Elastic bending of semiconductor wafer revisited and comments on Stoney's equation

Since Stoney derived a simple relationship between the stress of a thin film and the curvature of its substrate for an electrolytically deposited metallic film on a thick substrate plate in 1909, the equation has been widely used by electrochemists to calculate stresses in electrolytically deposited films. Although the equation seems to work well for thin films with negligible thickness in comparison to the substrate, the underlying assumption in his derivation is found to be questionable. Various works have been published which calculate more precisely the stresses in the bending of a two-layer composite plate, where a thin layer on a thick substrate is an extreme case. However, there is some confusion in differentiating elastic bending due to internal stress from the externally applied moments (including our previous work). We re-examine the two bending conditions and point out that Stoney's equation was derived based on a neutral axis for zero bending moment which does not exist in the pure bending of a steel ruler on which his derivation was based. We also show that a single neutral axis for stresses does not exist in a two-layer system bent by internal stresses. Finally we rederive the formula for the relationship between the wafer curvature and the lattice mismatch or differential thermal expansion far a binary composite plate using the proper bending moment calculation and examine the case of wafer bending of GaAs on Si substrate.