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[Makoto Tachibana](https://orcid.org/0000-0002-5907-5563), [Cédric Bourgès](https://orcid.org/0000-0001-9056-0420), [Takao Mori](https://orcid.org/0000-0003-2682-1846)

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[Thermal conductivity of BaZrO<sub>3</sub> and KTaO<sub>3</sub> single crystals](https://mdr.nims.go.jp/datasets/c1309da3-73ca-4075-84d7-1c5a34a69dfd)

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Thermal conductivity of BaZrO3 and KTaO3 single crystalsThermal conductivity of BaZrO3 and KTaO3 single crystalsMakoto Tachibana1* , Cédric Bourgès2 , and Takao Mori1,31Research Center for Materials Nanoarchitechtonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan2International Center for Young Scientists, National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044, Japan3Graduate School of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba 305-8577, Japan*E-mail: TACHIBANA.Makoto@nims.go.jpReceived June 10, 2024; accepted June 26, 2024; published online July 11, 2024BaZrO3 and KTaO3 are two rare examples of perovskite oxides that retain the ideal cubic structure down to the lowest temperature. In this paper,we report thermal conductivity (κ) between 300 and 773 K on single crystals of these compounds. For BaZrO3, the κ of 7.5 Wm−1K−1 at 300 K is∼40% larger than the previously reported polycrystalline values. For KTaO3, our value of 13.1 Wm−1K−1 at 300 K clarifies the sources of error insome of the previously reported data. These results underscore the importance of high-quality experimental data in benchmarking the accuracy ofadvanced first-principles κ calculations. © 2024 The Author(s). Published on behalf of The Japan Society of Applied Physics by IOP Publishing LtdSupplementary material for this article is available onlineFrom the viewpoint of both physical properties anddevice applications, perovskite oxides ABO3 are animportant group of compounds.1,2) The perfect per-ovskite structure has a simple cubic structure (space groupPm3̅m), with the B ion at the center of the corner-sharingoxygen octahedron, and the A ion occupying the 12-foldcoordinated site between the octahedra. In most perovskites,the cubic structure is realized only at high temperatures, andthey undergo one or several structural phase transitions oncooling.1,2) For example, BaTiO3 and KNbO3 show three setsof ferroelectric (FE) transitions, from cubic to tetragonalP4mm (at 402 and 708 K, respectively) to orthorhombicAmm2 (278 and 498 K) to rhombohedral R3m (183 and263 K). Each of these FE transitions is driven by thecondensation of a soft zone-center transverse optic (TO)phonon mode, which displaces the Ti4+/Nb5+ ion into aspecific direction.3) Another type of structural transformationis the antiferrodistortive (AFD) rotation of oxygen octahedra,which is associated with the softening and condensation of azone-boundary mode.3) SrTiO3 exhibits one such example,where the oxygen octahedra rotate about cubic [001] to forma tetragonal I4/mcm structure below 105 K.3) Many otherperovskites undergo this type of transition, often at very hightemperatures (T> 800 K).1,2)Even when the structure remains cubic down to T→0 K, asin BaZrO3 and KTaO3, some form of lattice instability isusually found in perovskites. For example, recent neutronand X-ray inelastic scattering measurements4) on BaZrO3found softening of the zone-boundary R-point optic mode oncooling, confirming the incipient AFD instability predicted infirst-principles calculations.5,6) Similarly, KTaO3 showsstrong softening of the zone-center TO mode, but theanticipated FE transition is suppressed by quantumfluctuations.7,8) It is interesting to point out that SrTiO3shows both of these AFD and FE instabilities, the formerleading to the transition at 105 K as described above, and thelatter remaining incipient due to quantum fluctuations.7,9)Because the soft modes arise from strong latticeanharmonicity,3) a detailed understanding of phonon proper-ties in perovskites remains a challenging problem. Phononthermal conductivity (κ) is a good example. At roomtemperature and above, κ in nonmetallic crystals is largelylimited by Umklapp phonon-phonon scattering.10) Whilebasic first-principles κ calculations11,12) [using interatomicforce constants at 0 K from density functional theory (DFT)and three-phonon scattering to solve the Boltzmann transportequation] have been successful in many solids, it is not thecase for perovskites.13) To overcome this problem, recenttheoretical studies14–17) have focused on advanced techni-ques, such as the self-consistent phonon theory18) for treatingfinite temperature effects, incorporation of both three-phononand four-phonon scattering,19) and calculation of the off-diagonal terms in the heat flux operators20) to describe thecoherence effects. BaZrO3 and KTaO3 are particularlyimportant materials in this respect, as the structures remaincubic at all temperatures and experimental data are availablefor checking the κ calculations.In this study, we report κ between 300 and 773 K on thesingle crystals of BaZrO3 and KTaO3, based on the flashmeasurements of thermal diffusivity (D) and the use ofreliable heat capacity (Cp) data. Previously, κ of BaZrO3has been reported on polycrystalline samples,21–23) and manyrecent efforts have been aimed at understanding these datafrom first-principles calculations.15–17) However, the κ on asingle crystal shows a large (∼40%) increase from theprevious values, indicating that polycrystalline data do notrepresent intrinsic κ in BaZrO3. For KTaO3, there aredisagreements among the reported κ on single crystals,24–28)and our present data clarify the sources of error in some of theprevious measurements. We discuss the implications of theseresults for studying the κ of perovskites.High-quality, colorless, and transparent single crystals ofBaZrO3 (Refs. 6, 29, 30) and KTaO3 (Ref. 31) were obtainedfrom Crystal Base Co. Ltd. Our crystals were square plates,with 6× 6 mm2 faces and a thickness of 1.01 and 1.03 mm,respectively. The direction of the large faces was {110} forBaZrO3 and {100} for KTaO3. The κ between 300 and 773 Kwas determined through the relation κ=DCpρ, where ρ isdensity. The D was obtained in a nitrogen atmosphere by theflash method, using Netzsch LFA 467. In this method, heat issupplied by a flash of light to the front face of a platespecimen, and the temperature at the rear face is recorded as aContent from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of thiswork must maintain attribution to the author(s) and the title of the work, journal citation and DOI.071003-1© 2024 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdApplied Physics Express 17, 071003 (2024) LETTERhttps://doi.org/10.35848/1882-0786/ad5c26https://crossmark.crossref.org/dialog/?doi=10.35848/1882-0786/ad5c26&domain=pdf&date_stamp=2024-07-11https://orcid.org/0000-0002-5907-5563https://orcid.org/0000-0002-5907-5563https://orcid.org/0000-0001-9056-0420https://orcid.org/0000-0001-9056-0420https://orcid.org/0000-0003-2682-1846https://orcid.org/0000-0003-2682-1846mailto:TACHIBANA.Makoto@nims.go.jphttps://doi.org/10.35848/1882-0786/ad5c26https://creativecommons.org/licenses/by/4.0/https://doi.org/10.35848/1882-0786/ad5c26function of time. For the measurements, the samples werecoated on both sides with a thin layer of graphite, and themodel of Mehling et al.32) was used in the analysis to accountfor ballistic radiative transfer (photon conduction). The Dvalues (see supplementary data) have an accuracy of ±3%.The Cp of KTaO3 up to 390 K was measured on a smallersingle crystal,24) using the relaxation method of a QuantumDesign Physical Property Measurement System (PPMS) withan accuracy of ±1%.33) The Cp values at higher T, and forBaZrO3 at all T, were determined as described below. Forboth compounds, literature X-ray values34,35) were used for ρat each T. The overall accuracy of our κ at 300 K is ±5%.We first discuss the Cp data, which are shown in Fig. 1. ForBaZrO3, Wang et al.36) recently provided a table of Cp below300 K, based on earlier data and their own results fromrelaxation calorimetry. These values are plotted in Fig. 1. Athigher T, the most accurate Cp is usually obtained from dropcalorimetry.37) Several sets of such data are available onBaZrO3, and Tsvetkov et al.38) gave the recommended valuesshown in Fig. 1. As these Cp values are deemed to be veryreliable, we use these data to obtain the κ for the singlecrystal. In the previous three κ studies on BaZrO3,21–23) Cpwas determined in each case by differential scanningcalorimetry (DSC).39) While two of these studies21,22) ob-tained Cp in reasonable agreement with the present values,the third study by Liu et al.23) reported Cp that is larger by 8%at 300 K, as shown in Fig. 1. As described below, thisdiscrepancy accounts for their excess κ around 300 K.For KTaO3, much fewer Cp data are available in theliterature. The Cp measured in the present study is plottedwith open circles in Fig. 1. Interestingly, the values matchthose of isostructural SrTiO3 (Ref. 40, from drop calorimetry)between 330 and 390 K, suggesting that we can use the Cp ofSrTiO3 up to 773 K as a proxy for KTaO3. With theadditional constraint that the Cp must be close to theDulong-Petit value of 125 Jmol−1K−1 at higher T, we expectthe accuracy of the present estimate to be better than ∼3%. Inany case, these data illuminate the errors in Cp used inprevious κ studies: (1) In Ref. 25 Wang et al. used DSC toobtain ∼100 Jmol−1K−1 in the entire T region, which isshown with a dotted line in Fig. 1. Although the source ofthis unusual result is not clear, DSC is known to dependstrongly on operator experience41,42) and can lead to an errorof 50% or more.41) (2) In Ref. 28, Hu et al. used the flashmethod to measure both D and Cp, the latter shown with opensquares in Fig. 1. The Cp obtained by this method often lackshigh precision,41,42) which is evident in this case from theunusual curvature.We now shift our focus to heat transport. The κ for BaZrO3is shown in Fig. 2, which includes both the present data on asingle crystal and the earlier data on polycrystallinesamples.21–23) As expected for a nonmetallic crystalline solid,each set of data shows a continuous drop in κ with increasingT. The data for the single crystal can be fitted by κ= AT−α,with α= 0.74 giving the best fit. A more significant feature,however, is the much larger κ at 300 K found in the singlecrystal: the value of 7.5Wm−1K−1 is a 47% and 32%increase from the previous data by Yamanaka et al.22) andLiu et al.,23) respectively. We take the comparison withYamanaka et al.’s data to be more meaningful since Liu et al.’s larger κ below 450 K arises mostly from the excessive Cp(see Fig. 1). Compared to these two polycrystalline data,Vassen et al.’s21) κ at T ⩾ 473 K exhibits much larger values,even overtaking the single crystalline κ above 773 K. Theorigin of this anomalously large κ is not clear. Nevertheless,its weak T dependence leads to an extrapolated value of∼6.5Wm−1K−1 at 300 K, which is still significantly lowerthan the single crystalline value.The polycrystalline samples of BaZrO3 were denseceramic pellets prepared using standard techniques,21–23)Fig. 1. Heat capacity of BaZrO3 and KTaO3. For BaZrO3, published datafrom Liu et al.,23) Wang et al.,36) and Tsvetkov et al.38) are shown. The databy Tsvetkov et al.38) follow Cp = a + bT + cT−2, which is shown with asolid line. For KTaO3, the present data and published data from Wang etal.25) and Hu et al.28) are shown. See supplementary data for the present databelow 200 K. Heat capacity of SrTiO3 from de Ligny and Richet40) is alsoshown with a solid line.Fig. 2. Thermal conductivity of BaZrO3. The solid line through the presentsingle crystalline data is fit by κ = AT−α, where α = 0.74. Polycrystallinedata from Vassen et al.,21) Yamanaka et al.,22) and Liu et al.23) are alsoshown.071003-2© 2024 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 17, 071003 (2024) M. Tachibana et al.and were composed of μm-sized grains.23) As the effect ofporosity on κ was either negligible21,23) or corrected,22) thesmaller κ in polycrystals can be ascribed to phonon scatteringat grain boundaries. Indeed, a study43) on SrTiO3 showed acontinuous reduction in κ as the grain size was reduced fromthe bulk single crystal to 20 μm to 55 nm, and ourstudies44,45) on BaTiO3 and LaAlO3 also exhibited smallerκ in polycrystals. In this context, it is intriguing to point outthat the recent first-principles studies15–17) on BaZrO3 treatedthe polycrystalline κ values as intrinsic, without consideringthe possibility of grain boundary effects. We believe that thisstate of affairs comes partly from the calculated values ofphonon mean free path (MFP): for BaZrO3 and many otherperovskites, the dominant MFP is of the order of several nmat 300 K,15,16) giving an expectation that μm-sized grains donot affect the heat transport. However, it must also be pointedout that the phonon MFP is intrinsically broadband innature,10) and there are always some phonons with longMFPs that can be scattered at grain boundaries. At present,much of the theoretical effort15–17) is aimed at finding themost accurate scheme to calculate κ, which requires reliableexperimental data as a benchmark. The present result showsthat there is a danger in using polycrystalline data for such apurpose, at least until the grain boundary effects areincorporated into the calculation.For KTaO3, Fig. 3 compares the present result with othersingle crystalline data in the literature. Fitting the present databy κ= AT−α yields α= 0.89, with the fit becoming slightlypoorer at the highest T. The first item to note in Fig. 3 is thegood agreement between the present data and some of theearlier data, whereas others show much larger differences.The present value of 13.1Wm−1K−1 at 300 K is nearlyidentical to 13.4Wm−1K−1 by Langenberg et al.,27) whoused the 3ω method to obtain the κ from 20 to 350 K. In the3ω method,46) metal lines are fabricated on the surface of thesample, and the third-harmonic voltage is measured whileapplying an alternating current. Like the flash method used inthe present study, the 3ω method does not suffer from largeerrors due to radiation loss.46) The excellent agreementbetween the two data sets attests to the high reliability ofthese measurement techniques.Hofmeister26) and Hu et al.28) also used the flash method toobtain D. Their D values are lower than our D by 9.7% and7.7% at 300 K, and 7.5% and 3.2% at 493 K, respectively(see supplementary data). These differences are slightlylarger than the combined accuracies, such that variations incrystal quality may also contribute to the difference.However, this level of difference should not be a source ofgreat concern, and we consider these data to be roughly inagreement with each other. In Ref. 28 Hu et al. used the flashmethod to obtain Cp (see Fig. 1), and the resultant κ is shownin Fig. 3. It is clear that the anomalous curvature in their κoriginates from the same feature in Cp. The figure also showsκ based on Hofmeister’s D and the present Cp. The goodagreement with our κ up to 773 K indicates that spuriouscontributions from ballistic radiative transfer26) have beencorrectly removed in both of these data.In contrast to these results, much smaller and much larger κis found in Wang et al.25) and Tachibana et al.,24) respec-tively: (1) Wang et al.25) used the flash method to obtain Dand DSC to obtain Cp. Their D values are lower than our Dby 35% at 300 K and 18% at 493 K (see supplementary data).The anomalously low D was already pointed out byHofmeister and attributed26) to the relatively thick (2 mm)sample used by Wang et al. As discussed above, Wang et al.’s Cp is also low (see Fig. 1). These values of D and Cp yieldthe κ shown in Fig. 3, which is evidently much lower thanother κ data. (2) Tachibana et al.24) used the pulse powermethod46) of the thermal transport option of the PPMS toobtain the κ between 2 and 370 K. In this method, a heater,two thermometers, and a heat sink are attached along thesample within a radiation shield, and the temperatures arerecorded upon application of periodic heat pulses. Like theclassic steady-state method, the pulse power method offers adirect κ measurement that can be used down to low T.46)However, these methods suffer from large radiation losses athigh T (>150 K), which are evidenced as a T3 tail in the raw κdata.42) Although the PPMS estimates and subtracts thiscontribution based on the emissivity and surface area of thesample, there is no simple way to verify this correction.Moreover, uncertainties in the sample geometry could alsoaffect the obtained κ. We believe these features of themeasurement led to the excessively large κ reported byTachibana et al.We have discussed in detail the anomalous results ofTachibana et al.24) and Wang et al.,25) because these datahappen to be the only ones mentioned in the first-principles κcalculations.47,48) For example, Fu and Singh47) performed κcalculations using the local density approximation (LDA) tothe DFT and the phonon frequencies fixed at 0 K. With thelattice parameter fully relaxed, they obtainedκ= 18Wm−1K−1 at 300 K and noted on the good agreementwith Tachibana et al.’s PPMS data. On the other hand, Fu etal.48) used the generalized gradient approximation (GGA) tothe DFT and calculated the κ under relaxation time approx-imation (RTA).11,12) These authors obtainedκ= 7.4Wm−1K−1 at 400 K, and remarked on the excellentFig. 3. Thermal conductivity of single crystalline KTaO3. The solid linethrough the present data is fit by κ = AT−α, where α = 0.89. Published datafrom Tachibana et al.,24) Wang et al.,25) Hofmeister,26) Langenberg et al.,27)and Hu et al.28) are also shown. The κ for Hofmeister is calculated by usingthe heat capacity from the present study.071003-3© 2024 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 17, 071003 (2024) M. Tachibana et al.agreement with Wang et al.’s data. In light of Fig. 3, it nowappears that these calculations47,48) did not adequatelycapture the heat transport in KTaO3. As recent first-principlescalculations8,9) succeeded in reproducing the phonon soft-ening behavior in this compound, it is of great interest toextend such works to calculate the κ and compare the resultswith those of Fig. 3.Finally, we provide additional comments on the T depen-dence and first-principles calculations of κ. In many perovs-kites, κ is shown to decrease more slowly than the usual T−1dependence.13) This is one sign that first-principles κ calcula-tions on perovskites require advanced treatments, beyond thebasic framework11,12) of harmonic approximation (HA) andthree-phonon (3 ph) scattering. Accordingly, Zhao et al.15)calculated the κ for many perovskites, with the HA replacedby the self-consistent phonon (SCP) method and four-phonon(4 ph) scattering added to 3 ph scattering. More recently,Zheng et al.16) focused on BaZrO3 and additionally calcu-lated the off-diagonal (OD) terms in the heat fluxoperators.20) In both studies, detailed discussions weremade on how the changes in the calculation scheme affectκ and the value of α in the power law κ∝ T−α. As notedabove, the single crystalline data for BaZrO3 obey α= 0.74up to 773 K, and this is more clearly presented with alogarithmic plot in Fig. 4. Interestingly, nearly the same α(0.75) was obtained by the SCP+3,4 ph+OD scheme,16)which also gave κ= 6.8Wm−1K−1 at 300 K that is in fairagreement with the experimental value. Thus, the agreementbetween the calculation and experiment is better than whatthe authors stated16) based on polycrystalline data.For KTaO3, Fig. 4 shows that the present data slightlydeviate from the simple T−α behavior; when the upper T limitof the fit is reduced from 773 to 500 K, the obtained αincreases from 0.89 to 0.92. If this behavior is an intrinsicproperty of KTaO3, it could perhaps be associated with thestrong T dependence of the soft mode.8) Alternatively, thisbehavior may be due to inaccuracies in the D measurement orCp estimation at high T. To gain a better perspective on thisissue, we plot two sets of single crystalline data43,49) forSrTiO3 in Fig. 4. Here, Wang et al.’s data43) follow α= 0.80only below 600 K, while Muta et al.’s data49) do not appear tofollow a simple power law at any T. 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