Solid State Chemistry and its Applications. Anthony R. West

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 undergo coupled rotations about the z axis.

Similar coupled rotations may or may not also occur about x and y axes. Within a given sheet of octahedra, the octahedral rotations cannot occur independently of each other since clockwise rotation of one octahedron demands anticlockwise rotation of the four adjacent, corner‐linked octahedra. However, between adjacent planes in, for instance, the z direction there is no automatic coupling between adjacent planes. This gives rise to two commonly observed possibilities in which the octahedral rotations in adjacent planes are identical or the exact opposite.

      These two possibilities are illustrated in Fig. 1.41(j) and (k). Detailed analysis has shown that 15 different tilt systems are theoretically possible and classification schemes have been developed by Glazer and Alexandrov. The widely used Glazer method is as follows:

      1 Starting with the undistorted cubic perovskite, the three axes of the octahedron are given the letters a, b and c. If the degree of any rotation about all three axes is the same, they are labelled. aaa; if rotation about one axis is different, for example about c, the labelling is aac.

      2 Each of the three letters carries a superscript; if there is no rotation about that axis, the superscript is 0; if the rotation between adjacent planes of octahedra is the same, and in‐phase, the superscript