Analysis of Asymmetric Warpage of Thin Wafers on Flat Plate Considering Bifurcation and Gravitational Force

Large deformation of thin silicon wafer on flat plate induced by film stress was analyzed by analytic method and finite element analysis (FEA) method considering bifurcation and gravitation. Bifurcation criterion was obtained by energy method, and resulting warpage was calculated from obtained curvature. Anisotropy of substrate and film material was considered. For simple cases where mismatch strain was isotropic but the silicon substrate was anisotropic, explicit expression of the bifurcation criterion was obtained. Material anisotropy made great effect on the criterion for (100) wafer. It was almost always in the bifurcation state when the wafer was thinned for packaging process. FEA models to analyze bifurcation were investigated. Gravitational force accompanying with the contact between the flat plate and the silicon wafer were applied. A carefully chosen quarter-model rotated by 45 degrees was sufficient to analyze the warpage behavior. Bending down by gravitational force and nonuniform curvature over the wafer were observed. Two FEA schemes were proposed. One was combination of buckling and postbuckling analysis. The other was the force or moment perturbation method. Both methods well described the warpage considering bifurcation and gravitation, and reached the same result.