Nonparametric Regression for Substrate Alignment Accuracy
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Solution Overview
Problem
Current exposure methods face challenges in achieving high alignment accuracy for miniaturized devices due to the tradeoff between overfitting and productivity, as the number of measurement points for alignment marks is limited, leading to increased measurement time and reduced productivity.
Innovation Solution
A method utilizing a nonparametric regression model, specifically a Gaussian process regression model, to estimate the position of non-measurement regions on a substrate, allowing for accurate alignment with a small number of measurement points, thereby improving alignment accuracy and reducing measurement time.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Manufacturing precision
If the number of measurement points to measure the positions of alignment marks is increased to suppress overfitting, then manufacturing precision is improved, but productivity deteriorates due to increased measurement time
Solution Approach 1:
The patent changes the fundamental parameter of the regression model from parametric (polynomial) to nonparametric (Gaussian process), enabling accurate prediction with fewer measurement points. This parameter change in the mathematical model allows achieving high alignment accuracy without increasing the number of measurement points, thus resolving the contradiction between manufacturing precision and productivity
Solution Approach 2:
The patent replaces the conventional parametric regression approach with a nonparametric Bayesian approach using Gaussian processes. This substitution of the mathematical modeling mechanism enables the system to achieve accurate predictions with limited data, resolving the tradeoff between measurement accuracy and measurement time
2Manufacturing precision
If the degree of freedom of the function model is increased by using a higher-order component, then manufacturing precision is improved, but reliability deteriorates due to overfitting when the number of measurement points is small
Solution Approach 1:
The patent changes the model parameter from fixed-degree polynomial to adaptive Gaussian process model, which automatically adjusts its complexity based on the available measurement data. This prevents overfitting while maintaining high prediction accuracy, resolving the contradiction between manufacturing precision and reliability
Solution Approach 2:
The Gaussian process regression model incorporates uncertainty quantification and adaptive learning from measurement data. The model provides feedback on prediction confidence and adjusts its behavior accordingly, preventing overfitting while maintaining high accuracy even with limited measurement points
Data Source
AI summary
A method of obtaining an array of a plurality of regions on a substrate, including obtaining position measurement data by measuring a mark assigned to each sample region among the plurality of regions on the substrate, and estimating a position of each non-measurement region excluding the sample region among the plurality of regions by using a regression model used to estimate the array from the position measurement data, wherein the regression model is a nonparametric regression model.