On-Chip Variation Modeling via Spatial Correlation
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Solution Overview
Problem
Current IC design simulations struggle to accurately model on-chip variation (OCV) with spatial effects, as local variations are not fixed across the IC chip and are influenced by distances between devices, making it challenging to satisfy correlation constraints between all pairs of devices, especially as the number of devices increases.
Innovation Solution
The method involves creating a spatial correlation matrix to derive a random number generation function that accounts for spatial correlations between devices, using techniques like Discrete Fourier Transformation or Principal Component Analysis to reduce calculation complexity and generate randomized device properties that follow a Gaussian distribution, allowing for more accurate and efficient modeling of OCV with spatial effects.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If local variations are assumed as fixed percentage changes for timing analysis, then the simulation process is simple and fast, but the modeling accuracy of on-chip variation spatial effects is insufficient
Solution Approach 1:
The patent transforms the OCV modeling problem by changing the parameter representation from fixed local variation percentages to spatially correlated random variations. A spatial correlation matrix is constructed where elements represent correlation coefficients between device property variations at different locations. This parameter transformation enables accurate modeling of spatial effects while managing complexity through structured mathematical representation.
Solution Approach 2:
The patent uses Principal Component Analysis (PCA) to create a simplified copy of the spatial correlation structure. By decomposing the spatial correlation matrix into eigenvalues and eigenvectors, the method generates random variations that replicate the essential spatial correlation patterns without requiring direct manipulation of all pairwise device correlations. This copying approach reduces computational complexity while preserving the critical spatial dependency characteristics.
2Measurement precision
If spatially correlated random variations are created to model OCV spatial effects, then the modeling realism is improved, but the calculation complexity increases exponentially with the number of devices
Solution Approach 1:
The patent changes the computational parameters by working with the spectral decomposition (eigenvalues and eigenvectors) of the spatial correlation matrix rather than directly computing all pairwise correlations. This parameter transformation reduces the computational burden from exponential to polynomial complexity, enabling realistic OCV modeling for large-scale IC designs with numerous devices.
Solution Approach 2:
The patent applies local quality by generating random variations at each device location based on its specific position in the spatial correlation structure. Each device's variation is locally determined by its coordinates and the spatial correlation function, rather than requiring global coordination of all device variations. This local generation approach significantly reduces computational complexity while maintaining global spatial correlation consistency.
3Quantity of substance
If the number of devices on IC increases, then the functional density is improved, but the difficulty of satisfying correlation constraints among all device pairs increases
Solution Approach 1:
The patent creates a universal spatial correlation model that can handle any number of devices on the IC chip through a single framework. The spatial correlation matrix and its spectral decomposition provide a multi-functional approach that works regardless of device count, eliminating the need for different correlation satisfaction methods for different scale IC designs. This universal model scales efficiently with increasing device density.
Solution Approach 2:
The patent transforms the correlation constraint satisfaction problem into a parameter generation problem using the spectral decomposition of the spatial correlation matrix. By changing from directly enforcing pairwise constraints to generating variations according to the decomposed eigenstructure, the method efficiently handles increasing device numbers without proportionally increasing computational difficulty.
Data Source
AI summary
An integrated circuit (IC) design method includes receiving a spatial correlation matrix, R, of certain property of post-fabrication IC devices; and deriving a random number generation function g(x, y) such that random numbers for a device at a coordinate (x, y) can be generated by g(x, y) independent of other devices, and all pairs of random numbers satisfy the spatial correlation matrix R. The method further includes receiving an IC design layout having pre-fabrication IC devices, each of the pre-fabrication IC devices having a coordinate and a first value of the property. The method further includes generating random numbers using the coordinates of the pre-fabrication IC devices and the function g(x, y); deriving second values of the property by applying the random numbers to the first values; and providing the second values to an IC simulation tool.