Anisotropic elasticity of multilayered crystals deformed by a biperiodic network of misfit dislocations

Abstract

We investigate the displacement and stress fields associated with a biperiodic misfit dislocation network located along a single interface in a multilayered crystal composite of $(N\ensuremath{-}1)$ thin bonded anisotropic elastic layers sandwiched between two semi-infinite anisotropic media. Specifically, dislocation networks of coplanar, biperiodic, hexagonal-based linear misfit are considered within continuum elasticity theory. While the homogeneous solutions are obtained by using the double Fourier series and the Stroh formalism, the solutions for multilayered structures are expressed in terms of a transfer matrix technique and the generalized Barnett-Lothe tensors. The transfer matrix technique lends itself to composites containing large numbers of bonded crystal layers because only a $3\ifmmode\times\else\texttimes\fi{}3$ matrix inversion is required. The use of the generalized Barnett-Lothe tensor facilitates the treatment of inherent elastic anisotropy in the constituent crystals. The correctness and the versatility of the method are illustrated by calculating the stress field associated with a multilayer formed by alternating GaAs and Si layers $(N=5)$ containing a single array of edge misfit dislocations along one interface. To further demonstrate the influence of the material anisotropy, numerical examples for the misfit dislocation induced stresses are given for the $(N=5)$ multilayered structure (formed by GaAs and Si) and for the induced surface displacements for an InAs thin film over a GaAs substrate. Both cubic and simplified isotropic materials are considered.