Feedback Control of Cluster Tools for Regulating Wafer Delays

Robotized cluster tools for semiconductor manufacturing have strict time constraints such that a wafer processed at a processing chamber should be unloaded within a specified time limit. Otherwise, it has a serious quality problem due to residual gases and heat within the chamber. Even though there have been studies on identifying a feasible tool operation schedule over such time constraints on wafer delays, such a schedule is subject to timing disruptions or time variation, and thus may violate the time constraints. In this study, we propose a more robust method of regulating wafer delays against timing disruptions not to exceed a specified limit. We first model the discrete-event behavior of a tool by a timed event graph. We then develop a feedback controller for single-armed and dual-armed cluster tools that can satisfy the time constraints by regulating wafer delays. To do this, we develop a feedback controller for the timed event graph by analyzing the timing behavior in a linear system model based on the max-plus algebra. The feedback controller postpones an event or firing of a transition, i.e., loading a wafer into a chamber, until a properly determined time elapses after an associated preceding event occurs. Finally, we present examples of feedback control and show that the feedback control is quite robust even under persistent time variation.