Our group has proposed several chip-scale CMP models, with key assumptions including the notion of planarization length in the pattern density model , and step height dependent polishing rate in the density step height model . In the effective density model, planarization length is the characteristic length of an elliptic weighting function based on the long-range pad deformation and pressure distribution during CMP. This semi-physical model is often adequate and usually gives a fitting error of a few hundred angstroms. As ever-shrinking device size pushes for tighter control of post CMP uniformity, however, we need a chip-scale CMP model with better accuracy.
In this work, we re-examine the physical basis for averaging weighting functions and step height dependence, particularly in the context of contact mechanics based model formulations. The comparison of the two models confirms that the analytical density and step height models can be viewed as approximations to the contact wear model. The study also suggests a new dependence of contact height on line space and pattern density.