High-temperature thermal conductivity of ferroelectric and antiferroelectric perovskites

We report thermal conductivity (κ) above 300 K for perovskite ferroelectrics BaTiO3 (T c ≈ 402 K) and PbTiO3 (763 K), as well as antiferroelectrics PbZrO3 (503 K) and PbHfO3 (476 and 433 K). BaTiO3 and PbTiO3 show similar κ in the paraelectric phase. In contrast, smaller and glasslike κ is found above T c for PbZrO3 and PbHfO3, signifying the presence of large anharmonic distortions in the paraelectric phase. Low-temperature heat capacity on PbZrO3 shows a lack of glasslike thermal behavior in the antiferroelectric phase.

The perovskite oxides BaTiO 3 , PbTiO 3 , PbZrO 3 , and PbHfO 3 are some of the most important compounds in the study of ferroelectricity and antiferroelectricity. 1) At high temperatures, these compounds have the same paraelectric cubic structure (space group Pm3 m), where the transition metal ions are located at the center of the oxygen octahedron, and the Ba/Pb ions occupy the 12-fold coordinated sites between the octahedra. Different sequences of first-order structural transitions are observed on cooling: (1) BaTiO 3 shows a series of three ferroelectric (FE) transitions, from the cubic to tetragonal (P4mm) at T c ≈ 402 K, tetragonal to orthorhombic (Amm2) at 278 K, and finally to rhombohedral (R3m) at 183 K. Each of these transitions is marked by uniform displacement of Ti ions, which is driven by the softening and condensation of the zone-center transverse optic (TO) mode. However, the soft mode is overdamped and the Ti ions show dynamic displacements along 〈111〉 in the cubic phase, leading to much debate on whether the orderdisorder mechanism plays important roles in BaTiO 3 . 2,3) (2) PbTiO 3 displays a single FE transition at T c ≈ 763 K to the tetragonal (P4mm) phase, where the displacement arises mostly from Pb ions. The FE soft mode in this case is underdamped (well-defined), such that PbTiO 3 is often regarded as showing a "textbook" example of displacive FE transition. (3) PbZrO 3 undergoes an antiferroelectric (AFE) transition 4,5) at T c ≈ 503 K to an orthorhombic (Pbam) phase, which contains eight formula units per cell. The AFE structure is described by the antiparallel shifts of Pb ions along the cubic [110] and the antiferrodistortive (AFD) tilting of oxygen octahedra around [110]; these displacements correspond to lattice modes at the Σ and R points, respectively, of the Brillouin zone. In addition, the dielectric susceptibility shows Curie-Weiss behavior in the cubic phase (as in BaTiO 3 and PbTiO 3 ), implying the softening of zonecenter TO mode. Recent studies [6][7][8][9] are divided on which mode takes the role of the primary driving force for the AFE transition. The importance of order-disorder character has also been discussed. 9) (4) PbHfO 3 shows two AFE transitions at T c ≈ 476 and 433 K, the former to an incommensurate (IC) phase 10) and the latter to an orthorhombic (Pbam) phase that is isostructural with PbZrO 3 . It has been suggested that PbHfO 3 holds a key to understanding PbZrO 3 , as a similar intermediate IC phase is found in PbZrO 3 under high pressure. 11) Because the FE and AFE instabilities interact with the anharmonic scattering of heat-carrying acoustic phonons, thermal conductivity (κ) measurements should provide important insights into these compounds. Indeed, the highquality κ data for perovskite SrTiO 3 , obtained from both experiments 12,13) and first-principles calculations, 14) contributed much to the recent discussion on its phonon dynamics. In contrast, κ data [15][16][17][18] for BaTiO 3 , PbTiO 3 , PbZrO 3 , and PbHfO 3 are still mostly lacking, though κ of thin films has been examined in recent studies. [19][20][21] To remedy this situation, we provide bulk high-temperature (T ⩾ 300 K) κ data for the four perovskites. The results reveal a strong difference between the FE and AFE compounds: only the latter exhibit distinct glasslike behavior in their cubic phase. We also present low-T heat capacity (C p ) on PbZrO 3 single crystals, which shows a lack of glasslike thermal behavior in the AFE phase.
Polycrystalline samples of BaTiO 3 , PbTiO 3 , PbZrO 3 , and PbHfO 3 were prepared by solid-state reactions. High purity (⩾ 99.9%) powders of BaCO 3 , PbO, TiO 2 , ZrO 2 , and HfO 2 , with particle sizes of ∼1-20 μm, were dried and weighed to either the stoichiometric ratio (for BaTiO 3 ) or 1% excess PbO. The powders were then mechanically ground for 30 min and calcined in air for 3 h at 1373 K (BaTiO 3 ) or 1123 K (the others). After the calcined products were ground again, they were divided into two portions. To the first portion was added 2 wt% polyvinyl alcohol (PVA) binder, which was then pelletized under an isostatic pressure of 200 MPa and heated at 723 K for 3 h to remove the binder. For PbTiO 3 , PbZrO 3 , and PbHfO 3 , these pellets were buried in the second portion in a tightly covered Pt crucible, which was in turn sealed in an Al 2 O 3 crucible using alumina cement. This procedure minimized the loss of PbO during the 3 h sintering at 1523 K (PbTiO 3 ) or 1573 K (PbZrO 3 and PbHfO 3 ). For BaTiO 3 , the pellets were sintered directly in the air at 1623 K for 4 h. Powder X-ray diffraction confirmed that the products were single-phase.
The κ data were obtained using the relation κ = DρC p , where D is thermal diffusivity and ρ is density. The D values were determined in a nitrogen atmosphere on heating direction by the flash technique, using either Netzsch LFA 467 for 300-773 K or LFA 467HT for 300-993 K. For each measurement, a 10 × 10 mm 2 square plate of ∼1.5 mm thickness was coated on both sides with a thin layer of graphite. The C p values were taken from the literature, as described below. The ρ was determined by the Archimedes method, and its T dependence was calculated using thermal expansion data. [22][23][24][25] As the ρ of sintered pellets were 88%-95% of the theoretical values, the raw κ values (κ raw ) were converted to the κ of a fully dense solid through the relation κ raw /κ = 1-4f/3, where f is the porosity of the specimen. 26) The overall uncertainty in κ is estimated to be less than ±10% for BaTiO 3 , PbZrO 3 , and PbHfO 3 , and slightly larger for PbTiO 3 (see below).
Single crystals of PbZrO 3 with < 2 mm in size were grown by the flux method using excess PbO as the flux. The powder X-ray diffraction pattern was indistinguishable from the polycrystalline data, and differential scanning calorimetry showed a sharp peak with the onset T of 506.5 K on heating and 505.5 K on cooling. A collection of three single crystals was used to measure C p between 0.5 and 2 K using the relaxation method of the Physical Properties Measurement System by Quantum Design.
As the accuracy of our κ is constrained by that of C p , we first examine the C p values used in this study. The inset of Fig. 1 shows published C p for PbTiO 3 , PbZrO 3 , and PbHfO 3 , in units of J K −1 mol −1 . 27 The high quality of the data can be ascertained by the observations that (1) they join smoothly with the lower-T data obtained from other measurements, 27) and (2) a clean, sharp peak is observed for each of the firstorder transitions. Also, in the cubic phase of each compound, C p saturates at a value slightly below the Dulong-Petit limit of 125 J K −1 mol −1 , and this continues at least up to 1050 K for PbTiO 3 . 28) In other perovskites, including BaTiO 3 , 29,30) SrTiO 3 , 31) and BaZrO 3 ,32) high-T C p exceeds the Dulong-Petit limit due to anharmonic lattice expansion. We associate the unusual behavior of Pb-based perovskites with the easily polarizable Pb ions, although further studies are certainly needed to confirm this idea. In any case, the data provide reliable C p for the present purpose, except in the vicinity of T c where even small sample dependence leads to a large difference in C p . For this reason, we exclude κ data in such T regions.
The C p for BaTiO 3 is shown in the main panel of Fig. 1. The data up to 420 K were obtained by Moriya using an adiabatic calorimeter, 33) which has an accuracy of 0.2% above 300 K. (The sharp peak at T c climbs up to 210.35 J K −1 mol −1 at 401.72 K.) The figure also shows Coughlin and Orr's C p , 29,30) which lacks the sharp peak but exhibits larger values just above T c . Accordingly, we represent the data by the C p of SrTiO 3 (Ref. 31) multiplied by 1.03, which joins smoothly with Moriya's data. It is noted that more recent 34) C p on BaTiO 3 appears to be too large by ∼5%.
The main panel of Fig. 2 shows the κ of BaTiO 3 , for which the present result is labeled as polycrystalline. Also shown is κ for a single crystal, obtained from Hofmeister's D data 35) using the same C p . (As it shows a lower T c , C p in the cubic phase was extrapolated.) These results exhibit several important features. First, polycrystalline values are lower than those of single crystal by ∼20%, which is similar to the case 13) of SrTiO 3 shown in the inset. Such reduction in κ should result from the scattering of thermal phonons at grain boundaries. Second, the single crystal shows strongly anisotropic κ in the tetragonal phase, and their weighted average is closely matched in its slope by the polycrystalline data. The polycrystalline κ of 2.8 W m −1 K −1 at 300 K agrees well with the published room temperature values. 36) Third, κ in the cubic phase is less than half the values in SrTiO 3 , and it decreases almost linearly with T, unlike the more typical T −1 The single crystal values are obtained from reported thermal diffusivity data. 35) Inset: κ of single crystal and polycrystalline SrTiO 3 . 13) The polycrystalline data come from a sample with an average grain size of 10 μm. 13) drop in SrTiO 3 . From the kinetic expression, κ = (1/3)Cvl, where C is the heat capacity per unit volume, v is the averaged phonon group velocity, and l is the mean free path of the phonons. As comparable v for BaTiO 3 and SrTiO 3 can be deduced from the acoustic dispersion curves, 37) the smaller κ in BaTiO 3 is attributed to a shorter l, arising from stronger phonon scattering by anharmonic phonon-phonon interactions.
Possible sources of additional phonon scattering in cubic BaTiO 3 are (i) the 〈111〉 disorder of Ti ions, 2,3) which is most dynamic in character, 3) and (ii) the FE soft mode. (For SrTiO 3 , the former is practically absent 38,39) and the latter occurs at much lower T. 1) ) To further evaluate these contributions, we next compare the κ of BaTiO 3 and PbTiO 3 , shown in Fig. 3; for PbTiO 3 , the 〈111〉 Ti disorder is negligible, 38,39) and the FE soft mode is reported to interact strongly with the heat-carrying phonons. 40) Moreover, PbTiO 3 shows much flatter acoustic dispersion curves than BaTiO, 37) leading to a lower v. (Vibration energies at zone boundaries are less than half the values in BaTiO 3 ). Considering these differences, the observation of similar cubic κ for BaTiO 3 and PbTiO 3 (∼2-2.5 W m −1 K −1 ) implies shorter l in BaTiO 3 , which in turn can be attributed to the 〈111〉 Ti disorder. Recently, this disorder in BaTiO 3 was incorporated into the calculation of phonon dispersion curves. 39) It would be of great interest to extend such works to obtain κ from the first-principles. Figure 3 also reveals stronger anomaly 15) across T c for PbTiO 3 , reflecting its larger polarization and tetragonal strain 1) (c/a = 1.06 in PbTiO 3 and 1.01 in BaTiO 3 at room temperature). Also, when the measurements were repeated after the sample was cooled from the cubic phase, only PbTiO 3 showed significant difference (4% larger) from the first measurements; similar difference (3% smaller) occurred on another PbTiO 3 sample heated only to 773 K. This feature suggests that κ in the FE phase depends on how the large tetragonal strain is accommodated in the sample, which varies each time it experiences the transition. Perhaps related to this, the present κ shows a stronger slope in the FE phase than the previous polycrystalline data, 15) and κ = 3 W m −1 K −1 at 300 K is much lower than 5 W m −1 K −1 for a multidomain single crystal 17) [see Fig. 4(a)]. Thus, measurements on a single domain crystal are probably needed to assess the intrinsic κ in the FE phase, and to make a close comparison with the results of first-principles κ calculations. 40,42) The κ of AFE compounds PbZrO 3 and PbHfO 3 are also shown in Fig. 3. Their small values compared to BaTiO 3 and PbTiO 3 are obvious, and the smaller κ for PbHfO 3 is ascribed to the heavier mass of Hf compared to Zr. Furthermore, both the magnitude and T dependence in the cubic phase resemble those of glasses, as evidenced through comparison with the κ of amorphous silica 41) (a-SiO 2 ). Such glasslike κ occurs when l is limited to the order of interatomic distance, 45) where heat is carried by a random walk of energy between localized oscillators. 43) Empirically, this condition is satisfied in amorphous solids and a range of chemically disordered crystals. 43) On the other hand, PbZrO 3 and PbHfO 3 do not possess chemical disorder, so that, the glasslike κ should be related to large anharmonic distortions, which are presumably stronger than the Ti disorder in BaTiO 3 . The evidence for such distortions can be found in structural studies, 46,47) where  large displacements of Pb ions and oxygen octahedral tilting are reported for the cubic phase. Moreover, the dynamic nature of such distortions is seen through the relaxation polar mode, 48) and these results have been interpreted as signatures of order-disorder mechanism for the AFE transition. 9,48) Interestingly, the present results provide additional support for the order-disorder mechanism, as κ changes to crystallike dκ/dT < 0 below T c .
We view the dκ/dT < 0 behavior in the AFE phase as an important result, since previous data on PbZrO 3 showed either a constant 15) κ or a weakly positive 21) dκ/dT below T c : only the present result strongly questions Lawless's conclusion 44) that PbZrO 3 exhibits a glasslike thermal behavior in the AFE state. To discuss this problem in sufficient detail, we now focus on the low-T thermal properties. Figure 4(a) shows the κ of PbTiO 3 , 17) PbMg 1/3 Nb 2/3 O 3 (PMN), 17) a-SiO 2 , 43) and PbZrO 3 , plotted in the double logarithmic scale. Here, both the present result and Lawless's data 44) are plotted for polycrystalline PbZrO 3 , while the data for PbTiO 3 and PMN come from single crystals. As an example of normal ferroelectrics, PbTiO 3 exhibits the usual crystalline κ behavior, with a peak at 65 K. In contrast, a FE relaxor PMN shows the prototypical glasslike behavior, 49) with a signature plateau at ∼10 K. The glasslike behavior in PMN can be attributed to the presence of randomly oriented polar nanoregions. 17,50) As for the AFE compound PbZrO 3 , Lawless suggested that a glasslike plateau should occur above 40 K, the upper limit of the data. 44) However, the glasslike plateau should not occur at such a high-T. 43,49) Moreover, the present data suggest the presence of a broad peak above 40 K.
To make a more definitive statement on PbZrO 3 , we have measured the low-T C p on single crystals. The result plotted as C p /T versus T 2 , is shown with the other perovskites 17) in Fig. 4(b). Here, the linear lines are the fit to C p = γT + βT 3 , where the intercept γ arises from the glasslike two-level tunneling systems 49) and the slope β is due to phonons. For PbTiO 3 , γ = 0 and β agrees with the value obtained from elastic constants, 17) which are the expected behavior of a crystalline insulator. In contrast, PMN has a glasslike γ and β is much larger than the elastic value, 17) indicating the presence of glasslike nonacoustic contributions. As for PbZrO 3 , an even larger γ (=0.8 mJ K −1 mol −1 ) is seen in Lawless's data, and it was used as evidence for glasslike behavior. 44) However, this result is not reproduced by our data on single crystals, which show a lack of γ but similar β. In particular, our C p above ∼1.1 K closely follows that of PbTiO 3 , which is consistent with similar elastic Debye temperatures 17,51) (396 K for PbZrO 3 and ∼360-370 K for PbTiO 3 ). There is an additional weak upturn below 1.1 K in our data, and this can be attributed to extrinsic contributions from lattice defects and/or impurities. 50) Thus, these observations lead us to suggest that PbZrO 3 exhibits crystalline thermal behavior in the AFE phase. It should be noted that the accuracy of Lawless's C p for other compounds has already been questioned. 50,52) In summary, we have examined the high-T κ for BaTiO 3 , PbTiO 3 , PbZrO 3 , and PbHfO 3 . In contrast, κ in the cubic phase is found between the FE and AFE perovskites, where only the latter shows glasslike behavior. The AFE transition is accompanied by a change to the crystalline thermal behavior, which is also seen in the low-T C p of PbZrO 3 . With the recent advances in first-principles κ calculations, the present results are expected to serve as an important database for anharmonic lattice dynamical studies of these technologically important materials.