臺灣博碩士論文加值系統

晶圓允收參數(Wafer)相關資料通常呈現多重混和的線性模式，然而決定資料中線性模式的指標(Model Indicator)卻往往不出現在良率分析的資料欄位，我們稱這些不出現在資料欄位中的數據為「隱藏因子」，如何偵測這些「隱藏因子」進而估測已被混和的資料線性模式，是半導體業界極欲解決的問題。
針對「隱藏因子」之偵測與多重混和的WAT線性模式之估測問題，以迴歸分析搭配最大期望(Expectation Maximum; EM)演算法之叢聚式迴歸分析方法(EM-Based Clusterwise Regression; EMCR)已被提出，並已在少許半導體業界案例獲得驗證。EMCR雖然可以處理多重資料模式，但其仍存在著落入區域最佳解之問題，它能夠保證下一代的參數值估測比此代好，但卻無法保證其最終結果為最佳之參數估測的狀態。本論文針對區域最佳解問題，進行三項議題之研究：
(1)EMCR演算法之起始解探討(EMCR_Initial)：本研究主要模擬兩種真實模型，兩平行(Parallel)模型與兩交叉(Cross-Over)模型，發現僅有真實模型為兩平行之模型才會落入區域最佳解之情況，因此本研究將EMCR演算法之起始解給與兩平行之模型，驗證其效果後得知，確實能夠有效避免落入嚴重區域最佳解的情形發生，觀察其迴歸參數的誤差率，大約比使用隨意給定歸屬權重當成EMCR演算法起始解，提升了10％以上
(2)EMCR演算法落入區域最佳解之型態探討(EMCR_Convergence)：本研究觀察當EMCR落入區域最佳解時的模式歸屬權重頻率分配特徵，發現若是資料點的模式歸屬權重皆非常接近0.5或是歸屬權重在0附近及1附近佔的比例很少，則其非常有可能為陷入區域最佳解之情況，因此本研究即以此特徵來偵測是否落入區域最佳解的情形。模擬結果後發現，在原始資料為平行的情況之下，確實在EMCR起始解為隨機值時，偵測出區域最佳解的情況，而降低迴歸參數的誤差率。
(3)EMCR演算法混合PSO(Particle Swam Optimization)演算法之探討(PSO-EMCR)：因為PSO演算法能夠將參數(粒子)更新速度較為快速，且有跳離區域最佳解的能力，但並無法確保下一代的解一定比上一代更好；而EMCR演算法必須經過迴歸技術的運算，運算時間長但能確保下一代的解比上一代更好，但可能有落入區域最佳解的情況，因此嘗試以PSO演算法來跳離區域最佳解，及更新速度快，合併EMCR演算法下一代比上一代更好的優點，開發出PSO-EMCR演算法。但是在本研究經過驗證比較後，PSO_EMCR演算法並無如預期的出現最佳解，反而比EMCR演算法起始解給予隨機值時，迴歸參數估測的更不準確。
針對具有隱藏類別因子的半導體廠晶圓允收測試資料，本研究以四種方法進行EMCR的效能測試： 1. EMCR演算法給予隨機起始解、2. EMCR演算法給予平行起始解、3.EMCR演算法給予隨機起始解並加上落入區域最佳解偵測、4. PSO_EMCR演算法，因為實際案例的真實模型無從得知，因此本研究以重複取樣技術估測EMCR迴歸參數的信賴區間，發現四種方法都能偵測出顯著的迴歸因子，其中EMCR演算法搭配平行起始解有最佳收斂效率，而PSO_EMCR演算法達到收斂所需時間之平均值與變異數都較大。

In semiconductor manufacturing, Wafer Acceptance Test (WAT) data measured at the end of production line manifests multiple liner models with respect to in-line data. Unfortunately, the model indictor for modeling WAT data is usually a “hidden variable”. EM-Based Clusterwise Regression( EMCR) technique has been applied to the modeling of WAT data with multiple linear models. However, EMCR adopts greedy search strategy for iterative parameter updating and therefore very often falls into local optimum. To cope with the local optimum problem, this thesis studies two issues:
(1) The selection of EMCR initial solutions (EMCR initial): Two types of data model, parallel and cross-over, are designed to study various strategies for setting EMCR initial solutions. Our simulation studies show that the setting of EMCR initial solution as parallel regression models performs a smaller model mismatch rate, which descends about 15％ compared to the model mismatch rate performed by EMCR with random initialization strategy.
(2) The detection of EMCR local-optimum(EMCR Convergence)：The probabilistic membership of individual data with respect to each model is studied when EMCR falls into local optimum. It is observed that, for a specific model, if the distribution of probabilistic membership has no occurrences near zero or one, it may fall into the local optimum. Simulation results show that the EMCR adopting both the local-optimum detection strategy and random initialization strategy indeed descends the model-mismatch rate 10%
A data set with 23 samples from a semiconductor foundry is selected for performance evaluation. Due to the fact that the sample size is small and the true model is unknown, the re-sampling technique is applied to estimate the 95％ confidence interval of EMCR parameters. The results show that the setting of EMCR initial solution as parallel regression models has the smallest confidence interval of EMCR parameters, while the EMCR with random initializations performs the largest confidence interval of EMCR parameters.

目錄 V
圖目錄 VI
表目錄 IX
第一章 緒論 1
1.1 研究背景與動機 1
1.2 研究目標與方法 4
1.3 論文架構 6
第二章 問題描述與文獻探討 7
2.1 問題描述 7
2.2 相關文獻 9
第三章 以最大期望演算法進行晶圓允收測試資料之叢聚式迴歸EMCR演算法 13
3.1 呈現混和模式的晶圓允收測試資料 13
3.2 混和模式的概似函數(Likelihood Function) 15
3.3 混和模式的最大概似估計值 (Maximum Likelihood Estimation) 16
第四章 區域最佳解與EMCR演算法起始解討論 19
4.1 區域最佳解各種形式及可能形成原因 19
4.2 EMCR演算法之起始解給予平行模式說明(Initial Solution with Parallel) 25
第五章 起始解的收斂問題再討論 28
5.1 收斂(Convergence)的各種樣式 28
5.2 系統中的參數更新(Estimation Convergence) 30
5.3 系統中的參數更新(Estimation Convergence)作法說明 36
5.4 系統中的參數更新法則例子： 39
第六章 實驗設計與實際案例探討 41
6.1 實驗方法目的與衡量標準 41
6.2 模擬資料結果與實際案例探討 48
第七章 結論 73
參考文獻 76
附錄 79
A. PSO-EMCR演算法 79
B. PSO-EMCR實驗設計-起始解為隨機值 VS. PSO_EMCR參數模擬 96
C. PSO-EMCR實際案例驗證 105
D. 結論 110

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