Extension of the Stoney formula for the incremental stress of thin films

Abstract

The curvature-based technique is widely used to monitor stresses in thin film-substrate structures. The fundamental theory is based on the Stoney formula and its extensions, which mostly represents the average stress in thin films. However, the characterization of incremental stresses has not been clearly described, which leads to significant limitations in its application scope. Here, a formula for the incremental stress of thin films is proposed by discretizing the film growth layer for use in arbitrary modulus and thickness ratios. In specific cases, this formula can be manifested as Stoney's and Freund's formulas. The stress of electrodeposited Ni with different current densities was measured in situ for substrates at different elastic moduli and thicknesses. The formula was proven to be a more accurate expression of the incremental stress compared with averaging using Freund's formula or from the slope method. Furthermore, the proposed formula can be applied to arbitrary modulus and thickness ratios. This method provides valuable insight to monitor stresses in thin films and paves the way for further understanding the mechanisms of stress generation.