Modeling, optimization and control of spatial uniformity in manufacturing processes

Spatial uniformity, or the uniformity of product output characteristics at different locations in a batch of product is modeled, optimized, and controlled using a methodology called multiple response surfaces, which may be used to characterize the results of an experimental design. Multiple, low-order polynomial models are used to model the output characteristics at each of several sites within a batch of product. The uniformity model is then obtained by manipulating these multiple models. The approach is compared to the traditional method of fitting a single high-order polynomial to the calculated uniformity. Experimental results confirm that similar or improved modeling accuracy is obtained with fewer data points using the new method due to the use of low-order models. Characteristics of the approach are examined both analytically and in application to plasma etching, silicon epitaxy, tungsten chemical vapor deposition (CVD) and the simulation of polysilicon low-pressure CVD (LPCVD).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>