Stability of ferroic phases in the highly piezoelectric Pb(Zr_xTi_1−x)O₃ ceramics

The morphotropic phase boundary in the phase diagram of the technologically important Pb(Zr_xTi_1−x)O₃ (PZT) ceramics has been traditionally believed to separate ferroelectric tetragonal and rhombohedral phase regions. This old picture has come under close scrutiny during the last eight years following the discovery of new monoclinic phases in the Cm and Cc space groups. This article presents a brief overview of these discoveries in which the use of multiple diffraction probes (X-ray, electron, neutron diffraction) in conjunction with physical property measurements has played a crucial role. A new phase diagram of PZT showing the stability fields of these structures below room temperature is also presented.

1. Introduction

PbTiO₃ (PT) is a well known ferroelectric material with perovskite structure. The cubic paraelectric phase of PT transforms to a tetragonal ferroelectric phase below 763 K (Jaffe et al., 1971). This phase transition is accompanied by a large transformation strain (tetragonality η = = 0.0635) which leads to cracking of ceramic PT bodies on cooling from the sintering temperature to the room temperature through the Curie point (763 K). In order to develop sinterable PT-based ceramics, chemical substitutions at the Pb and/or Ti sites have been attempted in the past. This search has led to the discovery of the well known Pb(Zr_xTi_1−x)O₃ (PZT), Pb(Mg_1/3Nb_2/3)_xTi_1−xO₃ (PMN-xPT), Pb(Zn_1/3Nb_2/3)_xTi_1−xO₃ (PZN–xPT) and other similar solid-solution systems with extraordinary piezoelectric properties (Jaffe et al., 1971). These piezoelectric materials are of tremendous technological importance for use as sensors and actuators (Jaffe et al., 1971).

The tetragonal structure of PT in the PZT, PMN–xPT and PZN–xPT solid-solution systems is stable at room temperature up to a composition (x) corresponding to a nearly vertical phase boundary in the temperature ( T) – composition (x) phase diagram. This phase boundary is commonly known as the morphotropic phase boundary (MPB) (Jaffe et al., 1971). The dielectric constant, piezoelectric coefficients and the electromechanical coupling coefficients of these ceramics are maximized for the composition closest to the MPB. To date, there is no satisfactory explanation for this extreme response although three different models based on coexistence of phases (Isupov, 1968), lattice instability near room temperature (Mishra et al., 1996) and polarization rotation (Fu & Cohen, 2000; Bellaiche et al., 2000; Vanderbilt & Cohen, 2001) have been proposed.

Until 1999, the MPB was believed to separate the stability phase fields of the ferroelectric tetragonal (F_T) and the ferroelectric rhombohedral (F_R) phases with the coexistence of these phases in the MPB region (Jaffe et al., 1971; Mishra et al., 1997). Several new monoclinic phases have recently been discovered in the MPB region of these solid-solution systems with Cm (Noheda et al., 1999; Noheda, Gonzalo et al., 2000; Noheda, Cox et al., 2000), Cc (Hatch et al., 2002; Ranjan et al., 2005) and Pm (Singh & Pandey, 2001, 2003; Kiat et al., 2002) space groups. The sixth-order Landau theory, which successfully explained the stability of the ferroelectric tetragonal, orthorhombic and rhombohedral phases of BaTiO₃ and other ABO₃ perovskites, has been extended to eighth order (Vanderbilt & Cohen, 2001), to account for the newly discovered monoclinic phases. The eighth-order Landau theory has revealed three different monoclinic phases, commonly labelled as M_A, M_B and M_C types, with space groups Cm, Cm and Pm, respectively (Vanderbilt & Cohen, 2001). The M_A- and M_B-type phases have the same space group (Cm) but in the former P_x = P_y > P_z while in the latter P_x = P_y < P_z, where P_x, P_y and P_z are polarization components along the pseudocubic axes. First-principles calculations on PZT have not only confirmed the presence of the monoclinic phases but have also shown that they are responsible for the maximum electromechanical response (Fu & Cohen, 2000; Bellaiche et al., 2000). A unique feature of the structure of the monoclinic phases is that their polarization vector can lie anywhere in a symmetry plane in contrast to that in the tetragonal and rhombohedral phases where it could lie only along the specified crystallographic directions [001] and [111], respectively (Noheda, Gonzalo et al., 2000; Noheda, Cox et al., 2000; Singh & Pandey, 2003). The polarization vector of the monoclinic phase can therefore adjust itself easily to the external electric field direction and may lead to larger electromechanical response (Fu & Cohen, 2000; Bellaiche et al., 2000; Noheda, Gonzalo et al., 2000; Noheda, Cox et al., 2000; Vanderbilt & Cohen, 2001). The presence of these monoclinic phases also minimizes the elastic energies at the various interphase boundaries (Jin et al., 2003).

The purpose of this article is to present a brief overview of these discoveries in relation to PZT in which the use of multiple diffraction probes (X-ray, electron, neutron diffraction) in conjunction with physical property measurements has played a crucial role in establishing the structure and the stability fields of the two newly discovered monoclinic phases in the Cm and Cc space groups.

2. Structure of the room-temperature phases

2.1. The old picture

The structure of PZT has until recently been believed to be tetragonal (F_T) and rhombohedral (F_R) on the Ti- and Zr-rich sides of the morphotropic phase boundary (MPB) (Jaffe et al., 1971) while the two phases were reported to coexist over a range of composition across the MPB. As per the description given in the book by Jaffe et al. (1971), the MPB lies around x = 0.520. The F_T phase is therefore stable for x < 0.520 while F_R is stable for 0.52 < x < 0.62. On increasing the Zr content beyond x = 0.62, a new rhombohedral phase appears in which the neighbouring oxygen octahedra are rotated in an anti-phase manner about the threefold axis. The space group of this rhombohedral phase is reported to be R3c and the corresponding tilt system is a⁻a⁻a⁻ in Glazer's notation (Glazer, 1972). This rhombohedral phase in the R3c space group transforms to the rhombohedral phase in the R3m space group on heating above room temperature. Accordingly, the two ferroelectric rhombohedral phases in the R3m and R3c space groups are labelled and (Jaffe et al., 1971). The phase is stable up to x ≃ 0.94 beyond which an antiferroelectric phase in the space group Pbam is stabilized, which is commonly labelled as the A_O phase. The cubic paraelectric phase to which F_T, , and A_O phases transform is labelled as the P_C phase. Thus the old PZT phase diagram contains F_T (P4mm), (R3m), (R3c), A_O (Pbam/Pba2) and P_C () phases.

The composition width (Δx) of the MPB depends on the preparation conditions. For example, in samples prepared by the conventional solid-state route, this width has been reported to be as small as ~0.05 (Noheda, Cox et al., 2000; Schönau et al., 2007) and as large as ~0.15 (Ari-Gur & Benguigui, 1974). The intrinsic width of the MPB region was shown to be much smaller, Δx ≃ 0.01, by Mishra et al. (1996, 1997) in chemically homogeneous samples prepared by a semi-wet route (Singh et al., 1993). Fig. 1 depicts the synchrotron powder XRD profiles of the pseudocubic 111, 200 and 220 reflections of PZT prepared by the semi-wet route for different Zr content across the MPB. For the tetragonal (F_T) structure, the 111 profiles should be a singlet whereas the 200 and 220 are doublets. For the rhombohedral () structure, the 200 is a singlet while 220 and 111 are doublets. The doublet character of 200 and 220 and singlet nature of 111 confirms the tetragonal structure for x = 0.515 in Fig. 1. On the other hand, the structure is apparently rhombohedral for x > 0.525, as 200 is now a singlet while 111 is a doublet. For x = 0.520 and 0.525, the diffraction profiles do not correspond to either F_T or phases, and this was interpreted as being due to a coexistence of the two phases (Mishra et al., 1996, 1997).

Figure 1 Powder synchrotron XRD profiles of 111, 200 and 220 pseudocubic reflections of PZT in the composition range 0.515 to 0.60 (after Singh et al., 2007a).

2.2. The current picture

2.2.1. Structure of PZT with x = 0.525

The composition x = 0.525 was earlier regarded as consisting of coexisting T and R phases (Singh et al., 1993) on the basis of the doublet nature of the 200 and 222 profiles. Recently, Ragini et al. (2001) carried out a Rietveld refinement for this composition using rotating-anode powder XRD data and concluded that this composition consists of a coexistence of the F_T and a monoclinic phase in the Cm space group, earlier discovered by Noheda et al. (1999) below room temperature for x = 0.520. Full-pattern Rietveld refinements of x = 0.525 carried out using synchrotron powder XRD data considering coexistence of and F_T phases and pure monoclinic phase reveal that the structure of PZT for this composition is nearly pure monoclinic in the Cm space group (Singh et al., 2007a). The results of Rietveld refinements are illustrated using a few selected profiles in Fig. 2, from which it is evident that the pure monoclinic phase in the Cm space group accounts very well for the observed profiles for this composition. The refinement for the model based on the coexistence of F_T and phases leads to very poor fit. Consideration of the coexisting F_T phase with the monoclinic phase does not lead to any statistically significant improvement in the fits (Singh et al., 2007a). The consideration of coexisting and monoclinic phases, as proposed by Frantti et al. (2002), on the other hand, deteriorates the fits and leads to higher χ² (2.22) as compared to that for pure Cm (χ² = 2.08) even though the number of refinable structural parameters has increased from 15 for pure Cm to 24 for the Cm + R3m model (Singh et al., 2007a). All these facts establish the structure of PZT for x = 0.525 as pure monoclinic in the Cm space group. We label this ferroelectric monoclinic phase as the phase, in analogy with the labelling of the ferroelectric rhombohedral phase in R3m space group as by Jaffe et al. (1971). The monoclinic cell parameters (a_m, b_m, c_m) of are related to the elementary perovskite cell parameters (a_p, b_p, c_p) as: a_m ≃ √2 a_p, b_m ≃ √2 b_p and c_m ≃ c_p. The equivalent perovskite cell parameters of the phase for x = 0.525 are a_p ≃ 4.0681, b_p ≃ 4.0552, c_p ≃ 4.0975, which implies that this phase is of M_A type (P_X = P_Y < P_Z) in the notation of Vanderbilt & Cohen (2001).

Figure 2 Observed (dots), calculated (continuous line) and difference (bottom line) profiles of 110, 111 and 200 pseudocubic reflections of Pb(Zr0.525Ti0.475)O3 using (a) coexistence of rhombohedral and tetragonal and (b) pure monoclinic (Cm) structural models. The vertical tick marks above the difference profiles give the positions of the Bragg reflections (after Singh et al., 2007a).

2.2.2. Structure of PZT with x ≤ 0.520

Rietveld refinements for x ≤ 0.515 have confirmed the tetragonal structure in the P4mm space group but the location of the Pb atoms has been found to be disordered along the pseudocubic 〈110〉 directions perpendicular to the tetragonal c ([001]) axis (Noheda, Gonzalo et al., 2000; Ragini et al., 2002). These directions correspond to the [100] and [010] directions of the monoclinic phase suggesting that, within the framework of the average tetragonal structure of PZT for x < 0.520, the local structure at the unit-cell level is indeed monoclinic like. The results of the Rietveld refinements for the intermediate composition 0.515 < x < 0.525, i.e. x = 0.520, show that tetrag­onal and monoclinic phases coexist for this composition (Singh et al., 2007b), which gives a composition width of Δx ≃ 0.01 for the phase coexistence region.

2.2.3. Structure of PZT with 0.525 < x < 0.62

On increasing the Zr content beyond x = 0.525, the splitting of the 200 peak in Fig. 1 disappears but there is an anomalous broadening of this peak, as previously noted by Ragini et al. (2002) even in the rotating-anode powder XRD data. Fig. 3 shows the composition dependence of the ratio of the full widths at half-maximum (FWHM) of 200 and 111 reflections (using rhombohedral indices) obtained from the synchrotron powder XRD data. This ratio is nearly equal to 1 for the tetragonal compositions, as expected. For the rhombohedral compositions also this ratio should have been nearly equal to one but it is much larger for x ≥ 0.535. This anomalous broadening has in the past been accounted for in the Rietveld refinements (Noheda, Gonzalo et al., 2000) using anisotropic peak broadening functions proposed by Stephens (1999). However, we have recently shown that the use of the anisotropic peak broadening functions to account for the anomalous broadening of the h00- and hh0-type reflections leads to mismatch in the observed and calculated synchrotron XRD profiles for other reflections, like 111 (Singh et al., 2007a). This is illustrated in Fig. 4(a). If one constrains the refinement to account for the observed width of 111 profiles, large mismatch appears for the 200 and 220 reflections (see Fig. 4b). For PZT compositions much richer in Zr content, Corker et al. (1998) have proposed that acceptable agreement factors in the Rietveld refinements using the rhombohedral space group could be obtained only if one considers off-centred displacements of the Pb atoms along the 〈110〉 pseudocubic directions. However, consideration of such a local disorder model in the Rietveld refinements for x = 0.535 did not improve the fits (see Fig. 4c). All these comprehensively rule out the R3m space group, with or without local disorder of Pb, for x = 0.535. Use of the Cm space group, on the other hand, leads to much lower χ² and excellent fits for both the 200- and 111-type reflections (see Fig. 4d). The results shown in Fig. 4 thus clearly favour the Cm space group for x = 0.535 in agreement with the earlier results of Rietveld refinements using rotating-anode XRD data (Ragini et al., 2002). Similar refinements using synchrotron XRD data for higher Zr compositions revealed the inadequacy of the rhombohedral structure and the correctness of the Cm space group. It is thus clear that the structure of PZT at 300 K is pure monoclinic () in the Cm space group for the composition range 0.525 ≤ x ≤ 0.62 (Singh et al., 2007a) and not rhombohedral, as hitherto believed for decades [see e.g. books by Jaffe et al. (1971), Jona & Shirane (1962) and Lines & Glass (1977)].

Figure 3 Variation of the ratio of the FWHM of the profiles of the 200 and 111 pseudocubic reflections for Zr-rich compositions of PZT (after Singh et al., 2007a).

Figure 4 Observed (dots), calculated (continuous line) and difference (bottom line) profiles of 111, 200 and 220 pseudocubic reflections of PZT for x = 0.535 using various structural models. The vertical tick marks above the difference profiles give the positions of the Bragg reflections (after Singh et al., 2007a).

The variation of the equivalent elementary perovskite cell parameters with composition at 300 K for the monoclinic compositions (x ≥ 0.520) is depicted in Fig. 5. The a_p and b_p lattice parameters of the monoclinic phase are found to increase with composition for 0.520 ≤ x < 0.545 and then show saturation for x ≥ 0.545. The monoclinic angle β also increases with increasing x up to x < 0.545 but starts decreasing for x ≥ 0.545. The value of c_m is highest for x = 0.520 and it decreases rather fast with x before showing saturation for x ≥ 0.535. The c/a ratio decreases sharply with increasing Zr content and becomes very close to one for x ≥ 0.545 as can be seen from the inset in Fig. 5. The unit-cell volume increases sharply for x < 0.545 and then increases slowly for higher Zr content (see the inset).

Figure 5 Variation of the equivalent elementary perovskite cell parameters with composition (x). The inset shows the cell volume and c/a ratio against composition (after Singh et al., 2007a).

2.2.4. Structure of PZT with 0.62 < x < 0.90

The structure of PZT is known to change for x > 0.62 due to a tilt transition, driven by R point (q = ½½½) anomaly, occurring above room temperature. Traditionally, the rhombohedral R3c space group (Jona & Shirane, 1962) in the a⁻a⁻a⁻ tilt system (Clarke & Glazer, 1976; Glazer et al., 1978; Corker et al., 1998) has been assigned to such compositions. On heating above the room temperature, this phase (labelled by Jaffe et al., 1971) transforms to (space group R3m). However, Corker et al. (1998), while refining the structure using powder neutron diffraction data, found that, unless one considers local disordered displacements of Pb in the 〈110〉 pseudocubic directions, the R3c space group cannot give satisfactory agreement factors. For x = 0.90 and 0.95, monoclinic phases in the Pc and Pm space groups have been postulated on the basis of electron diffraction patterns (Woodward et al., 2005). For these compositions, superlattice reflections due to anti-phase octahedral tilting associated with R (q = ½½½) point instability are observed in the powder neutron diffraction data (Glazer et al., 1978; Corker et al., 1998). Our synchrotron XRD data for compositions with 0.62 < x ≤ 0.90 reveal anomalous broadening of the 200 peak, similar to that observed in the composition range 0.530 ≤ x < 0.62, as can be seen from Fig. 3 for x = 0.70, 0.80 and 0.90. This anomalous broadening of the h00 and hh0 reflections and the presence of superlattice reflections can be accounted for in the Cc space group (tilt system a⁻a⁻c⁻), which has been discovered recently at low temperatures for x = 0.520 (see Ragini et al., 2001; Hatch et al., 2002; Ranjan et al., 2005; see §3.6 for more details). Fig. 6 depicts the observed and calculated profiles of the 111, 200 and 220 pseudocubic reflections, obtained after full-pattern Rietveld refinements using synchrotron XRD data in the 2θ range 18 to 130°. In these refinements, we have used anisotropic thermal parameters and anisotropic peak broadening functions, which account for the anomalous broadening of the 200 and 220 reflections, as can be seen from Fig. 6. However, the mismatch for the 111 peak is quite noticeable for the R3c space group, similar to that shown in Fig. 4(a) for R3m. Consideration of the Cc space group, on the other hand, accounts for all the reflections very satisfactorily. Thus, we conclude that the structure of PZT for 0.62 < x ≤ 0.90 is also monoclinic but in the Cc space group, which we label as phase in analogy with the phase given by Jaffe et al. (1971) for this composition range. It is worth mentioning that the neutron data used by Corker et al. (1998) did not have sufficient resolution to capture the anomalous broadening shown in Fig. 3. Our refinements for the Cc space group, which gives a better fit than the R3c space group between the observed and calculated profiles, suggest that the monoclinic order is not local at the unit-cell level as envisaged by Corker et al. (1998) but longer ranged, enabling the choice of a monoclinic space group in the Rietveld refinements.

Figure 6 Observed (dots), calculated (continuous line) and difference (bottom line) profiles of 111, 200 and 220 pseudocubic reflections of PZT for x = 0.70 using (a) R3c and (b) Cc space groups (after Singh et al., 2007b).

3. The low-temperature phase transitions

3.1. Background

Using the piezoelectric resonance frequency (f_r), planar electromechanical coupling coefficient (k_P), and X-ray diffraction measurements, Mishra et al. (1997) established the following sequence of phase transitions near MPB: F_T to P_C for x ≤ 0.520, F_T + F_R → F_T → P_C for x = 0.525, F_R → F_T + F_R → F_T → P_C for 0.530 ≤ x ≤ 0.545, and F_R → P_C for x ≥ 0.550 on successively increasing the temperatures above 300 K. These studies clearly suggested that the phase coexistence at room temperature in the MPB region is due to a first-order phase transition between the low-temperature rhombohedral (stable below room temperature) and higher-temperature tetragonal phases, since even the pure F_R phase for 0.530 ≤ x ≤ 0.545 passed through a two-phase region consisting of F_R and F_T phases before transforming fully into the F_T phase. Mishra et al. (1996) accordingly proposed that the enhanced electromechanical response near the MPB compositions is linked with the lattice instability in the vicinity of the F_R to F_T phase transition. Evidence for such an instability was found by Thapa (1998) in our laboratory who observed a dielectric anomaly for PZT with x = 0.520 below the room temperature.

3.2. XRD evidence for a tetragonal to monoclinic phase transition

The first structural evidence for a low-temperature phase transition was reported by Noheda et al. (1999) who discovered a tetragonal to monoclinic phase transition below room temperature in PZT compositions (x = 0.520) close to the MPB. They also confirmed that the space group of this monoclinic phase is Cm (Noheda, Gonzalo et al., 2000). This phase transition is evidenced by the appearance of new peaks as illustrated in Fig. 7, which depicts synchrotron powder XRD profiles of the pseudocubic 200, 220 and 222 reflections of PZT with x = 0.520 in the temperature range 10 to 400 K. It is evident from this figure that the structure is nearly tetragonal at 400 K. On cooling to 300 K, the 220 and 222 reflections show anomalous asymmetric broadening. On further cooling below room temperature, new peaks emerge. For example, the singlet 222 profile of the tetragonal phase becomes a triplet of 402, 042 and reflections of the monoclinic phase in the Cm space group. As pointed out by Noheda, Gonzalo et al. (2000), this monoclinic phase can be pictured as providing a `bridge' between the F_R and F_T structures in the region of the MPB since the polarization vector of the monoclinic phase lies in the pseudocubic (10) plane, whereas the polarization vectors of the and F_T phases are along the [111] and [001] pseudocubic directions. This is illustrated in Fig. 8. The polar axis of the monoclinic phase for x = 0.520 is tilted about 24° from the [001] direction towards the [111] direction. It is interesting to note that evidence for a tetrag­onal-to-monoclinic phase transition in the ferroelectric PbFe_0.5Nb_0.5O₃ was reported by Bonny et al. (1997) and Lampis et al. (1999) using single-crystal and powder diffraction data, respectively, prior to the work of Noheda et al. (1999).

Figure 7 Temperature evolution of the powder synchrotron XRD profiles of 200, 220 and 222 pseudocubic reflections of PZT with x = 0.520. The Miller indices of the reflections shown at 10 K correspond to the monoclinic structure in the Cm space group.

Figure 8 Direction of polarization vector in the [111], [in the plane ()] and FT [001] phases of PZT.

3.3. Evidence for two phase transitions in elastic modulus and dielectric studies

Soon after the report by Noheda et al. (Noheda et al., 1999; Noheda, Gonzalo et al., 2000; Noheda, Cox et al., 2000) about the low-temperature tetragonal-to-monoclinic phase transition in PZT, Ragini et al. (2001) found evidence for one more low-temperature phase transition in poled PZT ceramics with x = 0.515 and 0.520 on the basis of the temperature dependence of the piezoelectric resonance frequency (f_r) and dielectric constant (ɛ′). The piezoelectric resonance frequency is related to the elastic modulus ( 1/S₁₁) (see Jaffe et al., 1971, p. 293). The elastic modulus of normal solids which expand on heating is known to increase with decreasing temperature. However, it is evident from Fig. 9, which depicts the temperature variation of 1/S₁₁ (as obtained from f_r values) and ɛ′ for x = 0.520, that 1/S₁₁ decreases with decreasing temperature up to about 260 K. Such an anomalous temperature dependence of the elastic modulus is a signature of a lattice instability due to softening of some phonon modes due to an impending structural phase transition. The temperature dependence of 1/S₁₁ becomes normal in the temperature range 230 to 260 K, as evidenced by its increasing behaviour with decreasing temperature, revealing a phase transition around 260 K. This temperature is close to the tetragonal-to-monoclinic phase transition temperature of 250 K reported by Noheda et al. (1999) for x = 0.520 using XRD data. In­triguingly, the 1/S₁₁ starts decreasing again anomalously on further cooling below ~230 K up to 210 K. Below 210 K, the temperature variation of 1/S₁₁ becomes once again normal. This reveals yet another phase transition below 210 K in PZT with x = 0.520 for which there is no evidence in the synchrotron XRD data of Noheda et al. (Noheda et al., 1999; Noheda, Gonzalo et al., 2000; Noheda, Cox et al., 2000) or the synchrotron XRD profiles on our sample shown in Fig. 7. Corresponding to these two phase transitions around 260 and 210 K, one also observes anomalies in the real part of the dielectric constant (ɛ′), as shown in Fig. 9.

Figure 9 Temperature variation of the elastic modulus ( 1/S11) (open circles) and real part of the dielectric constant () (filled circles) for a poled Pb(Zr0.52Ti0.48)O3 ceramic (after Ragini et al., 2001).

3.4. Evidence for a superlattice phase by electron and neutron diffraction

The observation of only one phase transition in the powder XRD study of PZT with x = 0.520 by Noheda et al. (Noheda et al., 1999; Noheda, Gonzalo et al., 2000; Noheda, Cox et al., 2000) and Ragini et al. (2001) and two phase transitions in the elastic modulus and dielectric studies was quite enigmatic. The first structural evidence for the lower-temperature phase transition in tetragonal PZT compositions came through electron diffraction studies by Ragini et al. (2001) on a PZT composition with x = 0.515, for which the two low-temperature phase transitions, as revealed by the elastic modulus and dielectric measurements, occur around 200 and 260 K. Fig. 10 depicts the [001] and [10] zone selected-area electron diffraction (SAD) patterns for x = 0.515. It is evident from this figure that weak superlattice reflections with pseudocubic indices of the type ½ [hkl] with h, k, l = 2n + 1 (n any integer) are present in the SAD patterns recorded at 189 and 98 K. The appearance of these superlattice reflections below 219 K clearly suggests that the pseudocubic unit cell is doubled between 219 and 189 K. Since the second low-temperature anomaly in the elastic modulus and dielectric constant in PZT with x = 0.515 occurs at 201 K (see Ragini et al., 2001), these anomalies, in the light of the electron diffraction results, are associated with a cell-doubling transition. It was shown by Ragini et al. (2001) that the c parameter of the superlattice phase is twice the c parameter of the monoclinic phase discovered by Noheda et al. (Noheda et al., 1999; Noheda, Gonzalo et al., 2000; Noheda, Cox et al., 2000). Since the first report by Ragini et al. (2001), this superlattice phase has been subsequently confirmed in the TEM studies by Noheda et al. (2002) and Woodward et al. (2005).

Figure 10 The evolution of [001] and zone-axis selected-area electron diffraction patterns with temperature for PZT with x = 0.515. Superlattice reflections due to a cell doubling transition are seen in the 189 and 98 K patterns (after Ragini et al., 2001).

Ranjan et al. (2002) carried out a powder neutron diffraction study of PZT with x = 0.520 in the temperature range 300 to 10 K. This study also confirmed the appearance of superlattice reflections which are marked with arrows in Fig. 11. These superlattice peaks cannot be indexed with respect to the Cm monoclinic cell. They appear at T ≤ 210 K as can be seen from the inset to Fig. 11 which depicts the temperature evolution of one of these reflections. This temperature (~210 K) coincides with the second phase-transition temperature of PZT with x = 0.520 shown in Fig. 9. Indexing of all the reflections in Fig. 11, including the weak ones, requires a monoclinic cell whose c parameter is twice that of the monoclinic phase reported by Noheda et al. (Noheda et al., 1999; Noheda, Gonzalo et al., 2000; Noheda, Cox et al., 2000). Subsequent to the work of Ranjan et al. (2002), superlattice reflections in the low-temperature powder neutron diffraction patterns of tetragonal PZT compositions close to MPB were reported by Frantti et al. (2002) and Cox et al. (2005) also.

Figure 11 Powder neutron diffraction pattern of PZT with x = 0.520 at 10 K. The inset shows the evolution of the 311 superlattice reflection in the temperature range 270 to 10 K (after Ranjan et al., 2002).

3.5. Role of antiferrodistortive phase transition

In oxide perovskites like SrTiO₃ (Bruce & Cowley, 1980), LaAlO₃ (Howard et al., 2000) and CaTiO₃ (Kennedy et al., 1999), antiferrodistortive (AFD) structural phase transitions, involving `anti-phase' and `in-phase' octahedral tilts (rotations) due to the softening and freezing of the R (q = ½½½) and M (q = ½½0) point modes of the cubic phase, are known to lead to unit-cell doubling and appearance of superlattice reflections. These superlattice structures can be described in terms of a doubled pseudocubic cell of the type 2a_p × 2b_p × 2c_p, where a_p, b_p and c_p are the unit-cell param­eters of the elementary perovskite cell. With respect to such a doubled pseudocubic cell, the Miller indices of the main perovskite reflections correspond to three even (e) numbers (i.e., the Miller indices are of eee type) while the superlattice reflections are represented by all-odd (ooo) or two-odd and one-even (ooe) type indices depending on whether they have resulted from anti-phase (or negative) or in-phase (or positive) tilts of the neighbouring O-atom octahedra, respectively (Glazer, 1972). The Miller indices of the superlattice reflections in Fig. 11 are 311, 511 and 535, all of which are of ooo type indicating that these reflections have arisen from anti-phase (−) tilting of the neighbouring TiO₆ octahedra. This analysis suggests that the cell doubling transition discovered by Ragini et al. (2001) and Ranjan et al. (2002) in their electron and neutron diffraction studies is an AFD phase transition involving an R-point instability. This conclusion is also supported by the results of first principles density functional calculations (Fornari & Singh, 2001) which reveals a substantial R-point AFD instability coexisting with Γ-point ferroelectric instability for MPB compositions.

3.6. Structure of the superlattice phase

Ranjan et al. (2002) have proposed a simple model shown in Fig. 12 for the structure of the low-temperature superlattice phase. In a stack of two monoclinic cells of the Cm type shown in this figure, an R-point instability leading to a^oa^oc⁻-type tilt will rotate the four O atoms labelled 3, 4, 5 and 6 in the z = ¼ plane and another set of four O atoms, labelled 3′, 4′, 5′ and 6′, in the z = ¾ plane in the anticlockwise and clockwise manners about the [001] direction by equal magnitude. As a result of such an anti-phase tilt about the [001] direction, the mirror planes at y = 0,½ of the Cm space group are destroyed while new a-glide planes appear at y = ¼,¾. Further, the doubled monoclinic cell shown in Fig. 12 is a body-centred monoclinic cell. Amongst the monoclinic space groups listed in International Tables for Crystallography, it was found that the space group corresponding to the structural model shown in Fig. 12 is I1a1. With a new choice of axes, the space group I1a1 can equivalently be written as Cc in the standard setting [see International Tables for Crystallography (2005), Vol. A, space group No. 9]. The unit-cell axes A, B, C of the Cc space group are related to those (a_m, b_m, c_m) of the Cm space group as A = a_m + 2c_m, B = b_m, C = −a_m. This space group can result from the space group by coupling the and R₄⁺ IR's, leading to the a⁻a⁻c⁻ tilt system in the Cc space group (Stokes & Hatch, 2000). It is interesting to note that a single anti-phase tilt a^oa^oc⁻ about the c direction of the Cm phase considered in Fig. 12 is sufficient to reduce the symmetry to Cc, but the Cc symmetry also allows tilts in the ab plane to appear.

Figure 12 Schematic diagram of the doubled monoclinic unit cell showing superposition of anti-phase rotation of octahedra (about [001] direction) and other atoms, i.e. O(1), O(2), Pb(1), Pb(2), Zr/Ti(1) and Zr/Ti(2), displaced as in the Cm space-group model. The open circles, filled big circles and filled small circles represent O, Pb and Zr/Ti atoms, respectively. The arrows indicate the sense of displacements from their respective undisplaced position (after Ranjan et al., 2002).

Ranjan et al. (2002) initially proposed a Pc space group for this superlattice phase which was subsequently corrected to I1a1 ≡ Cc, and confirmed by Rietveld refinement (Hatch et al., 2002). The Cc space group has since been confirmed by several workers in TEM studies (Noheda et al., 2002; Woodward et al., 2005). Further, a recent first-principles calculation has also confirmed that the Cc phase is the ground state of PZT with x = 0.520 (Kornev et al., 2006). This phase has also been reported in a high-pressure neutron diffraction study of PZT with x = 0.520 (Rouquette et al., 2005).

Subsequent to the above work by Ranjan et al. (2002) and Hatch et al. (2002), Frantti et al. (2002), while confirming the findings of Ranjan et al. (2002) on the appearance of the superlattice reflections in powder neutron diffraction data, proposed that the superlattice reflections are due to a rhombohedral phase in the R3c space group, which coexists with the monoclinic phase in the Cm space group. In view of this controversy, Ranjan et al. (2005) revisited the earlier Rietveld analysis using the coexistence of R3c and Cm phases and compared these results with those obtained for the pure Cc phase model (Hatch et al., 2002) and also a model based on the coexistence of Cc and Cm phases (Noheda et al., 2002). The results of these refinements are presented in Fig. 13 which depicts the observed, calculated and difference profiles for a few selected peaks of PZT with x = 0.520 at 10 K. As can be seen from this figure, the Cc space-group model accounts satisfactorily for the 311 and 511 superlattice reflections. The R3c model of superlattice phase along with a coexisting Cm phase leads to distinct mismatch between the observed and calculated peak positions for the 311 and 511 superlattice peaks. The calculated peak positions occur at lower 2θ angles as compared to the observed ones. This difference in the positions of superlattice reflections is also seen in the report of Frantti et al. (see inset to Fig. 3 of Frantti et al., 2002). It may be mentioned that Frantti et al. (2002) did not show the zoomed pattern of 511 superlattice reflection for which the mismatch between the observed and calculated pattern is more pronounced (see Fig. 13). Thus, the Cm + R3c phase model of Frantti et al. (2002) can be discarded. For the Cc phase model, although the calculated superlattice peak positions are in good agreement with those observed, the fit for the perovskite peaks like 200 and 222 is not very good. Consideration of a Cm phase coexisting with the Cc phase improves the fit for both the superlattice as well as the perovskite reflections. Thus, we can conclude that PZT with x = 0.520 at 10 K consists of a mixture of two phases with Cc and Cm space groups. The refined cell parameters and coordinates for the Cc phase obtained after the inclusion of the coexisting Cm phase in the Rietveld analysis are comparable to those given by Hatch et al. (2002). The equivalent Cm cell parameters obtained for the Cc space-group model are comparable to those of Noheda, Gonzalo et al. (2000). The cell parameters [a = 5.758 (1), b = 5.729 (1), c = 4.0930 (7) Å, β = 90.59 (1)°] of the coexisting Cm phase are on the other hand found to be quite different from those reported by Cox et al. (2005) in a similar but independent study. This coexisting Cm phase is similar to the room-temperature Cm phase reported by Ragini et al. (2002) for pseudorhombohedral PZT compositions with x ≥ 0.530. As shown elsewhere (Singh et al., 2007b), this Cm phase coexists with the tetragonal phase at room temperature also for x = 0.520 and its fraction remains constant down to 10 K, i.e. this coexisting phase does not take part in any of the two low-temperature phase transitions. We thus conclude that PZT with x = 0.520 has a monoclinic superlattice phase in the Cc space group, which coexists with a secondary monoclinic phase in the Cm space group. As shown by Ranjan et al. (2005), the equivalent of the refined x and z coordinates of all the atoms for the Cc space group when expressed in the I1a1 setting are very close to those obtained by Noheda, Gonzalo et al. (2000) for the Cm space group. This shows that the lower-temperature monoclinic phase in the Cc space group retains the main structural framework of the higher-temperature monoclinic phase with Cm space group. The only difference between the two phases is in the y displacements of the O atoms which are responsible for the superlattice reflections (Ranjan et al., 2002). Because of the low scattering factor of oxygen for X-rays, these small displacements do not lead to measurable intensities for the superlattice reflections in powder X-ray diffraction data, as a result of which Noheda et al. (Noheda et al., 1999; Noheda, Gonzalo et al., 2000; Noheda, Cox et al., 2000) missed the superlattice Cc phase in their synchrotron XRD studies.

Figure 13 Observed (dots), calculated (solid lines) and difference (lines at the bottom) profiles for the 311 and 511 superlattice and 200 and 222 perovskite reflections of PZT with x = 0.520 at 10 K (after Ranjan et al., 2005).

4. The outlook

It is evident from the foregoing that the structure of PZT is tetragonal (F_T) for x < 0.520 and monoclinic for x > 0.520 while the two phases coexist for x = 0.520. Thus the MPB of PZT lies around x = 0.520 and separates the tetragonal (F_T) and monoclinic () phase regions, and not the tetragonal (F_T) and rhombohedral () phase regions, believed so far in the literature (Jaffe et al., 1971). The results reviewed above also reveal that the tetragonal compositions (x ≤ 0.520) in the vicinity of the MPB undergo a succession of phase transitions from F_T (P4mm) to (Cm) to (Cc) below room temperature. Further, the phase stable at room temperature for 0.525 ≤ x ≤ 0.550 transforms to (Cc) phase on cooling below room temperature and F_T (P4mm) phase on heating above room temperature. The F_T ↔ F_M^HT ↔ transitions are accompanied by large coexistence regions indicative of their first-order character. Mishra et al. (1997) and Mishra & Pandey (1997) have shown that P_C ↔ F_T transition for x < 0.545 is also first order as confirmed by discontinuous change of cell parameters and thermal hysteresis. For x ≥ 0.550, the phase (which was considered as by Mishra et al., 1997) transforms directly to the P_C phase with no thermal hysteresis indicating second-order transition. These observations thus suggest a cross-over composition 0.545 ≤ x < 0.550 at which the paraelectric ↔ ferroelectric transition may exhibit tricritical behaviour, as was first pointed out by Mishra et al. (1997) and Mishra & Pandey (1997). Another tricritical point towards the Zr-rich end was earlier reported by Whatmore et al. (1978) around x = 0.94.

Because of the wide coexistence region, the exact phase boundaries between various PZT phases cannot be located using diffraction data. The temperature dependence of the piezoelectric resonance frequency (f_r) and the elastic modulus ( 1/S₁₁) (see for example Fig. 9 for x = 0.520) can, however, be used to define the phase boundaries for a heating or cooling cycle rather precisely. Combining the structural and physical property measurements data, a new phase diagram of PZT around the MPB region has been constructed by Pandey & Ragini (2003) which in a slightly modified form is shown in Fig. 14. The new features of this phase diagram are the F_T– and – phase boundaries below 300 K. The – phase boundary below room temperature appears as an extrapolation of the – phase boundary above room temperature. The – phase boundary above room temperature has been known prior to the recent discoveries but it was labelled as the – phase boundary (see Jaffe et al., 1971). It is interesting to note from this figure that the F_T → (Cm) → (Cc) transition on cooling below room temperature can occur up to about x ≃ 0.48. Further, a linear extrapolation of the and phase boundary up to T = 0 K suggests that the antiferrodistortive instability at 0 K may persist up to x ≃ 0.31, on the Ti-rich side of the MPB. In the composition range 0.31 ≤ x ≤ 0.48, the F_T phase may therefore transform under an R-point instability to a doubled cell structure directly without the intermediate phase. Whether this phase retains the Cc space group confirmed experimentally for x ≥ 0.520 or a new space group I4cm, predicted recently on the basis of first-principles calculations under virtual crystal approximation (VCA) (Kornev et al., 2006), needs to be examined experimentally. Further, the experimental results, reviewed in this article do not support the existence of R3m and R3c phase regions near the MPB shown in a recently proposed phase diagram (Kornev et al., 2006). The theoretical phase diagram by Kornev et al. (2006) contains P4mm, Cm, Cc and I4cm phase regions meeting at a point which is in obvious contradiction to the Gibbs phase rule which does not permit more than three phase regions meeting at a point in the T–x phase diagram. A careful experimental study is therefore required to settle the phase boundaries near x = 0.48 in Fig. 14. Woodward et al. (2005) have recently proposed a Pc space group for 0.90 ≤ x ≤ 0.94 instead of R3c and Pm instead of R3m on the basis of indexing of the selected-area electron diffraction patterns. However, these space groups are yet to be tested using profile-fitting techniques of powder diffraction and hence we have not shown their stability fields in Fig. 14.

Figure 14 New phase diagram of PZT (after Pandey & Ragini, 2003).

The powder diffraction profiles of various pseudocubic reflections for 0.525 < x < 0.94 resemble those for the rhombohedral structure, except for the anomalous broadening of the h00- (such as 200) and hh0- (such as 220) type reflections. The Cm and Cc monoclinic space groups for these compositions have been confirmed on the basis of the full-pattern Rietveld refinements using a rotating-anode source (Ragini et al., 2002) as well as synchrotron powder XRD data (Singh et al., 2007a,b). There is, however, no evidence for any splitting of the powder XRD profiles characteristic of the monoclinic Cm and Cc space groups even in the high-resolution synchrotron XRD data. This may be due to the fact that the coherently scattering domain size of the monoclinic phase in these compositions is rather small giving rise to large Scherrer broadening of various peaks, which in turn masks the monoclinic splitting of the powder XRD profiles. The fact that only certain reflections are anomalously broadened suggests that these domains are not only small but also thin-plate-like, as has recently been revealed in the TEM studies (Schönau et al., 2007). On application of an intense DC electric field, some of the monoclinic domains grow at the cost of others, as a result of which the XRD peaks showing anomalous broadening in the unpoled state start exhibiting splitting characteristic of the monoclinic phase in the poled state (see e.g. Guo et al., 2000). In the related PMN–xPT system, Singh et al. (2003) have reported characteristic monoclinic splitting of very high angle diffraction peaks in high-resolution powder neutron diffraction data for the so-called `rhombohedral' compositions exhibiting anomalous broadening of h00- and hh0-type reflections in the powder XRD data. The high-angle peaks (pseudocubic 440, 620) resolved in the neutron data have too low an intensity in the XRD data due to the dependence of the atomic scattering factor (f). No such study has so far been carried out on PZT samples with 0.525 < x ≤ 0.90.

Wang, Khachaturyan and co-workers (Jin et al., 2003; Wang, 2007) have recently advanced an adaptive phase model for the monoclinic phases of the MPB ceramics, based on the theory of martensitic transitions. In this model, the monoclinic structure observed in PZT and other MPB ceramics appears due to a conformal miniaturization of the tetragonal domains. If one accepts this picture, there should not be any tetragonal-to-monoclinic phase transition. However, Ragini et al. (2001) have obtained unambiguous evidence for a phase transition as revealed by anomalous softening of the elastic modulus (expressed in terms of piezoelectric resonance frequency in the original paper) as a precursor to the tetragonal-to-monoclinic Cm and monoclinic Cm to monoclinic Cc phase transitions followed by hardening below the phase-transition temperatures in the stability fields of the Cm and Cc phases. The Raman scattering studies also reveal appearance of new modes in confirmation of the monoclinic Cm and Cc phases (Filho et al., 2000; Rouquette et al., 2006). So the work of Jin et al. (2003) and Wang (2007) provides only a geometrical description of the monoclinic phase in terms of nanodomains of the tetragonal phase using a set of crystallographic orientation relationships; it does not preclude a phase transition for which there is ample evidence of being driven by lattice instabilities.

The results reviewed in the preceding sections also reveal an interesting trend, pointed out by Glazer et al. (2004). The local structure of PZT on the tetragonal side of the MPB corresponds to short-range-ordered monoclinic regions as Pb²⁺ ions are displaced from cube corners along the monoclinic 〈100〉 directions (Noheda, Gonzalo et al., 2000; Ragini et al., 2002). The structure becomes truly long ranged monoclinic in the MPB region as shown in §2.2.1 for x = 0.525. Finally, it becomes short-ranged monoclinic for the Zr-rich compositions (x > 0.525) as indicated by the anomalous broadening of the h00- and hh0-type peaks. The absence of the rhombohedral phase on the Zr-rich side of the MPB for x > 0.525 suggests that the original idea of the monoclinic phase stable only in a narrow composition range near MPB as a bridging phase [see e.g. the recent reviews by Noheda (2002) and Noheda & Cox (2006)] between the tetragonal and rhombohedral phases is untenable. Further, in the absence of the rhombohedral phases on the Zr-rich side, the polarization rotation model of high piezoelectricity (Fu & Cohen, 2000; Bellaiche et al., 2000) loses relevance for PZT. A similar situation seems to exist in PMN–xPT also (Singh & Pandey, 2005; Singh et al., 2006), where the so-called 'rhombohedral' phase turns out to be monoclinic (see Singh et al., 2006 for the latest phase diagram).