processing and properties of advanced ceramics and composites v (ceramic transactions) || the effect...

THE EFFECT OF A-SITE VACANCIES ON CELL VOLUME AND TOLERANCE FACTOR OF PEROVSKITES

Rick Ubic, Kevin Tolman, Kokfoong Chan, Nicole Lundy Boise State University Boise, Idaho, USA

Steven Letourneau and Waltraud Kriven University of Illinois at Urbana-Champaign Urbana, IL, USA

ABSTRACT Point defects like vacancies can have a profound effect on the structure of perovskite

ceramics, but the exact mechanisms by which they do this are unclear. A predictive model for the pseudocubic lattice constant of perovskites, based solely on published ionic radii data, has been developed and adapted as a model for tolerance factor. These models more consistently predict both pseudocubic lattice constant, hence cell volume, and the tolerance factor than existing methods, thus also more accurately modeling the temperature coefficient of resonant frequency (TCF). The relationship between tolerance factor and TCF is revisited.

INTRODUCTION Perovskites abound both in nature and in the laboratory, and their wide compositional

range renders a variety of useful properties such that perovskites are encountered in applications as disparate as electroceramics, superconductors, refractories, catalysts, magnetoresistors, and proton conductors. They are also of interest for use as substrates or buffer layers for compound semiconductor heteroepitaxy. The design of such advanced materials requires an understanding of the relationship between chemical composition and crystal structure. Perovskite oxides have the general formula ABO3, and complex perovskites of the type A(BB )03 are now well-established materials for microwave applications because of their high quality factors (Qf) and low temperature coefficients of resonant frequency (Tf). These materials are typically engineered with various dopants on both A and B sites and often contain unwanted defects such as vacancies, especially on the anion sublattice.

A recent study1 established an empirical model relating the ionic radii to the pseudocubic lattice constant with an average absolute error of just 0.60%:

apc = 0.06741 + 0.49052(rA + rx) + 1.29212(rB + rx) (1)

where rA, TB, and rx are the various ionic radii assuming sixfold coordination. A later study2

used a similar approach in modeling the lattice constants of orthorhombic perovskites in space group Pbnm, which accounts for >90% of observed orthorhombic perovskites. The equations so derived yield results with absolute relative errors of 0.616%, 1.089%, and 0.714% for lattice constants a, b, and c, respectively.

The perovskite tolerance factor is generally defined geometrically as:

t = r A + r ° (2) V2(rB+r0)

331

Processing and Properties of Advanced Ceramics and Composites V: Ceramic Transactions. Edited by Narottam P. Bansal, J. P. Singh, Song Won Ko, Ricardo H. R. Castro,

Gary Pickrell, Navin Jose Manjooran, K. M. Nair and Gurpreet Singh. © 2013 The American Ceramic Society. Published 2013 by John Wiley & Sons, Inc.

Effect of A-Site Vacancies on Cell Volume and Tolerance Factor of Perovskites

where rA, rB, and r0 are the ionic radii of the ionic species on the A, B, and O sites, respectively. By assuming that the cation and anion sublattices are closely packed, the tolerance factor can be viewed as a measure of distortion of oxygen octahedra. The relationship between tolerance factor and the pseudocubic perovskite lattice constant is discussed at some length in reference 1. Reaney et al.3 established a relationship between tolerance factor and the temperature coefficient of permittivity (xe) by which it was determined that changes in xe were closely correlated to octahedral tilt transitions in niobate and tantalate perovskites. They observed that perovskites with t < -0.985 contained axes about which oxygen octahedra were tilted in an anti-phase arrangement, causing cell doubling in the three pseudocubic directions. Similarly, perovskites for which t < -0.965 undergo a further tilt transition whereby octahedra are tilted in-phase about one or more axes as well. Perovskites for which t > -0.985 were not observed to contain a tilt superlattice.

Ubic et al.4 derived an equation for tolerance factor based solely on pseudocubic lattice constant (which can either be experimentally measured or calculated via equation 1) and published values of rB and rx:

apc -0.05444 0.66046036 l(rB+rx) (3)

This expression, with rx now in its proper twofold coordination, can be used to derive the correct tilt structure for complex perovskites like Ba(Smi/2Sbi/2)03, for which equation 2 fails, or simple perovskites like CaTi03 and MgSiC>3, the structures of both of which are incorrectly predicted via equation 2.

It has already been shown5'6 that charge compensation in rare-earth-doped SrTiC>3 occurs via A-site vacancy formation via the reaction:

Ln203 SrTi°- ) 2Ln" + Vs'r + 30* (4)

Ubic et al.6 have shown by electron and neutron diffraction that the structure of Sri.3 /2CexTi03 (0.1333 < x < 0.4) is R3c. Oxygen octahedra are tilted about the pseudocubic [111] by up to 4.7°. Extrapolation of these results suggests that octahedral tilting might start to occur at 0 < JC < 0.013. The incorporation of the trivalent species in particular is thought to stabilize the tilted structure despite the fact that the effective tolerance factor t > 0.9895 for the entire compositional range and only dips to 0.9833 at JC = 0.4, at which point there is a structural phase transition, possibly to C2/c.7

In theory, equation 1 is only limited by the accuracy of rA, rB, and rx values used, and equations 2 and 3 should yield equivalent results; however, the question then arises of how to calculate ionic radii when sites are shared by more than one species or are partially vacant. The concept of an "average" cation size seems at first meaningless; however, as the strains caused by each species will be averaged over the whole structure, it is not unreasonable to expect local relaxations to allow for the stabilization of an "average" structure. The assumption that vacancies are zero-dimensional defects inevitably leads to large errors in equations 1 and 2, especially at large vacancy concentrations. In order to understand and correct this problem, compositions have been engineered with exact concentrations of A-site vacancies. When Rietveld refinements of x-ray diffraction data are performed on these compounds, it is possible to back-calculate actual average bond lengths, from which the effective contribution (size) of vacancies to the average A-site ionic radius, rA, can be determined.

332 • Processing and Properties of Advanced Ceramics and Composites V


PROCEDURE Five compositions in the system Srj.^La^TiCb (x = 0.01, 0.05, 0.1667, 0.2222, and 0.25)

were prepared via the mixed-oxide route. Stoichiometric amounts of SrC03, Ti02 (99.9%, Aldrich Chemical Co., Milwaukee, WI), and La203 (99.9%, Alfa-Aesar, Ward Hill, MA) were ball milled in distilled water for four hours, using YSZ media in a high-density nylon pot. Slurries were dried, ground, and calcined at 1300°C for four hours. Approximately 2 wt% of polyethylene glycol (PEG 10,000, Alfa-Aesar, Ward Hill, MA) was added to the dried powders, which were then ground again into fine powder. Cylindrical pellets of about 3-4 mm in height and 10 mm in diameter were made by applying a pressure of 63 MPa. These compacts were then sintered for four hours at temperatures ranging from 1450 to 1650C.

In order to avoid the appearance of the parasitic La2Ti207 pyrochlore phase, compositions for which JC> 0.15 were produced in a three-step process whereby precursor phases La SrCh and SrTi03 were produced via the mixed-oxide route:

1300°C,12/irs

2xLa(OH)3 + 0.5jtSrCO3 -> 0.5jcLa4SrO7 + 0.5xCO2T 1300°C,4/ir5

(1 - 3.5JC)SI€03 + (1 - 3.5jc)Ti02 -> (l-3.5;c)SrTi03 + (1-3.5JC)C02T 1300°C,4/irs

0.5jcLa4SrO7 + (l-3.5x)SrTi03 + 3.5.xTi02 -> Sri.3jcLa2;cTi03

In order to achieve a phase-pure La4SrC>7 product, starting powders were first milled for four hours, dried, and calcined at 1300°C for four hours. Then the process was repeated twice with the same parameters. In the case of SrTi03, starting powders were milled for six hours prior to drying and calcining.

Powder samples were prepared for x-ray diffraction (D8 Discover, Bruker AXS, Madison, Wisconsin, USA) from post-calcined batches. Le Bail fits to x-ray diffraction data were conducted using DiffracPLUS TOPAS 4.2 (Bruker).

RESULTS All the compositions processed were single-phase perovskites, as illustrated in Fig. 1.

Although octahedral tilting was expected in all compositions, Le Bail fitting was conducted assuming an untilted, pseudocubic perovskite structure in Pm3m. In this setting, it is assumed that rB (rTi = 0.605)8 is unaffected by A-site doping and, consequently, both r0 and rA values can be extracted as simple functions of apc\ for example:

r0 = 0.5apc - rB

_ 4lapc-2r0 r. = p— S-l

When effective values of ro are calculated from refined apc values, only a negligible, nonsystematic change is observed, attributed to measuring error. Oxygen ions are coordinated to two B-site cations and four A-sites; therefore, an increase in the number of A-site vacancies will necessarily lower the coordination of second-nearest neighbors and so might be expected to cause a slight decrease in the average oxygen ionic radius; however, the maximum decrease observed throughout this series is only 1.03% from the ideal value given by Shannon8 for oxygen in twofold coordination (1.35 A), so this effect can fairly safely be ignored.

Processing and Properties of Advanced Ceramics and Composites V • 333


a D

<

25 30 35 40 45 50

Diffraction Angle (20)

JC=0.25

x= 0.2222

1667

AJJC=0.I 0500

x =0.0100 55

FIG. 1 - X-ray diffraction results of all compositions indexed according to the structure of pure cubic SrTiC>3.

Once values of rA are obtained, it is possible to back-calculate the effective size of vacancies, rv, from the stoichiometry by assuming the ionic sizes of Sr2+ (rsr = 1.44 A) and La3+

(/La = 1-36 A) in twelve-fold coordination are as published by Shannon.8 Two additional data points can be included from literature values of lattice constants corresponding JC = 0 (SrTiOa) and JC = lA (La^TiOs).10 Such calculations reveal that the effective vacancy size increases with JC and is negative for JC < 0.01 (Table 1).



Table I. Measured pseudocubic latticed constants and calculated ionic radii values JC 0 0.0100 0.0500 0.1667 0.2222 0.2500 0.3333

apc (A) 3.9050 3.9010 3.9015 3.8950 3.8823 3.8993 3.9384

'A (A) 1.4138 1.4129 1.4130 1.4117 1.4091 1.4126 1.4207

M A ) ~ -1.1076 1.0605 1.4301 1.4607 1.4903 1.5420

t (eqn. 3) 1.0011 0.9980 0.9984 0.9934 0.9836 0.9967 1.0270

apc calc (A) 3.8983 3.8961 3.8954 3.8937 3.8906 3.8879 3.8733

As Table 1 shows, vacancies have an appreciable effective size in this system. This effect might be explained by the electrostatic repulsion of the oxygen anions surrounding the vacancy. Considering that rsr = 1.44 A and rLa = 1.36 A, the values of ry are generally a considerable fraction of the average size of the A-site cations, climbing even to 113% of the cation size for JC = 0.3333 (although the unique layered structure of this compound almost certainly makes this value physically meaningless). The negative value for x = 01 corresponds to the relaxation of oxygen ions towards the vacant site, whereas the positive values correspond to mutual Coulombic repulsion of oxygen ions across the vacant site. These values fit well with the mathematical model published by Ubic et al.4 in 2009. Without accounting for the finite size of vacancies, the relative errors in apc values generated via equation 1 range from 0.55% < \Aapc\ < 3.42% for 0 < JC < lA; however, when one accounts for the finite sizes of vacancies, the new predicted values of apc (Table 1) have relative errors of just 0.03% < \Aapc\ < 0.29%.

Using these refinement results, equation 3 would yield tolerance factors 0.9836 < t < 1.0270 throughout the series, with the minimum occurring at JC = 0.2222. In all but the case of JC = 0.2222, the calculated tolerance factors would predict an untilted structure; however, it is known11 that the rhombohedral add tilt system is stabilized by highly charged A-site cations (e.g., La3+) and small tilt angles.

CONCLUSIONS Five compositions in the Sr2-3^/2La^Ti03 homologous series corresponding to x = 0.01,

0.05, 0.1667, 0.2222, and 0.25 have been produced and characterized via x-ray diffraction. LeBail refinements show that A-site vacancies in this system have an effective size due to both bond relaxation and mutual repulsion of coordinating oxygen ions. Such vacancies also cause a very slight reduction in the radius of oxygen anions as a result of the lowering of their secondary coordination.

Processing and Properties of Advanced Ceramics and Composites V • 335


REFERENCES ]R. Ubic, Revised Method for the Prediction of Lattice Constants in Cubic and Pseudocubic Perovskites, /. Am. Ceram. Soc, 90, 3326-30 (2007). 2R. Ubic and G. Subodh, The Prediction of Lattice Constants in Orthorhombic Perovskites, /. Alloys Compd., 488, 374-379 (2010). 3I.M. Reaney, E.L. Colla, and N. Setter, Dielectric and Structural Characteristics of Ba- and Sr-Based Complex Perovskites as a Function of Tolerance Factor, Jpn. J. Appl. Phys., Part 1, 33, 3984-90(1994). 4R. Ubic, G. Subodh, M.T. Sebastian, D. Gout, and T. Proffen, Effective Size of Vacancies in the Sri.3x/2CexTi03 Superstructure, Ceram. Trans., 204, 177-185 (2009). 5G. Subodh, J. James, M.T. Sebastian, R. Paniago, A. Dias, and R.L. Moreira, Structure and Microwave Dielectric Properties of Sr2+„Ce2Ti5+nOi5+3„ (n < 10) Homologous Series, Chem. Mater., 19, 4077-82 (2007). 6R. Ubic, G. Subodh, M.T. Sebastian, D. Gout, and T. Proffen, Structure of Compounds in the Sri.3x/2CexTi03 Homologous Series, Chem.Mater., 20, 3127-33 (2008). 7R. Ubic, G. Subodh, M.T. Sebastian, D. Gout, and T. Proffen, Structure of Sro.4Ceo.4nO3, Chem. Mater., 21, 4706-4710 (2009). 8R.D. Shannon, Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides, Acta Cryst., A32, 751-767 (1976). 9H.E. Swanson and R.K. Fuyat, Natl. Bur. Stand. (U.S.), Circ. 539, 3 44 (1954). 10 Z.S. Gonen, D. Paluchowski, P.Yu Zavalii, B.W. Eichhorn, and J. Gopalakrishnan, Reversible Cation/Anion Extraction from K2La2Ti30io: Formation of New Layered Titanates, KLa2Ti309.5 and La2Ti309, Inorg. Chem., 45, 8736-8742 (2006). nP.M. Woodward, Octahedral Tilting in Perovskites. II. Structure Stabilizing Forces, Acta Cryst., 653,44-66(1997).

ACKNOWLEDGEMENTS This work has been supported by the National Science Foundation through the Major Research Instrumentation Program, Award Number 0619795, and DMR 1052788.


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