a chemical mechanical polishing apparatus includes a platen , a polishing pad affixed to a surface of the platen , and a polishing head configured to retain and rotate a wafer while pressing a surface of the rotating wafer against the polishing pad . a first portion of the polishing pad that engages the polishing head proximate the edge of the wafer provides less rigidity than a second portion of the polishing pad that engages a portion of the surface of the wafer . for example , the polishing pad and / or the platen may have a recess or other cushioning structure positioned proximate a locus of movement of a portion of the polishing head that supports the edge of the wafer .