Asperity Filtering Regime

Finally, in the limit of very narrow trench widths, the trench bottom will see contact only from asperities that are both high enough and narrow enough to fit into the trench. All other asperities will simply span the gap between the sides of the feature. This is what might be termed the asperity filtering regime. 

We describe a model that combines rough surface contact mechanics with elastic pad mechanics. The theory is capable of describing planarization in both the classical contact and roughness-dominated regimes (see the solid curves in Fig. 6.7). We then introduce a simple approximation for the asperity filtering regime. 

In the asperity filtering regime, the Greenwood and Williamson theory no longer properly models the local contact pressure since the model contains no notion of asperity width. We describe a simple statistical method for incorporating width effects in two-dimensional polishing. A more sophisticated but more complex approach based on elasticity theory that takes into account asperity shape and the interaction of the asperity with trench structures of similar size can be found in Reference 27. The statistical approach assumes that... 

Equation 6.50 assumes that the asperity width is not correlated with the height that is, the joint probability density function for both height and width is separable. Equation 6.50 should be used in place of (6.41) in the topography polishing model when features have sizes in the filtering regime. 

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