Convex optimization of wafer temperature trajectories for rapid thermal processing

In this paper, we expand on the framework for achieving the tight control of the wafer temperature essential in rapid thermal processing (RTP) of semiconductor wafers. In our previous paper (1992), we established a method for identifying a state-space model of an RTP system at a processing condition of interest and designing a linear quadratic Gaussian (LQG) controller for disturbance regulation. In this paper we describe how convex optimization is used to obtain an approximation to the desired trajectory, close enough to allow high gain feedback controllers to reduce temperature nonuniformity. Temperature errors less than 30°C peak-to-peak, limited almost entirely by our system geometry, were achieved throughout a typical wafer recipe, which included ramps from room temperature to 900°C and from 900°C to 600°C, at the rate of 40°C per second. The benefits of convex optimization together with the LQG feedback control are demonstrated by experimental results obtained from an RTP system