Substrate Deformation Analysis Using Bessel Functions for Overlay Accuracy
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Solution Overview
Problem
In lithography processes for semiconductor manufacturing, non-linear deformations on substrates complicate the overlaying of patterns, which existing methods struggle to accurately account for, leading to positioning inaccuracies.
Innovation Solution
An analyzing method that uses a reference function, composed of Bessel functions multiplied by proportional coefficients, to fit measured position information and calculate optimal coefficients for correcting substrate deformations, combined with a measurement apparatus and exposure apparatus to align patterns accurately.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional deformation methods are used to express non-linear deformation on a substrate, then the substrate deformation can be modeled, but the positioning accuracy deteriorates due to inability to accurately account for non-linear deformations
Solution Approach 1:
The patent changes the parameter representation from conventional polynomial-based models to Bessel function-based models. The Bessel functions provide a more accurate mathematical representation of non-linear substrate deformations, allowing for precise positioning despite the complexity of the deformation patterns. This parameter change enables the system to capture the true nature of non-linear deformations while maintaining computational feasibility.
Solution Approach 2:
The patent replaces conventional mechanical/deformation-based modeling approaches with a mathematical substitution using Bessel functions. Instead of relying on traditional mechanical models that fail to capture non-linear behavior, the invention substitutes a more sophisticated mathematical framework that naturally describes non-linear deformation patterns, thereby improving positioning accuracy without requiring complex physical measurements.
2Manufacturing precision
If Bessel function-based reference functions are used to fit measured position information, then positioning accuracy is improved, but the calculation complexity increases
Solution Approach 1:
The patent transforms the alignment problem by changing the functional parameters from conventional polynomials to Bessel functions. This parameter change allows for more accurate fitting of non-linear deformation patterns, achieving superior pattern alignment accuracy. The Bessel function basis provides a natural framework for describing radial and angular deformation components, making the calculation more efficient despite the increased mathematical sophistication.
Solution Approach 2:
The patent segments the deformation analysis into distinct Bessel function components, each representing different aspects of the non-linear deformation. By decomposing the complex deformation field into manageable Bessel function terms with specific proportional coefficients, the system can calculate alignment corrections more efficiently. This segmentation allows for systematic calculation of each coefficient independently, reducing the overall computational burden.
Data Source
AI summary
An analyzing method includes preparing measured position information that is position information of a plurality of measured-parts formed on a substrate, and fitting a reference function, which is a sum of at least one function obtained by multiplying a criterion function expressed using a first type Bessel function by a proportional coefficient, to the measured position information and calculating an optimum value of at least one of the proportional coefficient.