A re-examination of antiferroelectric PbZrO_3 and PbHfO_3: an 80-atom Pnam structure
J. S. Baker, M. Pa?ciak, J. K. Shenton, P. Vales-Castro, B. Xu, J. Hlinka, P. Márton, R. G. Burkovsky, G. Catalan, A. M. Glazer, D. R. Bowler
AA re-examination of antiferroelectric PbZrO and PbHfO : an 80-atom P nam structure
J. S. Baker , , ∗ M. Pa´sciak , † J. K. Shenton , P. Vales-Castro , B. Xu , J. Hlinka ,P. M´arton , R. G. Burkovsky , G. Catalan , , A. M. Glazer , and D. R. Bowler , , London Centre for Nanotechnology, UCL, 17-19 Gordon St, London WC1H 0AH, UK Department of Physics & Astronomy, UCL, Gower St, London WC1E 6BT, UK Institute of Physics, Academy of Sciences of the Czech Republic,Na Slovance 1999/2, 182 21 Praha 8, Czech Republic Department of Materials, ETH Zurich, CH-8093 Z¨urich, Switzerland Catalan Institute of Nanoscience and Nanotechnology (ICN2),Campus Universitat Autonoma de Barcelona, Bellaterra 08193, Spain School of Physical Science and Technology, Soochow University, Suzhou 215006, China Peter the Great Saint-Petersburg Polytechnic University, Saint-Petersburg, Russian Federation Institut Catal`a de Recerca i Estudis Avan¸cats (ICREA), Barcelona 08010, Catalunya Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK and International Centre for Materials Nanoarchitectonics (MANA) National Institutefor Materials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan (Dated: February 23, 2021)First principles density functional theory (DFT) simulations of antiferroelectric (AFE) PbZrO and PbHfO reveal a dynamical instability in the phonon spectra of their purported low temperature P bam ground states. This instability doubles the c -axis of P bam and condenses five new smallamplitude phonon modes giving rise to an 80-atom
P nam structure. Compared with
P bam , thestability of this structure is slightly enhanced and highly reproducible as demonstrated through usingdifferent DFT codes and different treatments of electronic exchange & correlation interactions. Thissuggests that
P nam is a new candidate for the low temperature ground state of both materials.With this finding, we bring parity between the AFE archetypes and recent observations of a verysimilar
AFE phase in doped or electrostatically engineered BiFeO . Nearly seventy years have passed since the theory of the antiferroelectric (AFE) phenomenon was proposed by Kittel[1] and the observation in PbZrO (PZO) by Shirane, Sawaguchi and Takagi [2]. Today, although most consider PZOand PbHfO (PHO, PZO’s isoelectronic and isostructural partner) the AFE archetypes, many fundamental aspectsof these materials remain hotly contested. Indeed, recent years have brought the very nature of the AFE phasetransition into question with no clear consensus in sight [3–8]. Even the crystal structure of the low temperature AFEphase is a point of order. The majority of the community regard the structure as being best described with P bam symmetry [9–13], but the path to this agreement was a contentious one. Several different space group assignmentswere proposed (which are summarized in [9]) as well as suggestions of structural disorder [10]. Compounded withthis, the presence of complex twinning [10, 14, 15] and incommensurations [16, 17] are ubiquitous in these materials ∗ [email protected] † [email protected] a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b making the task of accurate structure refinement a challenging one. Recently, first principles calculations were usedto show that a large number of unique and increasingly incommensurate dynamical instabilities were present in thephonon dispersion relations of cubic PZO and ordered models of near-morphotropic PbZr x Ti − x O (PZT, x = 0 . P bam phase? Now that the technologicalimportance of AFE materials have been realized, it is crucial to return to address these fundamental issues to ensurethat new AFE technologies including solid state cooling (exploitative of the large negative electrocaloric effect [19–22])and energy storage devices [23] are built upon stable foundations.In this work, we use simulations based on density functional theory (DFT) to show that the phonon dispersionrelations of
P bam
PZO and PHO feature a single dynamical instability at the Z-point (of
P bam ), q Z = (0 , , ). Wereason that such instabilities should not persist within the low temperature ground state as they would condense intothe structure as a soft mode. Following the eigendisplacements of this instability, we demonstrate the emergence ofan 80-atom P nam phase slightly lower in energy than P bam and described by eleven distinct phonon modes .The phonon dispersion relations of
P m ¯3 m (Figure 1) and P bam (Figure 2) PZO/PHO are calculated using the im-plementation of density functional perturbation theory [24–26] (DFPT) implemented within
ABINIT [27, 28] ( v8.10.2 ).These calculations use the local density approximation of Perdew & Wang [29] (LDA-PW) and projector augmentedwave [30] (PAW) data sets from the
JTH library [31] ( v1.1 ). To correct for the undefined nature of long-range coulombinteractions as we approach the Γ-point [32], we apply the non-analytical correction [25] to the dispersions, correctlyaccounting for longitudinal-optical transverse-optical splitting [33]. To achieve high accuracy dispersions, we use a680.29 eV plane wave cutoff, relax the ionic positions to a stringent force tolerance of 1 × − eV/˚A and interpolatedispersions from exact frequencies calculated on dense Γ-centered 6 × × P m ¯3 m ) and 5 × × P bam ) q -point meshes. These meshes share dimensions and centering with the Monkhorst-Pack [34] (MP) k -point meshesused for Brillouin zone (BZ) integrals.While the phonon dispersion of P m ¯3 m PZO (Figure 1, blue lines) has been discussed elsewhere [18, 35], we brieflyremark on some important features and compare with the
P m ¯3m PHO dispersion (Figure 1, orange lines). With afocus on the imaginary branches (indicative of dynamical instabilities) we see that both materials are firmly unstablethroughout the entirety of the first BZ. In turn, this gives rise to a large number of unique unstable modes which arediscussed in [18]. We can see clearly the strong instabilities at the R and Σ-points known to comprise the majority ofthe distortion defining the
P bam
AFE phase [5, 36]. We also note that both materials feature exceptionally flat bands,especially in the vicinity of the R-point and along the most unstable R → M lines; a tell-tale sign that a material isprone to structural incommensurations. Comparing the two materials, we see that their imaginary spaces are stronglysimilar albeit PZO is slightly more unstable than PHO. Indeed, when analysing modes at the Γ, X, M Σ and R-points,the character of the instabilities of the two materials are identical (although, this comparison is imperfect for the M +2 mode as it has a real frequency in PHO).Looking now at the P bam dispersions (Figure 2), we see that PHO and PZO retain closely related dynamicalbehaviour. One particular feature the eye is drawn towards is the instability of an optical branch in the vicinity of the We choose to adopt a non-standard representation
P nam space group in place of (the equivalent)
P nma for two reasons. Firstly, using
P nam we retain the same orientation of axes as the well known
P bam structure. This way it is clear that the b-glide (of
P bam ) isreplaced by an n-glide operation in
P nam . Secondly, using
P nam avoids confusion with the prototype CaTiO -like P nma perovskitestructure, which is not similar to the AFE
P nam phase.
Z-point. Since
P bam is the purported low temperature ground state, this is surprising. The low temperature groundstate (or more precisely, the 0K ground state) should have no unstable modes . Exactly at the Z-point, this mode hasirreducible representation (irrep) Z +4 for PZO and PHO with wavenumbers 26 . i cm − and 24 . i cm − respectively.By convention, mode irreps are usually given as a decomposition of the P m ¯3 m phase, so, we unfold the single Z +4 irrep (of P bam ) to five irreps: T , T , Λ , Λ and ∆ where q T = ( , , ), q Λ = ( , , ) and q ∆ = (0 , , P m ¯3 m dispersions (Figure 1) andwere recorded in [18]. The distortion patterns (though exaggerated) for T and Λ are shown in Figure 3. We donot discuss the character of the remaining modes here since they only appear at a minuscule amplitude. T (Figure3a) is a long wavelength antiferrodistortive mode featuring octahedral rotations about the c axis. It is periodic overfour perovskite units with two octahedra rotating clockwise and two anticlockwise (a + + −− pattern). This modeis reminiscent of the super-tilting pattern observed in NaNbO [37] and AgNbO [38]. Λ (Figure 3b) is a Pb-Oantipolar mode. For the Pb displacements, it can be described as having a ‘two-up, two-down’ pattern in one PbOplane then a ‘two-left, two-right’ pattern in the next PbO plane. Within these planes, O moves antiparallel to the Pbdisplacements. Within the ZrO /HfO planes, O moves in a sinusoidal wave pattern with a period of four O sites.This pattern is reflected (about the Pb-O plane) in the next ZrO /HfO plane.We now introduce the eigendisplacements associated with these new irreps into the P bam structure, breaking thesymmetry and pushing the crystal into a new energy minimum. After relaxing the ionic positions, we arrive at an80-atom
P nam phase; a Klassengleiche maximal subgroup of
P bam with c -axis doubling. This structure is lower inenergy than P bam and described by eleven distinct irreps. This is the sum of the five mentioned in the previousparagraph and the six pre-existing in the
P bam structure [5]. To corroborate this energy lowering and to ensure thisnew phase is not a mere artefact of the LDA, we perform the same procedure with the PBESol [39] and SCAN [40]functionals . The relative stabilities are shown in Table I and the resulting P nam and
P bam crystal structures aregiven in Tables II and III respectively. In anticipation of small energy differences, the results given in Tables I-IIIare calculated with denser MP k -point grids (7 × × P bam , 7 × × P nam and 8 × × P m ¯3 m , allΓ-centered). To quantify the strength of each distortion, we calculate the primitive cell normalised mode amplitude A p for each irrep which we display in Table IV. A p is calculated by assigning atomic displacements (by symmetry)to an irrep and measuring the fractional displacements relative to the parent structure. We then normalise by afactor of (cid:112) V p /V s for primitive/supercell cell volumes V p /V s . A p is then the root sum squared (RSS) of each of thesedisplacements comprising the irrep. This is the format popularized by the ISODISTORT package [47].Table I shows that the new
P nam phase is lower in energy than
P bam for all three functionals used.
P nam is morestable by ∼ ≈ and Λ modes which appear with an amplitude similarto the S mode of P bam in the LDA-PW and PBESol calculations. For SCAN, the amplitudes of these modes aredegraded, explaining the narrowing of the energy difference between
P bam and
P nam for this functional. This effectis particularly apparent for
P nam
PHO where the amplitudes of these two modes are ≈ × smaller compared with PBESol calculations were performed using norm-conserving pseudopotentials generated by the ONCVPSP code [41] (v0.3) using inputfrom the PseudoDojo library [42]. A 1088.46 eV plane wave cutoff was used. SCAN calculations were performed with
VASP [43, 44]( v5.4.4 ) using PAWs (Zr sv 04Jan2005, Hf pv 06Sep2000, O 08Apr2002, Pb d 06Sep2000) [30] and a 700 eV plane wave cutoff. We alsoremark that the lower energy of
P nam compared with
P bam is reproducible with LDA and PBESol calculations with
VASP as well aswith Wu-Cohen functional [45] calculations with
SIESTA [46] ( v4.0 ). LDA-PW and PBESol. Even more interesting, the RSS of
P nam
PHO for the SCAN functional is lower than
P bam ;despite the introduction of five new modes, the total distortion decreases. This is the result of the reduced amplitude R +4 mode which competes with one of the new modes. It also worth mentioning that Tables II and III show that thelattice parameters (per ABO unit) of the two models are almost unchanged ( ∼ − ˚A difference). It is thereforeunlikely that we can experimentally distinguish between the two models by this simple comparison.There are three possibilities we can conceive for the origin of this new P nam phase which should all be considered.Firstly, it could be that what was observed as
P bam in experiment was
P nam all along. This could be forgivensince distinguishing between the two models in any given measurement could be difficult, especially without priorknowledge of the
P nam model. The new distortions are small in amplitude, so any measurement would likely haveto be performed with high resolution equipment at cryogenic temperatures while taking great care to account forthe possible presence complex twin domains [10, 14, 15]. We remark that if the low temperature ground state is
P nam , it would be unsurprising seeing as we know the vast majority of perovskites condense this symmetry atlow temperatures [48, 49] (although we reiterate that the 80-atom AFE
P nam struture is different to the commonperovskite prototype
P nma structure). The second origin we have conceived is that in some region below the measuredAFE phase transition temperature, the crystal is
P bam , but, at some point before 0K there is a second transitionto
P nam , previously undetected due to its small magnitude. This origin, however, is unlikely. Using dielectric lossmeasurements of single crystal PZO, we observe no dielectric anomalies from room temperature to 10K stronglysuggesting there is no such transition (unless of course the transition exists below 10K). While the presence of suchan anomaly would indicate the presence of a phase transition, it does not present the symmetry of the new phase. Itis perhaps only careful neutron diffraction/scattering experiments on single crystal samples which would allow properdetermination of
P nam over
P bam . Confirmation would arise from the observation of Λ, ∆ or T-point reflections. Inparticular, Λ-point reflections could be characterised with X-rays as we expect a strong Pb character in the distortion.The third origin is that the new distortions are artefacts of the theoretical approach. At the level of DFT, we relyon the available functionals used to approximate the electronic exchange & correlation interactions. While we havetried to minimize the possibility that these new distortions appear only with particular functionals (by using three atconsecutively higher rungs which all predict
P nam as a more stable phase than
P bam ), we cannot explicitly rule thisout. In addition, it is conceivable that while
P nam is indeed the proper ground state atomic configuration, quantumand thermal fluctuations of nuclei are sufficient to suppress the condensation of some phonon modes and thus force thethermodynamical equilibrium atomic positions towards
P bam at finite temperatures and pressures. Unfortunately,addressing the finite temperature properties of anharmonic crystals with unit cells as large as 80 atoms currentlyexceeds the computational tractability of ab initio simulations.While this
P nam phase appears exotic, it is more common than one might think. Similar 80-atom AFE
P nam phases are known to be metastable in BiFeO (BFO) [50] and have recently been experimentally stabilized under thecorrect electrostatic boundary conditions [51]. They are also known to appear in a whole host of BFO-based solidsolutions, including (Bi, La)FeO [52], BiFe . Mn . O [53], (Bi, Nd)FeO [54, 55] and BiFe . Sc . O [50, 56]. Wenote that in all of these cases, the magnitude of the distortions defining P nam over
P bam are much stronger than whatwe find for PZO and PHO (for example, the energy difference between
P bam and
P nam is found to be 47 meV/FUin BFO [51] while this difference is only ∼ or Pb [57–59] distinguishing them from BaTiO where polar distortions are drivenby Ti 3d-O 2p hybridization [59].To summarize, we have shown that the phonon dispersions of the purported P bam
AFE ground state of PZO andPHO are dynamically unstable. The eigendisplacements of the instability describe an 80-atom
P nam phase slightlylower in energy than
P bam . Given that AFE phases of this type seem to be ubiquitous in BFO and BFO-basedmaterials and it now appears (in DFT, at least) in the archetypal antiferroelectrics, the question must be asked: are80-atom
P nam phases the most common AFE arrangements in the perovskite oxides? While we cannot currentlyunequivocally declare that
P nam is the true low temperature ground state structure of PZO and PHO, from theperspective of DFT, it is clear that
P bam isn’t. The experimental verification (or invalidation) of
P nam symmetrywould surely be challenging but as we have demonstrated, it is now time to take another look.
ACKNOWLEDGEMENTS
J. S. Baker and D. R. Bowler are grateful for computational support from the UK Materials and Molecular ModellingHub, which is partially funded by EPSRC (EP/P020194), for which access was obtained via the UKCP consortiumand funded by EPSRC Grant Ref. No. EP/P022561/1. J. S. Baker and D. R. Bowler also acknowledge that thiswork used the ARCHER UK National Supercomputing Service funded by the UKCP consortium EPSRC Grant Ref.No. EP/P022561/1. J. K. Shenton acknowledges funding from the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovation programme Grant agreement No. 810451. Computationalresources were provided by ETH Zurich. We thank K. Roleder for supplying a single crystal PZO sample. We alsoacknowledge fruitful discussions with B. Grosso, N. Spaldin and N. Zhang. [1] C. Kittel, Phys. Rev. , 729 (1951).[2] G. Shirane, E. Sawaguchi, and Y. Takagi, Phys. Rev. , 476 (1951).[3] A. K. Tagantsev, K. Vaideeswaran, S. B. Vakhrushev, A. V. Filimonov, R. G. Burkovsky, A. Shaganov, D. Andronikova,A. I. Rudskoy, A. Q. R. Baron, H. Uchiyama, D. Chernyshov, A. Bosak, Z. Ujma, K. Roleder, A. Majchrowski, et al. , Nat.Commun. , 2229 (2013).[4] K. M. Rabe, in Functional Metal Oxides (Wiley-VCH Verlag GmbH & Co. KGaA, 2013) pp. 221–244.[5] J. ´I˜niguez, M. Stengel, S. Prosandeev, and L. Bellaiche, Phys. Rev. B , 220103(R) (2014).[6] Z. G. Fthenakis and I. Ponomareva, Phys. Rev. B , 184110 (2017).[7] P. Vales-Castro, K. Roleder, L. Zhao, J.-F. Li, D. Kajewski, and G. Catalan, Appl. Phys. Lett. , 132903 (2018).[8] B. Xu, O. Hellman, and L. Bellaiche, Phys. Rev. B , 020102(R) (2019).[9] H. Fujishita, Y. Shiozaki, N. Achiwa, and E. Sawaguchi, J. Phys. Soc. Japan , 3583 (1982).[10] A. M. Glazer, K. Roleder, and J. Dec, Acta Crystallogr. B Struct. Sci. , 846 (1993).[11] D. L. Corker, A. M. Glazer, J. Dec, K. Roleder, and R. W. Whatmore, Acta Crystallogr. B Struct. Sci. , 135 (1997).[12] V. Madigout, J. L. Baudour, F. Bouree, C. Favotto, M. Roubin, and G. Nihoul, Philos. Mag. A , 847 (1999).[13] H. Fujishita, Y. Ishikawa, S. Tanaka, A. Ogawaguchi, and S. Katano, J. Phys. Soc. Japan , 1426 (2003).[14] B. A. Scott and G. Burns, J. Am. Ceram. Soc. , 331 (1972). [15] O. E. Fesenko and V. G. Smotrakov, Ferroelectrics , 211 (1976).[16] R. G. Burkovsky, I. Bronwald, D. Andronikova, B. Wehinger, M. Krisch, J. Jacobs, D. Gambetti, K. Roleder, A. Ma-jchrowski, A. V. Filimonov, A. I. Rudskoy, S. B. Vakhrushev, and A. K. Tagantsev, Sci. Rep. , 41512 (2017).[17] A. Bosak, V. Svitlyk, A. Arakcheeva, R. Burkovsky, V. Diadkin, K. Roleder, and D. Chernyshov, Acta Crystallogr., Sect.B: Struct. Sci., Cryst. Eng. Mater , 7 (2020).[18] J. S. Baker and D. R. Bowler, Phys. Rev. B , 224305 (2019).[19] Y. Bai, G.-P. Zheng, and S.-Q. Shi, Mater. Res. Bull. , 1866 (2011).[20] W. Geng, Y. Liu, X. Meng, L. Bellaiche, J. F. Scott, B. Dkhil, and A. Jiang, Adv. Mater. , 3165 (2015).[21] M. Guo, M. Wu, W. Gao, B. Sun, and X. Lou, J. Mater. Chem. C , 617 (2019).[22] P. Vales-Castro, R. Faye, M. Vellvehi, Y. Nouchokgwe, X. Perpin`a, J. M. Caicedo, X. Jord`a, K. Roleder, D. Kajewski,A. Perez-Tomas, E. Defay, and G. Catalan, (2020), arXiv:2009.02184 [cond-mat.mtrl-sci].[23] Z. Liu, T. Lu, J. Ye, G. Wang, X. Dong, R. Withers, and Y. Liu, Adv. Mater. Technol. , 1800111 (2018).[24] X. Gonze, Phys. Rev. A , 1096 (1995).[25] X. Gonze and C. Lee, Phys. Rev. B , 10355 (1997).[26] S. Baroni, S. de Gironcoli, A. D. Corso, and P. Giannozzi, Rev. Mod. Phys. , 515 (2001).[27] X. Gonze, B. Amadon, P.-M. Anglade, J.-M. Beuken, F. Bottin, P. Boulanger, F. Bruneval, D. Caliste, R. Caracas, M. Cˆot´e,T. Deutsch, L. Genovese, P. Ghosez, M. Giantomassi, S. Goedecker, et al. , Comput. Phys. Commun. , 2582 (2009).[28] X. Gonze, F. Jollet, F. A. Araujo, D. Adams, B. Amadon, T. Applencourt, C. Audouze, J.-M. Beuken, J. Bieder,A. Bokhanchuk, E. Bousquet, F. Bruneval, D. Caliste, M. Cˆot´e, F. Dahm, et al. , Comput. Phys. Commun. , 106(2016).[29] J. P. Perdew and Y. Wang, Phys. Rev. B , 13244 (1992).[30] P. E. Bl¨ochl, Phys. Rev. B , 17953 (1994).[31] F. Jollet, M. Torrent, and N. Holzwarth, Comput. Phys. Commun. , 1246 (2014).[32] C. H. Henry and J. J. Hopfield, Phys. Rev. Lett. , 964 (1965).[33] W. Zhong, R. D. King-Smith, and D. Vanderbilt, Phys. Rev. Lett. , 3618 (1994).[34] H. J. Monkhorst and J. D. Pack, Phys. Rev. B , 5188 (1976).[35] P. Ghosez, E. Cockayne, U. V. Waghmare, and K. M. Rabe, Phys. Rev. B , 836 (1999).[36] J. Hlinka, T. Ostapchuk, E. Buixaderas, C. Kadlec, P. Kuzel, I. Gregora, J. Kroupa, M. Savinov, A. Klic, J. Drahokoupil,I. Etxebarria, and J. Dec, Phys. Rev. Lett. , 197601 (2014).[37] J. Chen and D. Feng, Phys. Status Solidi (a) , 171 (1988).[38] M. Yashima, S. Matsuyama, R. Sano, M. Itoh, K. Tsuda, and D. Fu, Chem. Mater. , 1643 (2011).[39] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Phys.Rev. Lett. , 136406 (2008).[40] J. Sun, A. Ruzsinszky, and J. Perdew, Phys. Rev. Lett. , 036402 (2015).[41] D. R. Hamann, Phys. Rev. B , 085117 (2013).[42] M. van Setten, M. Giantomassi, E. Bousquet, M. Verstraete, D. Hamann, X. Gonze, and G.-M. Rignanese, Comput. Phys.Commun. , 39 (2018).[43] G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169 (1996).[44] G. Kresse and J. Furthm¨uller, Comput. Mater. Sci. , 15 (1996).[45] Z. Wu and R. E. Cohen, Phys. Rev. B , 235116 (2006).[46] J. M. Soler, E. Artacho, J. D. Gale, A. Garc´ıa, J. Junquera, P. Ordej´on, and D. S´anchez-Portal, J. Phys. Condens. Matter , 2745 (2002).[47] B. J. Campbell, H. T. Stokes, D. E. Tanner, and D. M. Hatch, J. Appl. Crystallogr. , 607 (2006). [48] M. W. Lufaso and P. M. Woodward, Acta Crystallogr. B Struct. Sci. , 725 (2001).[49] N. A. Benedek and C. J. Fennie, J. Phys. Chem. C , 13339 (2013).[50] S. A. Prosandeev, D. D. Khalyavin, I. P. Raevski, A. N. Salak, N. M. Olekhnovich, A. V. Pushkarev, and Y. V. Radyush,Phys. Rev. B , 054110 (2014).[51] J. A. Mundy, C. A. Heikes, B. F. Grosso, D. F. Segedin, Z. Wang, B. H. Goodge, Q. N. Meier, C. T. Nelson, B. Prasad, L. F.Kourkoutis, W. D. Ratcliff, N. A. Spaldin, R. Ramesh, and D. G. Schlom, (2018), arXiv:1812.09615 [cond-mat.mtrl-sci].[52] D. A. Rusakov, A. M. Abakumov, K. Yamaura, A. A. Belik, G. V. Tendeloo, and E. Takayama-Muromachi, Chem. Mater. , 285 (2011).[53] A. A. Belik, A. M. Abakumov, A. A. Tsirlin, J. Hadermann, J. Kim, G. V. Tendeloo, and E. Takayama-Muromachi, Chem.Mater. , 4505 (2011).[54] S. Karimi, I. M. Reaney, I. Levin, and I. Sterianou, Appl. Phys. Lett. , 112903 (2009).[55] I. Levin, M. G. Tucker, H. Wu, V. Provenzano, C. L. Dennis, S. Karimi, T. Comyn, T. Stevenson, R. I. Smith, and I. M.Reaney, Chem. Mater. , 2166 (2011).[56] D. D. Khalyavin, A. N. Salak, N. M. Olekhnovich, A. V. Pushkarev, Y. V. Radyush, P. Manuel, I. P. Raevski, M. L.Zheludkevich, and M. G. S. Ferreira, Phys. Rev. B , 174414 (2014).[57] R. Seshadri and N. A. Hill, Chem. Mater. , 2892 (2001).[58] N. A. Hill and K. M. Rabe, Phys. Rev. B , 8759 (1999).[59] R. E. Cohen, Nature , 136 (1992). Γ X M Σ Γ R X i ¯ ν ( q ) [ c m − ] P m ¯3 m PZO
P m ¯3 m PHO
S R M
FIG. 1. The phonon dispersions of
P m ¯3 m PZO (blue) and PHO (orange).
Γ X S Y Γ Z U R T Z i ¯ ν ( q ) [ c m − ] P bam
PZO
Γ X S Y Γ Z U R T Z
P bam
PHO
FIG. 2. The phonon dispersions of
P bam
PZO (blue) and PHO (orange). a) T mode b) Λ mode a bc cb abca Pb OZr/Hf
FIG. 3. The two most important modes defining the difference between
P bam and
P nam
PZO/PHO. The magnitude of thedisplacement is exaggerated. a) The T mode. ZrO /HfO octahedra rotate in antiphase about the c axis in pairs (++ −− rotations). b) The Λ mode. Antipolar Pb displacements are grouped by a common displacement direction with coloured boxes(red: up, blue: down, orange: left, green: right). Zr/Hf (inactive in this mode) has been removed for clarity. TABLE I. The relative stability ∆ E (in meV/FU) of the P bam and
P nam phases compared to cubic
P m ¯3 m for PZO andPHO. ∆ E = E ( P bam/P nam ) − E ( P m ¯3 m ). LDA-PW PBESol SCAN P bam
PZO -310.744 -262.240 -258.943
P nam
PZO -311.878 -263.096 -259.138
P bam
PHO -95.309 -175.014 -205.946
P nam
PHO -96.026 -175.426 -206.186 TABLE II. The (fractional) Wyckoff positions ( x , y , z ) and orthorhombic lattice parameters for 80 atom P nam
PZO and PHOas calculated with the LDA-PW, PBESol and SCAN functionals.Site LDA-PW PBESol SCAN
P nam
PZOPb 8d (0.7946, 0.8764, -0.0057) (0.7963, 0.8757, -0.0061) (0.8044, 0.8760, -0.0035)Pb 4c (0.7865, 0.8699, 0.2500) (0.7883, 0.8689, 0.2500) (0.7947, 0.8691, 0.2500)Pb 4c (0.2150, 0.1258, 0.2500) (0.2122, 0.1260, 0.2500) (0.2049, 0.1281, 0.2500)Zr 8d (0.7549, 0.6248, 0.1252) (0.7554, 0.6246, 0.1252) (0.7585, 0.6243, 0.1251)Zr 8d (0.2576, 0.8760, 0.6245) (0.2589, 0.87636, 0.6246) (0.2607, 0.8763, 0.6247)O 8d (0.5094 0.4965, 0.8994) (0.5105, 0.4962, 0.8978) (0.5044, 0.4987, 0.8985)O 8d (0.4538, 0.7314, 0.8645) (0.4571, 0.7326, 0.8649) (0.4615, 0.7351, 0.8630)O 8d (0.4780, 0.7434, 0.3534) (0.4815, 0.7451, 0.3542) (0.4727, 0.7410, 0.3569)O 8d (0.7246, 0.6582, 0.4990) (0.7267, 0.6569, 0.4990) (0.7241, 0.6558, 0.4994)O 8d (0.0140, 0.5067, 0.6150) (0.0152, 0.5071, 0.6150) (0.0072, 0.5033, 0.6158)O 4c (0.6968, 0.5870, 0.2500) (0.7010, 0.5887, 0.2500) (0.6978, 0.5923, 0.2500)O 4c (0.3025, 0.3995, 0.2500) (0.2967, 0.3985, 0.2500) (0.3004, 0.4001, 0.2500) a (˚A) 5.8065 5.8671 5.9015 b (˚A) 11.6707 11.7505 11.8087 c (˚A) 16.2246 16.3833 16.4237 P nam
PHOPb 8d (0.7883, 0.8763, -0.0049) (0.7895, 0.8758, -0.0049) (0.7998, 0.8746, -0.0010)Pb 4c (0.7835, 0.8698, 0.2500) (0.7846, 0.8692, 0.2500) (0.7920, 0.8695, 0.2500)Pb 4c (0.2183, 0.1263, 0.2500) (0.2161, 0.1263, 0.2500) (0.2078, 0.1298, 0.2500)Hf 8d (0.7548, 0.6247, 0.1253) (0.7551, 0.6246, 0.1251) (0.7591, 0.6237, 0.1249)Hf 8d (0.2556, 0.8759, 0.6245) (0.2571, 0.8761, 0.6246) (0.2597, 0.8766, 0.6248)O 8d (0.5086, 0.4963, 0.8969) (0.5092, 0.4963, 0.8958) (0.5011, 0.4997, 0.8952)O 8d (0.4599, 0.7336, 0.8649) (0.4639, 0.7350, 0.8646) (0.4714, 0.7394, 0.8625)O 8d (0.4851, 0.7455, 0.3549) (0.4862, 0.7463, 0.3561) (0.4742, 0.7409, 0.3611)O 8d (0.7298, 0.6556, 0.4991) (0.7317, 0.6546, 0.4992) (0.7290, 0.6522, 0.4999)O 8d (0.0118, 0.5061, 0.6150) (0.0128, 0.5063, 0.6150) (0.0018, 0.5008, 0.6166)O 4c (0.7059, 0.5896, 0.2500) (0.7085, 0.5918, 0.2500) (0.7064, 0.5988, 0.2500)O 4c (0.2945, 0.3981, 0.2500) (0.2901, 0.3983, 0.2500) (0.2932, 0.3994, 0.2500) a (˚A) 5.7566 5.8147 5.8333 b (˚A) 11.5587 11.6482 11.6571 c (˚A) 16.1342 16.2886 16.2713 TABLE III. The (fractional) Wyckoff positions ( x , y , z ) and orthorhombic lattice vectors for 40 atom P bam
PZO and PHOcalculated with the LDA-PW, PBESol and SCAN functionals. We also include a comparison with 10K neutron diffractiondata. Site LDA-PW PBESol SCAN Exp (10K) [13] [12]
P bam
PZOPb 4g (0.7035, 0.8770, 0.0000) (0.7017, 0.8764, 0.0000) (0.6951, 0.8762, 0.0000) (0.6991, 0.8772, 0.0000)Pb 4h (0.2868, 0.1275, 0.5000) (0.2893, 0.1282, 0.5000) (0.2952, 0.1297, 0.5000) (0.2944, 0.1294, 0.5000)Zr 8i (0.2431, 0.8754, 0.2497) (0.2420, 0.8756, 0.2497) (0.2401, 0.8760, 0.2497) (0.2414, 0.8752, 0.2486)O 4e (0.0000, 0.0000, 0.7713) (0.0000, 0.0000, 0.7714) (0.0000, 0.0000, 0.7689) (0.0000, 0.0000, 0.7707)O 4f (0.0000, 0.5000, 0.7999) (0.0000, 0.5000, 0.7969) (0.0000, 0.5000, 0.7974) (0.0000, 0.5000, 0.7974)O 4h (0.6961, 0.0928, 0.5000) (0.7010, 0.0941, 0.5000) (0.6985, 0.0958, 0.5000) (0.6989, 0.0956, 0.5000)O 4g (0.7230, 0.1587, 0.0000) (0.7250, 0.1575, 0.0000) (0.7237, 0.1560, 0.0000) (0.7244, 0.1560, 0.0000)O 8i (0.2431, 0.8754, 0.2497) (0.5316, 0.7616, 0.7188) (0.5328, 0.7619, 0.7197) (0.5317, 0.7378, 0.7202) a (˚A) 5.8098 5.8716 5.9028 5.8736 b (˚A) 11.6864 11.7651 11.8129 11.7770 c (˚A) 8.0993 8.1776 8.2078 8.1909 P bam
PHOPb 4g (0.7101, 0.8769, 0.0000) (0.7092, 0.8764, 0.0000) (0.7003, 0.8748, 0.0000) (0.7114, 0.8768, 0.0000)Pb 4h (0.2840, 0.1281, 0.5000) (0.2855, 0.1285, 0.5000) (0.2920, 0.1300, 0.5000) (0.2928, 0.1298, 0.5000)Hf 8i (0.2442, 0.8754, 0.2497) (0.2434, 0.8756, 0.2497) (0.2406, 0.8764, 0.2498) (0.2421, 0.8745, 0.2455)O 4e (0.0000, 0.0000, 0.7714) (0.0000, 0.0000, 0.7714) (0.0000, 0.0000, 0.7669) (0.0000, 0.0000, 0.7650)O 4f (0.0000, 0.5000, 0.7951) (0.0000, 0.5000, 0.7927) (0.0000, 0.5000, 0.7905) (0.0000, 0.5000, 0.7932)O 4h (0.7049, 0.0950, 0.5000) (0.7083, 0.0959, 0.5000) (0.7065, 0.0995, 0.5000) (0.6996, 0.0983, 0.5000)O 4g (0.7284, 0.1560, 0.0000) (0.7303, 0.1551, 0.0000) (0.7290, 0.1522, 0.0000) (0.7329, 0.1561, 0.0000)O 8i (0.5282, 0.7609, 0.7195) (0.2434, 0.8756, 0.2497) (0.5271, 0.7599, 0.7234) (0.5280, 0.7410, 0.7190) a (˚A) 5.7585 5.8181 5.8330 5.8404 b (˚A) 11.5690 11.6580 11.6575 11.7057 c (˚A) 8.0581 8.1344 8.13517 8.1751 TABLE IV. The total decomposed mode amplitudes A p (described in the text) for each irrep using the P m ¯3 m phase as theparent and the P bam / P nam phase as the daughter. Data is presented in the format “LDA-PW PBESol SCAN”. RSS = (cid:113)(cid:80) i A p,i . Mode P bam
PZO
P bam
PHO
P nam
PZO
P nam
PHOR +4 − +5 − - - 0.1337 0.1420 0.0641 0.1283 0.1261 0.0160Λ - - 0.1275 0.1278 0.0735 0.1116 0.1007 0.0181Λ - - 0.0264 0.0263 0.0155 0.0227 0.0201 0.0046T - - 0.0046 0.0046 0.0028 0.0030 0.0026 0.0007∆5