Elastic strain in layered epitaxial materials

If two single crystals share a plane interface, and if every boundary atom on one side of the interface occupies a natural lattice site of the crystal on the other side, then the interface is said to exhibit bonding, as described 

For the results shown in this table, the X3 axis of reference is taken parallel to the c-axis and with the x and X2 reference axes taken on the basal (0001) plane of the crystal. All data, which represent room temperature elastic properties, are taken from the list provided in Kelly and Grove (1970). Individual references to the data are J.F. Smith and C.L. Arbogast, J. 

In the case of heteroepitaxy, the natural or stress-free lattice spacing of the film material Uf in some direction along the interface is different from the natural lattice parameter of the substrate Ug in that direction, in general. As a result, the film is said to have a lattice mismatch with respect to the substrate in that direction with the magnitude of the mismatch represented 

Denote the equi-biaxial mismatch elastic strain in the film material by 6f and in the substrate by Cg- Compatibility of deformation requires that 

The equi-biaxial stress associated with Cf and Cg are cTf = MfCf and as = MgCs, respectively, where Mf and Ms are the appropriate biaxial moduli for the materials at hand. Equihbrium requires that the net force on any cross-section must be zero, so that 

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