The mismatch stress field in a graded epilayer and its substrate was analyzed by analogy with thermal stresses. In particular, a straight dislocation line and a half-rectangular dislocation loop in a graded epilayer were investigated. In the case of the straight dislocation line, there were two positions in the epilayer where the resultant force which acted on the dislocation was zero. If a dislocation were generated from the surface and passed the first zero-force position, it would continue to move towards the second zero-force position, where the dislocation was at a stable equilibrium point. It was the stable equilibrium position that determined the dislocation formation energy and thus the critical thickness of the graded epilayer. In the case of the half-rectangular dislocation loop, its optimum shape occurred at a saddle point of the total energy profile as a function of dislocation width and length. The activation energy for the emission of a perfect or partial dislocation loop, the dislocation width and the aspect ratio of dislocation length to width, were all related to the graded-layer thickness. The activation energy, the dislocation width and the aspect ratio all decreased quickly as the thickness increased from its critical value and then gradually approached its extreme value as the thickness approached infinity. The extreme values of the activation energy, dislocation width and aspect ratio corresponded to those for a dislocation loop which was emitted from the surface of an epilayer having a uniform composition. The results indicated that dislocation emission was more difficult in a compositionally graded layer than in a compositionally uniform layer, and that the aspect ratio of the threading dislocation density to the mismatch dislocation density was lower in a compositionally graded layer than in a compositionally uniform layer. These theoretical results agreed with experimental observations.

A Dislocation in a Compositionally Graded Epilayer. T.Zhang: Physica Status Solidi A, 1995, 148[1], 175-89