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disorder in complex perovskites
Journal of Physics and Chemistry of Solids 61 (2000) 1519–1527 www.elsevier.nl/locate/jpcs
Relationship between ionicity, ionic radii and order/disorder in complex perovskites A.A. Bokov a, N.P. Protsenko b, Z.-G. Ye a,* a
Department of Chemistry, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 b Institute of Physics, Rostov State University, Rostov-on-Don 344090, Russian Federation Received 23 June 1999; accepted 5 December 1999
Abstract Analysis of the available structural data on the ordering of B-site cations in complex perovskites with the general formula A2B VB IIIO6 has been performed. Low ionization potentials of B-site cations are found to promote their ordering. A linear relation between the order–disorder phase transition temperature Tt and the sum of ionization potentials of B-site cations is observed in Pb2B VB IIIO6 perovskites. It can be used for predicting Tt of other complex compounds. The electronegativity equalization method (EEM) has been used to determine the effective ionic charges and the ordering energies. The results of simulation support the correlation disclosed. In contrast to what was previously postulated, the influence of the size of B-site cations on the ordering is shown to be negligible. The determining driving mechanism responsible for the ordering is found to be the electrostatic, rather than the elastic, interactions between the B-site ions. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keywords: D. Crystal structure; D. Phase transitions; A. Ceramics; A. Oxides
1. Introduction The perovskite structure (with general formula ABO3) can be described as a framework of oxygen octahedra sharing corners [1,2]. The B-cations are located in the centres of octahedra and the interstitial sites are occupied by the A-cations. In complex perovskites two (or more) cations of different valences are located in the equivalent crystallographic positions. Among the A2B 0 B 00 O6 complex perovskite oxides, where B 0 and B 00 are the cations of different valences, the representatives with ordered as well as disordered arrangement of B 0 and B 00 are frequently encountered [1,2]. In some oxides the degree of disorder can be changed by thermal annealing [3–7]. The perovskite materials designed for dielectric, piezo-, ferroelectric, optical and other applications exhibit a large range of interesting properties resulting from the intricate ordered/disordered structures. To control the properties it is important to know the factors determining the degree of disorder. It * Corresponding author. Tel.: ⫹ 1-604-291-3351; fax: ⫹ 1-604291-3765. E-mail address: [email protected] (Z.-G. Ye).
was generally believed that the ordered state is established primarily due to the charge difference between the two Bsite cations but the difference in their radii is also important. This viewpoint was formulated in the early papers [8,9] and has been shared by the majority of authors for many years [1,2,10–13]. Complex perovskites with a large difference in the valence between B 0 and B 00 are strongly driven towards an ordered state by electrostatic forces (only the ordered arrangement of cations agrees with the Pauling’s fourth rule [1]). Such compounds are always completely ordered. In the isovalent solid solutions the ordering was not observed as a rule because of the small electrostatic ordering energy. This energy calculated in the case of 1:1 ordering [14] appeared to be proportional to the square of B-site cationic charge difference. The other factor expected to be important is the difference in ionic radii between B 0 and B 00 cations DR 兩RB 0 ⫺ RB 00 兩: It was found [8,9] for A2B VB IIIO6 perovskites, where A Ba, Ca, Sr, B V B 0 stands for pentavalent Ta and Nb cations, and B III B 00 stands for all possible trivalent cations, that the critical difference in radius for ordering is about 9%. The influence of DR on the ordered superstructure arises from the requirement of
0022-3697/00/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(00)00004-4
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closest packing for oxygen octahedra populated by Bcations of different size [1]. Thus the driving forces for ordering by this mechanism are elastic in nature. Recently on the basis of an analysis of order–disorder phase transition temperatures in some Pb2B VBIIIO6 perovskites, we proposed that the difference in ionic radius is not an important factor for ordering, but the degree of ionicity should be taken into account instead [15]. Another indication of the negligible role of elastic interactions in the ordering process comes from the work by Bellaiche and Vanderbilt [16], who successfully explained various types of compositional order observed in a large class of complex perovskites, by assuming that the driving mechanism responsible for ordering is simply the electrostatic one. To elucidate the possible contribution from different mechanisms to the ordering in complex perovskites, we have reviewed the published literature and analyzed the structural data for the perovskites of the general formula A2B VB IIIO6 in this work. It has been found that the influence of size difference of B-type cations on the ordering is negligible. A correlation between the ordering state and the ionization potentials of cations has been proposed. Calculations of the ionic charges and the electrostatic ordering energy have been performed using the electronegativity equalization method to explain the correlation.
2. Analysis of the ordered and disordered perovskites The structural data for more than 300 complex perovskites with the general formula A2B VB IIIO6 are collected from the literature [8,17–46]. Table 1 presents a small section of the database. Most of the references are taken from the books by Fesenko [1] and Galasso [2]. The data for some compounds classified as “unreliable” in these books are excluded from further consideration if more recent data are not available. In the majority of works the structural data were obtained from X-ray diffraction. The Xray detection of ordering can be difficult in the case of small difference in scattering factor between the two B-site cations. This sometimes leads to the discrepancy between the data obtained by different authors. For example, Brixner [45] found a disordered structure in barium rare-earth niobates. Galasso and Darby [8] showed, however, that the disordered structure was due to the exposure time of X-ray photographs which was not long enough. They observed instead an ordering in all these compounds, except Ba2NbCeO6 which was not investigated in their work. Thus we consider all other reported barium rare-earth niobates as ordered ones, except for Ba2NbCeO6 which is excluded from our consideration because of unreliable data. We deal with other cases of discrepancy in the same manner. Fig. 1 illustrates the relation between the ordering and the size of B-site cations for all the A2B VB IIIO6 perovskites considered. Shannon’s ionic radii [47,48] are used. Every horizontal row of the points in the figure represents the
relative difference in ionic radii DR/RBmin (RBmin is the radius of the smaller of the two B-site cations) for compounds with the same A and B V cations. It can be seen that in most cases, the ordered compounds possess large DR. However, among the compounds which have small (or even zero) DR, ordered representatives can be found alongside with the disordered ones. In these compounds the sizes of oxygen octahedra around different B-site ions should be approximately the same and the elastic driving forces towards ordering should be small. Ba2PuInO6 is a good example. The interatomic Pu–O and In–O distances were found by X-ray diffraction ˚ ) in this perovskite [28], method to be the same (2.12 A nevertheless the compound is ordered. This means that the electrostatic energy is high enough to force the structure to order. The role of the elastic interactions in the ordering thus becomes questionable. As electrostatic and elastic energy increases with disordering, all the A2B VB IIIO6 materials should have the minimal potential energy in the ordered state. But the thermal motion is capable of destroying the ordering and the structural order–disorder phase transition occurs at some nonzero temperature Tt. This kind of phase transition was observed in several lead-containing complex perovskites, as listed in Table 2. Since the temperature Tt is proportional to the ordering energy [56], an increase of Tt with DR could be expected if the influence of ionic size on the ordering processes was essential. However, no such correlation can be observed (see Table 2 and Fig. 2). Furthermore, the highest Tt has been found in Pb2TaScO6 which has the smallest DR. Therefore, it is clear that some factors other than ionic size exert effects on the ordering. Based on our previous work [15], we have analyzed the correlation between the order/disorder state, the third ionization potential of the trivalent B-site cation (I3) and the fifth ionization potential of the pentavalent B-site cation (I5). The values of I3 and I5 are taken from Refs. [48,57]. The values of ionization potential for Re, Os, Pa, U, Pu and Am are not available in the literature. They are estimated by means of the method described in Appendix A and listed in Table 1. The dependence of Tt on the sum of ionization potentials I3 ⫹ I5 for the Pb-containing compounds is shown by squares in Fig. 3. The same compounds as in Fig. 2 are considered. The dependence can be interpolated as Tt Tt0 ⫺ at I3 ⫹ I5
1
where the temperature and ionization potentials are expressed in K and eV, respectively, Tt0 5648 K; at 56 K=eV: Relation (1) is shown by solid line in Fig. 3. It can be used for predicting order–disorder phase transition temperature Tt in lead-containing Pb2B VB IIIO6 perovskites. The values of Tt calculated with the help of Eq. (1) are presented in Table 2. The only significant disagreement between the calculated and experimental results is found in Pb2NbLuO6. The possible reasons for this discrepancy will be discussed in the next section, but here we underline that it could not be related to the size effect. The DR/RBmin
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Table 1 Data on ordering state at room temperature, preparation temperatures (Ts1 and Ts2) and parameters of B-site cations for selected A2B VB IIIO6 perovskites: ionic radii of pentavalent (RB 0 ) and trivalent (RB 00 ) cations, relative difference in ionic radii (DR/RBmin), fifth ionization potentials for pentavalent cations (I5) and third ionization potentials (I3) for trivalent cations Compound
˚) RB 0 (A
˚) RB 00 (A
DR/RBmin
I5 (eV)
I3 (eV)
Ordering
Ts1 (⬚C)
Ts2 (⬚C)
Ref.
Ca2TaCrO6 Ca2TaVO6 Ca2TaMnO6 Ca2NbVO6 Sr2OsScO6 Sr2ReHoO6 Sr2ReGaO6 Sr2SbGaO6 Sr2SbFeO6 Sr2SbMnO6 Sr2NbCrO6 Sr2UInO6 Ba2NbCoO6 Ba2NbNiO6 Ba2BiCrO6 Ba2BiYbO6 Ba2BiYO6 Ba2BiDyO6 Ba2BiTbO6 Ba2PuInO6 Ba2UScO6 Ba2PaInO6
64 64 64 64 58 58 58 60 60 60 64 76 64 64 76 76 76 76 76 74 76 78
62 64 65 64 75 90 62 62 64.5 65 62 80 61 60 62 87 90 91 92 80 75 80
0.032 0 0.016 0 0.29 0.55 0.069 0.033 0.075 0.083 0.032 0.053 0.049 0.067 0.23 0.145 0.184 0.197 0.211 0.081 0.013 0.026
45 45 45 50 52.7 50.6 50.6 56 56 56 50 40.1 50 50 56 56 56 56 56 42.5 40.1 42.5
30.95 29.31 33.69 29.31 24.75 22.84 30.7 30.7 30.64 33.69 30.95 28.03 33.49 35.16 30.95 25.03 20.5 22.8 21.91 28.03 24.75 28.03
Disordered Ordered Disordered Ordered Ordered Ordered Ordered Ordered Ordered Disordered Ordered Ordered Disordered Disordered Disordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered
1100 1150 1100 1150 1000 1000 1000 1100 1000 1000 1100 1000 1000 1000 1000 800 900 900 800 750 1000 1350
1400
[17] [18] [17] [18] [19] [20] [19] [21] [22] [22] [22] [23,24] [22] [22] [22] [25] [26,27] [26] [25] [28] [23] [29]
1400
1300 1300 1300 1300 1300 1300 1000 1075 1075 1000 950
Fig. 1. Relative difference in radii of B III and B V cations (DR/RBmin) for ordered (triangles) and disordered (circles) A2B VB IIIO6 complex perovskites.
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Table 2 Temperatures of order-disorder phase transition (Tt) calculated using Eq. (1) and experimentally found in perovskite compounds, and parameters of B-site cations: ionic radii of pentavalent (RB 0 ) and trivalent (RB 00 ) cations, relative difference in ionic radii (DR/RBmin) and ionization potentials for pentavalent (I5) and trivalent (I3) cations Compound
Pb2NbScO6 Pb2NbInO6 Pb2NbYbO6 Pb2NbLuO6 Pb2NbTmO6 Pb2NbErO6 Pb2TaScO6 Pb2TaInO6 Pb2TaYbO6 Pb2TaLuO6 Pb2TaTmO6
˚) RB 0 (A
64 64 64 64 64 64 64 64 64 64 64
˚) RB 00 (A
75 80 87 86 88 89 75 80 87 86 88
DR/RBmin
0.17 0.25 0.36 0.34 0.38 0.39 0.17 0.25 0.36 0.34 0.38
value is comparatively large in Pb2NbLuO6. This should lead to an enhanced value of Tt, but in fact the experimental Tt is smaller than the calculated one. In general the decrease of ionic radius leads to an increase of ionization potentials. That is why the analysis of data on a limited number of A2B VBIIIO6 perovskites containing B V cations with close I5 values (Ta and Nb) performed in earlier works [8–10] revealed some dependence of ordering state on DR. However, the ionization potentials do not correlate with atomic radii exactly (see, for example, Tables 1 and 2). Thus if DR influenced ordering considerably, relation (1) would be violated. The validity of this relation shows that the effect of the ionic size on Tt is negligibly small, and the elastic interactions associated with the difference in size between B-site cations do not contribute considerably to the interactions responsible for order/disorder behavior. Let us now consider the materials in which the ordering temperature Tt has not been measured experimentally. The degree of order at room temperature should depend
Fig. 2. Temperatures of order–disorder phase transition (Tt) versus relative difference in radii of B III and B V cations for Pb2B VB IIIO6 complex perovskites.
I5 (eV)
50 50 50 50 50 50 45 45 45 45 45
I3 (eV)
24.75 28.03 25.05 20.96 23.68 22.74 24.75 28 25.05 20.96 23.68
Tt (⬚C) Calc.
Exp.
Ref.
1189 1005 1172 1401 1249 1302 1469 1287 1452 1681 1529
1210 1020 1120 1130–1140 ⬎ 1200 ⬎ 1050 1450–1500 ⬎ 1070 ⬎ 1260 ⬎ 1200 ⬎ 1100
[49] [7,50] [51] [52] [53] [53] [49] [6,54] [53,55] [53] [53]
both on Tt and the temperature of ceramic preparation Ts [3,15] (all the materials considered here were prepared in ceramic form). If Ts p Tt the material should be completely ordered. If Ts ⬎ Tt ; it should be disordered at high temperature immediately after formation. During the subsequent cooling below Tt it can remain disordered or become ordered, depending on the rate of ordering inherent in the particular material. If this material dose not change the order degree during the cooling process, it remains disordered at room temperature forever. The reason for this is that the ordering implies the site exchange between B III and B V cations. It is a relaxation process with practically infinite
Fig. 3. Temperatures of order–disorder phase transition (Tt) for Pb2B VB IIIO6 complex perovskites (squares) and synthesis temperatures (Ts) for all other ordered (triangles) and disordered (circles) A2B VB IIIO6 complex perovskites versus sum of ionization potentials of B-site ions I3 ⫹ I5 :
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characteristic time at low (room) temperature. Thus the properties of materials at temperatures around Ts should be taken into account to determine the ordering state. The temperatures of synthesis are presented in Table 1 and plotted against I3 ⫹ I5 in Fig. 3 for all considered A2B VB IIIO6 compounds. In some works, the preparation of materials was performed by sequential heating up to two different temperatures Ts1 and Ts2 (⬎Ts1) in order to ensure complete solid state reactions. It is known [15,51] that the ordered (if Ts1 ⬍ Tt ) structure formed during synthesis at Ts1 becomes, after subsequent firing at Ts2, less ordered in some cases or remains unchanged in other cases. It depends on the rate of ordering process in specific material and the duration of firing at Ts2. Thus the degree of disorder in the final material is hardly predictable even if Tt can be estimated. It can be expected to be between the values corresponding to the equilibrium state resulting from the thermal treatments at Ts1 and Ts2, respectively. When plotting Fig. 3, for convenience in further discussion we took Ts Ts1 for ordered materials (shown by triangles) and Ts Ts2 for disordered ones (shown by circles). We can interpret the data in Fig. 3 by assuming that Tt follows relation (1) for all A2B VB IIIO6 compounds. As discussed above, only those materials can be disordered which were heated in the course of preparation above Tt. In agreement with this statement all disordered states (circles) are located in the top right part of Fig. 3 (in Ts ⬎ Tt area). Synthesis at temperatures below Tt should lead to an ordered structure. Therefore, the ordered states (triangles) can only be located in the bottom left part of Fig. 3 (in Ts ⬍ Tt area). Some triangles are found in the Ts ⬎ Tt area. It is possible that these compounds had been synthesized in disordered states, but the ordering process occurred during slow cooling after formation, leading to an ordered structure. Many peculiarities of order/disorder behavior can be clarified by taking into account the values of ionization potentials of B-site cations. For example, Ba2BiCrO6 is a disordered material in spite of its DR/RBmin value larger than that of several ordered Ba2BiB IIIO6 materials (see Table 1). This can be explained by the larger value of I3 in Ba2BiCrO6. The disordered state in Sr2SbMnO6, which has a larger DR/RBmin value than the ordered Sr2SbGaO6 and Sr2SbFeO6, can be explained in the same manner, so does the ordering in Ca2TaVO6 which has smaller DR/RBmin than the disordered Ca2TaCrO6 and Ca2TaMnO6. All the compounds containing Pu, Pa and U were found to be ordered [23,24,28,29], even those having small DR. According to our analysis this is due to the small values of I5 for these elements. In fact, Awasthy and co-workers [24,28] compared Ba2UB IIIO6, Sr2UB IIIO6 and Ba2PuB IIIO6 oxides (where B III are different trivalent cations), and found that in plutonium compounds the degree of ordering was lower than in uranium ones. We can explain this fact by the smaller I5 value of U than Pu. In the next section we give the theoretical justification for
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the correlation between ionization potentials and the order/ disorder state disclosed in complex perovskites.
3. Calculations and modeling We propose that the electrostatic interactions between the differently charged ions are the determining driving forces responsible for the ordering. Accordingly, the ordering energy Uod is merely the difference between the Madelung energies in the ordered and the disordered states. The effective charges of ions with the same oxidation state are different in unlike compounds (in other words the ionicities are different). This leads to the difference in Madelung energies. Thus to compare the ordering energies Uod in some compounds, it is necessary to estimate the ionic charges. We use semiempirical density functional electronegativity equalization method (EEM) to determine the effective ionic charges and Madelung energies and to confirm theoretically the correlation between Uod and the ionization potentials of constituting cations. The EEM is based on the definition of electronegativity x as the negative of the chemical potential of electronic cloud [58,59]. The electronegativity of an isolated atom a can be written as
x0a ⫺
2Ea0 2Ea0 2Na 2qa
2
where Ea0 is the electronic energy of the atom, Na Za ⫺ qa is the number of electrons on the atom, Za and qa are the atomic number and ionic charge, respectively. The formation of a molecule is accompanied by the transfer of appropriate, not necessarily integral, charge from one atom to another. The driving force for the charge transfer is the chemical potential (or electronegativity) difference [58,60]. In the final state the electronegativities of all the participating atoms are equalized. After that, the expression for the effective electronegativities of the atoms in the molecule (crystal) is:
xa
2Ea x ; 2qa
a 1; 2; …; k
3
where x is the average molecular electronegativity, k is the number of unlike atoms and Ea is the electronic energy of ion a in the molecule. Ea in the EEM approximation can be expressed [61,62] as a sum of the energy of the isolated atom Ea0 ; the additional energy E 0a associated with the changes of size and shape of electron cloud due to the confinement of the atom in the molecule (it includes the covalent bonding effects), and the energy of interaction with surrounding atoms E 0 0a : Ea Ea0 ⫹ E 0a ⫹ E 00a
4
The first two terms of this equation can be approximated
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Table 3 Parameters for the expression of electronegativity of isolated atom (in eV)
c0a b0a a0a a
Pb
Nb
Ta
Fe
In
Yb
Sc
Oa
0 3.805 3.615
⫺0.1683 8.145 ⫺12.487
0.1833 3.750 0.0667
1.0298 1.0533 5.819
⫺0.6539 8.5033 ⫺2.063
1.1587 0.515 5.610
0.9531 0.2597 5.348
0 8.7836 9.3187
Parameters a0a ⫹Daa ; b0a ⫹ Dba and c0a ⫹ Dca are given for oxygen.
by a smooth function of the atomic charge [62–64] Ea0 qa
a0a qa
⫹
b0a q2a
⫹
5
c0a q3a
Ea0 qa ⫹ E 0a qa a0a ⫹ Daa qa ⫹ b0a ⫹ Dba q2a ⫹ c0a ⫹ Dca q3a
6
In the case of ionic crystals (which is considered in this work) E 00a is a Madelung-type energy. The Madelung potential at the position of atom a can be written in a suitable form for further calculations [65]:
wa
X 2E 00a ql Fal 2qa l
7
where Fa l are independent of charge coefficients and the summation is extended over all the different sublattices (ql are the charges of ions located in sublattice l). By substituting Eqs. (4), (6) and (7) for the terms in Eq. (3) we obtain the final set of equations X a0a ⫹ Daa ⫹ 2 b0a ⫹ Dba qa ⫹ 3 c0a ⫹ Dca q2a ⫹ ql Fal l
Fal Aal =a;
x ;
a 1; 2; …; k
(8)
The self-consistent solutions of these k simultaneous equations alongside with the electroneutrality condition X qa 0 9 a
give the values of effective charges qa for k unlike ions in the studied structure and the value of their common electronegativity x : To obtain the solution, the polynomial coefficients should be known. The coefficients Fa l can be found for a specific crystal structure by Ewalds’ method [56]. The parameters a0a ; b0a and c0a can be calculated using the known values of ionization potentials Ina and electron affinities A a of atom a , and taking into account the charge q for integer values q n; we have Ea0 n Ea0 n ⫺ 1 ⫹ Ina ;
Ea0 ⫺1 Aa
This procedure has not been applied to perovskite-type compounds up to now. Chibotaru et al. [65] recently performed the EEM calculations for several cubic ABO3 perovskites under the condition Daa Dba Dca 0 for all cations. The only ion subjected to parameterization was oxygen. The authors obtained the values of ionic charges, which are in agreement with the results of self-consistent quantum chemistry calculations. The same approach is applied to the cubic complex perovskites A2B VB IIIO6 in this work. To calculate the Madelung potentials we use the superposition method developed for complex perovskites by Rosenstein and Schor [66]. The ordered structure A2B 0 B 00 O6 with B-ions having different charges qB 0 and qB 00 is considered as a superposition of a perovskite structure with the average charges qav qB 0 ⫹ qB 00 =2 in B-positions upon a rocksalt structure with qB 0 ⫺ qav and qB 00 ⫺ qav charges. In the case of disordered structure the Madelung energy is the same as for ABO3 perovskite with qB qav : The parameters for the calculation of Madelung potentials in perovskite structure can be found as [65]:
10
The corrections Daa , Dba and Dca which represent changes in the isolated atom coefficients due to the confinement of the atom in a crystal, can be calibrated so as to reproduce the results of quantum-mechanical ab initio calculations [61].
Aal Ala ; Aaa 0;
where a is the lattice parameter expressed in angstroms, AAB 58:57; ABO 78:886; AAO AoO 0 64:884; O and O 0 denote two of the three oxygen sublattices. For rocksalt structure the Madelung constant is AB 0 B 00 50:33: Thus we use the following expressions for calculating the Madelung potentials in A2B 0 B 00 O6 structure:
wA
1 q A ⫹ 3qO AAO a av AB
1 q A ⫹ 3gB 0 qO ABO ⫹ qB 00 ⫺ qav AB 0 B 00 a A AB 1 a qA AAB ⫹ 3qO ABO ⫹ qB 0 ⫺ qav AB 0 B 00 a a ⫹ 2x
wB 0 wB 00
wO
1 q ⫹ 2qO AOO ⫹ qav ABO a A
The factors gB 0 and gB 00 are introduced to take into account the shift x of oxygen ions from the symmetric positions due to the difference of B 0 and B 00 cations in sizes. We assume as a first approximation that gB 0 a= a ⫺ 2x; gB 00 a= a ⫹ 2x: The values of x are taken from Ref. [67]. The coefficients a0a ; b0a and c0a for cations are calculated
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Table 4 Lattice parameters (a), shifts of oxygen ions from the symmetric positions (x), effective ionic charges (q), Madelung potentials (w ), ordering energies (Uod) and sums of ionization potentials of B-site cations I3 ⫹ I5 for selected ordered complex perovskites Compound
˚) a (A
˚) x (A
qA
qB 0
qB 0
qO
w A (eV)
w B 0 (eV)
w B 00 (eV)
w O (eV)
Uod (kJ/mol)
I3 ⫹ I5 (eV)
Pb2NbFeO6 Pb2NbInO6 Pb2NbYbO6 Pb2TaScO6
8.04 8.22 8.3 8.16
0 0.084 0.106 0.047
1.44 1.13 1.14 1.04
3.20 3.42 3.27 4.52
2.47 2.89 2.24 3.09
⫺1.42 ⫺1.43 ⫺1.30 ⫺1.62
⫺13.83 ⫺11.36 ⫺11.05 ⫺11.24
⫺33.73 ⫺36.44 ⫺34.10 ⫺44.93
⫺29.18 ⫺29.81 ⫺24.05 ⫺33.92
16.43 16.64 14.81 19.39
79.80 85.32 250.08 381.22
80.64 78.04 75.04 49.5
from the fitting of ionization energies (10) with the function (5) using least-squares method. The results of fitting are presented in Table 3. The parameters Daa , Dba and Dca are considered to be zero. For oxygen these parameters are calibrated so as to obtain qualitatively correct values for effective ionic charges and ordering energy. The effective ionic charges, Madelung potentials, and electrostatic ordering energies obtained from the solutions of Eqs. (8) and (9) with oxygen parameters listed in Table 3 for some perovskites are given in Table 4. It appears impossible to adjust the unique oxygen parameters suitable for all compounds. For some of them, the calculated charges appear to be larger than the cation’s valence (as in the case of Pb2TaScO6 presented in Table 4). Evidently this discrepancy arises because the corrections of the isolated atomic electronegativity as a consequence of bonding (Daa , Dba and Dca ) were neglected. We do not intend to calculate the true ionic charges, but to find out the general trends in charge variation when the atomic parameters such
Fig. 4. Calculated ordering energy (Uod) for a model A2B VB IIIO6 cubic perovskite versus (a) the third ionization potential of B III cation and (b) the fifth ionization potential of B V cation.
as ionization potentials and ionic radii change. For the compounds presented in Table 4, the calculated trends are in agreement with the experimental observations, namely, the ordering energy increases with the decrease of the average ionization potentials of B-site cations. The calculated changes of Uod in real compounds are associated with the change not only in ionization potentials but also in structural parameters (a and x). Therefore, a model-study of the influence of ionization potential itself on the ordering process is performed in the next step. Pb2NbInO6 is chosen as a model compound. It can be considered a typical representative of A2B VB IIIO6 perovskites because of its nonzero but not very large DR/RBmin. The electronegativity parameters a0a ; b0a and c0a for one of the atoms, In or Nb, are changed arbitrarily around the real values presented in Table 3, while the parameters for all other atoms were kept equal to its real values. The structural parameters a and x also remain unchanged. The effective ionic charges and ordering energies are derived for every set of electronegativity parameters. Then the ionization potentials are calculated from Eqs. (5) and (10) using the same values of parameters. The relations found are depicted in Fig. 4. It can be seen that the rise of ionization potential of trivalent as well as pentavalent cations causes a decrease of ordering energy, as is in full agreement with the experimental observations reported in the previous section of this paper. It should be noted that the same simulation performed with some other values of electronegativity parameters could lead to qualitatively different results. As discussed above, the values of corrections Daa , Dba and Dca are not known, thus it is not possible to perform the calculations with the real ionic electronegativities. Meanwhile the variation of these corrections as a result of ordering (in particular due to the covalent bonding effects) might be expected to contribute to ordering energy. Recent first-principle supercell calculations in A3B 0 2B 00 O9 materials [68] showed that the formation of short A–O bonds due to the covalent effects can sometimes influence the type of 1:2 ordering, but the similar effects on Tt were not studied. Our simulation shows the possibility to obtain the results consistent with most experimental data using the same values of Daa , Dba and Dca parameters for ordered and disordered states, in other words, neglecting the covalent bonding effects. Some rare cases of disagreement between the experimental data and the predictions obtained in terms of our approach are
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understanding of the crystal chemical origin of order/disorder effects, and the microstructure–property relations in complex perovskite materials.
Acknowledgements This work was supported by the Natural Sciences and Engineering Council of Canada (Research Grant) and Simon Fraser University (President Research Grant). Appendix A. Estimation of ionization potentials Fig. 5. Ionization potentials (In) of elements versus their atomic numbers (Z).
It is known that the ionization potentials of elements with atomic numbers close to each other approximately obey the relation
probably related to the covalent bonding effects which appear unusually high in some particular compounds. Therefore, the decrease of ordering energy with increasing ionization potentials of B-site cations, as discovered in the previous section, can be explained in terms of the difference in electronegativities, leading to the difference in ionicities, ionic charges and hence electrostatic interactions between ions.
Ii⫹j Z ⫹ j gj ⫹ h2
4. Conclusions A comprehensive analysis of the available literature data on the ordering of different type B-site cations in complex perovskites with general formula A2B VBIIIO6 has been performed in this work. In contrast to what was previously postulated, the influence of the size of B-site cations on the ordering of complex perovskites is shown to be negligible. Thus the contribution of elastic interactions to the ordering energy is small. A linear relationship between the temperatures of order–disorder phase transitions Tt and the sum of ionization potentials of B-site cations I3 ⫹ I5 has been found in lead-containing oxides. It can be used to estimate Tt for those compounds in which the order-disorder transition has not been observed so far. The value of Tt as well as the ceramic preparation conditions should be taken into account to predict the degree of ordering in complex perovskite materials at room temperature. The calculations of effective ionic charges and ordering energies using the electronegativity equalization method have confirmed our suggestion that the electrostatic interaction between ions is the determining driving mechanism responsible for the ordering in complex perovskites A2B VB IIIO6. Due to the difference in ionicities, the effective charges of the ions with the same oxidation state are different in unlike compounds, leading to the difference in ordering energies, and hence in the temperatures of order– disorder phase transitions. The analysis performed in this work provides a better
A1
where g and h are parameters, i and j are integers, Ii⫹j Z ⫹ j is the i ⫹ jth ionization potential of the element with atomic number Z ⫹ j: For elements with small Z this relation is reported, for example, in Ref. [69]. Fig. 5 illustrates the validity of this relation for the elements with large Z. The dots in this figure are the known ionization potentials, the straight lines are determined by the Eq. (A1) with g and h parameters found by least-squares method (only those lines are shown which are used in further calculations). Thus if the potentials of neighboring elements are known, the unknown ionization potential of an element can be approximated from these equations. The results of approximation of I5 for Re Z 75; Os Z 76; Pa Z 91; U Z 92 and Pu Z 94 are listed in Table 1. It is impossible to find I3 for Am Z 95 because of the absence of data on its neighboring elements.
References [1] E.D. Fesensko, Perovskite Family and Ferroelectricity, Atomisdat, Moscow, 1971 (in Russian). [2] F.S. Galasso, Structure, Properties and Preparation of Perovskite-type Compounds, Pergamon Press, Oxford, 1969. [3] A.A. Bokov, I.P. Rayevsky, Ferroelectrics 90 (1989) 125. [4] N. Setter, L.E. Cross, J. Appl. Phys. 51 (1980) 4350. [5] L.E. Cross, Ferroelectrics 151 (1994) 305. [6] A. Kania, M. Pawelczyk, Ferroelectrics 124 (1991) 261. [7] N. Yasuda, H. Ohwa, J. Oohashi, K. Nomura, H. Terauchi, M. Iwata, Y. Ishbashi, J. Phys. Soc. Jpn 67 (1998) 3952. [8] F. Galasso, M. Darby, J. Phys. Chem. 66 (1962) 131. [9] E.G. Fesenko, V.S. Filipyev, M.F. Kupriyanov, Izv. AN SSSR Ser. Fiz. 28 (1964) 669 (in Russian). [10] N. Setter, L.E. Cross, J. Mater. Sci. 15 (1980) 2478. [11] V.A. Isupov, Izv. AN SSSR Ser. Fiz. 47 (1983) 559. [12] C.A. Randall, A.S. Bhalla, T.R. Shrout, L.E. Cross, J. Mater. Res. 5 (1990) 829. [13] I.W. Chen, P. Li, Y. Wang, J. Phys. Chem. Solids 57 (1996) 1525.
A.A. Bokov et al. / Journal of Physics and Chemistry of Solids 61 (2000) 1519–1527 [14] V.M. Yudin, A.G. Tutov, A.B. Sherman, V.A. Isupov, Izv. AN SSSR Ser. Fiz. 31 (1967) 1798 (in Russian). [15] A.A. Bokov, Ferroelectrics 183 (1996) 65. [16] L. Bellaiche, D. Vanderbilt, Phys. Rev. Lett. 81 (1998) 1318. [17] V.S. Filipjev, E.G. Fesenko, Sov. Phys. Crystallogr. 10 (1966) 243. [18] C. Chauvel, J.-C. Barnier, M. Andre, Comptes Rendus Ser. C 262 (1966) 1424. [19] A.W. Sleight, J. Longo, R. Ward, Inorg. Chem. 1 (1962) 245. [20] G. Baud, M. Capestan, Comptes Rendus Ser. C 267 (1968) 314. [21] A.W. Sleight, R. Ward, Inorg. Chem. 3 (1964) 292. [22] G. Blasse, J. Inorg. Nucl. Chem. 27 (1965) 993. [23] A.W. Sleight, R. Ward, Inorg. Chem. 1 (1962) 790. [24] S.K. Awasthy, D.M. Chachraburtty, V.K. Tondon, J. Inorg. Nucl. Chem. 29 (1967) 1225. [25] W.T.A. Harrison, K.P. Reis, A.J. Jacobson, L.F. Schneemeyer, V. Waszczak, Chem. Mater. 7 (1995) 2161. [26] A. Lenz, Hk. Mueller-Buschbaum, J. Less-Common Met. 161 (1990) L15. [27] S.K. Blower, C. Greaves, Solid State Commun. 68 (1988) 765. [28] S.K. Awasthy, D.M. Chachraburtty, V.K. Tondon, J. Inorg. Nucl. Chem. 30 (1968) 819. [29] C. Keller, J. Inorg. Chem. 27 (1965) 321. [30] E.G. Fesenko, et al., Neorgan. Mater. 6 (1970) 179 (in Russian). [31] V.S. Filipyev, E.G. Fesenko, Kristallografiya 10 (1965) 626 (in Russian). [32] M.F. Kupriyanov, E.G. Fesenko, Izv. AN SSSR Ser. Fiz. 29 (1965) 925 (in Russian). [33] L.I. Shvorneva, Yu.N. Venevtsev, Sov. Phys. JETP 22 (1965) 722. [34] A.I. Agranovskaja, et al., Izv. AN SSSR Ser. Fiz. 24 (1960) 1275 (in Russian). [35] E.G. Fesenko, et al., Neorgan Mater. 6 (1970) 800 (in Russian). [36] F. Galasso, G. Laydan, D. Flinchbaugh, J. Chem. Phys. 44 (1966) 1203. [37] P.G. Casado, A. Mendiola, I. Rasines, Z. Anorg. Allg. Chem. 510 (1984) 194–198. [38] A. Lenz, Hk. Mueller-Buschbaum, J. Less-Common Met. 161 (1990) 141. [39] C. Keller, K.H. Walter, J. Inorg. Nucl. Chem. 27 (1965) 1253. [40] C. Chauvel, J.-C. Barnier, M. Andre, Comptes Rendus Ser. C 263 (1966) 1536. [41] R.J. Roy, J. Am. Ceram. Soc. 37 (1954) 581. [42] M.F. Kupriyanov, V.S. Filipyev, Kristallografiya 8 (1963) 356 (in Russian). [43] V.S. Filipyev, E.G. Fesenko, Kristallografiya 6 (1961) 770 (in Russian).
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[44] A.S. Viskov, AN SSSR Doklady 162 (1965) 323 (in Russian). [45] L.H. Brixner, J. Inorg. Nucl. Chem. 15 (1960) 352. [46] Yu.E. Roginskaya, Yu.N. Venevtsev, G.S. Zhnadov, Sov. Phys. JETP 21 (1965) 817. [47] R.D. Shannon, Acta Crystallogr. A 32 (1976) 751. [48] D.R. Lide (Ed.), CRC Handbook of Chemistry and Physics CRC Press, Boca Raton, FL, 1998. [49] C.G.F. Stenger, A.J. Burggraaf, Phys. Status Solidi (a) 61 (1980) 275. [50] A.A. Bokov, I.P. Rayevsky, V.G. Smotrakov, Sov. Phys. Solid State 26 (1984) 1708. [51] A.A. Bokov, V.Y. Shonov, I.P. Rayevsky, E.S. Gagarina, M.F. Kupriyanov, J. Phys.: Condens. Matter 5 (1993) 5491. [52] A.A. Bokov, I.P. Rayevsky, Ferroelectrics 144 (1993) 147. [53] V.Y. Shonov, Dissertation for the Degree of Candidate of Physical and Mathematical Sciences, Rostov State University, Rostov-on-Don, 1992 (in Russian). [54] A.A. Bokov, I.P. Rayevsky, V.V. Neprin, V.G. Smotrakov, Ferroelectrics 124 (1991) 271. [55] A.A. Bokov, I.P. Raevskii, Sov. Tech. Phys. Lett. 16 (1990) 660. [56] C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1967. [57] K.M. Mackay, R.A. Mackay, W. Henderson, Introduction to Modern Inorganic Chemistry, Blackie Academic and Professional, London, 1996. [58] R.G. Parr, R.A. Donnelly, M. Levy, W.E. Palke, J. Chem. Phys. 68 (1978) 3801. [59] R.A. Donnelly, R.G. Parr, J. Chem. Phys. 69 (1978) 4431. [60] W.J. Mortier, K. Van Genechten, J. Gasteiger, J. Am. Chem. Soc. 107 (1985) 829. [61] W.J. Mortier, S.K. Ghosh, S. Shankar, J. Am. Chem. Soc. 108 (1986) 4315. [62] K.A. Van Genechten, W.J. Mortier, J. Chem. Phys. 86 (1987) 5063. [63] R.P. Iczkowski, J.L. Margrave, J. Am. Chem. Soc. 83 (1961) 3547. [64] C.K. Jørgensen, Oxidation Numbers and Oxidation States, Springer, Berlin, 1969. [65] L.F. Chibotaru, N.N. Gorinchoi, A.O. Solonenko, V.A. Shlyapnikov, F. Cimpoesu, Chem. Phys. Lett. 274 (1997) 341. [66] R.D. Rosenstein, R. Schor, J. Chem. Phys. 38 (1963) 1789. [67] A.A. Bokov, A.G. Khasabov, I.P. Rayevsky, Izv. AN SSSR Ser. Fiz. 54 (1990) 732 (in Russian). [68] L. Bellaiche, J. Padilla, D. Vanderbilt, Phys. Rev. B 59 (1999) 1834. [69] R.T. Sanderson, Simple Inorganic Substances, Robert E. Krieger, Malabar, FL, 1989 (p. 14).
Relationship between ionicity, ionic radii and order/disorder in complex perovskites A.A. Bokov a, N.P. Protsenko b, Z.-G. Ye a,* a
Department of Chemistry, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 b Institute of Physics, Rostov State University, Rostov-on-Don 344090, Russian Federation Received 23 June 1999; accepted 5 December 1999
Abstract Analysis of the available structural data on the ordering of B-site cations in complex perovskites with the general formula A2B VB IIIO6 has been performed. Low ionization potentials of B-site cations are found to promote their ordering. A linear relation between the order–disorder phase transition temperature Tt and the sum of ionization potentials of B-site cations is observed in Pb2B VB IIIO6 perovskites. It can be used for predicting Tt of other complex compounds. The electronegativity equalization method (EEM) has been used to determine the effective ionic charges and the ordering energies. The results of simulation support the correlation disclosed. In contrast to what was previously postulated, the influence of the size of B-site cations on the ordering is shown to be negligible. The determining driving mechanism responsible for the ordering is found to be the electrostatic, rather than the elastic, interactions between the B-site ions. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keywords: D. Crystal structure; D. Phase transitions; A. Ceramics; A. Oxides
1. Introduction The perovskite structure (with general formula ABO3) can be described as a framework of oxygen octahedra sharing corners [1,2]. The B-cations are located in the centres of octahedra and the interstitial sites are occupied by the A-cations. In complex perovskites two (or more) cations of different valences are located in the equivalent crystallographic positions. Among the A2B 0 B 00 O6 complex perovskite oxides, where B 0 and B 00 are the cations of different valences, the representatives with ordered as well as disordered arrangement of B 0 and B 00 are frequently encountered [1,2]. In some oxides the degree of disorder can be changed by thermal annealing [3–7]. The perovskite materials designed for dielectric, piezo-, ferroelectric, optical and other applications exhibit a large range of interesting properties resulting from the intricate ordered/disordered structures. To control the properties it is important to know the factors determining the degree of disorder. It * Corresponding author. Tel.: ⫹ 1-604-291-3351; fax: ⫹ 1-604291-3765. E-mail address: [email protected] (Z.-G. Ye).
was generally believed that the ordered state is established primarily due to the charge difference between the two Bsite cations but the difference in their radii is also important. This viewpoint was formulated in the early papers [8,9] and has been shared by the majority of authors for many years [1,2,10–13]. Complex perovskites with a large difference in the valence between B 0 and B 00 are strongly driven towards an ordered state by electrostatic forces (only the ordered arrangement of cations agrees with the Pauling’s fourth rule [1]). Such compounds are always completely ordered. In the isovalent solid solutions the ordering was not observed as a rule because of the small electrostatic ordering energy. This energy calculated in the case of 1:1 ordering [14] appeared to be proportional to the square of B-site cationic charge difference. The other factor expected to be important is the difference in ionic radii between B 0 and B 00 cations DR 兩RB 0 ⫺ RB 00 兩: It was found [8,9] for A2B VB IIIO6 perovskites, where A Ba, Ca, Sr, B V B 0 stands for pentavalent Ta and Nb cations, and B III B 00 stands for all possible trivalent cations, that the critical difference in radius for ordering is about 9%. The influence of DR on the ordered superstructure arises from the requirement of
0022-3697/00/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(00)00004-4
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closest packing for oxygen octahedra populated by Bcations of different size [1]. Thus the driving forces for ordering by this mechanism are elastic in nature. Recently on the basis of an analysis of order–disorder phase transition temperatures in some Pb2B VBIIIO6 perovskites, we proposed that the difference in ionic radius is not an important factor for ordering, but the degree of ionicity should be taken into account instead [15]. Another indication of the negligible role of elastic interactions in the ordering process comes from the work by Bellaiche and Vanderbilt [16], who successfully explained various types of compositional order observed in a large class of complex perovskites, by assuming that the driving mechanism responsible for ordering is simply the electrostatic one. To elucidate the possible contribution from different mechanisms to the ordering in complex perovskites, we have reviewed the published literature and analyzed the structural data for the perovskites of the general formula A2B VB IIIO6 in this work. It has been found that the influence of size difference of B-type cations on the ordering is negligible. A correlation between the ordering state and the ionization potentials of cations has been proposed. Calculations of the ionic charges and the electrostatic ordering energy have been performed using the electronegativity equalization method to explain the correlation.
2. Analysis of the ordered and disordered perovskites The structural data for more than 300 complex perovskites with the general formula A2B VB IIIO6 are collected from the literature [8,17–46]. Table 1 presents a small section of the database. Most of the references are taken from the books by Fesenko [1] and Galasso [2]. The data for some compounds classified as “unreliable” in these books are excluded from further consideration if more recent data are not available. In the majority of works the structural data were obtained from X-ray diffraction. The Xray detection of ordering can be difficult in the case of small difference in scattering factor between the two B-site cations. This sometimes leads to the discrepancy between the data obtained by different authors. For example, Brixner [45] found a disordered structure in barium rare-earth niobates. Galasso and Darby [8] showed, however, that the disordered structure was due to the exposure time of X-ray photographs which was not long enough. They observed instead an ordering in all these compounds, except Ba2NbCeO6 which was not investigated in their work. Thus we consider all other reported barium rare-earth niobates as ordered ones, except for Ba2NbCeO6 which is excluded from our consideration because of unreliable data. We deal with other cases of discrepancy in the same manner. Fig. 1 illustrates the relation between the ordering and the size of B-site cations for all the A2B VB IIIO6 perovskites considered. Shannon’s ionic radii [47,48] are used. Every horizontal row of the points in the figure represents the
relative difference in ionic radii DR/RBmin (RBmin is the radius of the smaller of the two B-site cations) for compounds with the same A and B V cations. It can be seen that in most cases, the ordered compounds possess large DR. However, among the compounds which have small (or even zero) DR, ordered representatives can be found alongside with the disordered ones. In these compounds the sizes of oxygen octahedra around different B-site ions should be approximately the same and the elastic driving forces towards ordering should be small. Ba2PuInO6 is a good example. The interatomic Pu–O and In–O distances were found by X-ray diffraction ˚ ) in this perovskite [28], method to be the same (2.12 A nevertheless the compound is ordered. This means that the electrostatic energy is high enough to force the structure to order. The role of the elastic interactions in the ordering thus becomes questionable. As electrostatic and elastic energy increases with disordering, all the A2B VB IIIO6 materials should have the minimal potential energy in the ordered state. But the thermal motion is capable of destroying the ordering and the structural order–disorder phase transition occurs at some nonzero temperature Tt. This kind of phase transition was observed in several lead-containing complex perovskites, as listed in Table 2. Since the temperature Tt is proportional to the ordering energy [56], an increase of Tt with DR could be expected if the influence of ionic size on the ordering processes was essential. However, no such correlation can be observed (see Table 2 and Fig. 2). Furthermore, the highest Tt has been found in Pb2TaScO6 which has the smallest DR. Therefore, it is clear that some factors other than ionic size exert effects on the ordering. Based on our previous work [15], we have analyzed the correlation between the order/disorder state, the third ionization potential of the trivalent B-site cation (I3) and the fifth ionization potential of the pentavalent B-site cation (I5). The values of I3 and I5 are taken from Refs. [48,57]. The values of ionization potential for Re, Os, Pa, U, Pu and Am are not available in the literature. They are estimated by means of the method described in Appendix A and listed in Table 1. The dependence of Tt on the sum of ionization potentials I3 ⫹ I5 for the Pb-containing compounds is shown by squares in Fig. 3. The same compounds as in Fig. 2 are considered. The dependence can be interpolated as Tt Tt0 ⫺ at I3 ⫹ I5
1
where the temperature and ionization potentials are expressed in K and eV, respectively, Tt0 5648 K; at 56 K=eV: Relation (1) is shown by solid line in Fig. 3. It can be used for predicting order–disorder phase transition temperature Tt in lead-containing Pb2B VB IIIO6 perovskites. The values of Tt calculated with the help of Eq. (1) are presented in Table 2. The only significant disagreement between the calculated and experimental results is found in Pb2NbLuO6. The possible reasons for this discrepancy will be discussed in the next section, but here we underline that it could not be related to the size effect. The DR/RBmin
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Table 1 Data on ordering state at room temperature, preparation temperatures (Ts1 and Ts2) and parameters of B-site cations for selected A2B VB IIIO6 perovskites: ionic radii of pentavalent (RB 0 ) and trivalent (RB 00 ) cations, relative difference in ionic radii (DR/RBmin), fifth ionization potentials for pentavalent cations (I5) and third ionization potentials (I3) for trivalent cations Compound
˚) RB 0 (A
˚) RB 00 (A
DR/RBmin
I5 (eV)
I3 (eV)
Ordering
Ts1 (⬚C)
Ts2 (⬚C)
Ref.
Ca2TaCrO6 Ca2TaVO6 Ca2TaMnO6 Ca2NbVO6 Sr2OsScO6 Sr2ReHoO6 Sr2ReGaO6 Sr2SbGaO6 Sr2SbFeO6 Sr2SbMnO6 Sr2NbCrO6 Sr2UInO6 Ba2NbCoO6 Ba2NbNiO6 Ba2BiCrO6 Ba2BiYbO6 Ba2BiYO6 Ba2BiDyO6 Ba2BiTbO6 Ba2PuInO6 Ba2UScO6 Ba2PaInO6
64 64 64 64 58 58 58 60 60 60 64 76 64 64 76 76 76 76 76 74 76 78
62 64 65 64 75 90 62 62 64.5 65 62 80 61 60 62 87 90 91 92 80 75 80
0.032 0 0.016 0 0.29 0.55 0.069 0.033 0.075 0.083 0.032 0.053 0.049 0.067 0.23 0.145 0.184 0.197 0.211 0.081 0.013 0.026
45 45 45 50 52.7 50.6 50.6 56 56 56 50 40.1 50 50 56 56 56 56 56 42.5 40.1 42.5
30.95 29.31 33.69 29.31 24.75 22.84 30.7 30.7 30.64 33.69 30.95 28.03 33.49 35.16 30.95 25.03 20.5 22.8 21.91 28.03 24.75 28.03
Disordered Ordered Disordered Ordered Ordered Ordered Ordered Ordered Ordered Disordered Ordered Ordered Disordered Disordered Disordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered
1100 1150 1100 1150 1000 1000 1000 1100 1000 1000 1100 1000 1000 1000 1000 800 900 900 800 750 1000 1350
1400
[17] [18] [17] [18] [19] [20] [19] [21] [22] [22] [22] [23,24] [22] [22] [22] [25] [26,27] [26] [25] [28] [23] [29]
1400
1300 1300 1300 1300 1300 1300 1000 1075 1075 1000 950
Fig. 1. Relative difference in radii of B III and B V cations (DR/RBmin) for ordered (triangles) and disordered (circles) A2B VB IIIO6 complex perovskites.
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Table 2 Temperatures of order-disorder phase transition (Tt) calculated using Eq. (1) and experimentally found in perovskite compounds, and parameters of B-site cations: ionic radii of pentavalent (RB 0 ) and trivalent (RB 00 ) cations, relative difference in ionic radii (DR/RBmin) and ionization potentials for pentavalent (I5) and trivalent (I3) cations Compound
Pb2NbScO6 Pb2NbInO6 Pb2NbYbO6 Pb2NbLuO6 Pb2NbTmO6 Pb2NbErO6 Pb2TaScO6 Pb2TaInO6 Pb2TaYbO6 Pb2TaLuO6 Pb2TaTmO6
˚) RB 0 (A
64 64 64 64 64 64 64 64 64 64 64
˚) RB 00 (A
75 80 87 86 88 89 75 80 87 86 88
DR/RBmin
0.17 0.25 0.36 0.34 0.38 0.39 0.17 0.25 0.36 0.34 0.38
value is comparatively large in Pb2NbLuO6. This should lead to an enhanced value of Tt, but in fact the experimental Tt is smaller than the calculated one. In general the decrease of ionic radius leads to an increase of ionization potentials. That is why the analysis of data on a limited number of A2B VBIIIO6 perovskites containing B V cations with close I5 values (Ta and Nb) performed in earlier works [8–10] revealed some dependence of ordering state on DR. However, the ionization potentials do not correlate with atomic radii exactly (see, for example, Tables 1 and 2). Thus if DR influenced ordering considerably, relation (1) would be violated. The validity of this relation shows that the effect of the ionic size on Tt is negligibly small, and the elastic interactions associated with the difference in size between B-site cations do not contribute considerably to the interactions responsible for order/disorder behavior. Let us now consider the materials in which the ordering temperature Tt has not been measured experimentally. The degree of order at room temperature should depend
Fig. 2. Temperatures of order–disorder phase transition (Tt) versus relative difference in radii of B III and B V cations for Pb2B VB IIIO6 complex perovskites.
I5 (eV)
50 50 50 50 50 50 45 45 45 45 45
I3 (eV)
24.75 28.03 25.05 20.96 23.68 22.74 24.75 28 25.05 20.96 23.68
Tt (⬚C) Calc.
Exp.
Ref.
1189 1005 1172 1401 1249 1302 1469 1287 1452 1681 1529
1210 1020 1120 1130–1140 ⬎ 1200 ⬎ 1050 1450–1500 ⬎ 1070 ⬎ 1260 ⬎ 1200 ⬎ 1100
[49] [7,50] [51] [52] [53] [53] [49] [6,54] [53,55] [53] [53]
both on Tt and the temperature of ceramic preparation Ts [3,15] (all the materials considered here were prepared in ceramic form). If Ts p Tt the material should be completely ordered. If Ts ⬎ Tt ; it should be disordered at high temperature immediately after formation. During the subsequent cooling below Tt it can remain disordered or become ordered, depending on the rate of ordering inherent in the particular material. If this material dose not change the order degree during the cooling process, it remains disordered at room temperature forever. The reason for this is that the ordering implies the site exchange between B III and B V cations. It is a relaxation process with practically infinite
Fig. 3. Temperatures of order–disorder phase transition (Tt) for Pb2B VB IIIO6 complex perovskites (squares) and synthesis temperatures (Ts) for all other ordered (triangles) and disordered (circles) A2B VB IIIO6 complex perovskites versus sum of ionization potentials of B-site ions I3 ⫹ I5 :
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characteristic time at low (room) temperature. Thus the properties of materials at temperatures around Ts should be taken into account to determine the ordering state. The temperatures of synthesis are presented in Table 1 and plotted against I3 ⫹ I5 in Fig. 3 for all considered A2B VB IIIO6 compounds. In some works, the preparation of materials was performed by sequential heating up to two different temperatures Ts1 and Ts2 (⬎Ts1) in order to ensure complete solid state reactions. It is known [15,51] that the ordered (if Ts1 ⬍ Tt ) structure formed during synthesis at Ts1 becomes, after subsequent firing at Ts2, less ordered in some cases or remains unchanged in other cases. It depends on the rate of ordering process in specific material and the duration of firing at Ts2. Thus the degree of disorder in the final material is hardly predictable even if Tt can be estimated. It can be expected to be between the values corresponding to the equilibrium state resulting from the thermal treatments at Ts1 and Ts2, respectively. When plotting Fig. 3, for convenience in further discussion we took Ts Ts1 for ordered materials (shown by triangles) and Ts Ts2 for disordered ones (shown by circles). We can interpret the data in Fig. 3 by assuming that Tt follows relation (1) for all A2B VB IIIO6 compounds. As discussed above, only those materials can be disordered which were heated in the course of preparation above Tt. In agreement with this statement all disordered states (circles) are located in the top right part of Fig. 3 (in Ts ⬎ Tt area). Synthesis at temperatures below Tt should lead to an ordered structure. Therefore, the ordered states (triangles) can only be located in the bottom left part of Fig. 3 (in Ts ⬍ Tt area). Some triangles are found in the Ts ⬎ Tt area. It is possible that these compounds had been synthesized in disordered states, but the ordering process occurred during slow cooling after formation, leading to an ordered structure. Many peculiarities of order/disorder behavior can be clarified by taking into account the values of ionization potentials of B-site cations. For example, Ba2BiCrO6 is a disordered material in spite of its DR/RBmin value larger than that of several ordered Ba2BiB IIIO6 materials (see Table 1). This can be explained by the larger value of I3 in Ba2BiCrO6. The disordered state in Sr2SbMnO6, which has a larger DR/RBmin value than the ordered Sr2SbGaO6 and Sr2SbFeO6, can be explained in the same manner, so does the ordering in Ca2TaVO6 which has smaller DR/RBmin than the disordered Ca2TaCrO6 and Ca2TaMnO6. All the compounds containing Pu, Pa and U were found to be ordered [23,24,28,29], even those having small DR. According to our analysis this is due to the small values of I5 for these elements. In fact, Awasthy and co-workers [24,28] compared Ba2UB IIIO6, Sr2UB IIIO6 and Ba2PuB IIIO6 oxides (where B III are different trivalent cations), and found that in plutonium compounds the degree of ordering was lower than in uranium ones. We can explain this fact by the smaller I5 value of U than Pu. In the next section we give the theoretical justification for
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the correlation between ionization potentials and the order/ disorder state disclosed in complex perovskites.
3. Calculations and modeling We propose that the electrostatic interactions between the differently charged ions are the determining driving forces responsible for the ordering. Accordingly, the ordering energy Uod is merely the difference between the Madelung energies in the ordered and the disordered states. The effective charges of ions with the same oxidation state are different in unlike compounds (in other words the ionicities are different). This leads to the difference in Madelung energies. Thus to compare the ordering energies Uod in some compounds, it is necessary to estimate the ionic charges. We use semiempirical density functional electronegativity equalization method (EEM) to determine the effective ionic charges and Madelung energies and to confirm theoretically the correlation between Uod and the ionization potentials of constituting cations. The EEM is based on the definition of electronegativity x as the negative of the chemical potential of electronic cloud [58,59]. The electronegativity of an isolated atom a can be written as
x0a ⫺
2Ea0 2Ea0 2Na 2qa
2
where Ea0 is the electronic energy of the atom, Na Za ⫺ qa is the number of electrons on the atom, Za and qa are the atomic number and ionic charge, respectively. The formation of a molecule is accompanied by the transfer of appropriate, not necessarily integral, charge from one atom to another. The driving force for the charge transfer is the chemical potential (or electronegativity) difference [58,60]. In the final state the electronegativities of all the participating atoms are equalized. After that, the expression for the effective electronegativities of the atoms in the molecule (crystal) is:
xa
2Ea x ; 2qa
a 1; 2; …; k
3
where x is the average molecular electronegativity, k is the number of unlike atoms and Ea is the electronic energy of ion a in the molecule. Ea in the EEM approximation can be expressed [61,62] as a sum of the energy of the isolated atom Ea0 ; the additional energy E 0a associated with the changes of size and shape of electron cloud due to the confinement of the atom in the molecule (it includes the covalent bonding effects), and the energy of interaction with surrounding atoms E 0 0a : Ea Ea0 ⫹ E 0a ⫹ E 00a
4
The first two terms of this equation can be approximated
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Table 3 Parameters for the expression of electronegativity of isolated atom (in eV)
c0a b0a a0a a
Pb
Nb
Ta
Fe
In
Yb
Sc
Oa
0 3.805 3.615
⫺0.1683 8.145 ⫺12.487
0.1833 3.750 0.0667
1.0298 1.0533 5.819
⫺0.6539 8.5033 ⫺2.063
1.1587 0.515 5.610
0.9531 0.2597 5.348
0 8.7836 9.3187
Parameters a0a ⫹Daa ; b0a ⫹ Dba and c0a ⫹ Dca are given for oxygen.
by a smooth function of the atomic charge [62–64] Ea0 qa
a0a qa
⫹
b0a q2a
⫹
5
c0a q3a
Ea0 qa ⫹ E 0a qa a0a ⫹ Daa qa ⫹ b0a ⫹ Dba q2a ⫹ c0a ⫹ Dca q3a
6
In the case of ionic crystals (which is considered in this work) E 00a is a Madelung-type energy. The Madelung potential at the position of atom a can be written in a suitable form for further calculations [65]:
wa
X 2E 00a ql Fal 2qa l
7
where Fa l are independent of charge coefficients and the summation is extended over all the different sublattices (ql are the charges of ions located in sublattice l). By substituting Eqs. (4), (6) and (7) for the terms in Eq. (3) we obtain the final set of equations X a0a ⫹ Daa ⫹ 2 b0a ⫹ Dba qa ⫹ 3 c0a ⫹ Dca q2a ⫹ ql Fal l
Fal Aal =a;
x ;
a 1; 2; …; k
(8)
The self-consistent solutions of these k simultaneous equations alongside with the electroneutrality condition X qa 0 9 a
give the values of effective charges qa for k unlike ions in the studied structure and the value of their common electronegativity x : To obtain the solution, the polynomial coefficients should be known. The coefficients Fa l can be found for a specific crystal structure by Ewalds’ method [56]. The parameters a0a ; b0a and c0a can be calculated using the known values of ionization potentials Ina and electron affinities A a of atom a , and taking into account the charge q for integer values q n; we have Ea0 n Ea0 n ⫺ 1 ⫹ Ina ;
Ea0 ⫺1 Aa
This procedure has not been applied to perovskite-type compounds up to now. Chibotaru et al. [65] recently performed the EEM calculations for several cubic ABO3 perovskites under the condition Daa Dba Dca 0 for all cations. The only ion subjected to parameterization was oxygen. The authors obtained the values of ionic charges, which are in agreement with the results of self-consistent quantum chemistry calculations. The same approach is applied to the cubic complex perovskites A2B VB IIIO6 in this work. To calculate the Madelung potentials we use the superposition method developed for complex perovskites by Rosenstein and Schor [66]. The ordered structure A2B 0 B 00 O6 with B-ions having different charges qB 0 and qB 00 is considered as a superposition of a perovskite structure with the average charges qav qB 0 ⫹ qB 00 =2 in B-positions upon a rocksalt structure with qB 0 ⫺ qav and qB 00 ⫺ qav charges. In the case of disordered structure the Madelung energy is the same as for ABO3 perovskite with qB qav : The parameters for the calculation of Madelung potentials in perovskite structure can be found as [65]:
10
The corrections Daa , Dba and Dca which represent changes in the isolated atom coefficients due to the confinement of the atom in a crystal, can be calibrated so as to reproduce the results of quantum-mechanical ab initio calculations [61].
Aal Ala ; Aaa 0;
where a is the lattice parameter expressed in angstroms, AAB 58:57; ABO 78:886; AAO AoO 0 64:884; O and O 0 denote two of the three oxygen sublattices. For rocksalt structure the Madelung constant is AB 0 B 00 50:33: Thus we use the following expressions for calculating the Madelung potentials in A2B 0 B 00 O6 structure:
wA
1 q A ⫹ 3qO AAO a av AB
1 q A ⫹ 3gB 0 qO ABO ⫹ qB 00 ⫺ qav AB 0 B 00 a A AB 1 a qA AAB ⫹ 3qO ABO ⫹ qB 0 ⫺ qav AB 0 B 00 a a ⫹ 2x
wB 0 wB 00
wO
1 q ⫹ 2qO AOO ⫹ qav ABO a A
The factors gB 0 and gB 00 are introduced to take into account the shift x of oxygen ions from the symmetric positions due to the difference of B 0 and B 00 cations in sizes. We assume as a first approximation that gB 0 a= a ⫺ 2x; gB 00 a= a ⫹ 2x: The values of x are taken from Ref. [67]. The coefficients a0a ; b0a and c0a for cations are calculated
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Table 4 Lattice parameters (a), shifts of oxygen ions from the symmetric positions (x), effective ionic charges (q), Madelung potentials (w ), ordering energies (Uod) and sums of ionization potentials of B-site cations I3 ⫹ I5 for selected ordered complex perovskites Compound
˚) a (A
˚) x (A
qA
qB 0
qB 0
qO
w A (eV)
w B 0 (eV)
w B 00 (eV)
w O (eV)
Uod (kJ/mol)
I3 ⫹ I5 (eV)
Pb2NbFeO6 Pb2NbInO6 Pb2NbYbO6 Pb2TaScO6
8.04 8.22 8.3 8.16
0 0.084 0.106 0.047
1.44 1.13 1.14 1.04
3.20 3.42 3.27 4.52
2.47 2.89 2.24 3.09
⫺1.42 ⫺1.43 ⫺1.30 ⫺1.62
⫺13.83 ⫺11.36 ⫺11.05 ⫺11.24
⫺33.73 ⫺36.44 ⫺34.10 ⫺44.93
⫺29.18 ⫺29.81 ⫺24.05 ⫺33.92
16.43 16.64 14.81 19.39
79.80 85.32 250.08 381.22
80.64 78.04 75.04 49.5
from the fitting of ionization energies (10) with the function (5) using least-squares method. The results of fitting are presented in Table 3. The parameters Daa , Dba and Dca are considered to be zero. For oxygen these parameters are calibrated so as to obtain qualitatively correct values for effective ionic charges and ordering energy. The effective ionic charges, Madelung potentials, and electrostatic ordering energies obtained from the solutions of Eqs. (8) and (9) with oxygen parameters listed in Table 3 for some perovskites are given in Table 4. It appears impossible to adjust the unique oxygen parameters suitable for all compounds. For some of them, the calculated charges appear to be larger than the cation’s valence (as in the case of Pb2TaScO6 presented in Table 4). Evidently this discrepancy arises because the corrections of the isolated atomic electronegativity as a consequence of bonding (Daa , Dba and Dca ) were neglected. We do not intend to calculate the true ionic charges, but to find out the general trends in charge variation when the atomic parameters such
Fig. 4. Calculated ordering energy (Uod) for a model A2B VB IIIO6 cubic perovskite versus (a) the third ionization potential of B III cation and (b) the fifth ionization potential of B V cation.
as ionization potentials and ionic radii change. For the compounds presented in Table 4, the calculated trends are in agreement with the experimental observations, namely, the ordering energy increases with the decrease of the average ionization potentials of B-site cations. The calculated changes of Uod in real compounds are associated with the change not only in ionization potentials but also in structural parameters (a and x). Therefore, a model-study of the influence of ionization potential itself on the ordering process is performed in the next step. Pb2NbInO6 is chosen as a model compound. It can be considered a typical representative of A2B VB IIIO6 perovskites because of its nonzero but not very large DR/RBmin. The electronegativity parameters a0a ; b0a and c0a for one of the atoms, In or Nb, are changed arbitrarily around the real values presented in Table 3, while the parameters for all other atoms were kept equal to its real values. The structural parameters a and x also remain unchanged. The effective ionic charges and ordering energies are derived for every set of electronegativity parameters. Then the ionization potentials are calculated from Eqs. (5) and (10) using the same values of parameters. The relations found are depicted in Fig. 4. It can be seen that the rise of ionization potential of trivalent as well as pentavalent cations causes a decrease of ordering energy, as is in full agreement with the experimental observations reported in the previous section of this paper. It should be noted that the same simulation performed with some other values of electronegativity parameters could lead to qualitatively different results. As discussed above, the values of corrections Daa , Dba and Dca are not known, thus it is not possible to perform the calculations with the real ionic electronegativities. Meanwhile the variation of these corrections as a result of ordering (in particular due to the covalent bonding effects) might be expected to contribute to ordering energy. Recent first-principle supercell calculations in A3B 0 2B 00 O9 materials [68] showed that the formation of short A–O bonds due to the covalent effects can sometimes influence the type of 1:2 ordering, but the similar effects on Tt were not studied. Our simulation shows the possibility to obtain the results consistent with most experimental data using the same values of Daa , Dba and Dca parameters for ordered and disordered states, in other words, neglecting the covalent bonding effects. Some rare cases of disagreement between the experimental data and the predictions obtained in terms of our approach are
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understanding of the crystal chemical origin of order/disorder effects, and the microstructure–property relations in complex perovskite materials.
Acknowledgements This work was supported by the Natural Sciences and Engineering Council of Canada (Research Grant) and Simon Fraser University (President Research Grant). Appendix A. Estimation of ionization potentials Fig. 5. Ionization potentials (In) of elements versus their atomic numbers (Z).
It is known that the ionization potentials of elements with atomic numbers close to each other approximately obey the relation
probably related to the covalent bonding effects which appear unusually high in some particular compounds. Therefore, the decrease of ordering energy with increasing ionization potentials of B-site cations, as discovered in the previous section, can be explained in terms of the difference in electronegativities, leading to the difference in ionicities, ionic charges and hence electrostatic interactions between ions.
Ii⫹j Z ⫹ j gj ⫹ h2
4. Conclusions A comprehensive analysis of the available literature data on the ordering of different type B-site cations in complex perovskites with general formula A2B VBIIIO6 has been performed in this work. In contrast to what was previously postulated, the influence of the size of B-site cations on the ordering of complex perovskites is shown to be negligible. Thus the contribution of elastic interactions to the ordering energy is small. A linear relationship between the temperatures of order–disorder phase transitions Tt and the sum of ionization potentials of B-site cations I3 ⫹ I5 has been found in lead-containing oxides. It can be used to estimate Tt for those compounds in which the order-disorder transition has not been observed so far. The value of Tt as well as the ceramic preparation conditions should be taken into account to predict the degree of ordering in complex perovskite materials at room temperature. The calculations of effective ionic charges and ordering energies using the electronegativity equalization method have confirmed our suggestion that the electrostatic interaction between ions is the determining driving mechanism responsible for the ordering in complex perovskites A2B VB IIIO6. Due to the difference in ionicities, the effective charges of the ions with the same oxidation state are different in unlike compounds, leading to the difference in ordering energies, and hence in the temperatures of order– disorder phase transitions. The analysis performed in this work provides a better
A1
where g and h are parameters, i and j are integers, Ii⫹j Z ⫹ j is the i ⫹ jth ionization potential of the element with atomic number Z ⫹ j: For elements with small Z this relation is reported, for example, in Ref. [69]. Fig. 5 illustrates the validity of this relation for the elements with large Z. The dots in this figure are the known ionization potentials, the straight lines are determined by the Eq. (A1) with g and h parameters found by least-squares method (only those lines are shown which are used in further calculations). Thus if the potentials of neighboring elements are known, the unknown ionization potential of an element can be approximated from these equations. The results of approximation of I5 for Re Z 75; Os Z 76; Pa Z 91; U Z 92 and Pu Z 94 are listed in Table 1. It is impossible to find I3 for Am Z 95 because of the absence of data on its neighboring elements.
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