以吉氏抽樣數值法鑑別影響半導體製造良率的關鍵機台組合

機台共同性分析（Tool Commonality Analysis; TCA）技術的穩健性對於半導體晶圓廠產品良率診斷的有效性有極大的影響。然而，所有現行演算法皆以貪婪搜尋策略為主，卻缺乏診斷機台組合效應的能力。當晶圓良率損失的根本原因是機台組合效應而非一個單機台效應時，以貪婪搜尋為導向的TCA演算法通常會導致高誤判率 (型I誤差) 和高漏判率 (型II誤差)。隨著半導體元件單位尺寸不斷縮小以及機台總數逐漸變大，晶圓良率損失趨向由特定的機台組合引起，貪婪搜尋導向的TCA演算法所引起的問題變得更加嚴峻。為了有效解決機台組合效應的問題，本論文首先針對每個機台建立二進位的機台健康指標，以判定一個特定機台是否應被納入為影響良率的機台組合中，並提出利用馬可夫鏈蒙地卡羅動態搜索技術的吉氏抽樣 (Gibbs Sampler) 數值方法於所有機台健康指標搜尋空間中，以重複疊代方式求取一個機台的健康指標在固定其他機台健康指標下的條件機率與模擬值。當模擬過程收斂時，統計出發生最頻繁的機台健康指標組合，即可判定影響良率的關鍵機台組合。本論文開發出以吉氏抽樣數值方法為基礎的TCA演算法，其中影響效能的機台健康指標條件機率分別採用有母數最大概似值與無母數假設檢定所得之p-value來估算。模擬和實際數據驗證結果顯示，以吉氏抽樣數值法為基之TCA演算法對於機台組合效應具有極佳的診斷能力，特別是在異常事件中，良率改變幅度較小、關鍵機台組合的使用率較低、或在機台使用率上具有與關鍵機台組合呈現高度相關的正常機台，吉氏抽樣數值法不僅在正確率上遠優於以貪婪搜尋策略為基之TCA，其運算效率也滿足應用之要求，能將計算複雜度從O（2n）減少到O（n2），其中n是機台的數量。

關鍵字

半導體製造良率；機台共同性分析；機台組合效應；馬可夫鏈蒙地卡羅；吉氏抽樣

並列摘要

In semiconductor manufacturing, the soundness of tool commonality analysis (TCA) technique has a high impact on the effectiveness of product yield diagnosis. However, all up-to-date TCA algorithms are based on greedy search strategies, which are naturally poor in identifying combinational root causes. When the root cause of wafer yield loss is tool combination instead of a single tool, the greedy-search-oriented TCA algorithm usually results in both high false and high miss identification rates. As the feature size of semiconductor devices continuously shrinks down, the problem induced by greedy-search-oriented TCA algorithm becomes severer because the total number of tools is getting large and wafer yield loss is more likely caused by a specific tool combination. To cope with the tool combination problem, a new TCA algorithm based on Gibbs Sampler, a Markov Chain Monte Carlo (MCMC) stochastic search technique, is proposed. In specific, a tool health indicator with binary value is defined for each tool to determine if it should be involved in the tool combination as root cause. With the Gibbs Sampler, the computation complexity is reduced from O(2n) to about O(n2), where n is the number of tools. Simulation and field data validation results show that the proposed TCA algorithm performs well in identifying the ill tool combination.

並列關鍵字

Semiconductor Manufacturing Yield Analysis ； Tool Commonality Analysis ； Tool Combination ； Markov Chain Monte Carlo ； Gibbs Sampler

參考文獻

Benjamini, Y. and D. Yekutieli (2001). "The control of the false discovery rate in multiple testing under dependency." Annals of statistics: 1165-1188.

Casella, G. and E. I. George (1992). "Explaining the Gibbs sampler." American Statistician: 167-174.

Chen, A. and A. Hong (2010). "Sample-Efficient Regression Trees (SERT) for Semiconductor Yield Loss Analysis." Semiconductor Manufacturing, IEEE Transactions on 23(3): 358-369.

Chien, C. F., W. C. Wang, et al. (2007). "Data mining for yield enhancement in semiconductor manufacturing and an empirical study." Expert Systems with Applications 33(1): 192-198.

de Vries, D. K. and Y. Chandon (2007). "On the False-Positive Rate of Statistical Equipment Comparisons Based on the Kruskal–Wallis H Statistic." Semiconductor Manufacturing, IEEE Transactions on 20(3): 286-292.

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以吉氏抽樣數值法鑑別影響半導體製造良率的關鍵機台組合

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