Accurate Pressure Calculation Method for CMP Modeling Using Fourier Analysis

Chemical-mechanical polishing/planarization (CMP) is one of the key processes used in electronic chip manufacturing. In CMP, a rotating wafer is pressed facedown onto a rotating polishing pad. A chemical “slurry” containing abrasive particles and chemical reagents is deposited on the pad during polishing, and flows between the wafer and the pad. The combined action of the polishing pad, abrasive particles, and chemical reagents results in material removal and planarization of the wafer surface.In recent years, modeling of the CMP process has become critical for detection of planarity defects in chips before manufacturing. One of the key parameters affecting the surface planarization is the pressure, with which the wafer is pressed against the pad. Calculation of the pressure distribution across the wafer surface is crucial for modeling the CMP process. This pressure calculation typically uses contact mechanics methods that include solving an integral equation using fast Fourier (FFT) and inverse fast Fourier (IFFT) transforms. However, the kernel of the integral equation has singularities that lead to numerical instability.In this paper, we propose a method to avoid numerical instabilities in pressure calculation by using an analytical expression for the Fourier transform of the kernel function.