Critical thickness during three-dimensional epitaxial growth—A self-consistent approach

Abstract

A self-consistent model is developed to describe the formation of misfit dislocation loops in hemispherical epilayers of finite size grown on thick substrates. The lattice mismatch between the substrate and the epilayer is described by virtual interfacial dislocation loops situated in the interface. The free surface boundary conditions are satisfied by the surface dislocation loops situated on the surface of the hemispherical island. A misfit dislocation loop is nucleated and the changes in the energy of the configuration are used to determine if a lowering in the total energy of the configuration can occur. The model consisting of the primatic dislocation loops in the epilayer is replaced by a simpler dislocation model consisting of glide dislocation loops so that the stress fields and interaction energy terms between dislocations in a two-phase medium could be evaluated. The energy associated with the coherent epilayer and that with the misfit dislocation are evaluated by the minimization of the total energy of the configuration with respect to the Burgers vectors of the surface dislocation loops. The position of the misfit dislocation loop from the interface is changed for a given size of the hemispherical island and the energy terms are determined. The formation of a misfit dislocation loop is considered favorable when the energy of the configuration in the presence of the misfit dislocation is lower than that of the coherent epilayer. The numerical analysis is carried out for hemispherical islands of GaAs grown on (100) Si with a misfit dislocation of Burgers' vector 3.84 Å. It has been found energetically favorable to nucleate a misfit dislocation loop at a distance of 3 Å from the interface when the radius of the hemispherical island is equal to or greater than 40 Å. In addition, a misfit dislocation loop could be nucleated at a larger distance from the interface when the size of the epilayer is larger. The results are interpreted in terms of the favorable position for nucleation of the misfit dislocation and its glide motion towards the interface.