Pressure calculation using obstacle problem for CMP modeling

Chemical mechanical polishing/planarization (CMP) is one of the most important fabrication technologies in the semiconductor industry. CMP is used to achieve planar surfaces and remove excess deposited material from the wafer. During CMP, a chemical "slurry" containing abrasive particles and chemical reagents is deposited on the polishing pad. The polishing pad is then pressed against the rotating wafer. The combined action of the polishing pad and chemical slurry results in material removal and planarization of the wafer surface. Modeling of the CMP process allows detection of potential planarization defects in chips before manufacturing. Accurate computation of pressure distribution across the wafer surface is crucial for accurate modeling of the CMP process. The pressure distribution calculation is usually done using the Hertz contact theory and Chekina model, which involves computations of direct and inverse fast Fourier transforms (FFT and IFFT) and pad displacements updates. In this paper, we adopted an obstacle problem approach for calculating the pressure distribution across the wafer/die surface for CMP modeling. In an obstacle problem, the pad is assumed to be an elastic membrane with fixed boundaries that is displaced by the wafer surface. The goal is to find the equilibrium position for the pad and calculate the pressure distribution over the surface in contact. The main advantage of this approach is that computations of FFT and IFFT and recalculations of the pad displacements can be avoided. This can lead to a more accurate model that is independent of pad displacement update method. The approach is tested on several examples. The results show correct physical behaviour and are in agreement with expectations.