Simulation of the deformation behaviour of large thin silicon wafers and comparison with experimental findings

Abstract

The deformation of large thin uncoated silicon wafers without remaining intrinsic misfit stresses resting on a ring is investigated. We use both, Finite Element simulations and THz tomography mapping. Specific attention is given the scaling of the warping for increasing slenderness of those wafers. We follow the approach of starting with a known solution for a compact wafer and increase the slenderness, i.e. increase the radius and decrease the thickness, using simulation models. Then, we measure the warping by THz mapping for some slender wafers and compare the data to simulation results. We compare the maximum warpage for given loadings and we compare the deflected shapes. Due to the geometric ratio radius/thickness of over 1000∶1 and the anisotropic material behaviour, simulations can only be done effectively using shell element modelling of a spatial plate. And due to large warpages in the order of 10 times of the thickness, only incremental update Lagrange nonlinear calculations give reliable results. Simulations using the available shell elements overestimate slightly the values measured by tomography, but still yield acceptable values with errors less than 10% for very slender wafers and below for more compact ones. For invariable loading conditions, a logarithmic scaling function gives an acceptable estimate for the maximum warpage for increasing slenderness. An additional important observation was that the warpage of thin wafers is heavily affected by the size of the contact radius of a weight.