Determination of thin-film stresses on round substrates

Thin films deposited on wafers show, in some cases, large variations in stress. The local curvature is often used in the Stoney equation [ G. G. Stoney , Proc. R. Soc. London, Ser. A 82 , 172 ( 1909 ) ] to calculate the local stress. This practice leads to false stress results. A general method for the calculation of the film stress, for films with isotropic stress, is presented here. The stress is calculated from both the local and overall shape of the substrate. Simple examples of isotropic stress were simulated and the substrate deformation calculated. The film stress was calculated back from the substrate shape with good agreement with the initial stress distribution. The substrate shape under films with centrosymmetric stress distribution can be analytically calculated. The current method successfully calculates the stress distribution from the substrate shape. Some guidelines for the deviation of the calculated stress from that of Stoney's are given. Finally, the case of nonisotropic stressed film cannot be generally solved since there is no unique stress distribution for any given substrate shape.