A Theoretical Analysis for Arbitrary Residual Stress of Thin Film/Substrate System With Nonnegligible Film Thickness

Stoney formula is widely used in advanced devices to estimate the residual stress of thin film/substrate system by measuring surface curvature. Many hypotheses including that thin film thickness is ignored are required, thus bringing significant error in characterizing the inhomogeneous residual stress distribution. In this article, arbitrary residual stresses on thin film/substrate structures with nonnegligible film thickness are modeled and characterized. We introduce nonuniform misfit strain and establish the governing equations including mismatched strain, displacements, and interfacial stresses based on the basic elastic theory. The parameterization method and the method of constant variation are used in the process of equation decoupling. The expressions between displacements, surface curvatures, and misfit strain are determined through decoupling calculations. By substituting misfit strain, residual stresses are expressed by parametric equation related to surface curvature. It further indicates that there is a "non-local" part between the film stress and curvature at the same point. Compared to neglecting the film thickness, the proposed method eliminate relative errors up to 58.3%, which is of great significance for stress measurement of thin films and substrates.