Compressed Fourier Space for 2D Periodic Pattern Modeling
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Solution Overview
Problem
Modeling complex two-dimensional periodic structures in semiconductor manufacturing is challenging due to high memory and calculation requirements, especially when using rigorous numerical methods like RCWA, which expand exponentially with the number of harmonic orders in the Fourier space.
Innovation Solution
The Fourier expansion is selectively truncated by determining and retaining only harmonic orders with significant contributions, compressing the Fourier space to reduce memory and calculation times, allowing for real-time analysis or generation of a library for complex samples.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If full Fourier expansion is used for modeling two-dimensional periodic structures, then measurement accuracy is improved, but memory requirements and calculation time increase significantly
Solution Approach 1:
The patent extracts and retains only the significant harmonic orders from the full Fourier expansion that contribute meaningfully to the scatter signature. By identifying and removing redundant harmonic components below a threshold contribution level, the method reduces memory requirements while preserving measurement accuracy for two-dimensional periodic structures
Solution Approach 2:
The patent changes the parameter of Fourier expansion by selectively truncating harmonic orders based on their contribution threshold. This parameter modification transforms the complete Fourier series into a compressed representation that maintains accuracy for 2D periodic patterns while reducing computational burden
2Measurement precision
If full Fourier expansion is used for modeling two-dimensional periodic structures, then measurement accuracy is improved, but calculation time increases significantly
Solution Approach 1:
The patent extracts and retains only the significant harmonic orders from the full Fourier expansion that contribute meaningfully to the scatter signature. By identifying and removing redundant harmonic components below a threshold contribution level, the method reduces calculation time while preserving measurement accuracy for two-dimensional periodic structures
Solution Approach 2:
The patent applies partial action by using only the necessary subset of harmonic orders rather than the complete Fourier expansion. This partial approach retains sufficient information for accurate measurement while eliminating computationally expensive redundant calculations
3Adaptability or versatility
If modeling is performed for complex two-dimensional periodic structures, then measurement capability is improved, but device complexity increases
Solution Approach 1:
The patent changes the modeling approach by introducing selective truncation of Fourier harmonic orders based on contribution thresholds. This parameter modification simplifies the complex modeling process for two-dimensional periodic structures while maintaining the capability to handle diverse semiconductor device patterns
Solution Approach 2:
The patent applies local quality by treating different harmonic orders differently based on their individual contribution levels. Significant harmonics are retained with full detail while less significant ones are truncated, creating a non-uniform but optimized modeling approach that reduces complexity
Applied Scientific Principles
This section explains which scientific principles are used to turn an abstract innovation direction into a practical engineering solution.
Function Achieved in This Case
This approach significantly reduces memory and calculation requirements while maintaining desired accuracy, enabling efficient real-time analysis and library generation for complex two-dimensional periodic structures, with calculations being 28-150 times faster and requiring less than a third to one-tenth of the original memory.
Implementation Method 1
determining the contribution for harmonic orders in the Fourier transform of the permittivity function
Implementation Method 2
scatterometry measures structures using light that is scattered from the structure, sometimes referred to as a scatter signature
Data Source
AI summary
The process of modeling a complex two-dimensional periodic structure is improved by selectively truncating the Fourier expansion used in the calculation of resulting scatter signature from the model. The Fourier expansion is selectively truncated by determining the contribution for each harmonic order in the Fourier transform of the permittivity function and retaining the harmonic orders with a contribution that is above a threshold. The Fourier space may be compressed so that only the selected harmonic orders are used, thereby reducing the required memory and calculation times. The compressed Fourier space may be used in a real-time analysis of a sample or to generate a library that is used in the analysis of a sample.