Optimized Scatterometry for Diffracting Structures
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Solution Overview
Problem
Scatterometry computations for characterizing diffracting structures are time and resource intensive due to complex geometry, making them impractical for timely measurements in applications like semiconductor manufacturing.
Innovation Solution
The implementation of approximate spectral methods and symmetry-accelerated spectral methods, along with hybrid derivatives for computing the Jones matrix, reduces computational complexity and resource requirements, enabling faster evaluation of diffracting structures.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If rigorous diffraction modeling algorithms (RCWA) are used to determine reflectance from diffracting structures, then measurement precision is improved, but computation time and resource requirements increase significantly
Solution Approach 1:
The patent transforms the computational problem by changing the mathematical parameters and representation of the diffraction model. Instead of using traditional RCWA eigenvalue computations, the invention employs a spectral domain formulation with Taylor series approximations, fundamentally altering the computational parameters to achieve both accuracy and speed
Solution Approach 2:
The patent replaces the traditional mechanical/eigenvalue-based computational approach with a spectral domain method using Taylor series expansions. This substitution eliminates the need for iterative eigenvalue solvers while maintaining measurement precision through analytical derivative computations
2Manufacturing precision
If complex geometric models of diffracting structures are used to achieve accurate parameter characterization, then manufacturing precision is improved, but device complexity and resource requirements increase
Solution Approach 1:
The patent changes the mathematical representation of geometric parameters by using spectral domain formulations. Complex geometric shapes are represented through Fourier series expansions in the spectral domain, transforming complex spatial computations into simpler frequency-domain operations that maintain precision while reducing complexity
Solution Approach 2:
The patent creates simplified spectral copies of the complex geometric models. Instead of directly computing with complex geometric representations, the invention uses spectral domain approximations that capture the essential geometric features while enabling faster, less resource-intensive computations
3Measurement precision
If traditional eigenvalue-based diffraction algorithms are used, then measurement accuracy is maintained, but productivity and throughput are reduced due to time-intensive computations
Solution Approach 1:
The patent performs preliminary computations by pre-calculating Taylor series expansion coefficients and spectral domain representations before actual measurements. This preliminary action creates lookup tables and pre-computed parameters that dramatically accelerate real-time measurements while maintaining accuracy
Solution Approach 2:
The patent substitutes traditional iterative eigenvalue-solving mechanics with direct spectral domain computations using Taylor series. This replacement eliminates time-consuming iterative processes while preserving measurement accuracy through analytical solutions
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
These optimizations significantly reduce the time and resources needed for scatterometry computations, allowing for more efficient characterization of diffracting structures and improving the speed of analysis.
Implementation Method 1
light is directed onto a periodic grating formed in a workpiece and spectra of reflected light are measured and analyzed to characterize the grating
Implementation Method 2
spectra of reflected light are measured
Data Source
AI summary
Provided are optimized scatterometry techniques for evaluating a diffracting structure. In one embodiment, a method includes computing a finite-difference derivative of a field matrix with respect to first parameters (including a geometric parameter of the diffracting structure), computing an analytic derivative of the Jones matrix with respect to the field matrix, computing a derivative of the Jones matrix with respect to the first parameters, and computing a finite-difference derivative of the Jones matrix with respect to second parameters (including a non-geometric parameter). In one embodiment, a method includes generating a transfer matrix having Taylor Series approximations for elements, and decomposing the field matrix into two or more smaller matrices based on symmetry between the incident light and the diffracting structure.