Crystal structure of perovskite-type oxides Ideal perovskite-type structure

Formally, the “perovskite” designation is reserved for CaTiO3 mineral, discovered by Gustav Rose and named after another mineralogist Lev Perovski. Soon it has been found that other compounds possess the same cubic structure and ABX3 stoichiometry. These materials are referred as having perovskite-type structure (or simply as perovskites), although CaTiO3 was later determined to exhibit orthorhombic symmetry [116]. From the crystallographic point of view, ideal perovskite structure is cubic with Pm-3m space group (group no. 221 in Hermann-Mauguin notation). In ABX3 formula, A typically represents a large metal cation, usually belonging to lanthanide or alkaline earth metal groups. B-site cations are significantly smaller, and typically 3d metal cations are located in this position, however, commonly other cations can be also present, like cerium or niobium. Most typical X-site ion is the oxygen anion, but sulfide or halide anions can also occupy this site.

Combination of the oxidation state of ions in ABO3 is often A³⁺, B³⁺ and O^2-, however, various different options are also possible (e.g. □Mo⁶⁺O3, Li⁺Nb⁵⁺O3 or Ba²⁺Ti⁴⁺O3;

Chapter 4. Physicochemical properties of perovskite-type oxides 38 different proportion is possible and of scientific interest, since it allows to stabilize metal cations in B sublattice in unusual oxidation state such as Cu³⁺, Bi⁴⁺ or Ir⁵⁺.

Fig. 4.1 shows visualization of the crystal structure of the ideal perovskite unit cell.

In this structure X-site anions together with A cations form a face centered cubic (fcc) arrangement, in which ¼ of the formed octahedral voids is occupied by B-site cations. The A-site cations are surrounded by twelve anions in 12-fold (cubo-octahedral) coordination, while B cations are surrounded by six anions in octahedral coordination (BX6 octahedra).

Regarding X anions, their coordination sphere is created by four A cations and two B cations [118]. Alternatively, the structure can be described as created by corner-sharing BX6 octahedra, in which A cations are placed in all voids having 12-fold coordination.

Fig. 4.1. Visualization of structure of ideal, cubic perovskite. A-site can be described with Wyckoff notation as 1a (0,0,0), B-site as 1b (½,½,½) and X-site as 3c (0,½,½). Radii of ions not to scale.

Since only a few oxides from the ABX3 group crystallize in perfectly regular structure, in 1927 Goldschmidt introduced a structural tolerance parameter (tolerance factor) that allows for a geometric determination of a degree of deformation of the structure from the ideal, cubic one. With approximation of the ionic radii, the following equation (10) can be derived for the tolerance parameter :

(10)

where: , and are respectively the Shannon’s ionic radii [17] of the A and B cations, as well as the X anion. In the case of two or more different ions present in one sublattice, the respective average ionic radius should be used.

It was noticed that there is a certain correlation between the parameter and the crystal symmetry, as presented in Tab. 4.1, i.e. in most cases, the closer the value to unity, the higher the crystal symmetry. However, calculation of the tolerance factor for a selected chemical composition of ABX3-type oxide does not allow to predict the actual space group, in which the material would crystallize [118].

Tab. 4.1. Dependence of the crystal structure type and the Goldschmidt’s tolerance factor [117, 119].

tolerance factor symmetry

< 0.85 ilmenite-type structure (not perovskite)

0.85 < < 0.90 orthorhombic

0.90 < < 1 rhombohedral

≈ 1 cubic

1 < < 1.06 hexagonal (not perovskite)

The presented above relationship should be only treated as an approximation, since various other factors, such as nonstoichiometry in A or X lattice, degree of covalent bonding, metal-metal interactions, as well as Jahn-Teller effect (JTE) affect the actual structure. From this point of view it seems more adequate to use real distances between the atoms for calculation of the tolerance factor, which can be obtained from precise structural measurements [120, 121].

Distorted perovskite-type structures

Considering the above, it is not surprising that the actual ABX3 materials rarely adopt the ideal perovskite structure. More often they crystallize in a distorted structure, which arises from breaking of at least one of the possible symmetry elements present in the Pm-3m space group. This effect is mainly due to rotation or tilt of BX6 octahedra, displacement of B-site cation from the center of the octahedron, distortion of the octahedra e.g. due to the JTE or increase of the covalence bonding of A-X and/or B-X [118].

Rotation or tilt of BX6 octahedron is caused by mismatch, i.e. too large or too small size of the A cation with respect to the requirements for 12-fold coordination. To adapt, octahedron tilts, and therefore lowers the energy mode for the crystal. The rotation is manifested by a change in A-X bond lengths, which cease to be equal. Consequently, lowering of the symmetry occurs, and decrease of the coordination number of the A cation

Chapter 4. Physicochemical properties of perovskite-type oxides 40 is given as a rotation about each of the orthogonal Cartesian axes, and the rotation is specified as “ ”, “ ” and “ ” symbols. Moreover, superscript are used to describe direction of the rotation, whether it is in the same direction (so called in-phase tilt) or contrary (anti-phase tilt) in relation to the previous layer. Two basic types of octahedra tilts are visualized in Fig. 4.2.

a) b)

Fig. 4.2. Two main types of octahedral tilt: a) and b) , seen along z-axis. Based on [123].

In the case of the tilt system (Fig. 4.2a), generating P4/mbm space group (no.

127), octahedra are tilted only along the z-axis, and the tilt is in-phase. Lack of inclination along the x and y directions is given in double notation. The anti-phase arrangement (Fig. 4.2b) is described as , generating I4/mcm space group (no. 221). The octahedra tilt in this case is also along the z-axis only, but the following layer (in relation to one selected) is titled in the opposite direction. This is noted by symbol. In the notation repeating of the symbol indicates the same angle of rotation relative to the x-, y- or z-axis, while the superscript indicates a lack of rotation. Consequently, the regular structure of the ideal perovskite with no distortions present can be assigned with notation.

Initially, Glazer described 23 possible different combinations of the tilts. This number was eventually reduced by Howard and Stokes down to 15 [124]. Fig. 4.3a.

presents the theoretical relationship between the space groups in ABX3 perovskites, with possible phase transitions associated to a particular tilt system change. These transitions may occur due to change of the pressure, partial pressure of the oxidizer, temperature, as well as chemical composition. However, in literature there is no consensus, and somewhat different diagram for selected space groups was proposed by Thomas [125] (Fig. 4.3b).

Fig. 4.3. a) Classification of space groups and possible phase transitions between perovskite-type structures, as discussed in work [124]. b) Different diagram presented in work [125]. Solid lines indicate second order phase transition, while dashed lines specify first order phase transformations. Based on [124, 125].