Analysis of substrate curvature and film stress through a reformed mathematical description on mismatch strain

摘要

Stoney’s formula first related film stress to substrate curvature, but properties of film material were not contained in it, which was modified by Freund’s formula. In the derivation of Freund’s formula, a uniformly preassigned mismatch strain throughout film was adopted. For the epitaxial growth of film on substrate, the present paper suggests a different imagination that the deformation of the film-substrate system attributed to the mismatch between two crystals is produced in the following way: a coupled lattice constant emerged at the interface is shared by film and substrate, by which the interaction between film and substrate, and then the development of stress and strain fields in the bilayer structure are induced naturally. As a result, a theoretical model with a closed form for calculating the substrate curvature and the corresponding film stress has been established, the validity of which has been confirmed by compared with Stoney’s formula and Freund’s formula via examples. Our analysis is limited to the linear small deformation, but the new understanding on the mismatch strain can be extended to the nonlinear analysis on the large deformation.