Analyzing Multimodality in Density Distributions Using JMP Scripting

Explore variability sources hidden in density distributions through JMP scripting. The analysis focuses on identifying and filtering distribution modes in semiconductor fab electrical measurements using kernel estimation and empirical rules. Antonio D'Angelo and Felice Russo from Lfoundry S.r.l. Italy share methods for density function estimation, kernel smoothing, and Monte Carlo simulation to separate real signals from noise.

• Density Distributions
• Multimodality Analysis
• Semiconductor Fab
• Kernel Estimation
• JMP Scripting

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1. Multimodality distribution analysis Discover variability sources hidden in density distributions by using JMP scripting capability

2. Multimodality Distribution analysis Antonio D Angelo, Felice Russo Lfoundry s.r.l Italy Objective Abstract Methods JMP script, integrating the analytical capabilities of R: In a semiconductor Fab several electrical measurements are collected on different silicon wafers. Distributions associated to them typically are normal even though this is not always the case. Many times those distributions show multiple modes because of processes shift, tools deviation or offset 1. to estimate the probability density function using a suitable kernel estimator 2. to identify and filter the distribution modes according to an empirical rule Identify the distributions with multiples modes among the large number of registers measured on each silicon wafer Reduce the candidates for real multimodal distributions JMP addin to automate the entire process

3. Multimodality Distribution analysis Antonio D Angelo, Felice Russo Lfoundry s.r.l Italy Data Density function estimation Density function estimation Discrete bins Density func References estimation Density Estimation - S.J. Sheather - Statistical science 2004, vol19, No 4, 588-597 Density Estimation for Statistics and Data Analysis - B.W. Silverman - Monographs on Statistics and Applied Probability, London: Chapman and Hall, 1986 Kernel Smoothing - M. P. Wand & M. C. Jones - Monographs on Statistics and Applied Probability Chapman & Hall, 1995. Continous Modes Filtering filtering Modes Distance Sigma Index= Modes Filtering Result

4. Multimodality Distribution analysis Antonio D Angelo, Felice Russo Lfoundry s.r.l Italy Density function estimation General form for kernel estimation K : kernel function h : bandwidth = smoothing factor Kernel function satisfies:

5. Multimodality Distribution analysis Antonio D Angelo, Felice Russo Lfoundry s.r.l Italy Montecarlo simulation unbalanced samples Modes Filtering Problem: how to separate a real signal from noise? Simulation to identify a suitable figure of merit Success ratio vs Distance and Sigma Different Mean Same Sigma Figure of merit Different Sigma Same Mean Figure of merit

6. Multimodality Distribution analysis Antonio D Angelo, Felice Russo Lfoundry s.r.l Italy Input mask