基于广义线性模型的晶圆缺陷数据建模及仿真

Wafer Defect Data Modeling and Simulation Based on the Generalized Linear Model

晶圆缺陷数据是具有相邻空间关系的二维空间离散数据,通常会表现出一定的空间相关性。为了研究这类数据的空间分布和空间结构对空间信息的影响,以具体晶圆缺陷数据为例,基于泊松分布建立广义线性混合模型,并对空间相关项的协方差进行3种不同的空间建模,利用蒙特卡洛牛顿拉弗森算法进行估参,其中引用Metropolis-Hastings算法对空间相关项进行抽样。借助AIC指标,说明广义线性混合模型能够很好地模拟晶圆缺陷数据的分布特征,而且针对晶圆缺陷数据分布的稀疏程度,可调整模型从而得到更好的拟合效果。

Wafer defect data,which is a kind of two-dimensional spatial discrete data with adjacent spatial relationship,usually shows a certain spatial correlation.In order to study the influence of the spatial distribution and the spatial structure on spatial information,a generalized linear mixed model is established for spatial modeling based on poisson distribution by taking wafer defect data as an example in this paper.At the same time,three different covariance of the spatial correlation were proposed.Then it proposes the Monte Carlo Newton Raphson(MCNR)method to estimate the model parameters,in which the Metropolis-Hastings algorithm is used to sample the spatial random items.Next it develops corresponding monitoring schemes by utilizing the weighted likelihood ratio test(WLRT)procedure.The proposed schemes not only detect the shifts in coefficient but also the shifts in the spatial correlation variation.Last,a real example from a wafer manufacturing process is used to illustrate the implementation and effectiveness of the proposed approach.With AIC indicators,it is demonstrated that MLM model can well simulate distribution characteristics of wafer defect data.It could adjust the model to get a better fitting effect for the distribution sparsity of wafer defect data.