Spatial Variance Spectrum Analysis and Its Application to Unsupervised Detection of Systematic Wafer Spatial Variations

Abstract

Investigation of wafer spatial variations is critical for semiconductor process/equipment optimization and circuit design. The objective of spatial variation study is to differentiate the systematic variation component from the random component. This is usually done by contrasting with a set of known systematic patterns based on engineering knowledge. However, there could exist unknown systematic components remaining in the unexplained residuals and overlooked by the conventional spatial variation study. In this paper, we develop a novel spatial variance spectrum ( ) to analyze the systematic variations without any priori information of the systematic patterns. The is a series of spatial variations over a range of spatial moving window sizes from the smallest spatial moving window consisting of only two metrology sites to the largest one covering all metrology sites of the entire wafer. The can be used to characterize the wafer spatial variations and to detect existence of systematic variations by a proposed hypothesis test. We also propose an index to summarize from the the systematic proportion of the spatial variation. The proposed test and index of systematic variations will be demonstrated and validated through both hypothetical examples and actual cases of wafer critical dimension (CD) metrology data.Note to Practitioners-Wafer spatial variations study is critical to within-wafer control and optimization. Only through investigation of the spatial variations on the wafer surface do we get to learn more about the nature of the semiconductor process/equipment and the manufacturability of certain circuit design patterns. In this paper, we propose a novel analysis tool to help engineers characterize the wafer spatial variations and distinguish the systematic variation from the random variation. With the spatial systematic variations identified and described, the engineers can discover the opportunities to further improve the processing quality and thus the final yield.

Spatial variances of wind fields and their relation to second‐order structure functions and spectra

“…Mahadevan and Campbell [2002] use spatial variance in two dimensions to characterize spatial heterogeneity of tracers in oceanography, restricting their analysis to domains of size L, L=2, L=4, etc. A two-dimensional version of spatial variance is used by Blue and Chen [2011] for controlling and optimizing wafer spatial variations in integrated circuit fabrication. Lorenz [1979] introduced the ''poor man's spectral analysis,'' a method to construct the spectrum from spatial variances at scales that are a fraction 2 2n of the sample size, with n an integer, taking into account the Fourier transform properties when transforming from the frequency domain to the spatial domain [see also Cahalan et al, 1994] for a more complete description).…”

Deep Convolutional Generative Adversarial Networks-Based Data Augmentation Method for Classifying Class-Imbalanced Defect Patterns in Wafer Bin Map

“…J. Blue et al proposed a spatial analysis spectrum method by which to characterize spatial variations in wafers without prior knowledge of wafer patterns [19]. Youngji Yoo et al proposed an analytical methodology that predicts the results of package testing from the properties obtained by analyzing the defect patterns of wafer maps at the chip level [20].…”

Estimation of spatial boundaries with rolling variance and 2D KZA algorithm

“…However, the sample variance is substantially biased if the observations comprise clusters, which usually result in patterns as shown in Figure 1. In such case for a temporal series of data, Blue 4 recommended calculating the moving variance to reduce the bias. Similarly, instead of the moving variance of a temporal series, a series of variances are calculated for observations over the local area respectively, then form a spatial variance atlas ( Figure 1C & D).…”