Modelling the Elastic Energy of a Bifurcated Wafer: a Benchmark of the Analytical Solution vs. The ANSYS Finite Element Analysis

Abstract

The phenomenon of bifurcation significantly impacts the warpage in wafers and consequently the yield and reliability of final systems in packages (SiPs). In this work, an analytical model of the dependence of the bifurcation energy of the substrate on the residual stress has been developed for a metalized wafer. The analytical model has been validated by exploiting Finite Element Analysis (FEA) methods. In the specific, ANSYS® simulation experiments have been designed to determine the substrate energy for the use-case of an 8″ silicon 500 μm thick wafer metalized with a 4.5 μm aluminum layer, both in the spherical and bifurcation regime. In the simulations, the bifurcation case has been induced in the wafer by applying a pair of weak forces acting as perturbations along two perpendicular diameters. The applied forces and the energy of the substrate has been assessed within a theoretical framework. The resulting principal curvatures of the wafer have been analyzed according to the bifurcation diagram reported by the theory. Moreover, it is proven that the bifurcation energy is systematically lower with respect to the spherical case as the residual stress increases and has a second order power law predicted by the analytical model. The findings of the paper can be extended to other semiconductor substrates such as silicon carbide (SiC).