Lithography Simulation Using Polynomial Parameterization for Curved Mask Shapes
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Solution Overview
Problem
Current lithography simulation methods face challenges in accurately simulating mask shapes represented by curves due to the need to approximate curved shapes with straight lines, leading to increased data amounts and simulation errors.
Innovation Solution
The proposed method involves acquiring a mask shape to be transferred from a mask substrate to a wafer substrate, identifying control points on the contour figure, and using polynomial parameterization to perform a Fourier transform of the contour figure, allowing for direct processing of curved shapes without division into straight lines.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Manufacturing precision
If a curved mask shape is approximated by dividing it into straight lines to perform simulation, then the simulation can be performed using conventional methods, but the amount of data is excessively increased and simulation errors occur
Solution Approach 1:
The patent applies curvature by representing the mask shape as a continuous curve using polynomial parameterization instead of approximating it with straight lines. This allows the simulation to directly process curved geometries, maintaining simulation accuracy without increasing data amount through fine division.
Solution Approach 2:
The patent changes the parameterization method from piecewise linear segments to polynomial parameters that describe the continuous curve. By using polynomial coefficients to represent the mask shape, the system reduces data complexity while preserving the curved geometry information needed for accurate simulation.
2Manufacturing precision
If a curved mask shape is approximated by dividing it into straight lines to perform simulation, then the simulation can be performed using conventional methods, but the complexity of the method is excessively increased
Solution Approach 1:
The patent directly represents curved mask shapes using polynomial parameterization, eliminating the need to convert curves into multiple straight line segments. This approach simplifies the simulation method by working natively with curved geometries rather than requiring approximation and division operations.
Solution Approach 2:
The patent extracts the essential geometric information of the mask shape by representing it through polynomial parameters that define the curve. This extraction eliminates the need for complex discrete segmentation while retaining the necessary information for accurate optical simulation.
3Measurement precision
If the straight line is finely divided to reduce the difference from the curve, then the approximation error is reduced, but the amount of data is excessively increased
Solution Approach 1:
The patent transforms the representation from numerous discrete line segment parameters to a compact polynomial parameterization. This parameter transformation achieves high curve approximation accuracy using fewer parameters, as the polynomial function inherently captures the continuous curve behavior without requiring fine division into many segments.
Solution Approach 2:
By using polynomial parameterization, the patent directly models the curved geometry with a mathematical function that naturally describes continuous curves. This eliminates the need for discrete approximation with many small line segments, achieving accurate curve representation with reduced data quantity.
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 reduces data amounts and simulation errors by enabling the direct Fourier transformation of curved mask shapes, improving the accuracy of resist pattern prediction and enhancing the robustness of optical proximity effect correction.
Implementation Method 1
The arithmetic device Fourier-transforms the contour figure using the polynomial parameterization to predict a resist pattern
Data Source
AI summary
A lithography simulation method according to the present embodiment includes acquiring a mask shape to be transferred from a mask substrate to a wafer substrate using a projection exposure apparatus. The lithography simulation method also includes acquiring a control point on a contour figure included in the mask shape, a function form of a polynomial parameterization, and an order of the function form. The lithography simulation method also includes Fourier-transforming the contour figure using the polynomial parameterization based on the control point, the function form, and the order to predict a resist pattern.