Adaptive Feedforward for a Wafer Stage in a Lithographic Tool

Abstract

The wafer stage and reticle stage in a lithographic tool, used to manufacture integrated circuits (ICs), operate at nanometer accuracy during a scanning motion. The position accuracy and settling time after accelerating to the scanning velocity are largely determined by the feedforward controller. The feedforward controller calculates the required force for the stage to move according to its position profile by multiplying the reference acceleration with the known stage mass. Two effects limit the accuracy directly after the acceleration phase. First, the actual stage acceleration in response to the controller-calculated force depends on the position of the stage in its working range. A variation of 0.1% is observed. Second, dynamic resonances in the stage response require a higher-order feedforward model. Combined, these effects result in a 20-30 nm peak position error. This brief investigates online adaptation of the feedforward mass parameter, with the aim of reducing the position-dependent stage behavior. Adaptation of only the feedforward mass is shown not to be able to compensate for stage dynamics, additionally requiring a higher-order feedforward element. From the control force and the measured motion response, the feedforward mass parameter is estimated on line. Least-squares estimation is fast enough to update the mass parameter during the acceleration phase, which takes less than 100 ms. This brief addresses a number of practical aspects and shows that adaptive feedforward estimation in combination with higher-order feedforward reduces the peak position error consistently by creating position-independent behavior.

Feedforward Motion Control: From Batch-to-Batch Learning to Online Parameter Estimation

“…A linear least squares optimization problem with an analytic solution is developed to optimize the feedforward controller parameters. Furthermore, a detailed statistical analysis is provided in the case where measurement noise is present, indicating that biased estimates are obtained in earlier approaches [13]. An example confirms the theoretical conclusions, i.e., indicating the possible hazard of using biased estimates for feedforward control resulting in a deterioration of the positioning performance.…”

“…The following subcontributions are addressed in this work; 1) a detailed statistical analysis of the proposed framework is provided to show that biased estimates are obtained in earlier approaches, and, 2) a simulation study with a benchmark system is performed to confirm the theoretical conclusions, including the immediate benefit of direct learning. Indeed, theoretical conclusions in this paper may provide relevant new perspectives on earlier related approaches, including [13], which are further exploited.…”

“…this, together with Theorem 1 allows to further analyze the setting where input-output data is collected in a closedloop fashion, and online estimation is used to determine the feedforward parameter as in Fig. 2, which is also exploited in [13]. First, note that…”

Nonlinear Current Control for Reluctance Actuator with Hysteresis Compensation

“…The double differentiation may introduce a noise. However, when using a filter such as the least-square method and considering the noise level, it makes double differentiation acceptable as a method to calculate the stage acceleration [16]. In this paper, the acceleration is measured by the accelerometer.…”

Beyond Performance/Cost Tradeoffs in Motion Control: A Multirate Feedforward Design With Application to a Dual-Stage Wafer System

“…The present paper contains substantial original contributions including Contribution (I), Contribution (II), and Contribution (V). Related work on wafer stage control design includes feedback control [31], [32], feedforward control [33], LPV control [34], and sparse control [17]. In the present work, previously unexplored freedom in sampling is exploited, which makes the approach complementary to other approaches.…”