Minimum entropy based run-to-run control for semiconductor processes with uncertain metrology delay

A novel run-to-run control methodology for semiconductor processes with uncertain metrology delay which is developed by combining the minimum error entropy and the optimal control strategy is presented. In most semiconductor processes, the product quality data from the previous run are not often available before the start of the next run. Thus, the corrective step is often delayed by one batch or more, and the duration of the delay is uncertain with stochastic characteristics. Coupled with inaccurate process models, the delay may lead to significant variations of the process outputs even with the use of exponentially weighted moving average (EWMA) controllers. This paper proposes a new method of handling the uncertain metrology delay from a probability viewpoint. The fundamentals of the run-to-run control systems are first reexamined, and then an innovative performance index is given by incorporating the entropy (or information potential) and the mean value of tracking error with constraints on control input energy. The probability density function (PDF) based optimal control algorithm is proposed for processes where the disturbance and delay are non-Gaussian and the stability of the algorithm is analyzed. In addition, the methodology of the proposed control strategy is extended to include recursive PDF estimation and on-line real time implementation. The paper also includes a simulation example of minimum entropy control of a tungsten chemical-vapor deposition process to illustrate the methodology. Furthermore, comparisons between the conventional EWMA method and the proposed method are done to show the advantages of our newly proposed method.