Orthogonal Function Fitting for Nonlinear Variation Control
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Solution Overview
Problem
Current nonlinear overlay models in semiconductor manufacturing fail to accurately represent and control nonlinear spatial variations due to coupling among error components, leading to unstable coefficients and limited adaptability, which hinders robust diagnostics and control of patterning capability.
Innovation Solution
A system and method that dynamically configures and controls process parameters by measuring attributes, selecting orthogonal basis functions, fitting them to measured data, and adjusting coefficients to minimize variations, using a physically based model with orthogonal functions to project measurements onto a reference mesh.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If polynomial expansion with coupled terms is used to characterize nonlinear spatial variations, then the model can represent non-linear distribution over a domain, but the coefficients become unstable and coupling among error components precludes robust control
Solution Approach 1:
The patent segments the coupled error terms into separate orthogonal components. By transforming the polynomial basis functions into an orthogonal set (e.g., using Gram-Schmidt process or selecting orthogonal polynomials like Legendre or Chebyshev), each coefficient becomes independent and stable, eliminating the coupling problem while maintaining representation accuracy.
Solution Approach 2:
The patent changes the parameterization approach from coupled polynomial coefficients to orthogonal function coefficients. This parameter transformation stabilizes the coefficient determination by ensuring orthogonality conditions, making the coefficients insensitive to sampling density and measurement noise variations.
2Manufacturing precision
If current polynomial models are used, then variation can be minimized at sampled locations, but the models exhibit limited adaptability and proliferation of non-physical terms
Solution Approach 1:
The patent develops a universal orthogonal function framework that can adapt to different domain geometries, boundary conditions, and measurement configurations. The orthogonal basis functions can be selected or constructed based on specific physical constraints, making the model versatile across different lithographic tools and process conditions while maintaining manufacturing precision.
3Measurement precision
If high order polynomial terms are used to capture nonlinear variations, then the model can represent complex patterns, but the terms vary rapidly near domain boundaries causing instability
Solution Approach 1:
The patent applies local quality by selecting or constructing orthogonal basis functions that incorporate boundary conditions and domain-specific characteristics. This ensures that the functions are well-behaved at domain boundaries while still capturing nonlinear variations in the interior, improving both precision and stability simultaneously.
Data Source
AI summary
System, method and computer program product including instructions executed by a processor system for configuring and controlling a facility to perform a manufacturing process and updating a tool controlling the process according to a model employed for mapping calculated coefficients that characterize non-linear variations observed of a product to actual control parameters governing the processes/tools used by the facility during the manufacturing process. In a semiconductor manufacturing process, the method enables real-time control of variation in an exposure step of a patterning process using an exposure tool to minimize a nonlinear variation in one or more pattern attributes by adjusting the exposure tool or the patterning process corresponding to the calculated coefficients. In the method, measurements of product attributes, obtained by finite sampling over a well defined domain (e.g., a region such as a field/wafer in lithographic patterning), are projected onto a predefined reference mesh spanning the domain, using a physically based model comprised of functions constructed to be orthogonal and normalized over a discrete set of reference mesh locations.