Performance Analysis of EWMA Controllers Subject to Metrology Delay

1. Performance Analysis of EWMA Controllers Subject to Metrology Delay 報告者：碩研工管二甲 蔡依潾 原著：Ming-Feng Wu 中華民國 2008年 8月

2. Main point Preface Questions of this study EWMA controller Single EWMA controller Literature Reviews Metrology delay The Single EWMA Controller subject to Metrology delay Example Analysis conclusion

3. Preface • 半導體產業需要很高的投資成本 • 先進製程控制量測延遲是自然存在的問題 • 這篇研究主要分成二個部份 • 1.Exponentrally weighted moving average（EWMA） 控制器存在量測延遲下做探討 • 2.利用虛擬量測系統(virtual metrology system，VM) 來探討量測延遲的問題

4. Questions of this study 1.Is the investment in advanced metrology justified？ 2.How do we retune the controller parameters if the metrology delay is changed? 3.Can virtual metrology be used? 4.Do the above guidelines apply in case of variable delays？

5. EWMA Controller

6. The Single EWMA Controller(1/6) Model: is the observed process output at the run. is the process input at the run. is the intercept parameter. is the slope parameter. is the process disturbance.

7. The Single EWMA Controller(2/6)

8. The Single EWMA Controller(3/6)

9. The Single EWMA Controller(4/6)

10. The Single EWMA Controller(5/6) 調整截距 調整製程 : is the target of the process : the estimate of : the discount factor,

11. The Single EWMA Controller(6/6) 穩定條件 必要條件

12. Literature Reviews

13. Metrology delay • Lot-to-Lot metrology delay Production Line 1 2 3 4 5 6 7 8 d=0 1 2 3 4 5 6 7 8 d=1 1 2 3 4 5 6 7 d=2 1 2 3 4 5 6 d=3 1 2 3 4 5

14. The Single EWMA Controller subject to Metrology delay(1/2) Model: is the observed process output at the run. is the process input at the run. is the initial bias of process. is the process gain. is the disturbance input.

15. Process predicted model is the model offset parameter. is the model gain parameter. Disturbance is estimated to be The Single EWMA Controller subject to Metrology delay(2/2) delay

16. Formula of Process Output (公式1） Bias correction Time-correction noise reduce Model mismatch Backshift operator

17. Bias subject to Metrology delay (公式2）

18. Proof 1-1: Give From 公式2 when Proof (1/2)

19. Proof 1-2: For any ，We can find some value of So that From 公式2 when Proof (2/2)

20. The first property of bias subject to Metrology delay • For an overestimated process gain (i.e., ) • is a monotonic decreasing sequence. • For an underestimated process gain (i.e., ) • There exists some values of larger than which is oscillatory.

21. Proof The following can be easily derived from 公式1： Proof 2:

22. The second property of bias subject to Metrology delay • In case of , for all t>d • Then • Therefore • In case of and • The optimal value of is equal to one

23. The effects of time delay don the optimalλand SSE(1/2) N=20 N=200 We can found that the optimal value ofλdoes not change much with different run numbers.

24. The effects of time delay don the optimalλand SSE(2/2)

25. The effects of time delay d on the optimalλfor different noise to initial bias ratios A ratio of metrology noise to the magnitude of error initial bias estimate

26. The effects of time delay d on the optimalSSE for different noise to initial bias ratios

27. Time-correlated noise reduction

28. Non-stationary process disturbance IMA(1,1)

29. Stationary process disturbance ARMA(1,1)

30. Example Tungsten CVD process 沈澱速率 溫度 partial pressure of hydrogen d=3

31. Analysis(1/3)

32. Analysis(2/3) • We find H2 pressure is unchanged. • Due to the fact that C2 is • so much smaller than C1

33. Analysis(3/3) Optimal AMSE of the tungsten CVD process at difference metrology delay

34. conclusion 1.Is the investment in advanced metrology justified？ 2.How do we retune the controller parameters if the metrology delay is changed?

35. 心得 • 這篇報告主要探討SEWMA有delay的問題 • 探討有delay狀況下， 的影響 • 探探討有delay狀況下，分別探討穩定與不穩定干擾之情況控制器如何調整 • 我覺得新的發展方向可以針對DEWMA做delay的探討