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The paper examines the problem of film stress applying a correct three dimensional model. The results are compared with two different forms of Stoney s equation existing in the Literature and being widely used in the determination of stresses in thin films. It is shown theoretically that only one of the forms is based on an adequate model and yields accurate results whereas the other causes errors of about 30-40 % for typical substrate materials. In addition limits for the applicability of the correct Stoney equation are given. Introduction Internal stress in a film on a plate-like substrate causes the film-substrate compound to warp until mechanical equilibrium is reached, i.e. until both net force and bending moment are zero. A plate-like form of the substrate means that the substrate thickness, hs, is constant and small in comparison to its lateral dimensions. From the curvature of the elastically deformed coated substrate the average film stress, σ, can be calculated. This method is very popular since the curvature of the bent substrate can easily be measured and no information on the elastic parameters of the film material is necessary. As substrate material often silicon is used since its mechanical properties are well defined and well known. If necessary, small beams (cantilevers) can be made of single-crystal silicon using micromechanical technology (e.g. Elwenspoek and Jansen 1998) which allows to apply the method also to very thin films. When the thickness of the film, hf, is small compared to that of the substrate, a simple formula holds which was first published by STONEY 1909: 2 1 , 6 f s zz s f h E h R σ ≈ − (1) 1 Tel. ++493423656639, Fax. ++493423656666, E-mail: n.schwarzer@esae.de

On the determination of film stress from substrate bending: STONEY´s formula and its limits