form ability of abo3 perovskites

Journal of Alloys and Compounds 372 (2004) 40–48

Formability of ABO3 perovskites

Chonghe Lia, Kitty Chi Kwan Sohb, Ping Wua,∗a Institute of High Performance Computing, 1 Science Park Road, 01-01 The Capricorn, Singapore 117528, Singaporeb Department of Materials, National University of Singapore, Lower Kent Ridge Road, Singapore 119260, Singapore

Received 7 July 2003; received in revised form 6 October 2003; accepted 6 October 2003

Abstract

Regularities governing perovskite formability are investigated, by using empirical structure map methods and a total of 197 binary oxidesystems. Each of the 197 systems belongs to one of the three groups (A2O–B2O5, AO–BO2 and A2O3–B2O3), and among them only 121systems have a perovskite compound. It is found that the octahedral factor,rB/rO, is as important as the tolerance factor,(rA +rO)/

√2(rB +rO),

with regards to the formability of perovskite. On the structure map constructed by these two parameters, sample points representing systems offorming (perovskite) and non-forming, are distributed in distinctively different regions. Prediction criterions for the formabilities of perovskitesare obtained, by which only four systems are misclassified. The developed models can be used to search for new perovskites by screeningall possible elemental combinations in any of the above three groups. It may assist in the design of new substrate or buffer materials forsemiconductors where perovskite can play an important role.© 2003 Elsevier B.V. All rights reserved.

Keywords: Perovskite; Oxide; Tolerance factor; Octahedron; Formability

1. Introduction

Perovskite is one of the most frequently encounteredstructures in solid-state inorganic chemistry, and it accom-modates most of the metallic ions in the periodic table witha significant number of different anions. An ideal perovskitestructure has an ABX3 stoichiometry and a cubic crystalstructure, which is composed of a three-dimensional frame-work of corner-sharing BX6 octahedron. The A-site cationfills the 12 coordinate cavities formed by the BX3 networkand is surrounded by 12 equidistant anions[1], as seenin Fig. 1. Although majority of the perovskite compoundsare oxides or fluorides, other forms like heavier halides,sulphides, hydrides, cyanides, oxyfluorides and oxynitridesare also reported[2,3].

Perovskite and perovskite-related materials are importantcrystal structure due to their diverse physical/chemistryproperties[2,4,5] over a wide temperature range. For ex-ample, perovskite ceramics with ferroelectric and/or piezo-electric properties, such as BaTiO3 and Pb(ZrTi)O3 play adominant role in the electroceramics industry. Other indus-

∗ Corresponding author. Tel.:+65-6419-1212; fax:+65-6778-0522.E-mail address: [email protected] (P. Wu).

try interests of perovskites are: colossal magnetoresistance((SrCaLa)MnO3), PTC thermistor (BaTiO3), electroopti-cal modulator ((PbLa)(ZrTi)O3, BaTiO3), optical switch(LiNbO3), battery Material ((Li0.5−xLaxTiO3)), etc.

It is of interest to find out regularities governing the for-mation of perovskite-type compounds and use it to fur-ther guide the exploration of new materials. In early 1920s,Goldschmidt[6] has proposed a “tolerance factor” (t =rA + rO/

√2(rB + rO), whererA, rB and rO are the ionic

radii of A, B and O, respectively) to study the stability ofperovskites. Geometrically, for an ideal perovskite, the ra-tio of D(A–O), the bond length of A–O bond, toD(B–O),the bond length of B–O bond, is

√2 : 1. Thus, if the bond

length is roughly assumed to be the sum of two ionic radii,the t value of an ideal perovskite should be equal to 1.0.However, Goldschmidt found that, as an experimental fact,tvalues of most cubic perovskites are in the range of 0.8–0.9,and distorted perovskites occur in somewhere wider rangeof t.

Goldschmidt’s tolerance factort has been widely ac-cepted as a criterion for the formation of the perovskitestructure, a number of researchers have used it to discussthe perovskite stability, including oxides[7–11], fluorides[12–14], chlorides[15]. Up to now, almost all known per-ovskite compounds havet values in the range of 0.75–1.00.

0925-8388/$ – see front matter © 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.jallcom.2003.10.017

C.H. Li et al. / Journal of Alloys and Compounds 372 (2004) 40–48 41

Fig. 1. Ideal perovskite structure illustrated for ABO3. Note the corner-shared octahedra, extending in three dimensions.

However, it seems thatt = 0.75–1.00 is not a sufficientcondition for the formation of the perovskite structure, asindicated later, for some systems whoset are even within themost favourable range (0.8–0.9), no perovskite structure isstable.

Muller and Roy[1] proposed to plot “structural map”,which took the ionic radii of A and B as coordinates to studythe distribution of different structures for many ternarystructural families, including A2BX4, ABX4 and ABX3systems[1]. Furthermore, the schematic distribution mapof different crystal structure for A1+B5+O3, A2+B4+O3,and A3+B3+O3 systems separately, were given by the samemethod[1,16]. However, the criterion for perovskite formal-ity was not discussed, possibly due to the lack of accuratedata of crystal structure of some ABO3 compounds at thattime.

In China, a group of scientists[17] use a patternrecognition—atomic parameter method to study the regular-ities of formation of perovskite, including simple (ABO3-type compound) and complex oxides (A(B1B2)O3-typecompounds). Their model contains seven atomic parame-ters, including the radius of ion A and ion B, the electroneg-ativities of ion A and ion B, and the d electron number ofion B. The underline physics is not quite clear and a specialsoftware package is needed when predicting new compoundformation using their model.

In this paper, 197 ABO3-type compounds (of which, 121are perovskite, and the rests are non-perovskite) are collectedfrom Phase Diagram for Ceramics (I–XII) [18]. First of all,more recently available systems are used to test Muller andRoy’s results[1]. And a structural map spanned by ionicradius of cations A and B is obtained combining all three

groups (A1+B5+O3, A2+B4+O3, and A3+B3+O3) in onemap, which further indicated that the different charges of thesystems have little effect on the perovskite formability rule.The tolerance factor t and a newly defined octahedral factor(the radii ratio of the small cation B over the anion O) arefurther applied to construct a new and effective structuralmap leading to new criteria of the formability of perovskitecompounds.

2. Structural map method

The most important atomic parameter that dominates thecrystal structure of ionic compound is the ionic radius, whichmay be explained by a thermodynamic analysis. By the Sec-ond Law of thermodynamics, the Gibbs free energy of asolid is:

G = H − TS (1)

G = U + PV + 12 hν − TS (2)

Thus, Gibbs free energy= lattice energy+ terms which arevery small at lowP andT.

Thus, at low temperatures and pressures,G ≈ U, i.e.the lattice energy determines which structure is stable. Thelattice energyU for ionic compounds can be expressed as[1]:

U = −Mz2e2

rc + ra(3)

whereM is the Madelung constant, which is a constant forcertain structure,z the valence of ions,rc andra the radii ofanion and cation. So it can be seen that the crucial impor-tance of ionic radius is rooted directly in the fact that thelattice energy determines the Gibbs free energy and that inturn the lattice energy bears a simple inverse relationship tothe interatomic separation (sum of cation and anion radius).

In 1959, Mooser and Perason[19] applied a two-dimensiongraphic to study the stability of different compounds, thetwo factors they used were the difference of electroneg-ativity between the cation and the anion and the averageprinciple quantum number. They succeeded to discriminatethe crystal structures of AB-type compounds, of AX2-typehalides, and the metallic or non-metallic ternary fluoridesABX. Similar methods were called the structural map tech-nology, and more parameters were used to draw the graphic.In this paper, the formability of perovskite ABO3 will bestudied by a similar method.

3. Data collection

A total of 197 binary oxide systems are collected fromPhase Diagram for Ceramics [18], which can be dividedinto three groups: A2O–B2O5 system (corresponding to the

42 C.H. Li et al. / Journal of Alloys and Compounds 372 (2004) 40–48

A+B5+O3-type perovskites); AO-BO2 system (correspond-ing to A2+B4+O3-type perovskites) and A2O3–B2O3 system(corresponding to A3+B3+O3-type perovskites). Of which,121 systems (denoted ‘yes’) are found to have perovskitecompounds and 76 systems (denoted ‘no’) have no per-ovskite compounds.

According to Lufaso and Woodward[5], an ideal cu-bic perovskite can transform into other crystal structurethrough tilting the octahedral BO6, these resulted structureshave close relationship to the ideal perovskite. In general,all compounds with these 15 kinds of structures (one idealstructure plus 14 structures resulted by the tilting) belongto ‘perovskite’ catalog. It is well known that[1,2] crystalstructures can change with variation of temperature and/orpressure. Compounds with other structure can, therefore,transform into perovskite, and vice versa at different tem-perature and/or pressure. So the criterion for classifying aforming system (denoted ‘yes’ inTable 1) is that a com-pound of perovskite (any of the 15 crystal structures) isstable at room temperature and one atmosphere. Systemsdenoted ‘no’ represent at least one of the following threeconditions: (1) there is no new ternary compound; (2) thereare new ternary compounds but there is no oxide of thechemical formula ABO3; and (3) there is at least one ABO3compound but it is not a perovskite structure.

One hundred ninety-seven binary oxide systems with theirperovskite formability, ionic radius of constituent ions Aand B, tolerance factor and the octahedral factor (the ra-tio of radii of the small cation B over the radii of anionO) are listed inTable 1, part (a) (for A2O–B2O5 system),part (b) (for AO–BO2 system) and part (c) (A2O3–B2O3system). All perovskite structures in the tables are con-firmed by four references[1–3,20]. In addition, the ionicradii used here are the values for six coordination num-ber (Shannon’s value), most of them cited fromHandbookof Chemistry and Physics [21], some values which are notavailable in this handbook are complemented fromThe el-ements [22] and Perovskites and high TC Superconductor[2].

Table 1

No. System Formability rA

(A)rB

(A)Tolerance rB/rO

(a) Formability of perovskites in A2O–B2O5 system1 Li2O–Nb2O5 No 0.76 0.64 0.75 0.512 Li2O–Ta2O5 No 0.76 0.64 0.75 0.513 Li2O–V2O5 No 0.76 0.54 0.79 0.434 Rb2O–Nb2O5 No 1.52 0.64 1.03 0.515 Li2O–As2O5 No 0.76 0.46 0.83 0.376 Rb2O–Ta2O5 No 1.52 0.64 1.03 0.517 Cs2O–Nb2O5 No 1.67 0.64 1.09 0.518 K2O–As2O5 No 1.38 0.46 1.09 0.379 K2O–V2O5 No 1.38 0.54 1.04 0.43

10 Na2O–As2O5 No 1.02 0.46 0.94 0.3711 Na2O–P2O5 No 1.02 0.38 0.98 0.3012 Cs2O–V2O5 No 1.67 0.54 1.15 0.4313 Cu2O–P2O5 No 0.77 0.38 0.88 0.30

Table 1 (Continued )


(A)rB

(A)Tolerance rB/rO

14 K2O–P2O5 No 1.38 0.38 1.14 0.3015 Li2O–P2O5 No 0.76 0.38 0.87 0.3016 Tl2O–Sb2O5 No 1.50 0.60 1.05 0.4817 Na2O–Sb2O5 No 1.02 0.60 0.87 0.4818 Li2O–Sb2O5 No 0.76 0.60 0.77 0.4819 K2O–Sb2O5 No 1.38 0.60 1.00 0.4820 Na2O–Bi2O5 No 1.02 0.76 0.80 0.6021 Li2O–Bi2O5 No 0.76 0.76 0.71 0.6022 Ag2O–Bi2O5 No 1.15 0.76 0.84 0.6023 K2O–Nb2O5 Yes 1.38 0.64 0.98 0.5124 K2O–Ta2O5 Yes 1.38 0.64 0.98 0.5125 Na2O–Nb2O5 Yes 1.02 0.64 0.85 0.5126 Na2O–Ta2O5 Yes 1.02 0.64 0.85 0.5127 K2O–U2O5 Yes 1.38 0.76 0.92 0.6028 Na2O–U2O5 Yes 1.02 0.76 0.80 0.6029 Na2O–W2O5 Yes 1.02 0.62 0.86 0.4930 Na2O–V2O5 Yes 1.02 0.54 0.90 0.4331 Ag2O–V2O5 Yes 1.15 0.54 0.95 0.4332 Ag2O–Ta2O5 Yes 1.15 0.64 0.90 0.5133 Ag2O–Nb2O5 Yes 1.15 0.64 0.90 0.5134 Cs2O–I2O5 Yes 1.67 0.95 0.94 0.7535 Tl2O–I2O5 Yes 1.50 0.95 0.88 0.7536 Rb2O–I2O5 Yes 1.52 0.95 0.89 0.7537 K2O–I2O5 Yes 1.38 0.95 0.84 0.7538 Ag2O–Sb2O5 Yes 1.15 0.60 0.92 0.4839 K2O–Bi2O5 Yes 1.38 0.76 0.92 0.60

(b) Formability of perovskites in AO–BO2 system1 BaO–GeO2 No 1.35 0.53 1.03 0.422 CaO–SiO2 No 1.00 0.40 0.96 0.323 CoO–TiO2 No 0.65 0.61 0.72 0.484 MgO–GeO2 No 0.72 0.53 0.78 0.425 MgO–TiO2 No 0.72 0.61 0.75 0.486 MnO–TiO2 No 0.83 0.61 0.79 0.487 ZnO–TiO2 No 0.74 0.61 0.76 0.488 FeO–TiO2 No 0.61 0.61 0.71 0.489 NiO–TiO2 No 0.69 0.61 0.74 0.48

10 FeO–SiO2 No 0.61 0.40 0.80 0.3211 ZnO–SiO2 No 0.74 0.40 0.85 0.3212 MnO–GeO2 No 0.83 0.53 0.83 0.4213 MgO–SnO2 No 0.72 0.69 0.72 0.5514 SrO–SiO2 No 1.18 0.40 1.04 0.3215 EuO–SiO2 No 1.17 0.40 1.04 0.3216 SmO–SiO2 No 1.19 0.40 1.04 0.3217 BaO–SiO2 No 1.35 0.40 1.11 0.3218 CoO–SiO2 No 0.65 0.40 0.81 0.3219 SnO–PbO2 No 0.93 0.78 0.76 0.6220 NiO–SiO2 No 0.69 0.40 0.83 0.3221 PbO–SiO2 No 1.19 0.40 1.04 0.3222 MgO–SiO2 No 0.72 0.40 0.84 0.3223 SrO–GeO2 No 1.18 0.53 0.96 0.4224 PbO–GeO2 No 1.19 0.53 0.97 0.4225 CdO–GeO2 No 1.10 0.53 0.93 0.4226 CaO–GeO2 No 1.00 0.53 0.89 0.4227 BaO–MnO2 No 1.35 0.53 1.03 0.4228 CoO–MnO2 No 0.65 0.53 0.75 0.4229 NiO–MnO2 No 0.69 0.53 0.77 0.4230 SrO–MnO2 No 1.18 0.53 0.96 0.4231 CaO–VO2 Yes 1.00 0.58 0.87 0.4632 SrO–HfO2 Yes 1.18 0.71 0.88 0.5633 CaO–MoO2 Yes 1.00 0.65 0.84 0.5234 CaO–RuO2 Yes 1.00 0.62 0.85 0.4935 CaO–SnO2 Yes 1.00 0.69 0.82 0.55




(A)rB

(A)Tolerance rB/rO

36 SrO–CeO2 Yes 1.18 0.87 0.81 0.6937 SrO–RuO2 Yes 1.18 0.62 0.92 0.4938 SrO–SnO2 Yes 1.18 0.69 0.88 0.5539 BaO–CeO2 Yes 1.35 0.87 0.87 0.6940 SrO–ZrO2 Yes 1.18 0.72 0.87 0.5741 SrO–PbO2 Yes 1.18 0.78 0.85 0.6242 SrO–TiO2 Yes 1.18 0.61 0.92 0.4843 BaO–NbO2 Yes 1.35 0.68 0.95 0.5444 BaO–SnO2 Yes 1.35 0.69 0.95 0.5545 BaO–ZrO2 Yes 1.35 0.72 0.93 0.5746 SrO–MoO2 Yes 1.18 0.65 0.90 0.5247 SrO–VO2 Yes 1.18 0.58 0.94 0.4648 CaO–UO2 Yes 1.00 0.89 0.74 0.7149 BaO–ThO2 Yes 1.35 0.94 0.84 0.7550 BaO–UO2 Yes 1.35 0.89 0.86 0.7151 CaO–HfO2 Yes 1.00 0.71 0.81 0.5652 PbO–CeO2 Yes 1.19 0.87 0.81 0.6953 BaO–HfO2 Yes 1.35 0.71 0.94 0.5654 CaO–TiO2 Yes 1.00 0.61 0.85 0.4855 CaO–ZrO2 Yes 1.00 0.72 0.81 0.5756 BaO–TiO2 Yes 1.35 0.61 0.99 0.4857 PbO–TiO2 Yes 1.19 0.61 0.93 0.4858 PbO–ZrO2 Yes 1.19 0.72 0.87 0.5759 BaO–MoO2 Yes 1.35 0.68 0.95 0.5460 BaO–TbO2 Yes 1.35 0.76 0.91 0.6061 BaO–PrO2 Yes 1.35 0.85 0.87 0.6762 BaO–PbO2 Yes 1.35 0.78 0.90 0.6263 CaO–PbO2 Yes 1.00 0.78 0.78 0.6264 CaO–MnO2 Yes 1.00 0.53 0.89 0.42

(c) Formability of perovskites in A2O3–B2O3 system1 La2O3–B2O3 No 1.03 0.23 1.09 0.182 Sc2O3–B2O3 No 0.75 0.23 0.95 0.183 Sm2O3–B2O3 No 0.96 0.23 1.05 0.184 Al2O3–B2O3 No 0.54 0.23 0.85 0.185 Ga2O3–Al2O3 No 0.62 0.54 0.74 0.436 Eu2O3–Ln2O3 No 0.95 0.8 0.76 0.637 Dy2O3–B2O3 No 0.91 0.23 1.03 0.188 Er2O3–B2O3 No 0.89 0.23 1.02 0.189 Eu2O3–B2O3 No 0.95 0.23 1.05 0.18

10 Gd2O3–B2O3 No 0.94 0.23 1.04 0.1811 Ho2O3–B2O3 No 0.89 0.23 1.02 0.1812 Y2O3–B2O3 No 0.9 0.23 1.03 0.1813 Yb2O3–B2O3 No 0.86 0.23 1.01 0.1814 Bi2O3–Sm2O3 No 1.03 0.96 0.73 0.7615 V2O3–Cr2O3 No 0.64 0.62 0.71 0.4916 V2O3–Al2O3 No 0.64 0.54 0.75 0.4317 As2O3–B2O3 No 0.58 0.23 0.87 0.1818 Gd2O3–Y2O3 No 0.94 0.9 0.72 0.7119 Sm2O3–Y2O3 No 0.96 0.9 0.73 0.7120 Tm2O3–B2O3 No 0.88 0.23 1.02 0.1821 In2O3–Cr2O3 No 0.8 0.62 0.77 0.4922 In2O3–Fe2O3 No 0.8 0.55 0.80 0.4423 Sc2O3–Al2O3 No 0.75 0.54 0.79 0.4324 Sc2O3–Cr2O3 No 0.75 0.62 0.76 0.4925 La2O3–Cr2O3 Yes 1.03 0.62 0.86 0.4926 La2O3–Fe2O3 Yes 1.03 0.55 0.89 0.4427 La2O3–V2O3 Yes 1.03 0.64 0.85 0.5128 Sm2O3–Cr2O3 Yes 0.96 0.62 0.83 0.4929 Ce2O3–V2O3 Yes 1.01 0.64 0.84 0.5130 Er2O3–Fe2O3 Yes 0.89 0.55 0.84 0.4431 Er2O3–V2O3 Yes 0.89 0.64 0.80 0.5132 Eu2O3–Fe2O3 Yes 0.95 0.55 0.86 0.44



(A)rB

(A)Tolerance rB/rO

33 Gd2O3–Fe2O3 Yes 0.94 0.55 0.86 0.4434 Ho2O3–Fe2O3 Yes 0.89 0.55 0.84 0.4435 La2O3–Mn2O3 Yes 1.03 0.58 0.88 0.4636 La2O3–Ti2O3 Yes 1.03 0.67 0.84 0.5337 Nd2O3–Fe2O3 Yes 0.98 0.55 0.88 0.4438 Nd2O3–Ni2O3 Yes 0.98 0.56 0.87 0.4439 Nd2O3–Ti2O3 Yes 0.98 0.67 0.82 0.5340 Nd2O3–V2O3 Yes 0.98 0.64 0.83 0.5141 Pr2O3–Fe2O3 Yes 0.99 0.55 0.88 0.4442 Pr2O3–Mn2O3 Yes 0.99 0.58 0.86 0.4643 Pr2O3–Ni2O3 Yes 0.99 0.56 0.87 0.4444 Sm2O3–Fe2O3 Yes 0.96 0.55 0.87 0.4445 Sm2O3–Ni2O3 Yes 0.96 0.56 0.86 0.4446 Sm2O3–Ti2O3 Yes 0.96 0.67 0.81 0.5347 Tb2O3–Fe2O3 Yes 0.92 0.55 0.85 0.4448 Tb2O3–V2O3 Yes 0.92 0.64 0.81 0.5149 Tm2O3–Fe2O3 Yes 0.88 0.55 0.84 0.4450 Tm2O3–V2O3 Yes 0.88 0.64 0.80 0.5151 Yb2O3–Fe2O3 Yes 0.86 0.55 0.83 0.4452 Yb2O3–V2O3 Yes 0.86 0.64 0.79 0.5153 Y2O3–Cr2O3 Yes 0.9 0.62 0.81 0.4954 Y2O3–Fe2O3 Yes 0.9 0.55 0.84 0.4455 Lu2O3–Al2O3 Yes 0.86 0.54 0.83 0.4356 Y2O3–Ti2O3 Yes 0.9 0.67 0.79 0.5357 Gd2O3–Ti2O3 Yes 0.94 0.67 0.81 0.5358 Yb2O3–Al2O3 Yes 0.86 0.54 0.83 0.4359 La2O3–Al2O3 Yes 1.03 0.54 0.90 0.4360 Nd2O3–Al2O3 Yes 0.98 0.54 0.88 0.4361 Bi2O3–Fe2O3 Yes 1.03 0.55 0.89 0.4462 La2O3–Co2O3 Yes 1.03 0.55 0.89 0.4463 La2O3–Cu2O3 Yes 1.03 0.73 0.81 0.5864 La2O3–Ni2O3 Yes 1.03 0.56 0.89 0.4465 Dy2O3–Fe2O3 Yes 0.91 0.55 0.85 0.4466 Ce2O3–Al2O3 Yes 1.01 0.54 0.89 0.4367 Ce2O3–Cr2O3 Yes 1.01 0.62 0.85 0.4968 Eu2O3–Cr2O3 Yes 0.95 0.62 0.83 0.4969 Gd2O3–Cr2O3 Yes 0.94 0.62 0.83 0.4970 Lu2O3–Fe2O3 Yes 0.86 0.55 0.83 0.4471 Nd2O3–Cr2O3 Yes 0.98 0.62 0.84 0.4972 Sm2O3–V2O3 Yes 0.96 0.64 0.83 0.5173 Bi2O3–Al2O3 Yes 1.03 0.54 0.90 0.4374 Dy2O3–Cr2O3 Yes 0.91 0.62 0.82 0.4975 Er2O3–Cr2O3 Yes 0.89 0.62 0.81 0.4976 Yb2O3–Cr2O3 Yes 0.86 0.62 0.80 0.4977 Ho2O3–Cr2O3 Yes 0.89 0.62 0.81 0.4978 Pr2O3–Al2O3 Yes 0.99 0.54 0.88 0.4379 Ho2O3–Cr2O3 Yes 0.89 0.62 0.81 0.4980 Tm2O3–Al2O3 Yes 0.88 0.54 0.84 0.4381 La2O3–Y2O3 Yes 1.03 0.9 0.75 0.7182 Tm2O3–Cr2O3 Yes 0.88 0.62 0.80 0.4983 Lu2O3–Cr2O3 Yes 0.86 0.62 0.80 0.4984 Gd2O3–Al2O3 Yes 0.94 0.54 0.86 0.4385 Sm2O3–Al2O3 Yes 0.96 0.54 0.87 0.4386 La2O3–Ga2O3 Yes 1.03 0.62 0.86 0.4987 Eu2O3–Al2O3 Yes 0.95 0.54 0.87 0.4388 Er2O3–Al2O3 Yes 0.89 0.54 0.84 0.4389 Y2O3–Al2O3 Yes 0.9 0.54 0.85 0.4390 Dy2O3–Al2O3 Yes 0.91 0.54 0.85 0.4391 Nd2O3–Ga2O3 Yes 0.98 0.62 0.84 0.4992 Gd2O3–Ga2O3 Yes 0.94 0.62 0.83 0.4993 Eu2O3–Ga2O3 Yes 0.95 0.62 0.83 0.4994 Ho2O3–Al2O3 Yes 0.89 0.54 0.84 0.43


Fig. 2. (a) Perovskite formability in A+B5+O3 oxide systems. (b) Perovskite formability in A2+B4+O3 oxide systems. (c) Perovskite formability inA3+B3+O3 oxide systems.


Fig. 2. (Continued ).

4. rA–rB structural map for perovskite formability

According to Muller and Roy’s proposal[1], the rA–rBstructural map is a powerful tool in studying the ternaryoxide stability. First of all, we use more data which is cur-rently available to further test Muller and Roy’s regularitiesof perovskite formability[1]. To simplify the problem, these197 binary oxide systems are divided into three groups,A2O–B2O5 group, AO–BO2 group and A2O3–B2O3 group,based on the valence of ions A and B, and are investigatedby the structural map respectively as seen inFig. 2a–c. Itis shown that points representing perovskites and those ofnon-perovskites are located in distinctively different zones.

Encouraged by the above successes, three groups are com-bined into one as shown inFig. 3. It is astonishing that evenin this mixed group, the separation of compounds is reservedwell. This may imply that the stability of perovskite ABO3is mainly determined by the radii of cation A and B, andthe valence of ions A and B have less effect on the forma-bility of perovskites. All systems inFig. 3 has the commonanion O, effect of anion size on the perovskite stability maybe overlooked. On the other hand, the tolerance factor is awidely used parameter in perovskite study, which takes theanion radii into consideration. We further try to build up astructural map containing the tolerance factor that may lead

to simple and more effective new criteria for the perovskiteformability.

5. Tolerance factor and its limit

Goldschmidt’s tolerance factort has been widely acceptedas a criterion for the formation of the perovskite structure[1,2,6]. It can be defined as:

t = rA + rO√2(rB + rO)

And it is known that almost all perovskites have at valueranging from 0.75 to 1.00. However, it seems thatt =0.75–1.00 is a necessary but not a sufficient condition forthe formation of the perovskite structure. Even in the rangeof t = 0.8–0.9 which is the most favourable value for per-ovskites, there exist many systems that do not have anyperovskite structure[17], see Cu2O–P2O5, Li2O–As2O5,MnO–GeO2, Al2O3–B2O3 in Table 1.

Additional parameters may, therefore, be necessary for theprediction of perovskite formation. It is well known that[4]the octahedron BO6 is the basic mosaic or unit for perovskitestructure. If one cation and six anions form an octahedron,the ratio of their ionic radius (rB/rO) is reported within a


Fig. 3. Classification of perovskite compounds in 197 binary oxide systems.

limited range[23,24], this factor is defined as the octahedralfactor. So it is natural to try to construct a structural map bythe tolerance factor and the octahedral factor for perovskiteformability.

As shown inFig. 4, all perovskites and non-perovskitesare located in two different regions, and a clear border be-tween two kinds of compounds is identified. The criterion ofperovskite formability is, then, expressed by the followingequations:rB

rO= 0.425 (4)

rA + rO√2(rB + rO)

= 1 (5)

−4.317(rA + rO)√2(rB + rO)

+ 3.912= rB

rO(6)

For all 197 systems, only one system, MnO–GeO2,is wrongly discriminated as a “perovskite”, which is a“non-perovskite”.

6. Discussion

As noticed inFig. 3andTable 1, there exists a lowest limitof radii of cation B, 0.53 Å. If ion B is smaller than 0.53 Å,

one can never expect a stable perovskite even this systemhas a very favourable tolerance factor, i.e., Cu2O–P2O5 sys-tem (radius of P5+ is 0.38 Å), Li2O–As2O5 (radius of As5+is 0.46 Å) and MgO–SiO2 (radius of Si4+ is 0.40 Å). Thismay be explained as follows: in perovskites, the octahedronBO6 is the basic unit, if ion B is too small, this unit maybecome unstable. So does the perovskite. This can also bebetter explained by the ratio of cation and anion size,rc/ra,in the octahedron. The reported lowest limit ofrc/ra for oc-tahedron[23,24] is 0.414. In our structural map spannedby the tolerance factor and the octahedral factor (seen inFig. 4), the lowest limit of the octahedral factor (rB/rO) forperovskite formation is 0.425, these two values agree wellwith each other.

Although the criterion mentioned-above cannot correctlyclassify the CaO–MnO2 system, it is noticed in theFig. 4that this system locates at the boundary zone. Furthermore,some points near the boundary between perovskite andnon-perovskite inFig. 4, i.e. Eu2O3–In2O3, MnO–TiO2,MnO–GeO2, SrO–GeO2, PbO–GeO2, CdO–GeO2, andCaO–GeO2 system, can undergo perovskite–non-perovskitestructure transformation (all of them[1,25,26] can formperovskites under higher pressures). Also it is observed thatsome antimodides and bismuthades do not obey this regu-larity, seen inFigs. 2a, 3 and 4. Such as in the Na2O–Sb2O5,Na2O–Bi2O5 and Ag2O–Bi2O5 systems, where favourite


Fig. 4. Classification of perovskite compounds (octahedral factor vs. tolerance factor).

values of the tolerance factor and the octahedral factor areobserved, they do not form perovskite but ilmenite[1].It is not clear why our model does not work for systemscontaining Sb2O5, and Bi2O5.

It can also be seen fromFigs. 3 and 4that, whether per-ovskites can be formed in a binary oxide system is almostdecided only by the size of two cations, and their valencehave few effect on this formability. This result is consistentwith the previous report[27].

Goldschmidt’s tolerance factort has been widely acceptedas a criterion for the formation of the perovskite structure,and it is known that almost all perovskite compounds havet values ranging from 0.75 to 1.00. However, the tolerancefactor t is a necessary but not a sufficient condition for theformation of the perovskite structure. On the other hand, oc-tahedron BO6 is the basic unit of perovskites, so the octa-hedral factorrB/rO that related to the stability of octahedronBO6 is another governing parameter for the formability ofperovskites.

Although Muller and Roy’srA–rB structural map hasproven useful for gross structural separation, the regionsbordering different structural types are not well defined[28]. In order to solve this problem, in 1990s, Giaquintaand Loye[28] proposed a new structural map for predom-inantly A2O3–B2O3 group, which relies on the combina-tion of ionic radii and bond ionicities. With their model,

they successfully predicted two compound’s crystal struc-ture. Comparing to Giaquinta and Loye work, our modelhas the following features. (1) Two parameters that areused in our model (tolerance factor and octahedral factor)are physically meaningful. (2) Unlike predominately theA2O3–B2O3 group, our model works well for all threegroups (A2O3–B2O3, A2O–B2O5 and AO–BO2 group). (3)For perovskite formability, our model gives a simple and ef-fective prediction criterion, which may be easily adopted inthe design of advanced materials with perovskite structure.

7. Conclusions

In this research, 197 binary oxide systems, includingA2O–B2O5, AO–BO2 and A2O3–B2O3 groups, are col-lected, of which, only 121 systems can form perovskites.Using these samples, regularities of formability of per-ovskites are investigated by two structural maps. One mapis drawn by the ionic radii of cation A and B, another isspanned by the tolerance factor,(rA + rO)/

√2(rB + rO),

and the octahedral factor,rB/rO. Through this study, thefollowing conclusions are made:

1. In the rA–rB structural map, the systems with per-ovskite and those without perovskite are distributed


in distinctively different regions, there exists a clearboundary between these two kinds of samples.

2. The tolerance factor is a necessary but not sufficient con-dition for perovskite formability, another necessary pa-rameter is the octahedral factor. Using these two factorsthe perovskites formability can be reliably predicted.

3. In the structural map consisting of the tolerance factor,(rA + rO)/

√2(rB + rO), and the octahedral factor,rB/rO,

the points for perovskites and those for non-perovskitesare located in different zones via a clear boundary definedby:

rB

rO= 0.425 (4)

rA + rO√2(rB + rO)

= 1 (5)

−4.317(rA + rO)√2(rB + rO)

+ 3.912= rB

rO(6)

These equations form the criterions for perovskite forma-bility. Among 197 samples, only one system is mispre-dicted. This simple model may be applied to design newsubstrate or buffer materials in compound semiconductorepitaxy.

References

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http://www.chemistry.ohio-state.edu/~mlufaso/spuds/index.html

form ability of abo3 perovskites

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