半導體製程控制之晶圓允收測試及線上量測參數關係分析

Abstract

在半導體晶圓廠中，為了因應如今快速的良率提升及元件單位的縮小，有效的製程控制為一個重要環節。在次波長世代的晶圓製造，產業界已相繼提出及導入先進製程控制(Advanced Process Control, APC)的概念及方法，其中以調整後段製程參數來符合前段製程之設定來達到期望之良率，此為前饋控制(Feed-Forward Control)，亦為先進製程控制的應用之一。 晶圓允收測試(Wafer Acceptance Test, WAT)及線上量測(In-line)的關係分析模型為前饋控制的關鍵之一，有效運用此模型可達到製程良率控制及錯誤診斷之效益。迴歸分析(Regression Analysis)為使用於建構晶圓允收測試及線上量測關係的方法之一，透過迴歸分析可建立製程機台參數與晶圓允收測試量測值間的線性關係模型，此方法建立於單一線性模型的假設上。多重模型起因於模型的指標變數(Model Indicator)，指標變數可能是機台的設定或編號等等，並導致R-square的效能。然而實際上指標變數分離多個模型時並不會顯示在分析資料中。因此指標變數會是個隱藏變數。為了要解決多重模型的在關係分析上的問題，主要的挑戰就是隱藏的指標變數及迴歸估計參數在分析時無法事先知道。 在本研究中我們應用最大期望演算法(Expectation-Maximization, EM)來辨識多重迴歸模型。主要的構想是給定一個初始解的設定，然後反覆地估計隱藏變數及迴歸參數直到收斂為止。然而直接應用EM演算法還需要面對兩個挑戰：局部最佳解(Local Optimum)及模型過適(Overfitting)問題，局部最佳解出現於貪心法則策略而EM演算法就是其中一個，在此情況效率將會視其初始解而定，這時就很容易會收斂到大量的局部初始解得其中一種，為了解決局部最佳化問題，我們嘗試使用大量的初始解以期找到有效且最接近整體最佳解(Global Optimum)。模型過適則是一個在統計及資料探勘中常見的問題，也就是給予資料過多的解釋，這些解釋包含了許多不重要的或不常出現的錯誤。直觀來說，多重模型的效能一定會比單一模型好，但目前沒有量測的指標及方法來判斷哪種模型比較適合，因此我們用統計量來檢定評估單一模型與多重模型。最後我們發展了一個以EM為基礎的迴歸演算法(EM-Based-Regression, EMR) 含括上述的想法，並用來解決這兩個問題。 為了實驗驗證是否達到整體最佳解的逼近及模型過適避免，我們用模擬來評估其效能。我們一共設計三種模型：單一模型、平行的兩個模型、交叉的兩個模型。針對整體最佳解的逼近，我們使用可能性(Likelihood)來當作收斂估計和局部最佳解選取的準則，隨著可能性越來越大且變動的幅度越來越小，迴歸參數也會都收斂到一個數值，且其估計信賴區間包含原本的設定參數，表示估計到正確迴歸參數的機會很高。而模型過適避免的部分，我們比較了F檢定、貝式訊息準則(Bayesian Information Criterion, BIC)及經驗F檢定來評估統計顯著度。在三個模擬模型當中，經驗F檢定和貝式訊息準則表現了比較好的效能來評估哪種模型比較顯著，能成功夠將錯誤辨別率(False Identification Rate, 一般也稱為Type I Error)及遺漏察覺率(Miss Detection Rate, 一般也稱為Type II Error)都在0.05中。

Yield ramping and feature size shrinking continuously lead to tightening of process control. In semiconductor fabrication, process control is a key element for successful IC manufacturing. As IC technology advances towards the nanometer era, the concept of advanced process control (APC) has been proposed and implemented in semiconductor manufacturing. Feed-forward control is one of the APC strategies for manufacturing processes. Correlation analysis of Wafer Acceptance Test (WAT) data and In-line data is needed for yield control and fault diagnosis of the fabrication processes. Through the measurement data and correlation analysis, statistical methods are used to characterize the process status and establish a correlation model. Currently, regression analysis is one of the popular correlation analysis methods for fitting correlation models between WAT and In-line data with “single-model” assumption. However, “multiple models” due to model indicator which is an index to separate multiple models and could be the recipe names, equipment/chamber IDs, and results in the unsatisfied R-square performance. Unfortunately, the model indicator may not manifest itself on the data. Therefore, the model indicator always does not appear in the dataset and it is a “hidden variable.” To cope with the multi-model issue in correlation analysis, the main challenge is neither hidden variables nor regression parameters is pre-known. In this research, we extend the Expectation-Maximization (EM) algorithm for identifying the multiple regression models. The key ideas are: given the initial solution first, then iteratively estimate the hidden variables and regression parameters until convergence. However, direct application of the standard EM algorithm for regression has two challenges: local optimum and overfitting. Local optimum comes from a greedy search strategy such as EM algorithm whose performance depends on the initial solution and it is easy to converge with one of the numerous local minima. To solve the local optimal problems, we try many different initial solutions to find the effective local optimum which approximates global optimum. Overfitting is a common problem in statistics and data mining that gives data too much explanation including minor fluctuations or random error in the data. The outcome of the multi-model is definitely better than single-model, but now there is no measurement and metric capable of determining if the model is single-model or two-model. So we need to use statistical metric for evaluating the single-model against the multi-model. Finally, we develop an EM-Based-Regression (EMR) algorithm which contains above ideas to resolve these two problems. To validate global optimum approximation and overfitting avoidance of EMR algorithm, we conduct simulations to evaluate performance. Three types of simulation models are designed: single model, parallel two models, and cross two models. For global optimum approximation, we propose likelihood as the criteria of convergence estimation as the variation of likelihood is minor and local optimum selection when likelihood maximization. Then confident interval of the estimated regression parameters covers setting of model coefficients, so there is high probability to approach global optimum. For the overfitting problem, we compare the performance of several statistical metrics, which consists of F-statistic, Bayesian Information Criterion (BIC), and empirical F-statistic, to evaluate the statistic significance. In conclusion, the empirical F-test and the BIC-test achieve a better performance in distinguish model significance, and control the false identification rate (or Type I Error) and Miss Detection Rate (or Type II Error) less than 0.05.