Paper TF-ThP33
Stress-curvature Relationship for Configurations with Thin and Anisotropic Substrates undergoing Large Deformations

Thursday, October 22, 2015, 6:00 pm, Room Hall 3

In a thin film configuration, the film is often stressed to conform to the surface of the substrate, commonly due to epitaxial effects, difference in thermal expansion coefficients between the film and the substrate materials, or phase transformations accompanied with volume changes. This stress causes the film-substrate system to assume a curvature. The relation that relates this curvature to the stress in the film is referred to as the Stoney equation. One of the major assumptions of the Stoney equation is that the substrate is orders of magnitude thicker than the film, leading to small and purely elastic deformation of the substrate. Moreover, the well known equation also assumes that the substrate material is isotropic in nature.

At the moment, the traditional Stoney equation is being used to relate the film stress to the system curvature. However, in majority of the cases where the film stress is measured from the system curvature, Si wafers are used as substrates. Si wafers are anisotropic in nature as they are obtained by slicing single crystals. In this paper, a more generalized version of the Stoney equation is presented.

This form of the equation is derived by first relaxing the constraints that the film is very thin compared to the substrate and that the deformations are small. Next, this formula is modified specifically for the cases of Si(001) and Si(111) wafers. Ultimately, a modified version of the Stoney equation is presented which can be used for configurations with anisotropic substrates where the thicknesses of the film and substrate are comparable, and the deformations are large. Extension of this expression to systems assuming asymmetric elliptical configurations rather than symmetric spherical deformations, is also discussed.

Keywords: Thin film, Large deformation, Stoney, Silicon, Wafer curvature