Modeling of intra-die process variations for accurate analysis and optimization of nano-scale circuits

Abstract

This paper proposes the use of Karhunen-Loève Expansion (KLE) for accurate and efficient modeling of intra-die correlations in the semiconductor manufacturing process. We demonstrate that the KLE provides a significantly more accurate representation of the underlying stochastic process compared to the traditional approach of dividing the layout into grids and applying Principal Component Analysis (PCA). By comparing the results of leakage analysis using both KLE and the existing approaches, we show that using KLE can provide up to 4 − 5× reduction in the variability space (number of random variables) while maintaining the same accuracy. We also propose an efficient leakage minimization algorithm that maximizes the leakage yield while satisfying probabilistic constraints on the delay.

Within-die process variations: How accurately can they be statistically modeled?

“…The collection of random variables that represent the results of all test structures form a stochastic process or a random field that is spatially indexed by locations of the test structures. A number of recent efforts investigate how to model within-die variations [6,2,17,8]. Friedberg et al [6] design critical dimension test structures to capture the variations in gate length, and then model the correlations between the variables of the resultant random field using piece-wise linear functions.…”

“…While process variations are random in nature, within-die variations typically exhibit spatial correlations, i.e., devices that are spatially close to each other are likely to be more strongly correlated than devices that are spatially far from each other. This correlation has been the subject of a number of recent works [6,2,17,8].…”

A framework for statistical timing analysis using non-linear delay and slew models

“…In addition to treating the random variable corresponding to the parameter of each gate as a separate random variable, it can do so with a significantly small set of random variables {ξn(θ)} [4]. For example, consider the isotropic covariance function C(r1, r2) = exp(−cr|r1 − r2|) = C(r1, r2), where cr is the inverse of the correlation length in the radial direction.…”

“…Once we have independent non-Gaussian (say uniform) and Gaussian variables ζ and ξ respectively, we can construct the orthonormal basis functions corresponding to the distributions of the random variables ζ and ξ. The inner product in this case is also defined by (4), with the weight function w(·) being the joint distribution of ζ and ξ. Since the random variables are independent, their joint probability distribution will be the product of their marginal distributions.…”

“…In order to reduce the number of random variables in the analysis used for modeling the correlations due to intra-die variations, we use Karhunen-Loéve expansion (KLE) expansion. KLE provides a much more compact representation of the process variations and can provide the same degree of accuracy as the grid based approach with up to 4-5 times less number of random variables [4].…”