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

Abstract

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 SiN x-Al MEMS structure was studied by using the formula in this paper.

Mechanical Analyses and Structural Design Requirements for Flexible Energy Storage Devices

“…Stoney's formula [68][69][70][71][72][73] was extensively used to describe the residual stress of the film-substrate system produced by thermal change/phase transition/shrinkage of devices during the preparation and post-treatment process. The formula is the theoretical basis of curvature-based technique to evaluate the interface stress status in the film-substrate system.…”

Effects of surface and initial stresses on the bending stiffness of trilayer plates and nanofilms

“…Elastic and thermal mismatch of individual materials can result in significant internal residual stresses in multilayer systems. As internal residual stresses can influence the overall mechanical properties (such as the stiffness and the natural frequency) and hence the dynamic and stability characteristics of multilayer systems, their determination has long interested material scientists and engineers [Stoney 1909;Freund 2000;Hui et al 2000;Klein 2000;Dodin et al 2001;Freund and Johnson 2001;Thompson and Clyne 2001;Donadon et al 2002;Xu and Zhang 2003;Malzbender 2004;Huang and Zhang 2006;Huang and Zhang 2007]. Both the experimental measurement of internal stresses and the optimal design of multilayer systems rely heavily on the accuracy of the involved correlations.…”

Stress measurement in East Asian lacquer thin films owing to changes in relative humidity using phase-shifting interferometry

“…Provided that certain conditions are satisfied, s is independent of the Young modulus and Poisson's ratio of the film, which makes this method appealing for our purposes as these are currently unknown. The following assumptions (Huang & Zhang 2006) system has to be designed so that assumptions 1-4 are all met. Secondly, a sensitive method is required to measure the small deflections expected of the substrate ( §2b).…”