Optimal control of rapid thermal processing systems by empirical reductionof modes

In rapid thermal processing (RTP) of semiconductor wafers precise control o f wafer temperature is required throughout the process cycle to minimize do pant redistribution as well as wafer warpage. A low-dimensional dynamic mod el of RTP of semiconductor wafers is derived by using the Karhunen-Loeve-Ga lerkin procedure (Park, H. M.; Cho, D. H. Chem. Eng. Sci. 1996, 51, 81) to implement efficiently the optimal control scheme which ensures the transien t temperature uniformity in RTP systems. The optimal control scheme employi ng the low-dimensional dynamic model is compared with the conventional meth od using the original partial differential equation and is found to be very efficient without losing accuracy.