A Mixed-Lubrication Approach to Predicting CMP Fluid Pressure Modeling and Experiments

Chemical mechanical polishing (CMP) is a manufacturing process used to remove or planarize metallic, dielectric, or barrier layers on silicon wafers. During polishing, a wafer is mounted face up on a fixture and pressed against a rotating polymeric pad that is flooded with slurry. The wafer also rotates relative to the pad. The combination of load on the wafer fixture, relative speed of rotation, slurry chemistry, and pad properties influences polishing rates. Prior work has shown that an asymmetrical subambient pressure, which exceeds that expected from the applied load, can develop at the interface between the fixture and a plane pad. The spatial distribution of this pressure can be measured and then simulated using a specially designed fixture with water as the slurry. A mixed-lubrication approach to modeling the fluid pressure was developed by including the contact stress, frictional behavior, and fluid film thickness. For a given fixture/pad separation, the contact stress can be determined using a Winkler model approximation. The film thickness can be approximated as the distance from the fixture surface to the mean asperity plane. Once the fluid film thickness is known, the fluid pressure can be determined from the two-dimensional polar Reynolds equation using finite-differencing. The theoretical pressure solution was found to match the experimental pressures when the system of forces and moments were balanced. The iterative secant numerical method was employed to compute the appropriate fluid film thickness that accommodates a balanced system of forces and moments produced by the fluid/solid interactions. After the fluid pressure is determined from an initially assumed separation, all shear and normal forces are computed from the solid contact stress and hydrodynamic fluid pressure. The results agree with the experiments.