Statistical Transistor-Level Timing Analysis Using a Direct Random Differential Equation Solver

Abstract—To improve the accuracy of static timing analysis, the traditional nonlinear delay models are increasingly replaced by more physical gate models, such as current source models and transistor-level gate models. However, the extension of these accurate gate models for statistical timing analysis is still challenging. In this paper, we propose a novel statistical timing analysis method based on transistor-level gate models. The accuracy and efficiency are obtained by using an efficient random differential equation based solver. The correlations among signals and between input signals and delay are fully accounted for. In contrast to Monte Carlo simulation solutions, the variational waveforms for statistical delay calculation are obtained by simulating only once. At the end of statistical timing analysis, both the statistical delay moments and the variational waveforms are available. The proposed algorithm is verified with standard cells and ISCAS85 benchmark circuits in a 45-nm technology. The experimental results indicate that the proposed method can capture multiple input simultaneous switching for statistical delay calculation, and can provide 0.5 % error for delay mean and 2.7 % error for delay standard deviation estimation on average. The proposed statistical simulation introduces a small runtime overhead with respect to static timing analysis runtime. The MATLAB implementation of the proposed algorithm has two orders of magnitude speedup, compared to Spectre Monte Carlo simulation. Index Terms—Correlation, gate model, process variations, random differential equations, statistical timing analysis (STA). I