Celebrating the 100th anniversary of the Stoney equation for film stress: Developments from polycrystalline steel strips to single crystal silicon wafers
Stress in a thin film on a flexible substrate induces a curvature of the substrate. Usually the substrate is orders of magnitude thicker than the film, leading to small and purely elastic deformation of the substrate. In this case, the Stoney equation yields the stress in the film from the measured curvature of the substrate. The Stoney equation contains thickness of film and substrate and the elastic properties of the substrate. Typically the elastic properties of the substrate are specified by E (Young's modulus), and ν (Poisson's ratio). E and ν provide a valid description for elastically isotropic substrates, e.g. polycrystalline steel strips, as used by Stoney in 1909.
Today the Stoney equation is still used for relating substrate curvature to film stress. However, in the majority of thin film stress measurements by means of substrate curvature, Si wafers are used as the substrate. Silicon wafers are cut from single crystals and are thereby elastically anisotropic. In the present paper, a modified form of the Stoney equation, well known for elastic isotropic substrates, is derived for Si(001) and Si(111) wafers, using the elastic stiffness constants of silicon, cij, instead of the orientation averaged values E and ν, which do not have a meaning for elastically anisotropic single crystal materials.
Curvature measurements of thin films on Si(001) and Si(111) wafers are presented. The difference in film-stress-induced curvature of Si(001) and Si(111) wafers is discussed.