Critical thickness of an epilayer on a substrate with different elastic constants

The critical thickness of an epilayer on a substrate with different elastic constants is investigated by isotropic elasticity and anisotropic elasticity. Two closed formulas for the two approaches are derived to calculate the critical thickness. The results indicate that the self energy of the dislocation is controlled by the soft phase between the epilayer and the substrate, while the interaction energy depends only on the elastic constants of the thin film. It is easier for a dislocation to be formed if the substrate is softer than the film, and consequently the critical thickness is smaller. On the other hand, a soft epilayer can have a large thickness without any mismatch dislocation. For anisotropic cases, explicit equations are given here for the {100}, {110} and {111} epitaxial planes. The system of a Ge_xSi_1-x epilayer on a Si substrate is taken as an example to demonstrate the influence of the difference in elastic constants on the critical thickness. Even though the difference between the elastic constants of the epilayer and the substrate is not very large, ignoring this difference can cause a relative error over 20% in calculation of the critical thickness. For this system, a simplified equation yields sufficiently accurate results.