Statistical model for spatial correlation in thin film deposition and reactive growth

This paper presents a statistical method for the estimation of thickness variations present across a wafer lot in low pressure chemical vapor deposition (LPCVD) and reactively grown films. The method uses experimental thickness data to construct a unified Karhunen-Loève expansion based model that captures both deterministic and random thickness variations. The model uses a set of quadratic interpolation functions fitted to mean spatial data to approximate the deterministic nonuniformity and a few normalized random variables to represent run-to-run fluctuations. This model therefore retains the spatial correlations present between different deposition and growth steps in a process necessary for the estimation of parametric yield and permits the calculation of distribution functions over different lot populations (wafer, die, point, etc.). Models for spatial correlations in LPCVD oxide, nitride, polycrystalline silicon, and thermal oxide growth were constructed from a data set of 35000 thickness measurements recorded from a total of 40, 25-wafer runs. In each case, the model gives good predictions (90-95% confidence) with just one or two random variables.