We make a comparative analysis of rhombohedral perovskites (ABO3) with/without oxygen rotations and ionic shifts, within the framework of a generalised effective field approach. We analyse available data on LaAlO3 and LiTaO3 and new data on Zr-rich PZT, examples of three different ways of structural evolution with temperature of the rhombohedral perovskites. The coupling between polarization and tilt is studied in detail in PZT and a possible coupling between tilt and polarization in LiTaO3 is discussed.

Cereceda, Noé,

Duan, Ning,

Gonzalo, Julio A.,

Noheda, Beatriz,

We make a comparative analysis of rhombohedral perovskites (ABO3) with/without oxygen rotations and ionic shifts, within the framework of a generalised effective field approach. We analyse available data on LaAlO3 and LiTaO3 and new data on Zr-rich PZT, examples of three different ways of structural evolution with temperature of the rhombohedral perovskites. The coupling between polarization and tilt is studied in detail in PZT and a possible coupling between tilt and polarization in LiTaO3 is discussed

Noheda, Beatriz

Duan, Ning

Cereceda, Noé

Gonzalo, Julio A.

NARCIS 

Tilts and Ionic Shifts in Rhombohedral Perovskites

University of Groningen Digital Archive

University of Groningen

   University of GroningenTilts and Ionic Shifts in Rhombohedral PerovskitesNoheda, Beatriz; Duan, Ning; Cereceda, Noé; Gonzalo, Julio A.Published in:Journal of the Korean Physical SocietyIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1998Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noheda, B., Duan, N., Cereceda, N., & Gonzalo, J. A. (1998). Tilts and Ionic Shifts in RhombohedralPerovskites. Journal of the Korean Physical Society, 32, S162-S165.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019Journal of the Korean Physical Society, Vol. 32, Feburary 1998, pp. S162∼S164Tilts and Ionic Shifts in Rhombohedral PerovskitesBeatriz Noheda, Ning Duan*, Noe´ Cereceda and Julio A. GonzaloDept. F´ısica de Materiales, C-IV. Universidad Auto´noma de Madrid, 28049-Madrid, Spain* On leave from Shanghai Inst. of Ceramics, Shanghai, ChinaWe make a comparative analysis of rhombohedral perovskites (ABO3) with/without oxygenrotations and ionic shifts, within the framework of a generalised effective field approach. We analyseavailable data on LaAlO3 and LiTaO3 and new data on Zr-rich PZT, examples of three differentways of structural evolution with temperature of the rhombohedral perovskites. The couplingbetween polarization and tilt is studied in detail in PZT and a possible coupling between tilt andpolarization in LiTaO3 is discussed.I. INTRODUCTIONThe perovskite-based structures favour ferroelectricitydue to the large Lorentz contributions in the dipolar in-ternal field and to the intrinsic trend of the cations todisplace inside the oxygen octahedra On the other handthey show structural phase transitions related to the tiltof oxygen octahedra which allow a big variety of phases.The symmetry of the octahedra framework mainly deter-mines that of the crystal lattice, justifying the classifica-tion of the perovskites by their tilt systems. The cationshifts and the octahedra tilts are independent and due todifferent kind of interactions, but they can be coupled.A model to explain this coupling has been proposed re-cently by the authors.In the rhombohedral perovskites the tilt of the oxy-gen octahedra is about their triad axis and it can bedescribed as a−a−a−, according to Glazer’s notation [2],which means opposite rotation for adjacent octahedraalong each direction. Two space groups, R3c and R3c,can sustain this tilt system. The first space group has apolar symmetry allowing ferroelectricity, while the sec-ond one is non-polar keeping the cation in the inversioncentre. Rhombohedral perovskites in the space groupR3m can also be found which do not present octahedratilts and inversion centre.The sequence of space groups presented by rhombo-hedral perovskites with temperature before reaching thecubic (Pm3m) phase, can, in principle, show the differ-ent combinations of these space groups that are allowedby symmetry. Polar R3m and tilted R3c structures cantransform straight to the cubic phase by increasing tem-perature. Polar and tilted space group, R3c, contains theR3c and R3m symmetries and it has two possible waysto achieve the cubic phase, either R3c→R3c→Pm3mor R3c→R3m→ Pm3m. Examples of these two casesare LiTaO3 and PZT, respectively, although LiTaO3,as well as LiNbO3, does not show the cubic perovskitephase before melting. Examples, such as LaAlO3, ofrhombohedral perovskites showing only tilts in the lowtemperature phase, i.e. showing a R3c→Pm3m transi-tions, have been observed. Nevertheless, as far as weknow, it has not been found any rhombohedral per-ovskite having solely a R3m→Pm3m phase transition,which implies no oxygen rotations in the whole rangeof temperature. A candidate for checking this possi-bility would be PZT compositions close to the mor-photropic phase boundary. A Zr/Ti= 60/40 composi-tion of PZT presents the R3c→R3m transition as lowas 250 K, but rhombohedral compositions with higherZr content are difficult to characterise due to the highdiffusion and the mixture of phases (rhombohedral andtetragonal) in this region. It is known that the disap-pearance of the oxygen rotations in PZT through theFRL(R3c)-FRH(R3m) phase transition at TLH, producesa step in the polarization curve. It means that thereis some coupling of the octahedra tilts with the cationshifts. In this work we compare the three mentionedexamples: PZT (cation shifts+tilts→ cation shifts→ cu-bic), LiTaO3 (cation shifts+tilts→ tilts→ melting) andLaAlO3 (tilts→ cubic). We study the behaviour of thetwo independent order parameters, related to polariza-tion (ps) and rotation angle (ηs), and the coupling be-tween them.We have used available data of paramagnetic reso-nance in LaAlO3 by K. A. Mu¨ller et al. The spontaneouspolarization of LiTaO3 is from pyroelectric current databy A. M. Glass, while the structural data of LiTaO3 arefrom S. C. Abrahams et al. Experimental data of PZTare described in the next section.II. EXPERIMENTALCeramic PZT samples used were made at the Shang-hai Institute of Ceramics. The composition belongs to-S162-Tilts and Ionic Shifts in Rhombohedral Perovskites – Beatriz Noheda et al -S163-Fig. 1. Order parameters (ps and ηs) from pyroelectricdischarge and neutron data, respectively, are plotted versustemperature for rhombohedral PZT. The ps fit was made us-ing the generalised effective field approach. We label pstotal=ps+∆ps the sum of the polarization without tilt (ps) and theextra polarization (∆ps) appearing with the tilt.the rhombohedral region with a very high Zr content(Zr/Ti= 96.5/3.5). 1.5 wt% of Nb2O5 was added to im-prove the electrical properties. The constituent oxideswere ground, pressed into pellets, and fired at 1350 ◦Cfor 2 h. Cylinders of about 1 cm in height and 1 cmin diameter were sintered and hot pressed for the neu-tron diffraction runs [3]. The same procedure has beenused, more recently, for the preparation of thin platesof the same nominal composition, of 1 mm in thicknessand about 1 cm in diameter, which were used for thepyroelectric measurements.Neutron data used in this work were presented recentlyby the authors [3]. They were taken with neutron of 14.7meV energy at Brookhaven National Laboratory. The12 (311) superstructure peak, which practically dependsonly on the oxygen rotation angle, was followed withtemperature between 20 K and 360 K.Precise pyroelectric discharge data have been obtainedby means of a set-up specially designed to measure highFig. 2. (a) ∆ps vs. T, obtained subtracting ps fit frompstotal in Fig. 1. (b) ηs vs. T, from neutron data and predictedby Eq. (2). The arrow shows the place where the antiferro-electric phase could have taken place.Fig. 3. Order parameters (ps and ηs) from pyroelectriccurrent [8] and neutron data [9], respectively, are plotted ver-sus temperature for LiTaO3. The ps fit was made by thegeneralised effective field approach. The dashed line showthe prediction of the perfect coupling model (Eq. (3)).conductivity samples with a slow rate of temperaturechange. A detailed description of the system and itsworking scheme is given in a parallel work. Data shownin this work are taken during heating with a temperaturerate of 20 K/h between 160 K and 600 K.III. RESULTS AND DISCUSSIONVery accurate polarization versus temperature mea-surements has been obtained in PZT. Normalised spon-taneous polarization, ps= Ps/Pso, and tilt parameter,ηs=es/eso, have been plotted in Fig. 1, where Pso andeso are, respectively, the spontaneous polarization andthe tilt parameter at T=0 K. The structural parameter e,is related to the tilt angle, ω, by tanω = 4√3e. The stepin polarization at the FRL-FRH phase transition, TLH,can be clearly observed in the figure coinciding with thevanishing of the tilt.The generalised effective field approach has been usedFig. 4. Order parameter of LaAlO3 vs. temperature. Theexperimental data [7] are fitted with the generalised effectivefield approach.-S164- Journal of the Korean Physical Society, Vol. 32, Feburary 1998to analyse the data. The conjugated effective electricfield and staggered stress can be expressed as Eeff=E+βP+γP3+ δP5+... and |Xeff | = X + βtθ + γtθ3 +δtθ5 + ..., respectively, being E the external electric field,P the polarization, X the external stress and θ the sta-tistically resultant tilt. This theory leads to the followingequation of state with E= 0 (p= ps)TTCtanh−1 psps= 1 + gp2s + hp4s + ... (1)where the parameters g, h,... describe the relative im-portance (to the dipolar term, βP) of the higher orderterms (γP3, δP5..), and are proportional to the succes-sive [3] even powers of Pso(Pso2, Pso4...). A fit of theexperimental data of ps vs. T by means of Eq. (1) isshown in Fig. 1.The subtraction of the ps fit to the experimental datais shown in Fig. 2(a). It gives us the behaviour withtemperature of the normalised extra polarization, ∆ps,appearing due to the tilt. ∆ps reaches its maximum valueat TLH and should disappear at T= 0 K.A model to explain the tilt-polarization coupling wasintroduced in Ref. [3]. According to it, the tilt orderparameter can be expressed in terms of ps and ∆ps asηs =∆ps1− ps(ps + ∆ps) . (2)ηs is calculated from Eq. (2), being ps and ∆ps thoseof Figs. 1 and 2(a), respectively. ηs is represented inFig. 2(b) and compared to the experimental data. Itcan be seen the clear agreement between both structuraldata and model, in spite to the model simplicity. Thearrow in the figure shows a peculiar behaviour at 220K, which was attributed to the experimental error inprevious work [3]. Nevertheless this small anomaly hasbeen reproduced with the pyroelectric data through ourcoupling model. A new transition to an antiferroelectric(AFo) phase, at about the same temperature, has beenrecently proposed, but we have found here ferroelectricbehaviour until 160 K. Our composition has similar Nband Ti contents, and the phase diagram is still unknownin this region [10]. The tilt order parameter predicted byour model from the observed ∆ps is substantially smallerthan the experimental one. This might be due to thefact that in a ceramic material the octahedra tilt maynot be so efficiently coupled to the cation shifts as ina single crystal, as assumed for Eq. (2). On the otherhand, Eq. (2) assumes perfect coupling in the sense thatall the oxygen movement is reflected in the polarizationbehaviour. The validity of this assumption may dependin each particular case, on the cation charge and size,etc., as is manifested in LiTaO3 (see bellow). The secondcase to be examined is LiTaO3, which does not reach thecubic perovskite structure (Pm3m) before melting. Inthis case the high temperature phase is not ferroelectricbut possesses oxygen rotations. The low temperaturephase has the same space group (R3c) as that PZT, withboth tilts and cation shifts. This is shown in Fig. 3,where ps and ηs has been calculated from experimentaldata in Refs. [8] and [9], respectively.The symmetry of the respective equations of state [3]for P − E and θ − X allow us to follow, in the LiTaO3case, the same argument than before, interchanging ηsand ps. So we have∆ηs =1− η2s1 + psηsps. (3)The ∆ηs obtained from Eq. (3) with experimental datafor ps and ηs is plotted in Fig. 3 as a dashed line. In thiscase the coupling predicted by our model is not reflectedin the available neutron data, within the experimentaluncertainty. The high values of the tilt angle (about22◦) and cation shifts found in LiTaO3 is a result of theweak interaction between Li+ cations shifts and oxygenoctahedra tilts, due to the small Li charge and size. Thiswould leads to a weak, if any, observed tilt-polarizationcoupling. On the contrary, the large charge and size ofthe cations in PZT produces a strong coupling betweenthe two order parameters, allowing a smaller maximumtilt angle of about 6◦.The case of LaAlO3 is, however, an example of non-polar rhombohedral perovskite. Fig. 4 shows the ex-perimental data of the order parameter [7] and the fitobtained by means of the effective field approach, in thewhole temperature range, using the X − θ equation ofstate [3].In conclusion, we have contrasted the three observedpossible sequences of phases with temperature of rhom-bohedral perovskites. Our data on PZT allow a moreaccurate test of the coupling between tilt and polariza-tion. The possible weak coupling in LiTaO3 has beendiscussed.ACKNOWLEDGMENTSFinancial support from CICyT (PB96-0037) andNATO (CRG-0037) is acknowledged.REFERENCES[1] M. E. Lines and A. M. Glass, Principles and Applicationsof Ferroelectrics and Related Materials (Clarendon Press,1977).[2] A. M. Glazer, Acta Cryst. B28, 3384 (1972).[3] N. Cereceda, B. Noheda, T. Iglesias, J. R. Fdez-del-Castillo and J. A. Gonzalo, Phys. Rev. B 55, 6174 (1997).[4] H. Megaw and C. Darlington, Acta Cryst. A31, 161(1975).[5] A. Amin, R. E. Newnham, L. E. Cross and D. E. Cox,J. of Sol. Stat. Chem. 27, 248 (1981).[6] A. M. Glazer, S. A. Mabud and R. Clarke, Acta Cryst.B34, 1060 (1978).[7] K. Mu¨ller, W. Berlinger and F. Waldner, Phys. Rev.Lett. 21, 814 (1968).Tilts and Ionic Shifts in Rhombohedral Perovskites – Beatriz Noheda et al -S165-[8] A. M. Glass, Phys. Rev. 172, 564 (1968).[9] S. C. Abrahams, E. Buehler, W. C. Hamilton and S. J.Laplaca, J. Phys. Chem. Solids 34, 521 (1973).[10] B. Noheda, N. Duan, N. Cereceda and J. A. Gonzalo, J.Korean Phys. Soc. 32, S256 (1998).[11] Z. Ujma, J. Handerek and G. E. Kugel, Ferroelectrics198, 77 (1997).

Dissertations of the University of Groningen

   University of GroningenTilts and Ionic Shifts in Rhombohedral PerovskitesNoheda, Beatriz; Duan, Ning; Cereceda, Noé; Gonzalo, Julio A.Published in:Journal of the Korean Physical SocietyIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1998Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noheda, B., Duan, N., Cereceda, N., & Gonzalo, J. A. (1998). Tilts and Ionic Shifts in RhombohedralPerovskites. Journal of the Korean Physical Society, 32, S162-S165.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 04-06-2022Journal of the Korean Physical Society, Vol. 32, Feburary 1998, pp. S162∼S164Tilts and Ionic Shifts in Rhombohedral PerovskitesBeatriz Noheda, Ning Duan*, Noé Cereceda and Julio A. GonzaloDept. F́ısica de Materiales, C-IV. Universidad Autónoma de Madrid, 28049-Madrid, Spain* On leave from Shanghai Inst. of Ceramics, Shanghai, ChinaWe make a comparative analysis of rhombohedral perovskites (ABO3) with/without oxygenrotations and ionic shifts, within the framework of a generalised effective field approach. We analyseavailable data on LaAlO3 and LiTaO3 and new data on Zr-rich PZT, examples of three differentways of structural evolution with temperature of the rhombohedral perovskites. The couplingbetween polarization and tilt is studied in detail in PZT and a possible coupling between tilt andpolarization in LiTaO3 is discussed.I. INTRODUCTIONThe perovskite-based structures favour ferroelectricitydue to the large Lorentz contributions in the dipolar in-ternal field and to the intrinsic trend of the cations todisplace inside the oxygen octahedra On the other handthey show structural phase transitions related to the tiltof oxygen octahedra which allow a big variety of phases.The symmetry of the octahedra framework mainly deter-mines that of the crystal lattice, justifying the classifica-tion of the perovskites by their tilt systems. The cationshifts and the octahedra tilts are independent and due todifferent kind of interactions, but they can be coupled.A model to explain this coupling has been proposed re-cently by the authors.In the rhombohedral perovskites the tilt of the oxy-gen octahedra is about their triad axis and it can bedescribed as a−a−a−, according to Glazer’s notation [2],which means opposite rotation for adjacent octahedraalong each direction. Two space groups, R3c and R3c,can sustain this tilt system. The first space group has apolar symmetry allowing ferroelectricity, while the sec-ond one is non-polar keeping the cation in the inversioncentre. Rhombohedral perovskites in the space groupR3m can also be found which do not present octahedratilts and inversion centre.The sequence of space groups presented by rhombo-hedral perovskites with temperature before reaching thecubic (Pm3m) phase, can, in principle, show the differ-ent combinations of these space groups that are allowedby symmetry. Polar R3m and tilted R3c structures cantransform straight to the cubic phase by increasing tem-perature. Polar and tilted space group, R3c, contains theR3c and R3m symmetries and it has two possible waysto achieve the cubic phase, either R3c→R3c→Pm3mor R3c→R3m→ Pm3m. Examples of these two casesare LiTaO3 and PZT, respectively, although LiTaO3,as well as LiNbO3, does not show the cubic perovskitephase before melting. Examples, such as LaAlO3, ofrhombohedral perovskites showing only tilts in the lowtemperature phase, i.e. showing a R3c→Pm3m transi-tions, have been observed. Nevertheless, as far as weknow, it has not been found any rhombohedral per-ovskite having solely a R3m→Pm3m phase transition,which implies no oxygen rotations in the whole rangeof temperature. A candidate for checking this possi-bility would be PZT compositions close to the mor-photropic phase boundary. A Zr/Ti= 60/40 composi-tion of PZT presents the R3c→R3m transition as lowas 250 K, but rhombohedral compositions with higherZr content are difficult to characterise due to the highdiffusion and the mixture of phases (rhombohedral andtetragonal) in this region. It is known that the disap-pearance of the oxygen rotations in PZT through theFRL(R3c)-FRH(R3m) phase transition at TLH, producesa step in the polarization curve. It means that thereis some coupling of the octahedra tilts with the cationshifts. In this work we compare the three mentionedexamples: PZT (cation shifts+tilts→ cation shifts→ cu-bic), LiTaO3 (cation shifts+tilts→ tilts→ melting) andLaAlO3 (tilts→ cubic). We study the behaviour of thetwo independent order parameters, related to polariza-tion (ps) and rotation angle (ηs), and the coupling be-tween them.We have used available data of paramagnetic reso-nance in LaAlO3 by K. A. Müller et al. The spontaneouspolarization of LiTaO3 is from pyroelectric current databy A. M. Glass, while the structural data of LiTaO3 arefrom S. C. Abrahams et al. Experimental data of PZTare described in the next section.II. EXPERIMENTALCeramic PZT samples used were made at the Shang-hai Institute of Ceramics. The composition belongs to-S162-Tilts and Ionic Shifts in Rhombohedral Perovskites – Beatriz Noheda et al -S163-Fig. 1. Order parameters (ps and ηs) from pyroelectricdischarge and neutron data, respectively, are plotted versustemperature for rhombohedral PZT. The ps fit was made us-ing the generalised effective field approach. We label pstotal=ps+∆ps the sum of the polarization without tilt (ps) and theextra polarization (∆ps) appearing with the tilt.the rhombohedral region with a very high Zr content(Zr/Ti= 96.5/3.5). 1.5 wt% of Nb2O5 was added to im-prove the electrical properties. The constituent oxideswere ground, pressed into pellets, and fired at 1350 ◦Cfor 2 h. Cylinders of about 1 cm in height and 1 cmin diameter were sintered and hot pressed for the neu-tron diffraction runs [3]. The same procedure has beenused, more recently, for the preparation of thin platesof the same nominal composition, of 1 mm in thicknessand about 1 cm in diameter, which were used for thepyroelectric measurements.Neutron data used in this work were presented recentlyby the authors [3]. They were taken with neutron of 14.7meV energy at Brookhaven National Laboratory. The12 (311) superstructure peak, which practically dependsonly on the oxygen rotation angle, was followed withtemperature between 20 K and 360 K.Precise pyroelectric discharge data have been obtainedby means of a set-up specially designed to measure highFig. 2. (a) ∆ps vs. T, obtained subtracting ps fit frompstotal in Fig. 1. (b) ηs vs. T, from neutron data and predictedby Eq. (2). The arrow shows the place where the antiferro-electric phase could have taken place.Fig. 3. Order parameters (ps and ηs) from pyroelectriccurrent [8] and neutron data [9], respectively, are plotted ver-sus temperature for LiTaO3. The ps fit was made by thegeneralised effective field approach. The dashed line showthe prediction of the perfect coupling model (Eq. (3)).conductivity samples with a slow rate of temperaturechange. A detailed description of the system and itsworking scheme is given in a parallel work. Data shownin this work are taken during heating with a temperaturerate of 20 K/h between 160 K and 600 K.III. RESULTS AND DISCUSSIONVery accurate polarization versus temperature mea-surements has been obtained in PZT. Normalised spon-taneous polarization, ps= Ps/Pso, and tilt parameter,ηs=es/eso, have been plotted in Fig. 1, where Pso andeso are, respectively, the spontaneous polarization andthe tilt parameter at T=0 K. The structural parameter e,is related to the tilt angle, ω, by tanω = 4√3e. The stepin polarization at the FRL-FRH phase transition, TLH,can be clearly observed in the figure coinciding with thevanishing of the tilt.The generalised effective field approach has been usedFig. 4. Order parameter of LaAlO3 vs. temperature. Theexperimental data [7] are fitted with the generalised effectivefield approach.-S164- Journal of the Korean Physical Society, Vol. 32, Feburary 1998to analyse the data. The conjugated effective electricfield and staggered stress can be expressed as Eeff=E+βP+γP3+ δP5+... and |Xeff | = X + βtθ + γtθ3 +δtθ5 + ..., respectively, being E the external electric field,P the polarization, X the external stress and θ the sta-tistically resultant tilt. This theory leads to the followingequation of state with E= 0 (p= ps)TTCtanh−1 psps= 1 + gp2s + hp4s + ... (1)where the parameters g, h,... describe the relative im-portance (to the dipolar term, βP) of the higher orderterms (γP3, δP5..), and are proportional to the succes-sive [3] even powers of Pso(Pso2, Pso4...). A fit of theexperimental data of ps vs. T by means of Eq. (1) isshown in Fig. 1.The subtraction of the ps fit to the experimental datais shown in Fig. 2(a). It gives us the behaviour withtemperature of the normalised extra polarization, ∆ps,appearing due to the tilt. ∆ps reaches its maximum valueat TLH and should disappear at T= 0 K.A model to explain the tilt-polarization coupling wasintroduced in Ref. [3]. According to it, the tilt orderparameter can be expressed in terms of ps and ∆ps asηs =∆ps1− ps(ps + ∆ps). (2)ηs is calculated from Eq. (2), being ps and ∆ps thoseof Figs. 1 and 2(a), respectively. ηs is represented inFig. 2(b) and compared to the experimental data. Itcan be seen the clear agreement between both structuraldata and model, in spite to the model simplicity. Thearrow in the figure shows a peculiar behaviour at 220K, which was attributed to the experimental error inprevious work [3]. Nevertheless this small anomaly hasbeen reproduced with the pyroelectric data through ourcoupling model. A new transition to an antiferroelectric(AFo) phase, at about the same temperature, has beenrecently proposed, but we have found here ferroelectricbehaviour until 160 K. Our composition has similar Nband Ti contents, and the phase diagram is still unknownin this region [10]. The tilt order parameter predicted byour model from the observed ∆ps is substantially smallerthan the experimental one. This might be due to thefact that in a ceramic material the octahedra tilt maynot be so efficiently coupled to the cation shifts as ina single crystal, as assumed for Eq. (2). On the otherhand, Eq. (2) assumes perfect coupling in the sense thatall the oxygen movement is reflected in the polarizationbehaviour. The validity of this assumption may dependin each particular case, on the cation charge and size,etc., as is manifested in LiTaO3 (see bellow). The secondcase to be examined is LiTaO3, which does not reach thecubic perovskite structure (Pm3m) before melting. Inthis case the high temperature phase is not ferroelectricbut possesses oxygen rotations. The low temperaturephase has the same space group (R3c) as that PZT, withboth tilts and cation shifts. This is shown in Fig. 3,where ps and ηs has been calculated from experimentaldata in Refs. [8] and [9], respectively.The symmetry of the respective equations of state [3]for P − E and θ − X allow us to follow, in the LiTaO3case, the same argument than before, interchanging ηsand ps. So we have∆ηs =1− η2s1 + psηsps. (3)The ∆ηs obtained from Eq. (3) with experimental datafor ps and ηs is plotted in Fig. 3 as a dashed line. In thiscase the coupling predicted by our model is not reflectedin the available neutron data, within the experimentaluncertainty. The high values of the tilt angle (about22◦) and cation shifts found in LiTaO3 is a result of theweak interaction between Li+ cations shifts and oxygenoctahedra tilts, due to the small Li charge and size. Thiswould leads to a weak, if any, observed tilt-polarizationcoupling. On the contrary, the large charge and size ofthe cations in PZT produces a strong coupling betweenthe two order parameters, allowing a smaller maximumtilt angle of about 6◦.The case of LaAlO3 is, however, an example of non-polar rhombohedral perovskite. Fig. 4 shows the ex-perimental data of the order parameter [7] and the fitobtained by means of the effective field approach, in thewhole temperature range, using the X − θ equation ofstate [3].In conclusion, we have contrasted the three observedpossible sequences of phases with temperature of rhom-bohedral perovskites. Our data on PZT allow a moreaccurate test of the coupling between tilt and polariza-tion. The possible weak coupling in LiTaO3 has beendiscussed.ACKNOWLEDGMENTSFinancial support from CICyT (PB96-0037) andNATO (CRG-0037) is acknowledged.REFERENCES[1] M. E. Lines and A. M. Glass, Principles and Applicationsof Ferroelectrics and Related Materials (Clarendon Press,1977).[2] A. M. Glazer, Acta Cryst. B28, 3384 (1972).[3] N. Cereceda, B. Noheda, T. Iglesias, J. R. Fdez-del-Castillo and J. A. Gonzalo, Phys. Rev. B 55, 6174 (1997).[4] H. Megaw and C. Darlington, Acta Cryst. A31, 161(1975).[5] A. Amin, R. E. Newnham, L. E. Cross and D. E. Cox,J. of Sol. Stat. Chem. 27, 248 (1981).[6] A. M. Glazer, S. A. Mabud and R. Clarke, Acta Cryst.B34, 1060 (1978).[7] K. Müller, W. Berlinger and F. Waldner, Phys. Rev.Lett. 21, 814 (1968).Tilts and Ionic Shifts in Rhombohedral Perovskites – Beatriz Noheda et al -S165-[8] A. M. Glass, Phys. Rev. 172, 564 (1968).[9] S. C. Abrahams, E. Buehler, W. C. Hamilton and S. J.Laplaca, J. Phys. Chem. Solids 34, 521 (1973).[10] B. Noheda, N. Duan, N. Cereceda and J. A. Gonzalo, J.Korean Phys. Soc. 32, S256 (1998).[11] Z. Ujma, J. Handerek and G. E. Kugel, Ferroelectrics198, 77 (1997).

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   University of GroningenTilts and Ionic Shifts in Rhombohedral PerovskitesNoheda, Beatriz; Duan, Ning; Cereceda, Noé; Gonzalo, Julio A.Published in:Journal of the Korean Physical SocietyIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1998Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noheda, B., Duan, N., Cereceda, N., & Gonzalo, J. A. (1998). Tilts and Ionic Shifts in RhombohedralPerovskites. Journal of the Korean Physical Society, 32, S162-S165.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 31-10-2023Journal of the Korean Physical Society, Vol. 32, Feburary 1998, pp. S162∼S164Tilts and Ionic Shifts in Rhombohedral PerovskitesBeatriz Noheda, Ning Duan*, Noé Cereceda and Julio A. GonzaloDept. F́ısica de Materiales, C-IV. Universidad Autónoma de Madrid, 28049-Madrid, Spain* On leave from Shanghai Inst. of Ceramics, Shanghai, ChinaWe make a comparative analysis of rhombohedral perovskites (ABO3) with/without oxygenrotations and ionic shifts, within the framework of a generalised effective field approach. We analyseavailable data on LaAlO3 and LiTaO3 and new data on Zr-rich PZT, examples of three differentways of structural evolution with temperature of the rhombohedral perovskites. The couplingbetween polarization and tilt is studied in detail in PZT and a possible coupling between tilt andpolarization in LiTaO3 is discussed.I. INTRODUCTIONThe perovskite-based structures favour ferroelectricitydue to the large Lorentz contributions in the dipolar in-ternal field and to the intrinsic trend of the cations todisplace inside the oxygen octahedra On the other handthey show structural phase transitions related to the tiltof oxygen octahedra which allow a big variety of phases.The symmetry of the octahedra framework mainly deter-mines that of the crystal lattice, justifying the classifica-tion of the perovskites by their tilt systems. The cationshifts and the octahedra tilts are independent and due todifferent kind of interactions, but they can be coupled.A model to explain this coupling has been proposed re-cently by the authors.In the rhombohedral perovskites the tilt of the oxy-gen octahedra is about their triad axis and it can bedescribed as a−a−a−, according to Glazer’s notation [2],which means opposite rotation for adjacent octahedraalong each direction. Two space groups, R3c and R3c,can sustain this tilt system. The first space group has apolar symmetry allowing ferroelectricity, while the sec-ond one is non-polar keeping the cation in the inversioncentre. Rhombohedral perovskites in the space groupR3m can also be found which do not present octahedratilts and inversion centre.The sequence of space groups presented by rhombo-hedral perovskites with temperature before reaching thecubic (Pm3m) phase, can, in principle, show the differ-ent combinations of these space groups that are allowedby symmetry. Polar R3m and tilted R3c structures cantransform straight to the cubic phase by increasing tem-perature. Polar and tilted space group, R3c, contains theR3c and R3m symmetries and it has two possible waysto achieve the cubic phase, either R3c→R3c→Pm3mor R3c→R3m→ Pm3m. Examples of these two casesare LiTaO3 and PZT, respectively, although LiTaO3,as well as LiNbO3, does not show the cubic perovskitephase before melting. Examples, such as LaAlO3, ofrhombohedral perovskites showing only tilts in the lowtemperature phase, i.e. showing a R3c→Pm3m transi-tions, have been observed. Nevertheless, as far as weknow, it has not been found any rhombohedral per-ovskite having solely a R3m→Pm3m phase transition,which implies no oxygen rotations in the whole rangeof temperature. A candidate for checking this possi-bility would be PZT compositions close to the mor-photropic phase boundary. A Zr/Ti= 60/40 composi-tion of PZT presents the R3c→R3m transition as lowas 250 K, but rhombohedral compositions with higherZr content are difficult to characterise due to the highdiffusion and the mixture of phases (rhombohedral andtetragonal) in this region. It is known that the disap-pearance of the oxygen rotations in PZT through theFRL(R3c)-FRH(R3m) phase transition at TLH, producesa step in the polarization curve. It means that thereis some coupling of the octahedra tilts with the cationshifts. In this work we compare the three mentionedexamples: PZT (cation shifts+tilts→ cation shifts→ cu-bic), LiTaO3 (cation shifts+tilts→ tilts→ melting) andLaAlO3 (tilts→ cubic). We study the behaviour of thetwo independent order parameters, related to polariza-tion (ps) and rotation angle (ηs), and the coupling be-tween them.We have used available data of paramagnetic reso-nance in LaAlO3 by K. A. Müller et al. The spontaneouspolarization of LiTaO3 is from pyroelectric current databy A. M. Glass, while the structural data of LiTaO3 arefrom S. C. Abrahams et al. Experimental data of PZTare described in the next section.II. EXPERIMENTALCeramic PZT samples used were made at the Shang-hai Institute of Ceramics. The composition belongs to-S162-Tilts and Ionic Shifts in Rhombohedral Perovskites – Beatriz Noheda et al -S163-Fig. 1. Order parameters (ps and ηs) from pyroelectricdischarge and neutron data, respectively, are plotted versustemperature for rhombohedral PZT. The ps fit was made us-ing the generalised effective field approach. We label pstotal=ps+∆ps the sum of the polarization without tilt (ps) and theextra polarization (∆ps) appearing with the tilt.the rhombohedral region with a very high Zr content(Zr/Ti= 96.5/3.5). 1.5 wt% of Nb2O5 was added to im-prove the electrical properties. The constituent oxideswere ground, pressed into pellets, and fired at 1350 ◦Cfor 2 h. Cylinders of about 1 cm in height and 1 cmin diameter were sintered and hot pressed for the neu-tron diffraction runs [3]. The same procedure has beenused, more recently, for the preparation of thin platesof the same nominal composition, of 1 mm in thicknessand about 1 cm in diameter, which were used for thepyroelectric measurements.Neutron data used in this work were presented recentlyby the authors [3]. They were taken with neutron of 14.7meV energy at Brookhaven National Laboratory. The12 (311) superstructure peak, which practically dependsonly on the oxygen rotation angle, was followed withtemperature between 20 K and 360 K.Precise pyroelectric discharge data have been obtainedby means of a set-up specially designed to measure highFig. 2. (a) ∆ps vs. T, obtained subtracting ps fit frompstotal in Fig. 1. (b) ηs vs. T, from neutron data and predictedby Eq. (2). The arrow shows the place where the antiferro-electric phase could have taken place.Fig. 3. Order parameters (ps and ηs) from pyroelectriccurrent [8] and neutron data [9], respectively, are plotted ver-sus temperature for LiTaO3. The ps fit was made by thegeneralised effective field approach. The dashed line showthe prediction of the perfect coupling model (Eq. (3)).conductivity samples with a slow rate of temperaturechange. A detailed description of the system and itsworking scheme is given in a parallel work. Data shownin this work are taken during heating with a temperaturerate of 20 K/h between 160 K and 600 K.III. RESULTS AND DISCUSSIONVery accurate polarization versus temperature mea-surements has been obtained in PZT. Normalised spon-taneous polarization, ps= Ps/Pso, and tilt parameter,ηs=es/eso, have been plotted in Fig. 1, where Pso andeso are, respectively, the spontaneous polarization andthe tilt parameter at T=0 K. The structural parameter e,is related to the tilt angle, ω, by tanω = 4√3e. The stepin polarization at the FRL-FRH phase transition, TLH,can be clearly observed in the figure coinciding with thevanishing of the tilt.The generalised effective field approach has been usedFig. 4. Order parameter of LaAlO3 vs. temperature. Theexperimental data [7] are fitted with the generalised effectivefield approach.-S164- Journal of the Korean Physical Society, Vol. 32, Feburary 1998to analyse the data. The conjugated effective electricfield and staggered stress can be expressed as Eeff=E+βP+γP3+ δP5+... and |Xeff | = X + βtθ + γtθ3 +δtθ5 + ..., respectively, being E the external electric field,P the polarization, X the external stress and θ the sta-tistically resultant tilt. This theory leads to the followingequation of state with E= 0 (p= ps)TTCtanh−1 psps= 1 + gp2s + hp4s + ... (1)where the parameters g, h,... describe the relative im-portance (to the dipolar term, βP) of the higher orderterms (γP3, δP5..), and are proportional to the succes-sive [3] even powers of Pso(Pso2, Pso4...). A fit of theexperimental data of ps vs. T by means of Eq. (1) isshown in Fig. 1.The subtraction of the ps fit to the experimental datais shown in Fig. 2(a). It gives us the behaviour withtemperature of the normalised extra polarization, ∆ps,appearing due to the tilt. ∆ps reaches its maximum valueat TLH and should disappear at T= 0 K.A model to explain the tilt-polarization coupling wasintroduced in Ref. [3]. According to it, the tilt orderparameter can be expressed in terms of ps and ∆ps asηs =∆ps1− ps(ps + ∆ps). (2)ηs is calculated from Eq. (2), being ps and ∆ps thoseof Figs. 1 and 2(a), respectively. ηs is represented inFig. 2(b) and compared to the experimental data. Itcan be seen the clear agreement between both structuraldata and model, in spite to the model simplicity. Thearrow in the figure shows a peculiar behaviour at 220K, which was attributed to the experimental error inprevious work [3]. Nevertheless this small anomaly hasbeen reproduced with the pyroelectric data through ourcoupling model. A new transition to an antiferroelectric(AFo) phase, at about the same temperature, has beenrecently proposed, but we have found here ferroelectricbehaviour until 160 K. Our composition has similar Nband Ti contents, and the phase diagram is still unknownin this region [10]. The tilt order parameter predicted byour model from the observed ∆ps is substantially smallerthan the experimental one. This might be due to thefact that in a ceramic material the octahedra tilt maynot be so efficiently coupled to the cation shifts as ina single crystal, as assumed for Eq. (2). On the otherhand, Eq. (2) assumes perfect coupling in the sense thatall the oxygen movement is reflected in the polarizationbehaviour. The validity of this assumption may dependin each particular case, on the cation charge and size,etc., as is manifested in LiTaO3 (see bellow). The secondcase to be examined is LiTaO3, which does not reach thecubic perovskite structure (Pm3m) before melting. Inthis case the high temperature phase is not ferroelectricbut possesses oxygen rotations. The low temperaturephase has the same space group (R3c) as that PZT, withboth tilts and cation shifts. This is shown in Fig. 3,where ps and ηs has been calculated from experimentaldata in Refs. [8] and [9], respectively.The symmetry of the respective equations of state [3]for P − E and θ − X allow us to follow, in the LiTaO3case, the same argument than before, interchanging ηsand ps. So we have∆ηs =1− η2s1 + psηsps. (3)The ∆ηs obtained from Eq. (3) with experimental datafor ps and ηs is plotted in Fig. 3 as a dashed line. In thiscase the coupling predicted by our model is not reflectedin the available neutron data, within the experimentaluncertainty. The high values of the tilt angle (about22◦) and cation shifts found in LiTaO3 is a result of theweak interaction between Li+ cations shifts and oxygenoctahedra tilts, due to the small Li charge and size. Thiswould leads to a weak, if any, observed tilt-polarizationcoupling. On the contrary, the large charge and size ofthe cations in PZT produces a strong coupling betweenthe two order parameters, allowing a smaller maximumtilt angle of about 6◦.The case of LaAlO3 is, however, an example of non-polar rhombohedral perovskite. Fig. 4 shows the ex-perimental data of the order parameter [7] and the fitobtained by means of the effective field approach, in thewhole temperature range, using the X − θ equation ofstate [3].In conclusion, we have contrasted the three observedpossible sequences of phases with temperature of rhom-bohedral perovskites. Our data on PZT allow a moreaccurate test of the coupling between tilt and polariza-tion. The possible weak coupling in LiTaO3 has beendiscussed.ACKNOWLEDGMENTSFinancial support from CICyT (PB96-0037) andNATO (CRG-0037) is acknowledged.REFERENCES[1] M. E. Lines and A. M. Glass, Principles and Applicationsof Ferroelectrics and Related Materials (Clarendon Press,1977).[2] A. M. Glazer, Acta Cryst. B28, 3384 (1972).[3] N. Cereceda, B. Noheda, T. Iglesias, J. R. Fdez-del-Castillo and J. A. Gonzalo, Phys. Rev. B 55, 6174 (1997).[4] H. Megaw and C. Darlington, Acta Cryst. A31, 161(1975).[5] A. Amin, R. E. Newnham, L. E. Cross and D. E. Cox,J. of Sol. Stat. Chem. 27, 248 (1981).[6] A. M. Glazer, S. A. Mabud and R. Clarke, Acta Cryst.B34, 1060 (1978).[7] K. Müller, W. Berlinger and F. Waldner, Phys. Rev.Lett. 21, 814 (1968).Tilts and Ionic Shifts in Rhombohedral Perovskites – Beatriz Noheda et al -S165-[8] A. M. Glass, Phys. Rev. 172, 564 (1968).[9] S. C. Abrahams, E. Buehler, W. C. Hamilton and S. J.Laplaca, J. Phys. Chem. Solids 34, 521 (1973).[10] B. Noheda, N. Duan, N. Cereceda and J. A. Gonzalo, J.Korean Phys. Soc. 32, S256 (1998).[11] Z. Ujma, J. Handerek and G. E. Kugel, Ferroelectrics198, 77 (1997).

University of Groningen Research Database

  
 University of Groningen
Tilts and Ionic Shifts in Rhombohedral Perovskites
Noheda, Beatriz; Duan, Ning; Cereceda, Noé; Gonzalo, Julio A.
Published in:
Journal of the Korean Physical Society
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date:
1998
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Noheda, B., Duan, N., Cereceda, N., & Gonzalo, J. A. (1998). Tilts and Ionic Shifts in Rhombohedral
Perovskites. Journal of the Korean Physical Society, 32, S162-S165.
Copyright
Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the
author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).
Take-down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the
number of authors shown on this cover page is limited to 10 maximum.
Download date: 11-02-2018
Journal of the Korean Physical Society, Vol. 32, Feburary 1998, pp. S162∼S164
Tilts and Ionic Shifts in Rhombohedral Perovskites
Beatriz Noheda, Ning Duan*, Noe´ Cereceda and Julio A. Gonzalo
Dept. F´ısica de Materiales, C-IV. Universidad Auto´noma de Madrid, 28049-Madrid, Spain
* On leave from Shanghai Inst. of Ceramics, Shanghai, China
We make a comparative analysis of rhombohedral perovskites (ABO3) with/without oxygen
rotations and ionic shifts, within the framework of a generalised effective field approach. We analyse
available data on LaAlO3 and LiTaO3 and new data on Zr-rich PZT, examples of three different
ways of structural evolution with temperature of the rhombohedral perovskites. The coupling
between polarization and tilt is studied in detail in PZT and a possible coupling between tilt and
polarization in LiTaO3 is discussed.
I. INTRODUCTION
The perovskite-based structures favour ferroelectricity
due to the large Lorentz contributions in the dipolar in-
ternal field and to the intrinsic trend of the cations to
displace inside the oxygen octahedra On the other hand
they show structural phase transitions related to the tilt
of oxygen octahedra which allow a big variety of phases.
The symmetry of the octahedra framework mainly deter-
mines that of the crystal lattice, justifying the classifica-
tion of the perovskites by their tilt systems. The cation
shifts and the octahedra tilts are independent and due to
different kind of interactions, but they can be coupled.
A model to explain this coupling has been proposed re-
cently by the authors.
In the rhombohedral perovskites the tilt of the oxy-
gen octahedra is about their triad axis and it can be
described as a−a−a−, according to Glazer’s notation [2],
which means opposite rotation for adjacent octahedra
along each direction. Two space groups, R3c and R3c,
can sustain this tilt system. The first space group has a
polar symmetry allowing ferroelectricity, while the sec-
ond one is non-polar keeping the cation in the inversion
centre. Rhombohedral perovskites in the space group
R3m can also be found which do not present octahedra
tilts and inversion centre.
The sequence of space groups presented by rhombo-
hedral perovskites with temperature before reaching the
cubic (Pm3m) phase, can, in principle, show the differ-
ent combinations of these space groups that are allowed
by symmetry. Polar R3m and tilted R3c structures can
transform straight to the cubic phase by increasing tem-
perature. Polar and tilted space group, R3c, contains the
R3c and R3m symmetries and it has two possible ways
to achieve the cubic phase, either R3c→R3c→Pm3m
or R3c→R3m→ Pm3m. Examples of these two cases
are LiTaO3 and PZT, respectively, although LiTaO3,
as well as LiNbO3, does not show the cubic perovskite
phase before melting. Examples, such as LaAlO3, of
rhombohedral perovskites showing only tilts in the low
temperature phase, i.e. showing a R3c→Pm3m transi-
tions, have been observed. Nevertheless, as far as we
know, it has not been found any rhombohedral per-
ovskite having solely a R3m→Pm3m phase transition,
which implies no oxygen rotations in the whole range
of temperature. A candidate for checking this possi-
bility would be PZT compositions close to the mor-
photropic phase boundary. A Zr/Ti= 60/40 composi-
tion of PZT presents the R3c→R3m transition as low
as 250 K, but rhombohedral compositions with higher
Zr content are difficult to characterise due to the high
diffusion and the mixture of phases (rhombohedral and
tetragonal) in this region. It is known that the disap-
pearance of the oxygen rotations in PZT through the
FRL(R3c)-FRH(R3m) phase transition at TLH, produces
a step in the polarization curve. It means that there
is some coupling of the octahedra tilts with the cation
shifts. In this work we compare the three mentioned
examples: PZT (cation shifts+tilts→ cation shifts→ cu-
bic), LiTaO3 (cation shifts+tilts→ tilts→ melting) and
LaAlO3 (tilts→ cubic). We study the behaviour of the
two independent order parameters, related to polariza-
tion (ps) and rotation angle (ηs), and the coupling be-
tween them.
We have used available data of paramagnetic reso-
nance in LaAlO3 by K. A. Mu¨ller et al. The spontaneous
polarization of LiTaO3 is from pyroelectric current data
by A. M. Glass, while the structural data of LiTaO3 are
from S. C. Abrahams et al. Experimental data of PZT
are described in the next section.
II. EXPERIMENTAL
Ceramic PZT samples used were made at the Shang-
hai Institute of Ceramics. The composition belongs to
-S162-
Tilts and Ionic Shifts in Rhombohedral Perovskites – Beatriz Noheda et al -S163-
Fig. 1. Order parameters (ps and ηs) from pyroelectric
discharge and neutron data, respectively, are plotted versus
temperature for rhombohedral PZT. The ps fit was made us-
ing the generalised effective field approach. We label ps
total=
ps+∆ps the sum of the polarization without tilt (ps) and the
extra polarization (∆ps) appearing with the tilt.
the rhombohedral region with a very high Zr content
(Zr/Ti= 96.5/3.5). 1.5 wt% of Nb2O5 was added to im-
prove the electrical properties. The constituent oxides
were ground, pressed into pellets, and fired at 1350 ◦C
for 2 h. Cylinders of about 1 cm in height and 1 cm
in diameter were sintered and hot pressed for the neu-
tron diffraction runs [3]. The same procedure has been
used, more recently, for the preparation of thin plates
of the same nominal composition, of 1 mm in thickness
and about 1 cm in diameter, which were used for the
pyroelectric measurements.
Neutron data used in this work were presented recently
by the authors [3]. They were taken with neutron of 14.7
meV energy at Brookhaven National Laboratory. The
1
2 (311) superstructure peak, which practically depends
only on the oxygen rotation angle, was followed with
temperature between 20 K and 360 K.
Precise pyroelectric discharge data have been obtained
by means of a set-up specially designed to measure high
Fig. 2. (a) ∆ps vs. T, obtained subtracting ps fit from
ps
total in Fig. 1. (b) ηs vs. T, from neutron data and predicted
by Eq. (2). The arrow shows the place where the antiferro-
electric phase could have taken place.
Fig. 3. Order parameters (ps and ηs) from pyroelectric
current [8] and neutron data [9], respectively, are plotted ver-
sus temperature for LiTaO3. The ps fit was made by the
generalised effective field approach. The dashed line show
the prediction of the perfect coupling model (Eq. (3)).
conductivity samples with a slow rate of temperature
change. A detailed description of the system and its
working scheme is given in a parallel work. Data shown
in this work are taken during heating with a temperature
rate of 20 K/h between 160 K and 600 K.
III. RESULTS AND DISCUSSION
Very accurate polarization versus temperature mea-
surements has been obtained in PZT. Normalised spon-
taneous polarization, ps= Ps/Pso, and tilt parameter,
ηs=es/eso, have been plotted in Fig. 1, where Pso and
eso are, respectively, the spontaneous polarization and
the tilt parameter at T=0 K. The structural parameter e,
is related to the tilt angle, ω, by tanω = 4
√
3e. The step
in polarization at the FRL-FRH phase transition, TLH,
can be clearly observed in the figure coinciding with the
vanishing of the tilt.
The generalised effective field approach has been used
Fig. 4. Order parameter of LaAlO3 vs. temperature. The
experimental data [7] are fitted with the generalised effective
field approach.
-S164- Journal of the Korean Physical Society, Vol. 32, Feburary 1998
to analyse the data. The conjugated effective electric
field and staggered stress can be expressed as Eeff=
E+βP+γP3+ δP5+... and |Xeff | = X + βtθ + γtθ3 +
δtθ
5 + ..., respectively, being E the external electric field,
P the polarization, X the external stress and θ the sta-
tistically resultant tilt. This theory leads to the following
equation of state with E= 0 (p= ps)
T
TC
tanh−1 ps
ps
= 1 + gp2s + hp
4
s + ... (1)
where the parameters g, h,... describe the relative im-
portance (to the dipolar term, βP) of the higher order
terms (γP3, δP5..), and are proportional to the succes-
sive [3] even powers of Pso(Pso2, Pso4...). A fit of the
experimental data of ps vs. T by means of Eq. (1) is
shown in Fig. 1.
The subtraction of the ps fit to the experimental data
is shown in Fig. 2(a). It gives us the behaviour with
temperature of the normalised extra polarization, ∆ps,
appearing due to the tilt. ∆ps reaches its maximum value
at TLH and should disappear at T= 0 K.
A model to explain the tilt-polarization coupling was
introduced in Ref. [3]. According to it, the tilt order
parameter can be expressed in terms of ps and ∆ps as
ηs =
∆ps
1− ps(ps + ∆ps) . (2)
ηs is calculated from Eq. (2), being ps and ∆ps those
of Figs. 1 and 2(a), respectively. ηs is represented in
Fig. 2(b) and compared to the experimental data. It
can be seen the clear agreement between both structural
data and model, in spite to the model simplicity. The
arrow in the figure shows a peculiar behaviour at 220
K, which was attributed to the experimental error in
previous work [3]. Nevertheless this small anomaly has
been reproduced with the pyroelectric data through our
coupling model. A new transition to an antiferroelectric
(AFo) phase, at about the same temperature, has been
recently proposed, but we have found here ferroelectric
behaviour until 160 K. Our composition has similar Nb
and Ti contents, and the phase diagram is still unknown
in this region [10]. The tilt order parameter predicted by
our model from the observed ∆ps is substantially smaller
than the experimental one. This might be due to the
fact that in a ceramic material the octahedra tilt may
not be so efficiently coupled to the cation shifts as in
a single crystal, as assumed for Eq. (2). On the other
hand, Eq. (2) assumes perfect coupling in the sense that
all the oxygen movement is reflected in the polarization
behaviour. The validity of this assumption may depend
in each particular case, on the cation charge and size,
etc., as is manifested in LiTaO3 (see bellow). The second
case to be examined is LiTaO3, which does not reach the
cubic perovskite structure (Pm3m) before melting. In
this case the high temperature phase is not ferroelectric
but possesses oxygen rotations. The low temperature
phase has the same space group (R3c) as that PZT, with
both tilts and cation shifts. This is shown in Fig. 3,
where ps and ηs has been calculated from experimental
data in Refs. [8] and [9], respectively.
The symmetry of the respective equations of state [3]
for P − E and θ − X allow us to follow, in the LiTaO3
case, the same argument than before, interchanging ηs
and ps. So we have
∆ηs =
1− η2s
1 + psηs
ps. (3)
The ∆ηs obtained from Eq. (3) with experimental data
for ps and ηs is plotted in Fig. 3 as a dashed line. In this
case the coupling predicted by our model is not reflected
in the available neutron data, within the experimental
uncertainty. The high values of the tilt angle (about
22◦) and cation shifts found in LiTaO3 is a result of the
weak interaction between Li+ cations shifts and oxygen
octahedra tilts, due to the small Li charge and size. This
would leads to a weak, if any, observed tilt-polarization
coupling. On the contrary, the large charge and size of
the cations in PZT produces a strong coupling between
the two order parameters, allowing a smaller maximum
tilt angle of about 6◦.
The case of LaAlO3 is, however, an example of non-
polar rhombohedral perovskite. Fig. 4 shows the ex-
perimental data of the order parameter [7] and the fit
obtained by means of the effective field approach, in the
whole temperature range, using the X − θ equation of
state [3].
In conclusion, we have contrasted the three observed
possible sequences of phases with temperature of rhom-
bohedral perovskites. Our data on PZT allow a more
accurate test of the coupling between tilt and polariza-
tion. The possible weak coupling in LiTaO3 has been
discussed.
ACKNOWLEDGMENTS
Financial support from CICyT (PB96-0037) and
NATO (CRG-0037) is acknowledged.
REFERENCES
[1] M. E. Lines and A. M. Glass, Principles and Applications
of Ferroelectrics and Related Materials (Clarendon Press,
1977).
[2] A. M. Glazer, Acta Cryst. B28, 3384 (1972).
[3] N. Cereceda, B. Noheda, T. Iglesias, J. R. Fdez-del-
Castillo and J. A. Gonzalo, Phys. Rev. B 55, 6174 (1997).
[4] H. Megaw and C. Darlington, Acta Cryst. A31, 161
(1975).
[5] A. Amin, R. E. Newnham, L. E. Cross and D. E. Cox,
J. of Sol. Stat. Chem. 27, 248 (1981).
[6] A. M. Glazer, S. A. Mabud and R. Clarke, Acta Cryst.
B34, 1060 (1978).
[7] K. Mu¨ller, W. Berlinger and F. Waldner, Phys. Rev.
Lett. 21, 814 (1968).
Tilts and Ionic Shifts in Rhombohedral Perovskites – Beatriz Noheda et al -S165-
[8] A. M. Glass, Phys. Rev. 172, 564 (1968).
[9] S. C. Abrahams, E. Buehler, W. C. Hamilton and S. J.
Laplaca, J. Phys. Chem. Solids 34, 521 (1973).
[10] B. Noheda, N. Duan, N. Cereceda and J. A. Gonzalo, J.
Korean Phys. Soc. 32, S256 (1998).
[11] Z. Ujma, J. Handerek and G. E. Kugel, Ferroelectrics
198, 77 (1997).


   University of GroningenTilts and Ionic Shifts in Rhombohedral PerovskitesNoheda, Beatriz; Duan, Ning; Cereceda, Noé; Gonzalo, Julio A.Published in:Journal of the Korean Physical SocietyIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1998Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noheda, B., Duan, N., Cereceda, N., & Gonzalo, J. A. (1998). Tilts and Ionic Shifts in RhombohedralPerovskites. Journal of the Korean Physical Society, 32, S162-S165.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 21-06-2022Journal of the Korean Physical Society, Vol. 32, Feburary 1998, pp. S162∼S164Tilts and Ionic Shifts in Rhombohedral PerovskitesBeatriz Noheda, Ning Duan*, Noé Cereceda and Julio A. GonzaloDept. F́ısica de Materiales, C-IV. Universidad Autónoma de Madrid, 28049-Madrid, Spain* On leave from Shanghai Inst. of Ceramics, Shanghai, ChinaWe make a comparative analysis of rhombohedral perovskites (ABO3) with/without oxygenrotations and ionic shifts, within the framework of a generalised effective field approach. We analyseavailable data on LaAlO3 and LiTaO3 and new data on Zr-rich PZT, examples of three differentways of structural evolution with temperature of the rhombohedral perovskites. The couplingbetween polarization and tilt is studied in detail in PZT and a possible coupling between tilt andpolarization in LiTaO3 is discussed.I. INTRODUCTIONThe perovskite-based structures favour ferroelectricitydue to the large Lorentz contributions in the dipolar in-ternal field and to the intrinsic trend of the cations todisplace inside the oxygen octahedra On the other handthey show structural phase transitions related to the tiltof oxygen octahedra which allow a big variety of phases.The symmetry of the octahedra framework mainly deter-mines that of the crystal lattice, justifying the classifica-tion of the perovskites by their tilt systems. The cationshifts and the octahedra tilts are independent and due todifferent kind of interactions, but they can be coupled.A model to explain this coupling has been proposed re-cently by the authors.In the rhombohedral perovskites the tilt of the oxy-gen octahedra is about their triad axis and it can bedescribed as a−a−a−, according to Glazer’s notation [2],which means opposite rotation for adjacent octahedraalong each direction. Two space groups, R3c and R3c,can sustain this tilt system. The first space group has apolar symmetry allowing ferroelectricity, while the sec-ond one is non-polar keeping the cation in the inversioncentre. Rhombohedral perovskites in the space groupR3m can also be found which do not present octahedratilts and inversion centre.The sequence of space groups presented by rhombo-hedral perovskites with temperature before reaching thecubic (Pm3m) phase, can, in principle, show the differ-ent combinations of these space groups that are allowedby symmetry. Polar R3m and tilted R3c structures cantransform straight to the cubic phase by increasing tem-perature. Polar and tilted space group, R3c, contains theR3c and R3m symmetries and it has two possible waysto achieve the cubic phase, either R3c→R3c→Pm3mor R3c→R3m→ Pm3m. Examples of these two casesare LiTaO3 and PZT, respectively, although LiTaO3,as well as LiNbO3, does not show the cubic perovskitephase before melting. Examples, such as LaAlO3, ofrhombohedral perovskites showing only tilts in the lowtemperature phase, i.e. showing a R3c→Pm3m transi-tions, have been observed. Nevertheless, as far as weknow, it has not been found any rhombohedral per-ovskite having solely a R3m→Pm3m phase transition,which implies no oxygen rotations in the whole rangeof temperature. A candidate for checking this possi-bility would be PZT compositions close to the mor-photropic phase boundary. A Zr/Ti= 60/40 composi-tion of PZT presents the R3c→R3m transition as lowas 250 K, but rhombohedral compositions with higherZr content are difficult to characterise due to the highdiffusion and the mixture of phases (rhombohedral andtetragonal) in this region. It is known that the disap-pearance of the oxygen rotations in PZT through theFRL(R3c)-FRH(R3m) phase transition at TLH, producesa step in the polarization curve. It means that thereis some coupling of the octahedra tilts with the cationshifts. In this work we compare the three mentionedexamples: PZT (cation shifts+tilts→ cation shifts→ cu-bic), LiTaO3 (cation shifts+tilts→ tilts→ melting) andLaAlO3 (tilts→ cubic). We study the behaviour of thetwo independent order parameters, related to polariza-tion (ps) and rotation angle (ηs), and the coupling be-tween them.We have used available data of paramagnetic reso-nance in LaAlO3 by K. A. Müller et al. The spontaneouspolarization of LiTaO3 is from pyroelectric current databy A. M. Glass, while the structural data of LiTaO3 arefrom S. C. Abrahams et al. Experimental data of PZTare described in the next section.II. EXPERIMENTALCeramic PZT samples used were made at the Shang-hai Institute of Ceramics. The composition belongs to-S162-Tilts and Ionic Shifts in Rhombohedral Perovskites – Beatriz Noheda et al -S163-Fig. 1. Order parameters (ps and ηs) from pyroelectricdischarge and neutron data, respectively, are plotted versustemperature for rhombohedral PZT. The ps fit was made us-ing the generalised effective field approach. We label pstotal=ps+∆ps the sum of the polarization without tilt (ps) and theextra polarization (∆ps) appearing with the tilt.the rhombohedral region with a very high Zr content(Zr/Ti= 96.5/3.5). 1.5 wt% of Nb2O5 was added to im-prove the electrical properties. The constituent oxideswere ground, pressed into pellets, and fired at 1350 ◦Cfor 2 h. Cylinders of about 1 cm in height and 1 cmin diameter were sintered and hot pressed for the neu-tron diffraction runs [3]. The same procedure has beenused, more recently, for the preparation of thin platesof the same nominal composition, of 1 mm in thicknessand about 1 cm in diameter, which were used for thepyroelectric measurements.Neutron data used in this work were presented recentlyby the authors [3]. They were taken with neutron of 14.7meV energy at Brookhaven National Laboratory. The12 (311) superstructure peak, which practically dependsonly on the oxygen rotation angle, was followed withtemperature between 20 K and 360 K.Precise pyroelectric discharge data have been obtainedby means of a set-up specially designed to measure highFig. 2. (a) ∆ps vs. T, obtained subtracting ps fit frompstotal in Fig. 1. (b) ηs vs. T, from neutron data and predictedby Eq. (2). The arrow shows the place where the antiferro-electric phase could have taken place.Fig. 3. Order parameters (ps and ηs) from pyroelectriccurrent [8] and neutron data [9], respectively, are plotted ver-sus temperature for LiTaO3. The ps fit was made by thegeneralised effective field approach. The dashed line showthe prediction of the perfect coupling model (Eq. (3)).conductivity samples with a slow rate of temperaturechange. A detailed description of the system and itsworking scheme is given in a parallel work. Data shownin this work are taken during heating with a temperaturerate of 20 K/h between 160 K and 600 K.III. RESULTS AND DISCUSSIONVery accurate polarization versus temperature mea-surements has been obtained in PZT. Normalised spon-taneous polarization, ps= Ps/Pso, and tilt parameter,ηs=es/eso, have been plotted in Fig. 1, where Pso andeso are, respectively, the spontaneous polarization andthe tilt parameter at T=0 K. The structural parameter e,is related to the tilt angle, ω, by tanω = 4√3e. The stepin polarization at the FRL-FRH phase transition, TLH,can be clearly observed in the figure coinciding with thevanishing of the tilt.The generalised effective field approach has been usedFig. 4. Order parameter of LaAlO3 vs. temperature. Theexperimental data [7] are fitted with the generalised effectivefield approach.-S164- Journal of the Korean Physical Society, Vol. 32, Feburary 1998to analyse the data. The conjugated effective electricfield and staggered stress can be expressed as Eeff=E+βP+γP3+ δP5+... and |Xeff | = X + βtθ + γtθ3 +δtθ5 + ..., respectively, being E the external electric field,P the polarization, X the external stress and θ the sta-tistically resultant tilt. This theory leads to the followingequation of state with E= 0 (p= ps)TTCtanh−1 psps= 1 + gp2s + hp4s + ... (1)where the parameters g, h,... describe the relative im-portance (to the dipolar term, βP) of the higher orderterms (γP3, δP5..), and are proportional to the succes-sive [3] even powers of Pso(Pso2, Pso4...). A fit of theexperimental data of ps vs. T by means of Eq. (1) isshown in Fig. 1.The subtraction of the ps fit to the experimental datais shown in Fig. 2(a). It gives us the behaviour withtemperature of the normalised extra polarization, ∆ps,appearing due to the tilt. ∆ps reaches its maximum valueat TLH and should disappear at T= 0 K.A model to explain the tilt-polarization coupling wasintroduced in Ref. [3]. According to it, the tilt orderparameter can be expressed in terms of ps and ∆ps asηs =∆ps1− ps(ps + ∆ps). (2)ηs is calculated from Eq. (2), being ps and ∆ps thoseof Figs. 1 and 2(a), respectively. ηs is represented inFig. 2(b) and compared to the experimental data. Itcan be seen the clear agreement between both structuraldata and model, in spite to the model simplicity. Thearrow in the figure shows a peculiar behaviour at 220K, which was attributed to the experimental error inprevious work [3]. Nevertheless this small anomaly hasbeen reproduced with the pyroelectric data through ourcoupling model. A new transition to an antiferroelectric(AFo) phase, at about the same temperature, has beenrecently proposed, but we have found here ferroelectricbehaviour until 160 K. Our composition has similar Nband Ti contents, and the phase diagram is still unknownin this region [10]. The tilt order parameter predicted byour model from the observed ∆ps is substantially smallerthan the experimental one. This might be due to thefact that in a ceramic material the octahedra tilt maynot be so efficiently coupled to the cation shifts as ina single crystal, as assumed for Eq. (2). On the otherhand, Eq. (2) assumes perfect coupling in the sense thatall the oxygen movement is reflected in the polarizationbehaviour. The validity of this assumption may dependin each particular case, on the cation charge and size,etc., as is manifested in LiTaO3 (see bellow). The secondcase to be examined is LiTaO3, which does not reach thecubic perovskite structure (Pm3m) before melting. Inthis case the high temperature phase is not ferroelectricbut possesses oxygen rotations. The low temperaturephase has the same space group (R3c) as that PZT, withboth tilts and cation shifts. This is shown in Fig. 3,where ps and ηs has been calculated from experimentaldata in Refs. [8] and [9], respectively.The symmetry of the respective equations of state [3]for P − E and θ − X allow us to follow, in the LiTaO3case, the same argument than before, interchanging ηsand ps. So we have∆ηs =1− η2s1 + psηsps. (3)The ∆ηs obtained from Eq. (3) with experimental datafor ps and ηs is plotted in Fig. 3 as a dashed line. In thiscase the coupling predicted by our model is not reflectedin the available neutron data, within the experimentaluncertainty. The high values of the tilt angle (about22◦) and cation shifts found in LiTaO3 is a result of theweak interaction between Li+ cations shifts and oxygenoctahedra tilts, due to the small Li charge and size. Thiswould leads to a weak, if any, observed tilt-polarizationcoupling. On the contrary, the large charge and size ofthe cations in PZT produces a strong coupling betweenthe two order parameters, allowing a smaller maximumtilt angle of about 6◦.The case of LaAlO3 is, however, an example of non-polar rhombohedral perovskite. Fig. 4 shows the ex-perimental data of the order parameter [7] and the fitobtained by means of the effective field approach, in thewhole temperature range, using the X − θ equation ofstate [3].In conclusion, we have contrasted the three observedpossible sequences of phases with temperature of rhom-bohedral perovskites. Our data on PZT allow a moreaccurate test of the coupling between tilt and polariza-tion. The possible weak coupling in LiTaO3 has beendiscussed.ACKNOWLEDGMENTSFinancial support from CICyT (PB96-0037) andNATO (CRG-0037) is acknowledged.REFERENCES[1] M. E. Lines and A. M. Glass, Principles and Applicationsof Ferroelectrics and Related Materials (Clarendon Press,1977).[2] A. M. Glazer, Acta Cryst. B28, 3384 (1972).[3] N. Cereceda, B. Noheda, T. Iglesias, J. R. Fdez-del-Castillo and J. A. Gonzalo, Phys. Rev. B 55, 6174 (1997).[4] H. Megaw and C. Darlington, Acta Cryst. A31, 161(1975).[5] A. Amin, R. E. Newnham, L. E. Cross and D. E. Cox,J. of Sol. Stat. Chem. 27, 248 (1981).[6] A. M. Glazer, S. A. Mabud and R. Clarke, Acta Cryst.B34, 1060 (1978).[7] K. Müller, W. Berlinger and F. Waldner, Phys. Rev.Lett. 21, 814 (1968).Tilts and Ionic Shifts in Rhombohedral Perovskites – Beatriz Noheda et al -S165-[8] A. M. Glass, Phys. Rev. 172, 564 (1968).[9] S. C. Abrahams, E. Buehler, W. C. Hamilton and S. J.Laplaca, J. Phys. Chem. Solids 34, 521 (1973).[10] B. Noheda, N. Duan, N. Cereceda and J. A. Gonzalo, J.Korean Phys. Soc. 32, S256 (1998).[11] Z. Ujma, J. Handerek and G. E. Kugel, Ferroelectrics198, 77 (1997).

Proceedings - University of Groningen

   University of GroningenTilts and Ionic Shifts in Rhombohedral PerovskitesNoheda, Beatriz; Duan, Ning; Cereceda, Noé; Gonzalo, Julio A.Published in:Journal of the Korean Physical SocietyIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1998Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noheda, B., Duan, N., Cereceda, N., & Gonzalo, J. A. (1998). Tilts and Ionic Shifts in RhombohedralPerovskites. Journal of the Korean Physical Society, 32, S162-S165.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 03-06-2022Journal of the Korean Physical Society, Vol. 32, Feburary 1998, pp. S162∼S164Tilts and Ionic Shifts in Rhombohedral PerovskitesBeatriz Noheda, Ning Duan*, Noé Cereceda and Julio A. GonzaloDept. F́ısica de Materiales, C-IV. Universidad Autónoma de Madrid, 28049-Madrid, Spain* On leave from Shanghai Inst. of Ceramics, Shanghai, ChinaWe make a comparative analysis of rhombohedral perovskites (ABO3) with/without oxygenrotations and ionic shifts, within the framework of a generalised effective field approach. We analyseavailable data on LaAlO3 and LiTaO3 and new data on Zr-rich PZT, examples of three differentways of structural evolution with temperature of the rhombohedral perovskites. The couplingbetween polarization and tilt is studied in detail in PZT and a possible coupling between tilt andpolarization in LiTaO3 is discussed.I. INTRODUCTIONThe perovskite-based structures favour ferroelectricitydue to the large Lorentz contributions in the dipolar in-ternal field and to the intrinsic trend of the cations todisplace inside the oxygen octahedra On the other handthey show structural phase transitions related to the tiltof oxygen octahedra which allow a big variety of phases.The symmetry of the octahedra framework mainly deter-mines that of the crystal lattice, justifying the classifica-tion of the perovskites by their tilt systems. The cationshifts and the octahedra tilts are independent and due todifferent kind of interactions, but they can be coupled.A model to explain this coupling has been proposed re-cently by the authors.In the rhombohedral perovskites the tilt of the oxy-gen octahedra is about their triad axis and it can bedescribed as a−a−a−, according to Glazer’s notation [2],which means opposite rotation for adjacent octahedraalong each direction. Two space groups, R3c and R3c,can sustain this tilt system. The first space group has apolar symmetry allowing ferroelectricity, while the sec-ond one is non-polar keeping the cation in the inversioncentre. Rhombohedral perovskites in the space groupR3m can also be found which do not present octahedratilts and inversion centre.The sequence of space groups presented by rhombo-hedral perovskites with temperature before reaching thecubic (Pm3m) phase, can, in principle, show the differ-ent combinations of these space groups that are allowedby symmetry. Polar R3m and tilted R3c structures cantransform straight to the cubic phase by increasing tem-perature. Polar and tilted space group, R3c, contains theR3c and R3m symmetries and it has two possible waysto achieve the cubic phase, either R3c→R3c→Pm3mor R3c→R3m→ Pm3m. Examples of these two casesare LiTaO3 and PZT, respectively, although LiTaO3,as well as LiNbO3, does not show the cubic perovskitephase before melting. Examples, such as LaAlO3, ofrhombohedral perovskites showing only tilts in the lowtemperature phase, i.e. showing a R3c→Pm3m transi-tions, have been observed. Nevertheless, as far as weknow, it has not been found any rhombohedral per-ovskite having solely a R3m→Pm3m phase transition,which implies no oxygen rotations in the whole rangeof temperature. A candidate for checking this possi-bility would be PZT compositions close to the mor-photropic phase boundary. A Zr/Ti= 60/40 composi-tion of PZT presents the R3c→R3m transition as lowas 250 K, but rhombohedral compositions with higherZr content are difficult to characterise due to the highdiffusion and the mixture of phases (rhombohedral andtetragonal) in this region. It is known that the disap-pearance of the oxygen rotations in PZT through theFRL(R3c)-FRH(R3m) phase transition at TLH, producesa step in the polarization curve. It means that thereis some coupling of the octahedra tilts with the cationshifts. In this work we compare the three mentionedexamples: PZT (cation shifts+tilts→ cation shifts→ cu-bic), LiTaO3 (cation shifts+tilts→ tilts→ melting) andLaAlO3 (tilts→ cubic). We study the behaviour of thetwo independent order parameters, related to polariza-tion (ps) and rotation angle (ηs), and the coupling be-tween them.We have used available data of paramagnetic reso-nance in LaAlO3 by K. A. Müller et al. The spontaneouspolarization of LiTaO3 is from pyroelectric current databy A. M. Glass, while the structural data of LiTaO3 arefrom S. C. Abrahams et al. Experimental data of PZTare described in the next section.II. EXPERIMENTALCeramic PZT samples used were made at the Shang-hai Institute of Ceramics. The composition belongs to-S162-Tilts and Ionic Shifts in Rhombohedral Perovskites – Beatriz Noheda et al -S163-Fig. 1. Order parameters (ps and ηs) from pyroelectricdischarge and neutron data, respectively, are plotted versustemperature for rhombohedral PZT. The ps fit was made us-ing the generalised effective field approach. We label pstotal=ps+∆ps the sum of the polarization without tilt (ps) and theextra polarization (∆ps) appearing with the tilt.the rhombohedral region with a very high Zr content(Zr/Ti= 96.5/3.5). 1.5 wt% of Nb2O5 was added to im-prove the electrical properties. The constituent oxideswere ground, pressed into pellets, and fired at 1350 ◦Cfor 2 h. Cylinders of about 1 cm in height and 1 cmin diameter were sintered and hot pressed for the neu-tron diffraction runs [3]. The same procedure has beenused, more recently, for the preparation of thin platesof the same nominal composition, of 1 mm in thicknessand about 1 cm in diameter, which were used for thepyroelectric measurements.Neutron data used in this work were presented recentlyby the authors [3]. They were taken with neutron of 14.7meV energy at Brookhaven National Laboratory. The12 (311) superstructure peak, which practically dependsonly on the oxygen rotation angle, was followed withtemperature between 20 K and 360 K.Precise pyroelectric discharge data have been obtainedby means of a set-up specially designed to measure highFig. 2. (a) ∆ps vs. T, obtained subtracting ps fit frompstotal in Fig. 1. (b) ηs vs. T, from neutron data and predictedby Eq. (2). The arrow shows the place where the antiferro-electric phase could have taken place.Fig. 3. Order parameters (ps and ηs) from pyroelectriccurrent [8] and neutron data [9], respectively, are plotted ver-sus temperature for LiTaO3. The ps fit was made by thegeneralised effective field approach. The dashed line showthe prediction of the perfect coupling model (Eq. (3)).conductivity samples with a slow rate of temperaturechange. A detailed description of the system and itsworking scheme is given in a parallel work. Data shownin this work are taken during heating with a temperaturerate of 20 K/h between 160 K and 600 K.III. RESULTS AND DISCUSSIONVery accurate polarization versus temperature mea-surements has been obtained in PZT. Normalised spon-taneous polarization, ps= Ps/Pso, and tilt parameter,ηs=es/eso, have been plotted in Fig. 1, where Pso andeso are, respectively, the spontaneous polarization andthe tilt parameter at T=0 K. The structural parameter e,is related to the tilt angle, ω, by tanω = 4√3e. The stepin polarization at the FRL-FRH phase transition, TLH,can be clearly observed in the figure coinciding with thevanishing of the tilt.The generalised effective field approach has been usedFig. 4. Order parameter of LaAlO3 vs. temperature. Theexperimental data [7] are fitted with the generalised effectivefield approach.-S164- Journal of the Korean Physical Society, Vol. 32, Feburary 1998to analyse the data. The conjugated effective electricfield and staggered stress can be expressed as Eeff=E+βP+γP3+ δP5+... and |Xeff | = X + βtθ + γtθ3 +δtθ5 + ..., respectively, being E the external electric field,P the polarization, X the external stress and θ the sta-tistically resultant tilt. This theory leads to the followingequation of state with E= 0 (p= ps)TTCtanh−1 psps= 1 + gp2s + hp4s + ... (1)where the parameters g, h,... describe the relative im-portance (to the dipolar term, βP) of the higher orderterms (γP3, δP5..), and are proportional to the succes-sive [3] even powers of Pso(Pso2, Pso4...). A fit of theexperimental data of ps vs. T by means of Eq. (1) isshown in Fig. 1.The subtraction of the ps fit to the experimental datais shown in Fig. 2(a). It gives us the behaviour withtemperature of the normalised extra polarization, ∆ps,appearing due to the tilt. ∆ps reaches its maximum valueat TLH and should disappear at T= 0 K.A model to explain the tilt-polarization coupling wasintroduced in Ref. [3]. According to it, the tilt orderparameter can be expressed in terms of ps and ∆ps asηs =∆ps1− ps(ps + ∆ps). (2)ηs is calculated from Eq. (2), being ps and ∆ps thoseof Figs. 1 and 2(a), respectively. ηs is represented inFig. 2(b) and compared to the experimental data. Itcan be seen the clear agreement between both structuraldata and model, in spite to the model simplicity. Thearrow in the figure shows a peculiar behaviour at 220K, which was attributed to the experimental error inprevious work [3]. Nevertheless this small anomaly hasbeen reproduced with the pyroelectric data through ourcoupling model. A new transition to an antiferroelectric(AFo) phase, at about the same temperature, has beenrecently proposed, but we have found here ferroelectricbehaviour until 160 K. Our composition has similar Nband Ti contents, and the phase diagram is still unknownin this region [10]. The tilt order parameter predicted byour model from the observed ∆ps is substantially smallerthan the experimental one. This might be due to thefact that in a ceramic material the octahedra tilt maynot be so efficiently coupled to the cation shifts as ina single crystal, as assumed for Eq. (2). On the otherhand, Eq. (2) assumes perfect coupling in the sense thatall the oxygen movement is reflected in the polarizationbehaviour. The validity of this assumption may dependin each particular case, on the cation charge and size,etc., as is manifested in LiTaO3 (see bellow). The secondcase to be examined is LiTaO3, which does not reach thecubic perovskite structure (Pm3m) before melting. Inthis case the high temperature phase is not ferroelectricbut possesses oxygen rotations. The low temperaturephase has the same space group (R3c) as that PZT, withboth tilts and cation shifts. This is shown in Fig. 3,where ps and ηs has been calculated from experimentaldata in Refs. [8] and [9], respectively.The symmetry of the respective equations of state [3]for P − E and θ − X allow us to follow, in the LiTaO3case, the same argument than before, interchanging ηsand ps. So we have∆ηs =1− η2s1 + psηsps. (3)The ∆ηs obtained from Eq. (3) with experimental datafor ps and ηs is plotted in Fig. 3 as a dashed line. In thiscase the coupling predicted by our model is not reflectedin the available neutron data, within the experimentaluncertainty. The high values of the tilt angle (about22◦) and cation shifts found in LiTaO3 is a result of theweak interaction between Li+ cations shifts and oxygenoctahedra tilts, due to the small Li charge and size. Thiswould leads to a weak, if any, observed tilt-polarizationcoupling. On the contrary, the large charge and size ofthe cations in PZT produces a strong coupling betweenthe two order parameters, allowing a smaller maximumtilt angle of about 6◦.The case of LaAlO3 is, however, an example of non-polar rhombohedral perovskite. Fig. 4 shows the ex-perimental data of the order parameter [7] and the fitobtained by means of the effective field approach, in thewhole temperature range, using the X − θ equation ofstate [3].In conclusion, we have contrasted the three observedpossible sequences of phases with temperature of rhom-bohedral perovskites. Our data on PZT allow a moreaccurate test of the coupling between tilt and polariza-tion. The possible weak coupling in LiTaO3 has beendiscussed.ACKNOWLEDGMENTSFinancial support from CICyT (PB96-0037) andNATO (CRG-0037) is acknowledged.REFERENCES[1] M. E. Lines and A. M. Glass, Principles and Applicationsof Ferroelectrics and Related Materials (Clarendon Press,1977).[2] A. M. Glazer, Acta Cryst. B28, 3384 (1972).[3] N. Cereceda, B. Noheda, T. Iglesias, J. R. Fdez-del-Castillo and J. A. Gonzalo, Phys. Rev. B 55, 6174 (1997).[4] H. Megaw and C. Darlington, Acta Cryst. A31, 161(1975).[5] A. Amin, R. E. Newnham, L. E. Cross and D. E. Cox,J. of Sol. Stat. Chem. 27, 248 (1981).[6] A. M. Glazer, S. A. Mabud and R. Clarke, Acta Cryst.B34, 1060 (1978).[7] K. Müller, W. Berlinger and F. Waldner, Phys. Rev.Lett. 21, 814 (1968).Tilts and Ionic Shifts in Rhombohedral Perovskites – Beatriz Noheda et al -S165-[8] A. M. Glass, Phys. Rev. 172, 564 (1968).[9] S. C. Abrahams, E. Buehler, W. C. Hamilton and S. J.Laplaca, J. Phys. Chem. Solids 34, 521 (1973).[10] B. Noheda, N. Duan, N. Cereceda and J. A. Gonzalo, J.Korean Phys. Soc. 32, S256 (1998).[11] Z. Ujma, J. Handerek and G. E. Kugel, Ferroelectrics198, 77 (1997).

   University of GroningenTilts and Ionic Shifts in Rhombohedral PerovskitesNoheda, Beatriz; Duan, Ning; Cereceda, Noé; Gonzalo, Julio A.Published in:Journal of the Korean Physical SocietyIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1998Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noheda, B., Duan, N., Cereceda, N., & Gonzalo, J. A. (1998). Tilts and Ionic Shifts in RhombohedralPerovskites. Journal of the Korean Physical Society, 32, S162-S165.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 09-09-2023Journal of the Korean Physical Society, Vol. 32, Feburary 1998, pp. S162∼S164Tilts and Ionic Shifts in Rhombohedral PerovskitesBeatriz Noheda, Ning Duan*, Noé Cereceda and Julio A. GonzaloDept. F́ısica de Materiales, C-IV. Universidad Autónoma de Madrid, 28049-Madrid, Spain* On leave from Shanghai Inst. of Ceramics, Shanghai, ChinaWe make a comparative analysis of rhombohedral perovskites (ABO3) with/without oxygenrotations and ionic shifts, within the framework of a generalised effective field approach. We analyseavailable data on LaAlO3 and LiTaO3 and new data on Zr-rich PZT, examples of three differentways of structural evolution with temperature of the rhombohedral perovskites. The couplingbetween polarization and tilt is studied in detail in PZT and a possible coupling between tilt andpolarization in LiTaO3 is discussed.I. INTRODUCTIONThe perovskite-based structures favour ferroelectricitydue to the large Lorentz contributions in the dipolar in-ternal field and to the intrinsic trend of the cations todisplace inside the oxygen octahedra On the other handthey show structural phase transitions related to the tiltof oxygen octahedra which allow a big variety of phases.The symmetry of the octahedra framework mainly deter-mines that of the crystal lattice, justifying the classifica-tion of the perovskites by their tilt systems. The cationshifts and the octahedra tilts are independent and due todifferent kind of interactions, but they can be coupled.A model to explain this coupling has been proposed re-cently by the authors.In the rhombohedral perovskites the tilt of the oxy-gen octahedra is about their triad axis and it can bedescribed as a−a−a−, according to Glazer’s notation [2],which means opposite rotation for adjacent octahedraalong each direction. Two space groups, R3c and R3c,can sustain this tilt system. The first space group has apolar symmetry allowing ferroelectricity, while the sec-ond one is non-polar keeping the cation in the inversioncentre. Rhombohedral perovskites in the space groupR3m can also be found which do not present octahedratilts and inversion centre.The sequence of space groups presented by rhombo-hedral perovskites with temperature before reaching thecubic (Pm3m) phase, can, in principle, show the differ-ent combinations of these space groups that are allowedby symmetry. Polar R3m and tilted R3c structures cantransform straight to the cubic phase by increasing tem-perature. Polar and tilted space group, R3c, contains theR3c and R3m symmetries and it has two possible waysto achieve the cubic phase, either R3c→R3c→Pm3mor R3c→R3m→ Pm3m. Examples of these two casesare LiTaO3 and PZT, respectively, although LiTaO3,as well as LiNbO3, does not show the cubic perovskitephase before melting. Examples, such as LaAlO3, ofrhombohedral perovskites showing only tilts in the lowtemperature phase, i.e. showing a R3c→Pm3m transi-tions, have been observed. Nevertheless, as far as weknow, it has not been found any rhombohedral per-ovskite having solely a R3m→Pm3m phase transition,which implies no oxygen rotations in the whole rangeof temperature. A candidate for checking this possi-bility would be PZT compositions close to the mor-photropic phase boundary. A Zr/Ti= 60/40 composi-tion of PZT presents the R3c→R3m transition as lowas 250 K, but rhombohedral compositions with higherZr content are difficult to characterise due to the highdiffusion and the mixture of phases (rhombohedral andtetragonal) in this region. It is known that the disap-pearance of the oxygen rotations in PZT through theFRL(R3c)-FRH(R3m) phase transition at TLH, producesa step in the polarization curve. It means that thereis some coupling of the octahedra tilts with the cationshifts. In this work we compare the three mentionedexamples: PZT (cation shifts+tilts→ cation shifts→ cu-bic), LiTaO3 (cation shifts+tilts→ tilts→ melting) andLaAlO3 (tilts→ cubic). We study the behaviour of thetwo independent order parameters, related to polariza-tion (ps) and rotation angle (ηs), and the coupling be-tween them.We have used available data of paramagnetic reso-nance in LaAlO3 by K. A. Müller et al. The spontaneouspolarization of LiTaO3 is from pyroelectric current databy A. M. Glass, while the structural data of LiTaO3 arefrom S. C. Abrahams et al. Experimental data of PZTare described in the next section.II. EXPERIMENTALCeramic PZT samples used were made at the Shang-hai Institute of Ceramics. The composition belongs to-S162-Tilts and Ionic Shifts in Rhombohedral Perovskites – Beatriz Noheda et al -S163-Fig. 1. Order parameters (ps and ηs) from pyroelectricdischarge and neutron data, respectively, are plotted versustemperature for rhombohedral PZT. The ps fit was made us-ing the generalised effective field approach. We label pstotal=ps+∆ps the sum of the polarization without tilt (ps) and theextra polarization (∆ps) appearing with the tilt.the rhombohedral region with a very high Zr content(Zr/Ti= 96.5/3.5). 1.5 wt% of Nb2O5 was added to im-prove the electrical properties. The constituent oxideswere ground, pressed into pellets, and fired at 1350 ◦Cfor 2 h. Cylinders of about 1 cm in height and 1 cmin diameter were sintered and hot pressed for the neu-tron diffraction runs [3]. The same procedure has beenused, more recently, for the preparation of thin platesof the same nominal composition, of 1 mm in thicknessand about 1 cm in diameter, which were used for thepyroelectric measurements.Neutron data used in this work were presented recentlyby the authors [3]. They were taken with neutron of 14.7meV energy at Brookhaven National Laboratory. The12 (311) superstructure peak, which practically dependsonly on the oxygen rotation angle, was followed withtemperature between 20 K and 360 K.Precise pyroelectric discharge data have been obtainedby means of a set-up specially designed to measure highFig. 2. (a) ∆ps vs. T, obtained subtracting ps fit frompstotal in Fig. 1. (b) ηs vs. T, from neutron data and predictedby Eq. (2). The arrow shows the place where the antiferro-electric phase could have taken place.Fig. 3. Order parameters (ps and ηs) from pyroelectriccurrent [8] and neutron data [9], respectively, are plotted ver-sus temperature for LiTaO3. The ps fit was made by thegeneralised effective field approach. The dashed line showthe prediction of the perfect coupling model (Eq. (3)).conductivity samples with a slow rate of temperaturechange. A detailed description of the system and itsworking scheme is given in a parallel work. Data shownin this work are taken during heating with a temperaturerate of 20 K/h between 160 K and 600 K.III. RESULTS AND DISCUSSIONVery accurate polarization versus temperature mea-surements has been obtained in PZT. Normalised spon-taneous polarization, ps= Ps/Pso, and tilt parameter,ηs=es/eso, have been plotted in Fig. 1, where Pso andeso are, respectively, the spontaneous polarization andthe tilt parameter at T=0 K. The structural parameter e,is related to the tilt angle, ω, by tanω = 4√3e. The stepin polarization at the FRL-FRH phase transition, TLH,can be clearly observed in the figure coinciding with thevanishing of the tilt.The generalised effective field approach has been usedFig. 4. Order parameter of LaAlO3 vs. temperature. Theexperimental data [7] are fitted with the generalised effectivefield approach.-S164- Journal of the Korean Physical Society, Vol. 32, Feburary 1998to analyse the data. The conjugated effective electricfield and staggered stress can be expressed as Eeff=E+βP+γP3+ δP5+... and |Xeff | = X + βtθ + γtθ3 +δtθ5 + ..., respectively, being E the external electric field,P the polarization, X the external stress and θ the sta-tistically resultant tilt. This theory leads to the followingequation of state with E= 0 (p= ps)TTCtanh−1 psps= 1 + gp2s + hp4s + ... (1)where the parameters g, h,... describe the relative im-portance (to the dipolar term, βP) of the higher orderterms (γP3, δP5..), and are proportional to the succes-sive [3] even powers of Pso(Pso2, Pso4...). A fit of theexperimental data of ps vs. T by means of Eq. (1) isshown in Fig. 1.The subtraction of the ps fit to the experimental datais shown in Fig. 2(a). It gives us the behaviour withtemperature of the normalised extra polarization, ∆ps,appearing due to the tilt. ∆ps reaches its maximum valueat TLH and should disappear at T= 0 K.A model to explain the tilt-polarization coupling wasintroduced in Ref. [3]. According to it, the tilt orderparameter can be expressed in terms of ps and ∆ps asηs =∆ps1− ps(ps + ∆ps). (2)ηs is calculated from Eq. (2), being ps and ∆ps thoseof Figs. 1 and 2(a), respectively. ηs is represented inFig. 2(b) and compared to the experimental data. Itcan be seen the clear agreement between both structuraldata and model, in spite to the model simplicity. Thearrow in the figure shows a peculiar behaviour at 220K, which was attributed to the experimental error inprevious work [3]. Nevertheless this small anomaly hasbeen reproduced with the pyroelectric data through ourcoupling model. A new transition to an antiferroelectric(AFo) phase, at about the same temperature, has beenrecently proposed, but we have found here ferroelectricbehaviour until 160 K. Our composition has similar Nband Ti contents, and the phase diagram is still unknownin this region [10]. The tilt order parameter predicted byour model from the observed ∆ps is substantially smallerthan the experimental one. This might be due to thefact that in a ceramic material the octahedra tilt maynot be so efficiently coupled to the cation shifts as ina single crystal, as assumed for Eq. (2). On the otherhand, Eq. (2) assumes perfect coupling in the sense thatall the oxygen movement is reflected in the polarizationbehaviour. The validity of this assumption may dependin each particular case, on the cation charge and size,etc., as is manifested in LiTaO3 (see bellow). The secondcase to be examined is LiTaO3, which does not reach thecubic perovskite structure (Pm3m) before melting. Inthis case the high temperature phase is not ferroelectricbut possesses oxygen rotations. The low temperaturephase has the same space group (R3c) as that PZT, withboth tilts and cation shifts. This is shown in Fig. 3,where ps and ηs has been calculated from experimentaldata in Refs. [8] and [9], respectively.The symmetry of the respective equations of state [3]for P − E and θ − X allow us to follow, in the LiTaO3case, the same argument than before, interchanging ηsand ps. So we have∆ηs =1− η2s1 + psηsps. (3)The ∆ηs obtained from Eq. (3) with experimental datafor ps and ηs is plotted in Fig. 3 as a dashed line. In thiscase the coupling predicted by our model is not reflectedin the available neutron data, within the experimentaluncertainty. The high values of the tilt angle (about22◦) and cation shifts found in LiTaO3 is a result of theweak interaction between Li+ cations shifts and oxygenoctahedra tilts, due to the small Li charge and size. Thiswould leads to a weak, if any, observed tilt-polarizationcoupling. On the contrary, the large charge and size ofthe cations in PZT produces a strong coupling betweenthe two order parameters, allowing a smaller maximumtilt angle of about 6◦.The case of LaAlO3 is, however, an example of non-polar rhombohedral perovskite. Fig. 4 shows the ex-perimental data of the order parameter [7] and the fitobtained by means of the effective field approach, in thewhole temperature range, using the X − θ equation ofstate [3].In conclusion, we have contrasted the three observedpossible sequences of phases with temperature of rhom-bohedral perovskites. Our data on PZT allow a moreaccurate test of the coupling between tilt and polariza-tion. The possible weak coupling in LiTaO3 has beendiscussed.ACKNOWLEDGMENTSFinancial support from CICyT (PB96-0037) andNATO (CRG-0037) is acknowledged.REFERENCES[1] M. E. Lines and A. M. Glass, Principles and Applicationsof Ferroelectrics and Related Materials (Clarendon Press,1977).[2] A. M. Glazer, Acta Cryst. B28, 3384 (1972).[3] N. Cereceda, B. Noheda, T. Iglesias, J. R. Fdez-del-Castillo and J. A. Gonzalo, Phys. Rev. B 55, 6174 (1997).[4] H. Megaw and C. Darlington, Acta Cryst. A31, 161(1975).[5] A. Amin, R. E. Newnham, L. E. Cross and D. E. Cox,J. of Sol. Stat. Chem. 27, 248 (1981).[6] A. M. Glazer, S. A. Mabud and R. Clarke, Acta Cryst.B34, 1060 (1978).[7] K. Müller, W. Berlinger and F. Waldner, Phys. Rev.Lett. 21, 814 (1968).Tilts and Ionic Shifts in Rhombohedral Perovskites – Beatriz Noheda et al -S165-[8] A. M. Glass, Phys. Rev. 172, 564 (1968).[9] S. C. Abrahams, E. Buehler, W. C. Hamilton and S. J.Laplaca, J. Phys. Chem. Solids 34, 521 (1973).[10] B. Noheda, N. Duan, N. Cereceda and J. A. Gonzalo, J.Korean Phys. Soc. 32, S256 (1998).[11] Z. Ujma, J. Handerek and G. E. Kugel, Ferroelectrics198, 77 (1997).

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