Orthogonal Polynomial Overlay Error Modeling in Lithography
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Solution Overview
Problem
The precision of lithographic apparatus control is adversely impacted due to non-orthogonal basis functions used in modeling overlay errors across the wafer and field domains, leading to linear dependence and difficulty in estimating model parameters accurately.
Innovation Solution
Generating discrete orthogonal polynomials based on predetermined substrate measurement locations to model substrate properties, allowing for precise estimation and control of lithographic processing by using these polynomials as a basis function.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If non-orthogonal basis functions are used to model overlay errors, then the model can represent complex error patterns, but linear dependence occurs and parameter estimation precision deteriorates
Solution Approach 1:
The patent transforms the basis functions from non-orthogonal to orthogonal polynomials by changing the mathematical parameters of the model. This transformation maintains the ability to represent complex error patterns while eliminating linear dependence, thereby resolving the contradiction between model flexibility and parameter estimation precision
Solution Approach 2:
The patent combines orthogonal polynomial basis functions with a least-squares estimation framework to create a composite modeling approach. This composite structure ensures both the representational capability to model complex overlay errors and the mathematical stability required for precise parameter estimation
2Manufacturing precision
If non-orthogonal basis functions are used in the model, then the model can capture complex substrate property variations, but linear dependence makes parameter estimation difficult
Solution Approach 1:
The patent changes the mathematical parameters by using orthogonal polynomials instead of non-orthogonal basis functions. This parameter transformation maintains the ability to capture complex substrate variations while simplifying the parameter estimation process through the orthogonality property, which decouples the estimation equations
3Ease of manufacture
If simple parameterization models are used, then the model is easier to compute, but the precision of control moves deteriorates
Solution Approach 1:
The patent creates a composite modeling approach that combines orthogonal polynomial basis functions with least-squares estimation. This composite structure achieves an optimal balance between computational simplicity and control precision, as the orthogonality reduces computational complexity while maintaining high modeling accuracy
Data Source
AI summary
System and methods estimate model parameters of a lithographic apparatus and control lithographic processing by a lithographic apparatus. An exposure is performed using a lithographic apparatus across a wafer. A set of predetermined wafer measurement locations is obtained. Discrete orthonormal polynomials are generated using the predetermined substrate measurement locations. The overlay errors arising from the exposure are measured at the predetermined locations to obtain overlay measurements. The estimated model parameters of the lithographic apparatus are calculated from the overlay measurements by using the discrete orthogonal polynomials as a basis function to model the overlay across the wafer. Finally, the estimated model parameters are used to control the lithographic apparatus in order to provide corrected overlay across the wafer.