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# 統計在半導體產業的應用 -- Basic Statistic Methods

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Introduction of Engineering Data Analysis System for Industrial Statistics, Part II for statistical methods applied in semiconductor industrial.

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• N defects uniformly distribute in wafer with m chips, probability of chip contain defect n/m chip area A Pr{n/m; K=k}=[exp(-n/m)][(n/m)^k]/(k!) k=0, Y=exp(-n/m)
• ### 統計在半導體產業的應用 -- Basic Statistic Methods

1. 1. 半導體產業常用的統計方法
2. 2. 大綱 <ul><li>常用的統計方法 </li></ul><ul><li>各產業的挑戰 </li></ul><ul><li>進階分析用的方法 </li></ul>
3. 3. 基本統計量 <ul><li>Describe statistics </li></ul><ul><li>-- mean/median/std/Q1/Q3/percentile </li></ul><ul><li>Display </li></ul><ul><li>-- box/histogram/CDF/P-P/Q-Q </li></ul>
4. 4. 統計方法 <ul><li>假設檢定 </li></ul><ul><li>-- 要檢定什麼 ? Null=? Alternative=? </li></ul><ul><li>迴歸模型 </li></ul><ul><li>-- 自變數 ? 應變數 ? 連續 / 非連續 </li></ul>
5. 5. Design house <ul><li>資料來源 : </li></ul><ul><li>--WAT( 代工廠 ) </li></ul><ul><li>--CP and final test( 封測廠 ) </li></ul><ul><li>工程師的問題 : </li></ul><ul><li>-- 那些因素影響到 final yield </li></ul><ul><li>統計上的問法 ? </li></ul>
6. 6. 可用的統計方法 <ul><li>Correlation coefficient </li></ul><ul><li>Regression </li></ul><ul><li>Testing hypothesis </li></ul>
7. 7. 統計上的問題 <ul><li>那些變數與 final yield 有關 </li></ul>
8. 8. Correlation Coefficient
9. 9. Regerssion Residual standard error: 6.297 on 201 degrees of freedom Multiple R-Squared: 0.4439, Adjusted R-squared: 0.3111 F-statistic: 3.343 on 48 and 201 DF, p-value: 1.455e-09  Improve the model?
10. 10. Testing Hypothesis <ul><li>Divided into two group by yield </li></ul><ul><li>Null: mean equal </li></ul><ul><li>For all parameters </li></ul>
11. 11. Result:
12. 12. Display Graphics <ul><li>More than two group? </li></ul><ul><li>Other display way? </li></ul>
13. 13. Fab 廠資料來源 <ul><li>量測參數資料 (wafer) </li></ul><ul><li>製造時機台的監控資料 (equipment) </li></ul><ul><li>Defect inspection </li></ul><ul><li>WAT </li></ul><ul><li>CP </li></ul><ul><li>Wafer map( 由 Defect/WAT/CP 所衍生出來的 ) </li></ul>
14. 14. Fab 廠的挑戰 <ul><li>Time to market </li></ul><ul><li>-- yield </li></ul><ul><li>-- new technology </li></ul>
15. 15. Fab 廠常用的統計方法 Statistical Method Purpose Distribution Basic material for statistical tests. Used to characterize a population based upon a sample . Hypothesis testing Decide whether data under investigation indicates that elements of concern are the “same” or “different.” Experimental design and analysis of variance Determine significance of factors and models; Decompose observed variation into constituent elements. Categorical modeling Use when result or response is discrete (such as “very rough,” “ rough,” or “smooth”). Understand relationships, determine process margin, and optimize process. Statistical process control Determine if system is operating as expected. Regression Yield modeling. Yield impact Duane S. Boning, Jerry Stefani and Stephanie W. Butler: Statistical Methods for Semiconductor manufacturing
16. 16. Yield maintenance <ul><li>Process stable </li></ul><ul><li>-- statistical process control </li></ul><ul><li>Excursion resolve </li></ul><ul><li>-- finding root cause of yield drop </li></ul><ul><li>PM </li></ul><ul><li>-- Preventative maintenance </li></ul>
17. 17. Statistical Process Control Normal, +- 3 sigma ~ 99.7
18. 18. Yield drop <ul><li>process equipment malfunction </li></ul><ul><li>-- which process, which equipment(s) </li></ul><ul><li> group comparison for all possible process </li></ul>
19. 19. Group comparison <ul><li>Null: mean equal for all groups </li></ul><ul><li>Alternative: mean not equal </li></ul><ul><li>Group by equipment </li></ul><ul><li>Mean of measurement data </li></ul><ul><li> Other methods? </li></ul>
20. 20. PM <ul><li>Why? </li></ul><ul><li> 損耗 污染 </li></ul><ul><li>When </li></ul><ul><li> 經驗值 原始設定值 </li></ul><ul><li> better way? </li></ul><ul><li> first wafer effect </li></ul>
21. 21. Yield enhancement <ul><li>DOE for process improve </li></ul><ul><li>Finding key parameters for yield </li></ul><ul><li>Yield impact model </li></ul>
22. 22. Process improvement <ul><li>Material </li></ul><ul><li>Processing time </li></ul>
23. 23. Key parameters <ul><li>Domain knowledge </li></ul><ul><li> 物理性質 </li></ul><ul><li>Regression </li></ul><ul><li> variables </li></ul><ul><li> collinear </li></ul><ul><li>PCA </li></ul><ul><li> Other method? </li></ul>
24. 24. Yield impact model <ul><li>問題 </li></ul><ul><li>-- Defect item/Pattern 對 yield 的影響有多大 </li></ul><ul><li>資料 </li></ul>
25. 25. Yield impact model <ul><li>Logistic regression </li></ul><ul><li> coefficient as “kill probability” </li></ul><ul><li> impact value = pattern loss/total loss </li></ul><ul><li>每片 wafer 的 yield loss 依 kill probability 的比率分給每個有的 pattern, 總合所有的 wafer 就是該 pattern 的 pattern loss </li></ul>
26. 26. Yield Prediction <ul><li>Wafer Yield </li></ul><ul><li>Week/month </li></ul><ul><li>Next product </li></ul>
27. 27. Wafer Yield -- Poisson Model <ul><li>D : chip defect density </li></ul><ul><li>A : critical Area (chip area) </li></ul>Assumption: n defects randomly distribute in wafer with N chips The probability of one chip contains k defects m=n/N, K=0  Chip pass D = m/A Yield = Pass chip number/N
28. 28. Wafer Yield -- Other models <ul><li>Non-uniform defect density </li></ul><ul><li> </li></ul>f(D) is the defect density distribution Murphy model density formulation triangular Exponential Seeds
29. 29. Daily/Weekly/Monthly Yield <ul><li>Average of total wafers? </li></ul><ul><li>Regression </li></ul><ul><li>-- parameter selections </li></ul>
30. 30. Next product Yield <ul><li>Technology baseline </li></ul><ul><li>Previous product yield </li></ul>Time Yield Pilot Rump Mass production Phase out
31. 31. Summary of “basic statistical method” <ul><li>Statistics: description, display </li></ul><ul><li>Testing hypotheses: </li></ul><ul><li>root cause, important parameters </li></ul><ul><li>Regression: </li></ul><ul><li>yield modeling </li></ul>
32. 32. 新的挑戰 -- 進階分析的方法 <ul><li>資料量 </li></ul><ul><li>變數維度 </li></ul><ul><li>更好的控制 </li></ul><ul><li>演算速度的提昇 </li></ul><ul><li>跨領域的合作 </li></ul>
33. 33. Advanced topic -- testing <ul><li>Null hypothesis </li></ul><ul><li> 當 n 夠大時容易 reject </li></ul>
34. 34. Display – violin plot
35. 35. Display -- Correlogram
36. 36. Pattern classify <ul><li>Question </li></ul><ul><li>-- Which category is this wafer belong to? </li></ul><ul><li>Example </li></ul><ul><li>-- </li></ul>Ref: pattern recognition
37. 37. Pattern Classify <ul><li>如何描述 pattern </li></ul><ul><li>共有但可以區別的特性 </li></ul><ul><li>距離 </li></ul><ul><li>怎麼定 </li></ul>
38. 38. Clustering 分群 <ul><li>那幾片 wafer 可當成同一類 </li></ul><ul><li>怎麼分 </li></ul><ul><li>-- Hierarchical </li></ul><ul><li>up-down/bottom-up </li></ul><ul><li>-- Partitional </li></ul><ul><li>k-means/k-mediods </li></ul><ul><li>Reference: http://en.wikipedia.org/wiki/Cluster_analysis#Hierarchical_clustering </li></ul>
39. 39. Single Tool <ul><li>Testing? </li></ul><ul><li>Trend </li></ul>
40. 40. Golden Path P1 P2 P3 E1 E2 T1 T2 3*2*2 = 12 combination for 3 steps  Whole combination? Partition method Golden wafer/golden lot tracking
41. 41. Parameters >> observations <ul><li>Grouping parameters </li></ul><ul><li>Supersaturated design analysis </li></ul>
42. 42. Advanced Process Control <ul><li>Run-to-run </li></ul><ul><li>-- feedback control </li></ul><ul><li>Fault detection </li></ul><ul><li>-- Abnormal </li></ul><ul><li>Virtual Metrology </li></ul><ul><li>-- reduce metrology </li></ul><ul><li>-- feed for r2r </li></ul>
43. 43. Reference: <ul><li>http://www.siliconfareast.com/test-yield-models.htm </li></ul><ul><li>http://www.icyield.com/yieldmod.html </li></ul>