The energy of arrays of dislocations in an anisotropic half-space

Abstract

Exact closed-form solutions are obtained for the energy of arrays of dislocations lying at the interface between a strained epitaxial layer and substrate that are elastically anisotropic and have the same elastic constants. Closed-form solutions are obtained both when all dislocations in an array have the same Burgers vector and when the sign of one or more of the components of the Burgers vector alternates within an array. The method of solution makes the consideration of multiple arrays particularly simple (i.e. sets of more than one array in the interface, each inclined to the others at some non-zero angle, e.g. a pair of orthogonal arrays). Also, an approximate treatment is given of the case in which the elastic constants of the substrate differ from those of the layer by a small perturbation. The method is based on performing a perturbation expansion about the solution for the homogeneous body. A correction term, accurate to first order in the perturbation, is thus obtained to account for the inhomogeneous nature of the body. The given formulae will allow a thorough assessment of the significance of anisotropy in many areas of study, and in particular its significance for the mechanical stability of semiconductor heterostructure devices.