots do not appear in the ED pattern if viewed along the [1j10] direction. This study concluded that the R2 spots are observed in the 109° domain but not in the neighboring 71° domains. (26) Corker, D. L.; Glazer, A. M.; Whatmore, R. W.; Stallard, A.; Fauth, F. J. Phys.: Condens. Matter 1998, 10, 6251. (27) Glazer, A. M. Acta Crystallogr., Sect. A 1975, 31, 756. (28) Leung, K. Phys. ReV. B 2003, 67, 104108. (29) Strukov, B. A.; Levanyuk, A. P. Ferroelectric Phenomena in Crystals: Physical Foundations; Springer-Verlag: Berlin, 1998. (30) Kittel, C. Phys. ReV. 1946, 70, 965. (31) Lines, M. E.; Glass, A. M. Principles and Applications of Ferroelectric and Related Materials; Oxford University Press: New York, 2001. (32) Mitsui, T.; Furuichi, J. Phys. ReV. 1953, 90, 193. (33) Schilling, A.; Adams, T. B.; Bowman, R. M.; Gregg, J. M.; Catalan, G.; Scott, J. F. Phys. ReV. B 2006, 74, 024115. (34) Streiffer, S. K.; Eastman, J. A.; Fong, D. D.; Thompson, C.; Munkholm, A.; Ramana Murty, M. V.; Auciello, O.; Bai, G. R.; Stephenson, G. B. Phys. ReV. Lett. 2002, 89, 067601. (35) Nishida, K.; Osada, M.; Wada, S.; Okamoto, S.; Ueno, R.; Funakubo, H.; Katoda, T. Jpn. J. Appl. Phys. 2005, 44, L827. (36) Osada, M.; Nishida, K.; Wada, S.; Okamoto, S.; Ueno, R.; Funakubo, H.; Katoda, T. Appl. Phys. Lett. 2005, 87, 232902. (37) Dmowski, W.; Egami, T.; Faber, L.; Davies, P. K. Fundamental Physics of Ferroelectrics-Eleventh Williamsburg Ferroelectric Workshop; Cohen, R. E., Ed.; American Institute of Physics: Melville, NY, 2001; p 33. (38) Bell, A. J. J. Mater. Sci. 2006, 41, 13. (39) Glazer, A. M.; Thomas, P. A.; Baba-Kishi, K. Z.; Pang, G. K. H.; Tai, C. W. Phys. ReV. B 2004, 70, 184123. (40) Bellaiche, L.; Garcı´a, A. Phys. ReV. Lett. 2000, 84, 5427. (41) Fujioka, Y.; Kakihana, M. Trans. Mater. Res. Jpn. 2003, 28, 373. (42) Azad, A. K.; Ivanov, S.; Eriksson, S.-G.; Eriksen, J.; Rundlo¨f, H.; Mathieu, R.; Svedlindh, P. Mater. Res. Bull. 2001, 36, 2215. (43) Fujioka, Y.; Frantti, J.; Kakihana, M. J. Phys. Chem. B 2006, 110, 777. (44) Liegeois-Duychaerts, M.; Tarte, P. Spectrochim. Acta 1974, 30A, 1711. (45) Camargo, E. R.; Frantti, J.; Kakihana, M. J. Mater. Chem. 2001, 11, 1875. (46) Wilkinson, A. P.; Xu, J.; Pattanaik, S.; Billinge, S. J. L. Chem. Mater. 1998, 10, 3611. (47) Suchanek, W. L.; Lencka, M.; McCandlish, L.; Pfeffer, R. L.; Oledzka, M.; Mikulka-Bolen, K.; Rossetti, G. A., Jr.; Riman, R. E. Cryst. Growth Des. 2005, 5, 1715. (48) Cao, W.; Cross, L. E. Phys. ReV. B 1993, 47, 4825. (49) Ari-Gur, P.; Benguigui, L. Solid State Commun. 1978, 15, 1077. (50) Rane, M. V.; Navrotsky, A.; Rossetti, G. A., Jr J. Solid State Chem. 2001, 161, 402. (51) Rossetti, G. A.; Zhang, W.; Khachaturyan, A. G. Appl. Phys. Lett. 2006, 88, 072912. (52) Frantti, J.; Eriksson, S.; Hull, S.; Lantto, V.; Rundlo¨f, H.; Kakihana, M. J. Phys.: Condens. Matter 2003, 15, 6031. (53) Leineweber, A.; Mittemeijer, E. J. J. Appl. Crystallogr. 2004, 37, 123. (54) Stephens, P. W. J. Appl. Crystallogr. 1999, 32, 281.
Frantti (55) Zevin, L. S.; Kimmel, G. QuantitatiVe X-ray Diffractometry; Springer-Verlag: New York, 1995, p 263. (56) Larson, A. C.; Von Dreele, R. B. General Structure Analysis System; LANSCE MS-H805; Los Alamos National Laboratory: Los Alamos, NM, 2000. (57) Rossetti, G. A.; Cahill, P. F.; Biederman, R. R.; Sacco, A. Mater. Lett. 1999, 41, 72. (58) Glazer, A. M.; Roleder, K.; Dec, J. Acta Crystallogr., Sect. B 1993, 49, 846. (59) Corker, D. L.; Glazer, A. M.; Dec, J.; Roleder, K.; Whatmore, R. W. Acta Crystallogr., Sect. B 1997, 53, 135. (60) Michel, C.; Moreau, J. M.; Achenbach, G. D.; Gerson, R.; James, W. J. Solid State Commun. 1969, 7, 865. (61) Yamamoto, T. Jpn. J. Appl. Phys. 1998, 37, 6041. (62) Thomas, N. W.; Beitollahi, A. Acta Crystallogr., Sect. B 1994, B 50, 549. (63) Frantti, J.; Ivanov, S.; Eriksson, S.; Rundlo¨f, H.; Lantto, V.; Lappalainen, J.; Kakihana, M. Phys. ReV. B 2002, 66, 064108. (64) Corker, D. L.; Glazer, A. M.; Kaminsky, W.; Whatmore, R. W.; Dec, J.; Roleder, K. Acta Crystallogr., Sect. B 1998, 54, 18. (65) Whatmore, R. W.; Glazer, A. M. J. Phys. C: Solid State Phys. 1979, 12, 1505. (66) Teslic, S.; Egami, T. Acta Crystallogr., Sect. B 1998, 54, 750. (67) Noheda, B.; Cox, D. E.; Shirane, G.; Gonzalo, J. A.; Cross, L. E.; Park, S-E. Appl. Phys. Lett. 1999, 74, 2059. (68) Woodward, D. I.; Knudsen, J.; Reaney, I. M. Phys. ReV. B 2005, 72, 104110. (69) Though there have been TEM/ED reports according to which single-phase samples were prepared, it must be said that this method does not suit the phase purity determination due to the small volume probed by the electron beam. (70) Howard, C. J.; Stokes, H. T. Acta Crystallogr., Sect. B 1998, 54, 782. (71) Noheda, B.; Cox, D. E.; Shirane, G.; Guo, R.; Jones, B.; Cross, L. E. Phys. ReV. B 2000, 63, 014103. (72) Massa, W. Crystal Structure Determination; Springer-Verlag: Berlin, 2000, p 157. (73) Ari-Gur, P.; Benguigui, L. J. Phys. D: Appl. Phys. 1975, 8, 1856. (74) Ranjan, R.; Mishra, S. K.; Pandey, D.; Kennedy, B. J. Phys. ReV. B 2002, 65, 060102. (75) Hatch, D. M.; Stokes, H. T.; Ranjan, R.; Ragini; Mishra, S. K.; Pandey, D.; Kennedy, B. J. Phys. ReV. B 2002, 65, 212101. (76) Ranjan, R.; Singh, A. K.; Ragini; Pandey, D. Phys. ReV. B 2005, 71, 092101. (77) Ragini; Ranjan, R.; Mishra, S. K.; Pandey, D. J. Appl. Phys. 2002, 92, 3266. (78) Singh, A. K.; Pandey, D.; Yoon, S.; Baik, S.; Shin, N. Appl. Phys. Lett. 2007, 91, 192904. (79) Cox, D. E.; Noheda, B.; Shirane, G. Phys. ReV. B 2005, 71, 134110. (80) Smith, D. K.; Newkirk, H. W. Acta Crystallogr. 1965, 18, 983. (81) Rossetti, G. A.; Khachaturyan, A. G. Appl. Phys. Lett. 2007, 91, 072909. (82) International Tables for Crystallography. Volume A: Space-Group Symmetry; Hahn, T., Ed.; Springer: Dordrecht, The Netherlands, 2005; pp 78-85. (83) Glazer, A. M.; Mabud, S. A. Acta Crystallogr., Sect. B 1978, 34, 1065. (84) Noheda, B.; Gonzalo, J. A.; Cross, L. E.; Guo, R.; Park, S-E.; Cox, D. E.; Shirane, G. Phys. ReV. B 2000, 61, 8687. (85) Frantti, J.; Lappalainen, J.; Eriksson, S.; Lantto, V.; Nishio, S.; Kakihana, M.; Ivanov, S.; Rundlo¨f, H. Jpn. J. Appl. Phys. 2000, 39, 5697. (86) Megaw, H. D.; Darlington, C. N. W. Acta Crystallogr., Sect. A 1975, 31, 161. (87) Born, M.; Huang, K. Dynamical Theory of Crystal Lattices; Oxford University Press: Oxford, U.K., 1988. (88) Cowley, R. A. Acta Crystallogr. 1962, 15, 687. (89) Rouquette, J.; Haines, J.; Bornand, V.; Pintard, M.; Papet, Ph.; Marshall, W. G.; Hull, S. Phys. ReV. B 2005, 71, 024112. (90) Le Bellac, D.; Kiat, J. M.; Garnier, P.; Moudden, H.; Sciau, Ph.; Buffat, P. A.; Andre´, G. Phys. ReV. B 1995, 52, 13184. (91) Cohen, R. E. Nature 1992, 358, 136. (92) Lee, H. N.; Nakhmanson, S. M.; Chisholm, M. F.; Christen, H. M.; Rabe, K.; Vanderbilt, D. Phys. ReV. Lett. 2007, 98, 217602. (93) Grigoriev, A.; Sichel, R.; Lee, H. N.; Landahl, E. C.; Adams, B.; Dufresne, E. M.; Evans, P. G. Phys. ReV. Lett. 2008, 100, 027604. (94) Gonze, X.; Beuken, J.-M.; Caracas, R.; Detraux, F.; Fuchs, M.; Rignanese, G.-M.; Sindic, L.; Verstraete, M.; Zerah, G.; Jollet, F.; Torrent, M.; Roy, A.; Mikami, M.; Ghosez, Ph.; Raty, J.-Y.; Allan, D. C. Comput. Mater. Sci. 2002, 25, 478. Gonze, X.; Rignanese, G.-M.; Verstraete, M.; Beuken, J.-M.; Pouillon, Y.; Caracas, R.; Jollet, F.; Torrent, M.; Zerah, G.; Mikami, M.; Ghosez, Ph.; Veithen, M.; Raty, J.-Y.; Olevano, V.; Bruneval,
Review Article F.; Reining, L.; Godby, R.; Onida, G.; Hamann, D. R.; Allan, D. C. Z. Kristallogr. 2005, 220, 558. (95) Grinberg, I.; Ramer, N. J.; Rappe, A. M. Phys. ReV. B 2000, 62, 2311. (96) Rappe, A. M.; Rabe, K. M.; Kaxiras, E.; Joannopoulos, J. D. Phys. ReV. B 1990, 41, 1227. (97) Frantti, J.; Fujioka, Y.; Nieminen, R. M. J. Phys. Chem. B 2007, 111, 4287. (98) Warren, W. L.; Robertson, J.; Dimos, D.; Tuttle, B. A.; Pike, G. E.; Payne, D. A. Phys. ReV. B 1996, 53, 3080. (99) Robertson, J.; Warren, W. L.; Tuttle, B. A. J. Appl. Phys. 1995, 77, 3975. (100) King-Smith, R. D.; Vanderbilt, D. Phys. ReV. B 1994, 49, 5828. (101) Frantti, J.; Lantto, V.; Kakihana, M. Jpn. J. Appl. Phys. 1998, 37, 5406. (102) Lappalainen, J.; Frantti, J.; Lantto, V. J. Appl. Phys. 1997, 82, 3469. (103) Newnham, R. E. Properties of Materials: Anisotropy, Symmetry, Structure; Oxford University Press: New York, 2005. ¨ stu¨ndag, E.; Clausen, B.; Daymond, M. R. J. Appl. (104) Rogan, R. C.; U Phys. 2003, 93, 4104.
J. Phys. Chem. B, Vol. 112, No. 21, 2008 6535 ¨ stu¨ndag, E.; Bhattacharya, K. Nature (105) Li, J. Y.; Rogan, R. C.; U 2005, 4, 776. (106) Topolev, V.; Turik, A. J. Phys.: Condens. Matter 2001, 13, 771. (107) Fu, H.; Cohen, R. E. Nature 2000, 403, 281. (108) Noheda, B.; Cox, D. E.; Shirane, G.; Park, S.-E.; Cross, L. E.; Zhong, Z. Phys. ReV. Lett. 2001, 86, 3891. (109) Bellaiche, L.; Garcı´a, A. Phys. ReV. B 2001, 64, 060103. (110) Du, X.; Zheng, J.; Belegudu, U.; Uchino, K. Appl. Phys. Lett. 1998, 72, 2421. (111) Park, S.-E.; Shrout, T. J. Appl. Phys. 1997, 82, 1804. (112) Budimir, M.; Damjanovic, D.; Setter, N. Phys. ReV. B 2006, 73, 174106. (113) Nye, J. F. Physical Properties of Crystals; Clarendon Press: Oxford, U.K.; 1957; Chapter 10. (114) Kisi, E. H.; Piltz, R. O.; Forrester, J. S.; Howard, C. J. J. Phys.: Condens. Matter 2003, 15, 3631. (115) Kisi, E. H.; Forrester, J. S.; Knight, K. S. Acta Crystallogr. 2006, C62, i46.
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