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

This study addresses the problem of the critical thickness of an epilayer grown on a finite substrate with different elastic constants. The principle of superposition and Fourier integral methodology are used to solve the displacement and stress fields that satisfy the boundary conditions. The change in strain energy caused by the introduction of a misfit dislocation is defined as the dislocation formation energy Ef. Meanwhile, the epilayer thickness, corresponding to Ef=0, is the epilayer critical thickness hc. This investigation reveals a promising characteristic of using a thin substrate, namely that when the substrate is very thin and the shear modulus ratio of epilayer over substrate is 1/10, then if the corresponding hc is smaller than the substrate thickness, hc will decrease as the shear modulus ratio increases. However, if the corresponding hc is greater than the substrate thickness, hc markedly increases with an increase of the shear modulus ratio, becoming infinite. When the substrate is very thin, the hc also increases rapidly with the epilayer (substrate) Poisson ratio and finally reaches infinity; however, the pattern differs from that of the variation in the ratio of the shear modulus. If the substrate becomes thinner and transforms into a diaphragm structure, the epilayer critical thickness reaches infinity, regardless of the magnitude of the shear modulus ratio, epilayer, and substrate Poisson ratio. Results obtained when the epilayer and substrate share identical elastic constants are compared with those of Zhang et al. and Fruend and Nix. The present result lies between those obtained in these two earlier studies.