The ordering of crystallographic shear planes: Theory of regular arrays

In certain non-stoichiometric transition metal oxides, crystallographic shear planes are found instead of isolated oxygen vacancies. These planes tend to form arrays with regular spacings. The interactions responsible for this ordering have not been understood in the past. We propose that the major interaction comes from elastic strain, with smaller electrostatic terms: the configuration of the planes is one which minimises the elastic strain energy. Quantitative results are given for a realistic model, and agree with the main features observed experimentally. Thus the regular array of planes is stable, and the equilibrium separation of planes in the array is about half that predicted for an isolated pair of planes. The interaction between isolated planes proves an oscillatory function of their separation. One can understand from its detailed form why the observed mean spacing gets smaller as the planes come together to form larger groups.