Extending Stoney's equation to thin, elastically anisotropic substrates and bilayer films

• Stress–curvature relations for systems with thin and anisotropic substrates are derived.
• Relations are extended to accommodate bilayer films.
• Derived equations are applicable for systems with anisotropic film materials as well.

The Stoney equation has been a powerful tool for the thin film community to measure the residual stresses induced in a film through the measurement of curvature of a film–substrate system. Two of the main assumptions of the original Stoney equation are that the substrate is much thicker than the film and its material is isotropic in nature. However, in majority of the cases where the film stress is measured from the system curvature, Si wafers are used as substrates, which are anisotropic in nature. The anisotropic substrate problem was solved by Nix [1] for thick substrates. In this paper, a modified version of the Stoney equation is derived for configurations with thin anisotropic substrates, specifically for the cases of Si(001) and Si(111) wafers. The same methodology is then used to extend the Stoney formula to systems with bilayer films on thin substrates.