Electromagnetic Scattering Calculation for Finite Periodic Structures
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Solution Overview
Problem
Conventional methods for reconstructing critical dimension (CD) parameters on small, non-periodic structures in lithographic processes face challenges due to increased computational load and edge diffraction effects, making fast inline reconstruction or library generation impractical.
Innovation Solution
A method that approximates diffraction from finite, non-periodic gratings by using the solution from an infinitely periodic grating, involving numerical calculation of single-cell contrast current density and integrating a Green's function to determine scattered electric fields, allowing for efficient reconstruction of small metrology targets or on-product structures without significant computational increase.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional methods are used to compute diffraction from finite non-periodic gratings by solving over the whole grating, then measurement precision is improved, but device complexity and computational load increase prohibitively
Solution Approach 1:
The patent divides the finite non-periodic grating into multiple unit cells, each of which can be modeled using periodic boundary conditions. This segmentation allows the complex finite grating problem to be broken down into simpler periodic unit cell problems that can be solved independently and then combined, dramatically reducing computational load while maintaining measurement precision.
Solution Approach 2:
The patent introduces an intermediary approach by using the periodic grating solution as an approximation for the finite grating. Instead of directly solving the complex finite grating problem, the method uses periodic boundary conditions as an intermediary model that can be efficiently solved, then applies corrections to account for edge effects and finite size effects, achieving accurate CD reconstruction with reduced computational complexity.
2Measurement precision
If the finite structure is embedded in a much larger unit cell, then measurement precision is improved, but device complexity increases due to huge numerical complexity
Solution Approach 1:
Rather than embedding the finite structure in one large unit cell, the patent segments the structure into multiple smaller unit cells. Each small unit cell can be solved with periodic boundary conditions at a manageable numerical complexity level, and the overall diffraction spectrum is reconstructed by combining the solutions from all segments, avoiding the exponential increase in numerical complexity that would result from using a single large unit cell.
3Productivity
If the windowed approximation is used where scattered field is weighed with rectangular function, then computational load is reduced, but measurement precision deteriorates due to strong ringing effects
Solution Approach 1:
The patent modifies the approximation approach by changing the parameters of the unit cell model to account for edge effects. Instead of using a simple rectangular window function that causes ringing, the method adjusts the periodic boundary condition parameters and introduces correction terms that specifically address edge diffraction effects, thereby maintaining computational efficiency while improving measurement precision at the edges of the finite grating.
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
Enables efficient reconstruction of electromagnetic scattering properties and approximate structure determination for finite periodic structures, reducing computational load and edge diffraction errors, facilitating real-time CD reconstruction and accurate modeling of edge effects.
Implementation Method 1
calculating a scattered electric field outside the finite collection of unit cells, arising from the finite periodic structure
Implementation Method 2
computing the angular resolved diffraction spectrum from a grating profile
Data Source
AI summary
A method of determining electromagnetic scattering properties of a finite periodic structure has the steps: 1002: Calculating a single-cell contrast current density, within a unit-cell supporting domain of a single one of a finite collection of unit cells. 1004: Calculating a scattered electric field outside the finite collection of unit cells, by integrating, over the single unit cell's supporting domain, a Green's function with the determined single-cell contrast current density. 1006: The Green's function is obtained for observation points outside the finite collection of unit cells by summation across the finite collection of unit cells. The Green's function integrated with the determined single-cell contrast current density is obtained for observation points above the supporting domain with respect to a substrate underlying the finite periodic structure. 1008: Determining an electromagnetic scattering property, for example a diffraction pattern, of the finite periodic structure using the calculated scattered electric field.