structure and the framework of octahedra may be slightly twisted or distorted. An example is shown in Fig. 1.41(g) for the structure of GdFeO3. The reasons for the structural distortions are associated with the size requirements of the 12‐coordinate A and six‐coordinate B sites and whether adjustments to the structure are required to accommodate different‐sized cations. Also, more complex perovskite structures form in which two different cations may occupy either the A or B sites, giving a range of cation ordering possibilities.
1.17.7.1 Tolerance factor
The reason why structural distortions occur in many perovskites is that the A and/or B atoms are not exactly the right size to fit the sites generated by the remainder of the structure. In an oxide with the ideal, cubic perovskite structure, the bond lengths are related to the unit cell dimension, a, by
Since bond lengths for each element, oxidation state and coordination number usually fall within closely defined ranges (Appendix F), it is possible to use equation (1.6) to see how well the sizes of a particular A, B combination meet the requirements for an undistorted, ideal perovskite. The degree to which the sizes depart from equation (1.6) is given by a tolerance factor, t:
Table 1.18 Some compounds with the perovskite structure
| Compound | a/Å | Compound | a/Å | Compound | a/Å |
|---|---|---|---|---|---|
| KNbO3 | 4.007 | LaFeO3 | 3.920 | ||
| KTaO3 | 3.9885 | LaGaO3 | 3.875 | CsCaF3 | 4.522 |
| KIO3 | 4.410 | LaVO3 | 3.99 | CsCdBr3 | 5.33 |
| NaNbO3 | 3.915 | SrTiO3 | 3.9051 | CsCdCl3 | 5.20 |
| NaWO3 | 3.8622 | SrZrO3 | 4.101 | CsHgBr3 | 5.77 |
| LaCoO3 | 3.824 | SrHfO3 | 4.069 | CsHgCl3 | 5.44 |
| LaCrO3 | 3.874 | SrSnO3 | 4.0334 |
(1.7)
In practice, there is some flexibility over bond lengths and usually, a cubic perovskite forms with t in the range 0.9 < t < 1.0.
For t > 1, the B site is larger than required. If t is only slightly greater than 1.0, the structure distorts but is still basically a perovskite as in BaTiO3, t = 1.06. There may also be a change in the stacking sequence of the AX3 close packed layers from ccp to hcp to give the family of hexagonal perovskites typified by BaNiO3. For larger departures from t = 1.0, however, the B ion demands a smaller site, of lower coordination number, and the structure changes completely, as in BaSiO3 which has tetrahedral Si.
For smaller tolerance factors, 0.85 < t < 0.90, several different kinds of structural distortion occur because now, as in GdFeO3, the A cation is too small for its site. These distortions generally involve tilting and rotation of the BO6 octahedra as shown in Fig. 1.41(g). Consequently some, or all, of the B–O–B linkages are no longer linear but are zig‐zag, which has the effect of reducing the size of the A cation site.
1.17.7.2 BaTiO3
BaTiO3 is tetragonal at room temperature, a = 3.995, c = 4.034 Å, with the structure shown in projection on the ac plane in Fig. 1.41(h). Since Ti is slightly too small for its octahedral site, it displaces by about 6% of the Ti–O distance towards one of the corner oxygens; Ba2+ ions also undergo a smaller displacement in the same direction. This reduces the coordination of Ti to five (square pyramidal) and, to have reasonable Ti–O bond lengths, the structure also contracts slightly in the ab plane [not shown in (h)].
Ti atoms in adjacent unit cells undergo a similar displacement in the same direction and the resulting structure has a large dipole moment due to the separation of positive and negative charge centres. It is possible to flip the orientation of the dipoles: under the action of an applied electric field, the Ti atoms move through the centre of the octahedral site towards one of the other corner oxygens. This ready reversibility gives the structure high polarisability and a high permittivity (or dielectric constant) and is responsible for the property of ferroelectricity (see Section 8.7).
1.17.7.3 Tilted perovskites: Glazer notation
We saw in Fig. 1.41(g) how octahedra can tilt or rotate cooperatively leaving the BO6 octahedra essentially unchanged but reducing the size of the A site and its coordination number. Thus, in GdFeO3, the A site is eight‐coordinate instead of 12‐coordinate. Such structural distortions occur when the A cation is too small to occupy comfortably the 12 coordinate sites created by the array of corner‐sharing BO6 octahedra and, consequently, the octahedral rotations allow a reduction in the A–O bond length.
A wide variety of structural distortions occur in perovskites whose tolerance factor is less than unity. The most common distortion involves tilting or rotation of octahedra about one or more of the three axes of the octahedron. This is a cooperative process since octahedra link at their corners to adjacent octahedra in the 3D framework and, for instance, clockwise rotation of an octahedron about one axis causes anticlockwise rotation of the adjacent octahedra, Fig. 1.41(i). For the example shown, octahedra which form sheets in the xy plane
