Optimal control of conductive heating systems for microelectronics processing of silicon wafers and quartz photomasks

The interaction of optimal control and equipment design is analyzed for robust performance of conductive heating systems used in microelectronics processing applications. An optimal control scheme is designed to improve repeatability by minimizing the loading effects induced by the common processing condition of placement of a semiconductor wafer at ambient temperature on a bake plate at processing temperature. The optimal control strategy is a model-based method using linear programming to minimize the worst-case deviation from a nominal temperature set-point during the load disturbance condition. This results in a predictive controller that performs a predetermined heating sequence prior to the arrival of the wafer as part of the resulting feedforward/feedback strategy to eliminate the load disturbance. Experimental results are given for both a commercial conventional (large thermal mass) bake plate and a low thermal mass bake plate. Any temperature fluctuation due to the placement of a wafer and reticle is effectively eliminated.