Error analyses on some typically approximate solutions of residual stress within a thin film on a substrate

Abstract

Stoney's equation and subsequent modifications and some approximations are widely used to evaluate the macrostress within a film on a substrate, though some of these solutions are only applicable for thin films. The purpose of this paper is to review the considerable efforts devoted to the analysis of residual stresses in a single-layer film in the last century and recent years and to estimate the errors involved in using these formulas. The following are some of the important results that can be obtained. (1) The exact solution for the residual stress can be expressed in terms of Stoney's equation [Proc. R. Soc. London A82, 172 (1909)] and a correction factor, (1+Ση3)∕(1+η), where Σ,η are the ratios of the elastic modulus and the thickness of the film to those of the substrate, respectively. (2) When the thickness ratio of the film and the substrate is less than 0.1, Stoney's equation and Röll's approximation [J. Appl. Phys. 47, 3224 (1976)] do not cause serious errors. (3) The approximation proposed by Vilms and Kerps [J. Appl. Phys. 53, 1536 (1982)] is an improved modification for Stoney's equation and can be applicable when η≤0.3. (4) The approximations proposed by Brenner and Senderoff [J. Res. Natl. Bur. Stand. 42, 105 (1949)] and Teixeira [Thin Solid Films 392, 276 (2001)] can lead to serious errors and should be avoided. (5) The approximation based on the assumption of constant elastic modulus is only applicable for a ratio of η≤0.01 and can be very misleading.