Revisiting the valence stability and preparation of perovskite structure type oxides ABO3 with the use of Madelung electrostatic potential energy and lattice site potential

Valence stability of aliovalent ions is mostly correlated with lattice site potential in ionic crystals. Madelung electrostatic potential is obtained by adding all the lattice site potentials for all the ions present in a crystal structure. Therefore, valence stability and the stability of a crystal structure can be better understood with consideration of both the lattice site potential and Madelung electrostatic potential. This was first demonstrated more than four decades ago by one of the present authors. We revisit this situation by using re-calculated lattice site potential and Madelung electrostatic potential for perovskite structure type ABO3 compounds using a new computer program VESTA. We show that the formation of a perovskite structure type compound with the general formula ABO3 (where A and B are cations and O is an oxide ion) becomes energetically favorable when it has a higher Madelung electrostatic potential than the combined Madelung electrostatic potential of parent binary compounds AO and B2O3 or BO2. It is further shown that strong lattice site potential results in stability of high valence or high valence ions can be stabilized in a lattice site with strong lattice-site potential. It further follows that certain ions experience maximum lattice site potential at the B ion lattice site of the perovskite structure when compared to other structures such as fluorite BO2, rutile BO2 and corundum B2O3. Therefore, (i) the stability of an ion with a high (and uncommon) valence state at the B site being higher than that at the A site, (ii) occurrence of point defects at A or O sites with weak lattice site potentials, respectively and (iii) instability of perovskite A4+B2+O3, and A5+B1+O3 compounds, respectively can be rationalized by lattice site potential and Madelung electrostatic potential analysis.


Introduction
An enormous number of solids show crystallographic structural similarity to the mineral perovskite (CaTiO 3 ) and have the general formula ABX 3 . Metal oxides with chemical formula ABO 3 are a subclass of this large family of compounds. For many years, perovskite oxides have been studied extensively in the literature of chemistry, physics and geology for a large variety of solid state phenomena associated with electric, optical, thermal and magnetic properties. A large number of oxides with perovskite type structures are possible to synthesise in a laboratory for applications in vast areas spanning from electronics to catalysis. It is of immense importance to understand the factors that may correlate with the formation and stability of perovskite structure type ABO 3 compounds for the design of new compounds with novel structures containing desired valence states of B cations.
The reason for why many compounds would take perovskite structure type ABX 3 had been explained from ion-packing & tolerance factor, by Goldschmidt in 1926. 1 Even now, most of the research articles and text books on the assessment of stability of perovskite structure type ABO 3 oxides focus on the geometric parameters such as tolerance factor, and structure-eld map. [2][3][4][5] However, lattice and valence stabilities should be more related to thermodynamic energy and originate from thermodynamic principles rather than numerical geometrical parameters. Crystal chemistry and thermodynamic energy of ionic solids are reected into Madelung electrostatic potential as shown by Van Gool and Piken in 1969. 6,7 "The valence stability using lattice site-potential" of perovskite structure type ABO 3 was demonstrated by one of the authors of this article (M. Yoshimura) and was considered for the rst time in 1974. 8 This was referred by Rustam Roy in 1975 in a plenary lecture presented as the Robert B. Sosman Memorial Lecture at the 77th annual meeting of the American ceramics society and was published in 1977. 9 This is a novel and very less explored concept. We also note that not many researchers have been able to consider this concept perhaps due to the inaccessibility of the original article. 8 Here, the present article is written by re-visiting Madelung lattice energy and site potentials based upon recent data calculated using VESTA 10 program and recent crystallographic (structural) and thermodynamic data. We emphasize on the fact that ionic solids are stabilized due to the gain in lattice energy that is reected in Madelung electrostatic potential (E M ). Nevertheless, recently, Hoppe demonstrated that the enthalpy of formation of solids is reected in Madelung electrostatic potential in 1995 11 and Glasser has shown that for ionic solids the lattice energy and Madelung electrostatic potential energy (E M ) has a linear relationship in 2012. 12 The ideal perovskite structure type of ABO 3 compound has a cubic unit cell. This structure can be constructed from corner shared BO 6 octahedra with A cation being present at the 12 coordinated interstitial position within the octahedral network ( Fig. 1a). A and B cations are coordinated by 12 and six O 2À ions, respectively and O 2À ion is coordinated by four A and two B cations. Several structural modication of cubic perovskite structure has been identied such as tetragonal, orthorhombic, rhombohedral, hexagonal, and monoclinic, respectively either as polymorph of ABO 3 or upon chemical modication. The most important chemical fact of ABO 3 perovskite compound is that, apart from noble gasses and nonmetals, A or B in ABO 3 can be chosen from almost all elements and several aliovalent A or B ions such as A 1Àx A 0 x or B 1Ày B 0 y ; respectively can be combined in a single compound. Furthermore, non-ideal stoichiometry of oxygen or A such as ABO 3Àz or A 1Àx BO 3 , respectively are regularly observed. It is further interesting that many BO 3 compounds can adopt perovskite type structure where A cation is completely absent for example ReO 3 . Therefore, the structural skeleton formation by BO 3 sub lattice is generally regarded as prerequisite for the formation of perovskite structure type compounds. 2 Furthermore, the BO 3 sub lattice skeleton is primarily stabilized by the electrostatic potential energy gain by ionic arrangements in solid phase and additional stability of perovskite type ABO 3 compounds can arise from (i) the presence of large A cation within the interstitial position, (ii) crystal eld splitting and (iii) hybridization of orbitals. 2 The total internal energy of ionic solids may be expressed as where E M is Madelung electrostatic potential energy and E N is combination of all other energies (repulsive, zero point, crystal eld stabilization, vibrational, van der Waals and covalent, energies respectively). 13 The analysis of Madelung electrostatic potential and lattice site potential of a large number ABO 3 compounds (with varying combination of A and B) led to a set of fundamental criteria for the formation of perovskite structure type ABO 3 with varying combination and valence states of A and B and summarized by Yoshimura et al. 8,14 as follows: oxides of A can be reacted with oxides of B to form perovskite structure type ABO 3 compounds if: (i) B ion is smaller than A ion, (ii) valence of B ion is larger or equal to A ion and (iii) sum of valence of A and B ion is six (including mixed valence situation A 0 1Àx A 00 x or B 0 1Ày B 00 y ). An in depth analysis showed that the valence stability originates from gain of lattice site potential. Recently, there have been some excellent research work reported in the literature based on the use of Madelung electrostatic potentials energy for understanding the stability of perovskite structure type compounds. Shan et al. investigated Li ion insertion in perovskite structure type SrVO 3Àd , La 2/3 TiO 3Àd and (La, Li)TiO 3Àd , from the lattice site potential analysis and found that the insertion of lithium ion is limited by the effect of lattice site potential on the valence stability of V and Ti. 15 In these perovskite structure type oxides, the valence state of tetravalent V and Ti can be reduced to 3 upon insertion of monovalent Li ion at an interstitial site. It is interesting to note that for spinel structure type  electrostatic potential. 16 Furthermore, the perovskite structuretype is not limited to ABO 3 and a large number of organicinorganic halide crystallise into perovskite type structure with general formula ABX 3 . A bottle neck for the realization of full potential of so called 'perovskite solar cells' and related electronic devices has been the degradation of electronic devices fabricated using hybrid organic-inorganic ABX 3 compounds. 17 The degradation of these devices arise from instability of the perovskite structure type hybrid compounds and can also be understood from Madelung electrostatic potential analysis. 18 Furthermore, it has also been shown that the stability of oxides such as Li 4 SiO 4 towards the reaction Li 4 SiO 4 + CO 2 / Li 2 SiO 3 + Li 2 CO 3 has direct consequence on the CO 2 capture and storage abilities and recent study by Oh-ishi et al. had successfully employed lattice site potentials analysis of all Li + ions in the unit cells of Li 4 SiO 4 , Li 2 SiO 3 , and Li 2 CO 3 to evaluate the extraordinary stability. 19 On the other hand, lattice site potential can be used for the estimation of the oxygen to metal charge transfer and bandgap of perovskite type ABO 3 oxides using an ionic model developed by Zaanen et al. 20 and demonstrated in a large number of oxides by Torrance et al. 21 and Arima et al. 22 Kato et al. employed lattice site potential analysis to show that the valence band structure can be tailored by designing the stacking sequence of layers of a number of layered compounds BiOX (X ¼ Cl, Br, I), Bi 4 NbO 8 X (X ¼ Cl, Br), Bi 2 GdO 4 X (X ¼ Cl, Br), and SrBiO 2 X (X ¼ Cl, Br, I) and therefore, lattice site potential analysis may be useful for designing new photocatalysts. 23 Here we look back at the development of Madelung electrostatic potential and lattice site potential only for perovskite type ABO 3 oxides. We show that the lattice and valence stabilities of perovskite structure type ABO 3 compounds can be understood from simple Madelung electrostatic potential view point and valence stability is directly related to the lattice site potential. Based on the existing literatures, our hypothesis is that the stability and properties of perovskite type ABO 3 oxides can be rationalized better in terms of change in Madelung electrostatic potential energies rather than structural perturbation such as distortions and tolerance factors. 24,25 The Madelung electrostatic potential and lattice site potentials can be calculated using commercial rst principle calculation soware packages. We have used freely available soware VESTA 10 to recalculate the lattice site potentials and Madelung electrostatic potential directly from the known crystal structures. The Madelung electrostatic potential E M obtained from VESTA has the unit of MJ mol À1 asymmetric unit. In order to compare E M per mole of each compound with different crystal symmetry (space groups), the E M output of each compound from VESTA was multiplied by the multiplicity of general Wyckoff position and divided by the number of formula units in the unit cell.

Results and discussion
According to Van Gool and Piken, 6,7 the Madelung electrostatic potential energy E M can be expressed as where, q j is total charge number of lattice point j, p j is frequency of occurrence of j lattice point in the unit cell, f j is the lattice site potential ( A À1 ) at the lattice point j, and k is total number of molecules in the unit cell. The Madelung constant can be expressed as where M a is expressed in terms of the characteristic length a, such as the lattice parameter of unit cell. For compounds with complex structures an arbitrary characteristic length r such as nearest neighbor atomic distance can be dened and the Madelung constant can be expressed as From eqn (2)-(4) the Madelung electrostatic potential can be expressed by In VESTA, 10 lattice site potential f i and Madelung electrostatic potential E M of a given structure is calculated using following equations where Z j is valence of ion j, 3 0 is permittivity of vacuum and l ij is distance between ions i and j.
where For similar structures with xed atomic position (e.g. NaCl, CsCl), the lattice site potential f j for each site in the crystal structure is inversely proportional to the characteristic length. However, if the characteristic length is xed, for the rst time Yoshimura et al. 8 had analyzed that, lattice site potential f j , the Madelung constant M a and Madelung electrostatic potential energy E M varies with the valence states. Fig. 2 shows the relationship between valence state and lattice site potentials f j as well as Madelung electrostatic potential E M in ideal cubic perovskite type structure of ABO 3 with xed lattice constant of 3.881 A. Based on eqn (5) the Madelung constant M a also shows similar trend with valence. Fig. 2 shows that the lattice site potential at A site (f respectively. For the calculation of f and E M of A 0 B 6+ O 3 , a dummy A such as Na atom was placed at ( 1 2 1 2 1 2 ) and B cation Re 6+ at (0 0 0) atomic coordinates, respectively of space group Pm 3m (221) with both valence and occupancy, respectively of A were set to zero. The calculation for valence pairs (4-2) and (5-1) assumed hypothetical A 4+ B 2+ O 3 and A 5+ B 1+ O 3 perovskite structure type compounds because they rarely exist.
In Fig. 3 we show that, how the lattice site potential (f) of each ionic site and Madelung electrostatic potential (E M ) energy changes with lattice parameter when valence is xed, for each valence pair in each perovskite structure type ABO 3 . The variation observed in Fig. 3 is consistent with the observation of Sabry et al. in 2000 (ref. 24). From Fig. 2 and 3 it is observed that lattice site potential for all sites change signicantly with valence pairs. Lattice site potential for oxygen in A 3+ B 3+ O 3 is lowest. Therefore, it can be expected that oxygen can be easily removed from the perovskite type compounds with A 3+ B 3+ O 3 than A 0+ B 6+ O 3 . Indeed, the occurrence of oxygen vacancy is very common in A 3+ B 3+ O 3 and this may be of importance during designing catalysts for the availability of active lattice oxygen or designing materials for p-or n-type conductivity. 24,26 Valence stability is related to lattice energy which mainly arises from Madelung electrostatic potential and more directly related to lattice site potential.
Eqn (8) suggests that high valence of B in oxide BO 2 can be stable if its lattice energy is compensated by other energy terms. Since, the lattice energy is enthalpy at the energy terms such as DS and RT do not contribute at T ¼ 0 K), one can postulate that the stability of high valence ions is enhanced in crystal structure with large lattice energies. A similar analogy can be drawn for the valence stability in complex oxides where multiple ions are present such as in perovskite type ABO 3 compounds. Furthermore, cations provide large electrostatic potential contribution to the total Madelung electrostatic potential in crystal structure as shown in the plots of f A and f B in Fig. 3. This is in excellent with the trend of cation site potentials observed by Sabry et al. 24 Therefore, the valence stability of cation is rather directly related to the lattice-site potential of the cation site in a crystal structure and the stability enhances when they are present in lattice site with strong lattice site potential such as B site in perovskite structure type ABO 3 27 In order to explain this experimental fact, an in depth analysis of lattice site potential and Madelung site potential was presented in. 8 Here, we have recalculated the lattice site potential and Madelung electrostatic potential using VESTA 10 and presented in Table 1. Our recently calculated values show excellent matches with that using the procedure of Van Gool and Piken. 8 In Table 1 the lattice site potential of A ion in all perovskite structure type compounds appear to be close to those of AO but for B ion, the perovskite structure provides signicant lattice site potential gain than that in uorite structure. Except for BaTbO 3 which we believe due to not so accurate estimation of lattice parameter for TbO 2 .
Therefore, the experimental observation that B 4+ was not stable in uorite structure type B 4+ O 2 but stable in perovskite structure type A 2+ B 4+ O 3 is in accordance with the data presented in Table 1. The strong lattice site potential at B site results in large total Madelung electrostatic potential of perovskite structure. In general, Fig. 4 shows that lattice site potential of B site is stronger than A or O site in perovskite type ABO 3 in each valence state situation. Therefore, we nd that the formation of these perovskite type oxides are associated with signicant gain in Madelung electrostatic potential (DE M ) calculated by subtracting the combined E M of rock salt structure type AO and uorite structure type BO 2 from the E M of perovskite crystal structure type of nal ABO 3 (Table 1). For example, one can reasonably expect that Ba 2+ O + Ce 4+ O 2 / Ba 2+ Ce 4+ O 3 reaction may proceed as there is gain in Madelung electrostatic potential DE M of $ À43 kcal mol À1 . It should be noted that this discussion of lattice site potential and Madelung electrostatic potential is rst approximation but still can be useful to explain experimental observations. For the calculation of the E M and f, an ideal cubic structure for ABO 3 has been assumed and all the ions have been considered as point charges. In real crystals however, ions will have certain degree of polarization in their site, and various structural distortion generally leads to the deviation from cubic symmetry of the crystal structure of the perovskites. High structural distortion can lead to the decomposition of ABO 3 to component oxides. Therefore, the practical stability of these perovskite structure type ABO 3 oxides may not be very high.
It has also been shown that DE M correlates well with Goldschmidt tolerance factor is unavailable. In this situation the Madelung electrostatic potential and lattice site potential, can be calculated using a different characteristic length r ¼ r B + r O for each known A 2 O 3 , B 2 O 3 , BO, BO 2 , and ABO 3 using eqn (3)-(5) and graphically estimated for unknown compounds. 8 Further detailed discussion on the characteristic length and Madelung electrostatic potential can be found in. 8,12 However, for comparison purposes, here all lattice site  potentials were obtained using VESTA 10 that uses eqn (6) and (7) for lattice site potential and Madelung electrostatic potential and presented in Tables 1 and 2 3 have been reported to be made only under very high pressure conditions and these can be considered as very special valence combinations where stability is perhaps favored due to other energy factors such as covalency due to hybridization. 29,30 In Scheme 2 we show similar (see Table 2  In conclusion, we have revisited some fundamental aspects of synthesis, stability and properties of perovskite structure type ABO 3 . With the use of lattice site potential and Madelung electrostatic potential we have been able to demonstrate that (1) perovskite structure type ABO 3 compounds can be synthesized from their parent binary oxides because of gain in Madelung electrostatic potential which directly correlates with lattice energy, (2) Madelung electrostatic potential and lattice site potentials dictates the stability of perovskite structure and valence stability of cation in perovskite structure is direct consequence of strong cation lattice site potential and (3) the formation of defects such as oxygen vacancy or excess is driven by intrinsic lower lattice site potential of oxygen site. These results (1)-(3) can be seen various experimental data 13,32-34 however, have not been explained in terms of lattice site potential by the authors. Thus, the present article will contribute to understanding of synthesis, structures and properties of functional perovskite type oxides for various areas of applications.

Conflicts of interest
There are no conicts to declare.