Spatial Correlation Modeling for Semiconductor Variations
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Solution Overview
Problem
Conventional approaches to modeling spatial correlations in integrated circuit chip design, such as principal component analysis (PCA), fail to handle negative eigenvalues and result in inefficient circuit simulations due to increased terms for small grid sizes, leading to longer simulation times.
Innovation Solution
A method that develops specific forms of spatial correlations, selects the closest solution to a desired correlation, and models it using a system that minimizes terms by assigning stochastic variables and modifies unrealistic correlation coefficients to realistic ones, thereby improving accuracy and resource efficiency.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional PCA approaches are used to model spatial correlations, then the modeling framework is established, but negative eigenvalues arise and simulation time increases
Solution Approach 1:
The patent transforms the correlation matrix into the frequency domain using Fourier transform, changing the representation parameters from spatial domain coefficients to frequency domain spectral components. This parameter transformation allows negative eigenvalues to be identified and corrected in the frequency domain, where they manifest as negative spectral values that can be clipped to zero, resolving the reliability issue while maintaining computational efficiency
Solution Approach 2:
The patent replaces the conventional time-domain PCA mechanical approach with a frequency-domain spectral representation. By substituting the mechanical computation of spatial correlations with spectral analysis via Fourier transform, the system achieves both numerical stability (handling negative eigenvalues) and computational efficiency (reducing simulation time through compact spectral representations)
2Manufacturing precision
If PCA approaches are used for a large number of sub-regions (very small grid size), then spatial correlations are modeled, but the number of terms increases leading to longer circuit simulation time
Solution Approach 1:
The patent changes the parameter representation from a large number of spatial domain correlation coefficients to a compact set of frequency domain spectral components. This parameter transformation maintains the ability to model fine-grained spatial correlations (high manufacturing precision) while dramatically reducing the number of terms that need to be processed during circuit simulation (improving productivity)
3Adaptability or versatility
If PCA approaches are used, then spatial correlations are modeled, but there is no insight on how to extend the result to the limit of very small grid size
Solution Approach 1:
The patent substitutes the conventional spatial domain PCA framework with a frequency domain spectral framework using Fourier transform. This substitution provides natural insight into grid size extension because the spectral representation is scale-invariant and can be easily extended to continuous domains, offering theoretical insight that the discrete spatial PCA approach lacks
Data Source
AI summary
Modeling spatial correlations of semiconductor characteristic variations is disclosed. In one embodiment, a method includes developing a solution for each of a plurality of specific forms of spatial correlations of a characteristic of a circuit design and developing a plurality of solution methods for a given spatial correlation; selecting one of the solutions that is closest to a desired spatial correlation; and modeling the desired spatial correlation using the selected solution.