Adjoint Method for Semiconductor Metrology Derivative Computation
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Solution Overview
Problem
Current methods for determining sensitivities of reflection or transmission spectra to various grating parameters in semiconductor metrology are limited by the computational cost of finite difference approximations, which restrict the number of derivatives that can be computed and the information extracted from optical scattering data.
Innovation Solution
A method and apparatus using a model-based approach that involves illuminating a geometrical structure with polarized electromagnetic radiation, measuring spectral components, and determining parameter values that minimize an objective function using adjoint methods, which allow for faster and more accurate computation of derivatives compared to finite difference methods.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If finite difference approximation is used to compute derivatives, then measurement precision is improved, but computation time increases significantly
Solution Approach 1:
The patent pre-computes and stores sensitivity data in a lookup table before actual measurement processing. This preliminary action allows the system to avoid performing computationally expensive finite difference calculations during real-time operation, instead rapidly retrieving pre-computed derivative values that maintain measurement precision while dramatically reducing computation time.
Solution Approach 2:
The patent computes derivatives at selected parameter points and stores them in a lookup table, rather than computing all possible derivatives on-demand. This partial computation approach provides sufficient derivative information for accurate measurements while avoiding the excessive computation time that would result from calculating the complete derivative space in real-time.
2Measurement precision
If the number of computed derivatives is increased, then measurement precision is improved, but device complexity increases
Solution Approach 1:
The patent pre-computes and stores sensitivity derivatives in a lookup table structure, organizing complex derivative data in advance. This allows the measurement system to access multiple derivative values without performing complex real-time calculations, thereby improving feature profile detection accuracy while keeping the operational system complexity manageable through efficient data retrieval rather than complex computation.
3Measurement precision
If more wavelengths are processed, then measurement precision is improved, but productivity decreases
Solution Approach 1:
The patent pre-computes sensitivity derivatives for multiple wavelengths and stores them in the lookup table during an offline preparation phase. This allows the system to process and utilize optical scattering data across multiple wavelengths simultaneously during measurement without performing expensive real-time derivative calculations, thereby improving measurement precision through full spectral utilization while maintaining high processing throughput.
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 enables the computation of derivatives at a lower computational cost, allowing for higher-order parameterization of gratings and more efficient measurement setups, optimizing signal-to-noise ratios and providing precise estimates of critical dimensions.
Implementation Method 1
An area of a geometrical structure of dispersive materials is illuminated with incident electromagnetic radiation, wherein the incident electromagnetic radiation is polarized
Implementation Method 2
spectral components of the incident electromagnetic radiation reflected from the area are measured
Data Source
AI summary
One embodiment relates to a method of model-based optical metrology. An area of a geometrical structure of dispersive materials on a substrate is illuminated with polarized incident electromagnetic radiation using an illuminator of a scatterometer apparatus. Spectral components of the incident electromagnetic radiation reflected from the area are measured using a detector of the scatterometer apparatus. Using a computer for the scatterometer apparatus, parameter values are determined that minimize an objective function which represents a difference between the measured spectral components and computed spectral components based on a parameterized model of the geometrical structure. Steps for determining the parameter values that minimize the objective function include: computing a solution to state equations driven by a function representing the incident electromagnetic radiation, and computing a solution to an adjoint to the state equations. Other embodiments and features are also disclosed.