Longitudinal anisotropic stress and deformation in multilayered film heterostructures due to lattice misfit

A modified shear lag analysis based on the formulation of elastic anisotropy is proposed and developed to calculate the longitudinal stress in multilayered heterostructures of semiconductor materials with interfacial lattice misfit strain. The shear lag approach to analyze layered structures was adopted in Nakajima's model, in which the misfit strains were considered by imposing an internal force on each imaginary layer along the edges. Since lateral deflection and internal moment are not introduced in shear lag analysis, the moment balance cannot be satisfied even if the imaginary layer is very thin. Moreover, the internal force, imposed on the boundary of the layered structure, is determined based on the condition of isotropic elasticity; this method cannot evaluate the distributions of anisotropic longitudinal stress and deflection curvature in layered structures. In this study, a modified shear lag method, which is formulated based on the theory of anisotropic elasticity and also takes account of both the effects of lateral deflection and moment balance, is proposed. A realistic traction boundary condition, instead of imaginary internal force evaluated under the condition of infinite length, is imposed on the both edges of the structure. In convenience of formulation as well as to facilitate the efficiency of computation, the matrix and eigenvalue methods are also applied to decouple and solve the linear systems of differential equations. Since the moment balance, misfit strain and curvature are all considered in analysis, compared to the Nakajima's model, this method provides a more accurate and efficient approach to investigate the deflection and distributions of lateral stress induced by misfit strains and thermal loadings in multilayered heterostructures under the condition of anisotropic elasticity.