Extension of the Stoney formula for film–substrate systems with gradient stress for MEMS applications

Using the Stoney formula and its modifications, curvature-based techniques are gaining increasingly widespread application in evaluating the stress in a film on a substrate. In principle, the formula applies only when the stress is uniform throughout the film thickness. The main purpose of this paper is to extend the Stoney formula when the residual strain in the film is no longer uniform, but dependent on the z position. To achieve this goal, a general theory was introduced for the elastic deformation of an arbitrary, multilayered system. By practicing this general theory, we used a polynomial function to describe the gradient stress in a film, and contributions by different elements of the polynomial to both the curvature and the bending strain were derived. A finite element simulation for a typical film-substrate structure was then carried out, leading to the verification of the theory developed in this paper. In the discussion section, we explored the relation between the surface curvature and the bending curvature as well as the difference between the stress in the constrained planar state and that in the relaxed state. In addition, the accuracy of the simplified formula, using thin film approximation, was evaluated. Finally, a SiNx-Al MEMS structure was studied by using the formula in this paper.