Importance sampling in Gaussian random field EUV stochastic model for quantification of stochastic variability of EUV vias

Abstract

Importance Sampling methods allow to substantially reduce the number of trials in estimation of the rare failure probability or other stochastic metrics. These methods can be viewed as a rigorous generalization of quantitative "torture" or "stress" methods where the process is artificially modified to increase the probability of failure, and the failure probability estimations obtained for such modified process are extrapolated to the original process with rare failures. Applications of Importance Sampling methods are presented and demonstrated on computationally efficient estimations of via failure probability and via LCDU. The accuracy of the Importance Sampling LCDU estimates is verified by comparison with experimental results. Applications of Importance Sampling methods to experimental measurements are discussed.

Principal components and optimal feature vectors of EUVL stochastic variability: applications of Karhunen-Loève expansion to efficient estimation of stochastic failure probabilities and stochastic metrics

“…The parameters and the modeling assumptions of these test cases are as used in our earlier work 4,9 . In both test examples we use the simplified imaging model to calculate the EUV image in resist.…”

“…10) where ()  x is absorbance coefficient of the resist material, ()Ix is the intensity of EUV image in resist, () G x is a deprotection kernel of the resist process model, one of the functional parameters of the stochastic resist model, h N  is the average number of photons absorbed in the entire simulation area, V , for the given exposure dose, image in resist and the absorbance coefficient. Finally, x x is the normalization coefficient in the probability density function (PDF) of the EUV photon absorption site distribution.This covariance function can be used to calculate the KLE eigenvalues and eigenfunctions, without the need to perform Monte Carlo trials for its estimation.One additional important simplification of KLE in the case of EUVL stochastic model with a GRF deprotection comes from noticing that the random variables to the GRF deprotection through(9), form a joint multivariate normal distribution. Because these random variables are uncorrelated and have a unit variance (as guaranteed by KLE, per (8)), their covariance matrix and its inverse are both identity matrices.…”