多變量EWMA 回饋控制器最適設計之研究

Optimal Design and Analysis of Multivariate EWMA Controllers

周伯彥(Bo-Yan Jou)

指導教授 ： 曾勝滄

國立清華大學/理學院/統計學研究所/博士(2008年)

摘要

隨著國內IC產業的製造技術不斷地提升，製程管制工作就顯得日趨重要，其中批次控制 (run-to-run control) 更是扮演不可或缺的回饋監控的角色。它主要利用製程中產出值與目標值的差距，逐步調整投入變數，使得產出值能穩定收斂到目標值，其中又以EWMA (exponentially weighted moving average) 回饋控制器為業界普遍使用之model-based 控制策略。雖然文獻中對此控制器之製程穩定性條件已經廣泛地討論，但此控制器之績效表現仍有相當大的改進空間。因此，本論文將深入探討下列三個研究主題：

1. 在漂移 (drifted) 多投入多產出 (MIMO) 的系統中，起始投入變數 (input recipe) 之設定值存在無限多組可行解。由於起始投入變數之設定是否適切，將直接影響製程產出值之績效表現。因此，本論文提出最適起始投入變數之設定值，可使得製程起始偏差達極小化，進而使製程產出變數快速地接近特定目標值。此外，我們進一步提出一個新的dMEWMA修正控制器，並推導其製程穩定性條件。最後由一個實例說明新的dMEWMA修正控制器之績效表現顯著地優於傳統dMEWMA控制器。

2. 在執行批次控制之前，首要的工作就是建立製程的預測模型。然而，樣本數的選取將關係到模型中參數估計的準確度，進而影響製程的穩定條件。針對這個問題，Tseng et al. (2007) 提出一個簡便公式來決定執行dMEWMA控制器時線外階段所需之最適樣本數。由此公式可知，其投入變數的共變異數矩陣結構將嚴重地影響樣本數的決定。針對此難題，本論文提出之新的dMEWMA修正控制器，可有效地消除此干擾參數 (nuisance parameter) 所造成的效應，進而大幅地降低線外階段所需的實驗樣本數。同時在製程參數未知的情況下，我們也提出一個三階段方法來決定最適樣本數。

3. 傳統批次控制方法，主要建立在製程產出變數的量測數據可順利獲得下，方能執行製程回饋調整。然而實際半導體生產製程，若量測機臺存在著量測產能的限制，而必須執行產出值跳批量測時，如何執行一有效的回饋控制將格外重要。針對此跳批量測之批次控制器，本文首先推導其製程穩定性條件，並在控制 TMSE (total mean square error) 在給定的水準下，提出最大可容忍的跳批量測週期。

關鍵字

批次控制 ； EWMA控制器 ； 多投入多產出系統 ； 製程穩定條件

並列摘要

The thesis includes three research topics and the contents are given as follows:

1. For a drifted multiple-input and multiple-output (MIMO) system, the double multivariate exponentially weighted moving average (dMEWMA) controller is a popular run-to-run (RTR) controller for adjusting the process mean to a desired target. Although the stability and performance of conventional dMEWMA controller had been widely studied in literature, the issue of “how to choose an appropriate initial setting for the input recipe” had not been well discussed in detail. Obviously, if the initial recipe is not chosen appropriately, it usually requires a moderately large number of runs to bring the process output to approach its desired target. In this thesis, by using a Mahalanobis distance approach, we obtain an optimal initial setting for the input recipe in such a way that the process output will approach its desired target rapidly.
2. To implement the conventional dMEWMA controller, we need to build an input-output (I-O) predicted model at the off-line stage. Recently, Tseng et al. (2007) presented an explicit formula for determining a minimum sample size (which is needed to construct I-O predicted model) in such a way that the asymptotic stability of dMEWMA controller can be achieved with a guaranteed probability. This formula indicates that two key components on the sample size determination are: the canonical correlation of I-O variables and the condition number of the covariance matrix of input variables. Since this condition number is a nuisance parameter, the problem on how to minimize its effect on the sample size determination is of great practical importance. This thesis proposes a stable dMEWMA controller with which the sample size (required at the off-line stage) only depends on the canonical correlation of I-O variables. Hence, the sample size can be reduced significantly.
3. RTR feedback control methodologies in literature are implemented based on the assumption that process response variables can be successfully measured. In practical applications, due to the metrology capacity limitations, process response variables cannot be measured completely. Hence, “how to implement a skip-lot RTR feedback control scheme” becomes an important research topic. In this thesis, we investigate both the process long-term stability conditions and short-term performance. In addition, by controlling TMSE (total mean square error) less than a specific level, we also provide a simple rule to determine an allowable capacity for the metrology equipment.

並列關鍵字

run-to-run control ； EWMA controller ； multiple-input and multiple-output system ； stability conditions

參考文獻

[2] 林建華 (2006), “多變量 EWMA 控制器之績效及穩定性研究”, 國立清華大學統計學研究所博士論文.

[4] 詹雅婷 (2007), “Modified VEWMA 回饋控制器在兼具平移與漂移之IC製程的績效表現”, 國立清華大學統計學研究所碩士論文.

[7] 廖俊豪 (2006), “Adaptive VEWMA回饋控制器”, 國立清華大學統計學研究所碩士論文.

[10] Butler, S. W. and Stefani, J. A. (1994), “Supervisory run-to-run control of polysilicon gate etch using in situ ellipsometry,” IEEE Transactions on Semiconductor Manufacturing, 7, 193-201.

[11] Chen, A. and Guo, R. S. (2001), “Aged-based double EWMA controller and its application to CMP processes,” IEEE Transactions on Semiconductor Manufacturing, 14, 11-19.