Thermal conductivity of lead zirconate titanate PbZr_1−xTi_xO₃

Among the large variety of solid solutions formed by perovskite oxides, probably none has captured more attention than PbZr_1−x Ti_x O₃ (PZT), the most important material in the multi-billion dollar piezoelectric industry. ¹⁾ PZT shows a complex phase diagram (Fig. 1, after Jaffe et al. ²⁾), with the nearly vertical morphotropic phase boundary (MPB) at x ≈ 0.48 separating the ferroelectric (FE) rhombohedral region from the FE tetragonal region. The highest piezoelectric response is found as x approaches the MPB, but the detailed mechanism of this fascinating behavior remains controversial. ^3,4)

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At the left end of the phase diagram, PbZrO₃ shows a paraelectric (PE) cubic to antiferroelectric (AFE) orthorhombic transition at T_c ≈ 503 K. The AFE structure (space group Pbam) is described by antiparallel shifts of polar Pb ions along cubic [110] and antiferrodistortive (AFD) tilting of oxygen octahedra around [110]. ^5,6) Substitution of few% Ti for Zr leads to the FE rhombohedral region, which contains two phases: F_R(HT) (R3m) at high temperatures, which has the polarization along [111], and F_R(LT) (R3c) at low temperatures, which additionally shows AFD oxygen tilting around [111]. When the Ti content x exceeds the MPB, the structure becomes FE tetragonal (P4mm) with the polarization along [001]. This tetragonal region continues to PbTiO₃ (T_c ≈ 763 K) at the right end of the phase diagram.

An important update to these basic features of PZT occurred in 1999 when Noheda et al. found a narrow region (0.45 < x ≤ 0.48 at 300 K) of a FE monoclinic (Cm) phase on the left side of the MPB. ^7,8) As the low and "bridging" symmetry of this structure allows the polarization axis to rotate between the rhombohedral [111] and tetragonal [001] directions, the high piezoelectric response at MPB could be explained with the polarization rotation mechanism. ⁹⁾ However, the exact structure and stability region of the monoclinic phase continues to be disputed, with recent studies reporting the coexistence of rhombohedral and monoclinic orders in most of the rhombohedral region. ^10–14) Moreover, there remain the important questions on (1) competing FE, AFE, and AFD phonon instabilities, ^15–17) (2) disordered displacements of Pb ions and oxygen octahedra, ^18–21) and (3) anomalous lattice dynamics in the vicinity of MPB, ^16,22,23) which all have bearings on the functional performance of PZT.

Because thermal conductivity (κ) measurements probe the scattering of heat-carrying acoustic phonons by local distortions and anharmonic interactions, ²⁴⁾ they can provide important insights into the above issues on PZT. Indeed, our earlier κ study ²⁵⁾ on PbZrO₃ and PbTiO₃ for T ≥ 300 K revealed a low, glasslike κ (dκ/dT > 0) in the cubic phase for PbZrO₃, whereas PbTiO₃ was found to show a typical crystalline behavior. ^25,26) These results are intimately correlated with the local structures of these compounds, ²⁵⁾ and suggest that a systematic κ study on PZT will track the evolution of phonon dynamics across the phase diagram. With this objective, the present paper reports the κ of PZT (0 ≤ x ≤ 1) for T = 300−873 K. We find glasslike κ in both the FE rhombohedral phases and much of the cubic phase, indicating the presence of strong phonon damping in these regions of PZT. We also find that the κ shows a distinct minimum at x = 0.5. Although it is tempting to associate this minimum with the MPB for T < T_c, the persistence of this minimum in the cubic phase implies that conventional point-defect alloy scattering also plays an important role.

Polycrystalline samples of PZT with x = 0.2, 0.4, 0.5, 0.7, 0.8, and 0.9 were prepared by solid-state reactions. For each x, dried powders of PbO (99.99%), ZrO₂ (99.9%), and TiO₂ (99.999%) were weighed with 1% excess PbO. The powders were thoroughly mixed, pressed into pellets, covered in Pt sheets, and heated at 1123 K for 3 h in a closed Al₂O₃ crucible. These steps were then repeated to improve homogeneity. After the fired pellets were ground to fine powders and 2 wt% polyvinyl alcohol binder added, the mixture was ground, sieved, pelletized under an isostatic pressure of 200 MPa, and heated at 723 K for 3 h to remove the binder. The resulting pellets were then covered in Pt sheets and put on a powder bed of PbZrO₃ in a tightly covered Pt crucible, which was in turn sealed in an Al₂O₃ crucible using alumina cement. This procedure minimized the loss of PbO during the 3 h sintering at 1543 K. The sharp powder X-ray diffraction peaks (see supplementary data) and absence of extraneous peaks confirm that each sample is well crystallized without impurity phases.

The κ values were determined from the relation κ = DC_p ρ, where D is thermal diffusivity, C_p is heat capacity, and ρ is density. For C_p , we used Yoshida's published data ^27,28) (see below), which were obtained between 300 and 873 K by the enthalpy method using a Perkin Elmer differential scanning calorimeter. The data were obtained on the heating direction under N₂ atmosphere, and the accuracy is 0.3% at 400 K and ∼1% at 873 K. ^27,29) We mention here that Yoshida also used a relaxation technique to obtain C_p below 300 K, which join smoothly with the data above 300 K. ^27,28) D was obtained in the present study by the flash technique, using Netzsch LFA 467HT. The samples were square plates of 10 × 10 mm² faces and ∼1.0 mm thicknesses, with a thin layer of graphite coated on both sides. The measurements were taken for T ≥ 300 K on the heating direction under N₂. ρ was determined by the Archimedes method, and its T dependence was calculated from volumetric thermal expansion. ^30,31) For x = 0.4, we used the expansion data ³⁰⁾ of x = 0.36; this does not introduce significant error, since the volume changes by less than 1% between 300 and 873 K. As the ρ of sintered pellets were 89%−93% of the theoretical values, the raw κ values (κ_raw) were converted to the ρ of a fully dense solid through the relation κ_raw/κ = 1–4ϕ/3, where ϕ is the porosity of the specimen. ³²⁾ The overall accuracy in κ is estimated to be ±5%. Because the κ of PZT is low and the phonon mean free path (l) is of the order of ∼1 nm, ³³⁾ phonon scattering at grain boundaries and FE domain walls is expected to be insignificant in the present T region. ³³⁾

We first examine Yoshida's C_p data, which are reproduced in Fig. 2 for all 12 compositions. Here, the C_p values for x = 0 and 1 come from tabulated data, ³⁴⁾ whereas the values for other x were extracted from the figures in the PhD thesis ²⁷⁾ and conference abstract. ²⁸⁾ For each x, the most prominent feature is a peak at T_c, the position of which agrees well with the phase diagram. The peak is followed by a nearly constant C_p in the cubic phase (the Debye temperatures ²⁵⁾ are ∼360−400 K), at a value close to the Dulong–Petit limit of 125 J K⁻¹ mol⁻¹. For x = 0.05 and 0.1, another peak at the lower transition (see Fig. 1) can be seen at 370 and 365 K, respectively; for x = 0.2, such a peak is visible at ∼395 K under an expanded scale. ²⁷⁾ Focusing now on the shape of C_p at T_c, we find very sharp peaks at the first-order AFE and FE transitions in x = 0 and 1. As x moves away from these end members, however, the C_p peak is gradually suppressed and becomes more second-order like. The change from first- to second-order character (i.e. a tricritical point) appears to occur somewhere between x = 0.05 and 0.2 for the Zr-rich region, and between x = 0.6 and 0.7 for the Ti-rich region. Similar positions of tricritical points have been reported. ^35–37) Interestingly, an earlier C_p study ³⁶⁾ by Rossetti and Navrotsky also located the tricritical point as 0.6 ≤ x ≤ 0.7, despite their C_p for x = 0.6 and 0.7 showing sharper curvatures near T_c. The difference in peak shapes can be attributed to the synthesis method: while Yoshida used conventional solid-state reactions, ²⁷⁾ Rossetti and Navrotsky employed a metal-organic decomposition process to obtain more homogeneous samples. Although our present κ samples were prepared in a manner similar to Yoshida's samples, we expect some difference in C_p in the vicinity of T_c. For this reason, we exclude κ data in such T regions.

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The bottom and top panels of Fig. 3 show the κ of PZT for 0 ≤ x ≤ 0.5 and 0.5 ≤ x ≤ 1, respectively. The data for x = 0 and 1 are reproduced from our earlier study. ²⁵⁾ We first focus on 0 ≤ x ≤ 0.5. As noted earlier, the κ of PbZrO₃ (x = 0) above T_c is glasslike, with both the magnitude and T dependence (dκ/dT > 0) being comparable to those of amorphous silica. ²⁵⁾ Such glasslike κ occurs when the l of phonons is limited to the order of lattice spacing, at which heat is carried by a random walk of energy between localized oscillators. ³⁸⁾ Since PbZrO₃ is a stoichiometric compound without chemical disorder, the glasslike κ should be attributed to large anharmonic distortions, especially the off-center displacements of Pb and O ions reported in structural studies. ^19,21) Indeed, these displacements are known to diminish in the AFE state, ²¹⁾ and this is reflected in the κ as the observation of crystalline dκ/dT < 0 behavior below T_c.

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As we move to x = 0.2, the κ values in the cubic region increase slightly (∼10%) compared to x = 0; this could suggest that the FE lattice instability in x = 0.2 has a weaker effect in suppressing the κ than the AFE instability in x = 0. More importantly, the glasslike κ is now extended into the FE rhombohedral region below T_c, with a possible sign for a dip near 400 K due to the F_R(HT) to F_R(LT) transition. We take the glasslike κ below T_c to be a signature of structural complexity in the FE rhombohedral region, where the displacements of Pb and O ions are known to persist ²¹⁾ and phase coexistence of rhombohedral and monoclinic structures has been widely recognized. ^10–14) With further increase in x, κ remains very similar in shape and drops in value up to x = 0.5, indicating the persistence of structural complexity throughout this composition region. At x = 0.5, the κ is only slightly larger than the minimum theoretical limit calculated for PbZrO₃, ³⁹⁾ which is ∼0.85 and 0.9 W m⁻¹ K⁻¹ at 300 and 400 K, respectively. Although x = 0.5 is nominally in the tetragonal region, it still contains significant disorder due to its proximity to the MPB; according to a recent study, ¹¹⁾ this composition contains a mixture of tetragonal, rhombohedral, and monoclinic structures.

At this point, it should be mentioned that the κ of bulk PZT samples have been reported, ^33,40,41) which show 2.5, 1.3 and 1.8, and 0.9 W m⁻¹ K⁻¹, respectively, at 300 K. However, these studies used commercial "PZT" ceramics, which are intentionally doped with impurities to modify the piezoelectric and other properties. ¹⁾ (One study ³³⁾ found the composition to be Pb[Ti_0.37Zr_0.24Nb_0.25Ni_0.14]O₃). Such modification in composition can explain the discrepancies in κ data (especially the T dependence), ^33,40,41) both with each other and with our present data on pure PZT samples.

For 0.5 ≤ x ≤ 1 in the tetragonal regime, different behaviors are observed as a function of x. The end member PbTiO₃ (x = 1) exhibits a typical crystalline behavior, where κ drops approximately as T⁻¹ above 300 K; it implies that heat is carried by extended wavelike phonon excitations, with l being limited by the usual Umklapp phonon–phonon scattering. ^24,38) After a jump around T_c, the dκ/dT < 0 behavior continues in the cubic phase region, and this concurs with the negligible ionic displacements reported for PbTiO₃. ^19,42) With decreasing x from x = 1, the κ in the FE tetragonal region evolves continuously, without becoming truly glasslike until x = 0.5. This smooth transitional behavior supports the view that the tetragonal region is basically a single phase and structurally simpler than the rhombohedral region. ^8,9) Perhaps more interesting is the result for the cubic phase, where the glasslike T dependence is seen for all the compositions except x = 1. Previously, the synchrotron X-ray diffraction study ¹⁹⁾ on the cubic phase reported the off-center displacements of Pb ions to persist from x = 0 to at least up to x = 0.75, and subsequent neutron diffraction study ²⁰⁾ showed that O ions are also displaced at least up to x = 0.6 (compositions with higher x, except x = 1, were not examined in these studies). Considering the close correspondence between the structural and κ data, the glasslike κ in the cubic phase should be attributed to the local ionic displacements, which are prevalent in all PZT except in the close vicinity of x = 1.

Having examined the T dependence of κ, we now discuss its composition dependence. Previously, Foley et al. ³⁹⁾ have reported on the κ of PZT thin films (0 ≤ x ≤ 1) at room temperature. As shown in Fig. 4(a), κ of the thin films ³⁹⁾ are nearly constant at ∼1.3 W m⁻¹ K⁻¹ for 0 ≤ x ≤ 0.48, drop to ∼1.0 W m⁻¹ K⁻¹ across the MPB, and increase with x in the tetragonal region. In contrast, our bulk κ at 300 K shows a "V" shaped minimum at x = 0.5, with no apparent discontinuity across the MPB. Regardless of these differences, both sets of data show a minimum at x ≈ 0.5, and it is tempting to associate this feature with the anomalies in other physical properties of PZT. In particular, on approaching the MPB, (i) all the elastic moduli are alowered ^2,43) [the data ⁴⁴⁾ for 1/s₃₃ ^E are plotted in Fig. 4(b)], indicating a reduction of the phonon group velocity v, and (ii) the damping constant γ of the lowest frequency transverse optic mode in the THz range ²³⁾ increases significantly [Fig. 4(b)], suggesting that, through the optic-acoustic mode coupling, ^45,46) acoustic phonons also show damping and reduction of l near the MPB. From the kinetic expression κ = (1/3)Cvl, where C is the heat capacity per unit volume, such reduction in both v and l can lead to a minimum in κ at the MPB.

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On the other hand, this interpretation cannot explain the persistence of the minimum at x = 0.5 in the cubic phase [κ at 773 K is shown in Fig. 4(a)], where the MPB is no longer present. In the cubic phase, the minimum is simply attributed to the point-defect alloy scattering, ^38,47) which arises from Zr and Ti occupying the same structural site—in the PZT solid solution, mass fluctuations from the difference in mass (M_Zr = 91.22 u and M_Ti = 47.87 u) and strain fields from the difference in ionic radius (r_Zr = 0.72 Å and r_Ti = 0.605 Å) create point-defect phonon scattering, which is not present in the end compounds PbZrO₃ and PbTiO₃. Moreover, this contribution is usually most pronounced near 50% substitution, ^38,47) coinciding with the κ minimum at x = 0.5 in PZT. Since the same point defects are present below T_c, quantitative analysis of κ for ferroelectric PZT must consider the two critical features: the point-defect scattering and the changes in physical and structural properties around the MPB. We believe this is an important point to bear in mind, especially as first-principles calculations of κ for PZT are beginning to emerge in the literature. ⁴⁸⁾

To conclude, we have presented the systematic evolution of κ in PZT, for the composition and temperature ranges covering the classic phase diagram of Jaffe et al. ²⁾ In 1971, these authors wrote in the preface to their book ²⁾ that "thermal conductivity data are so scarce and self-contradictory that we have left them out." While this situation did not improve much in the last 52 years, it is felt that the present data can serve as a definite contribution to this topic. In particular, with the current interest in the electric-field switching of κ, ^33,48) the basic insights obtained in this study should become useful in further applications of PZT.

Acknowledgments