An exactly solvable model for calculating critical misfit and thickness in epitaxial superlattices - Layers of equal elastic constants and thicknesses

A parabolic interaction potential has been used to develop a model for calculating the misfit dislocation (MD) energy in the case of a superlattice of alternating layers of materials with equal elastic constants and thicknesses. The model, which is believed to be a good one for small misfits and to have some merit for covalent bonded materials, is exactly solvable for the critical thickness above which it is energetically favorable to lose coherency by the introduction of MDs into the interfaces. It was found, for a given misfit f, that the critical thickness for epitaxial superlattices free from their substrate is somewhat more than four times that for a single epilayer on a thick substrate. Furthermore, the critical thickness varies almost inversely with misfit to the power 1.22 when Poisson's ratio is 1/3. It was also shown that the critical misfit f(c) obtained by equating maximal misfit strain and MD energies is a significant overestimate of f(c). The results for a superlattice are compared with those of a thin layer on a thick substrate.