For CMOS feature size of 65 nm and below, local (or intra-die or within-die) variations in transistor Vt contribute stochastic variation in logic delay that is a large percentage of the nominal delay. Moreover, when circuits are operated at low voltage (Vdd ≤ 0.5V), the standard deviation of gate delay becomes comparable to nominal delay, and the Probability Density Function (PDF) of the gate delay is highly non-Gaussian. This paper presents a computationally efficient algorithm for computing the PDF of logic Timing Path (TP) delay, which results from local variations. This approach is called Non-linear Operating Point Analysis for Local Variations (NLOPALV). The approach is implemented using commercial STA tools and integrated into the standard CAD flow using custom scripts. Timing paths from a 28nm commercial DSP are analyzed using the proposed technique and the performance is observed to be within 5% accuracy compared to SPICE based Monte-Carlo analysis.

In order to achieve ultra-low power (ULP), ICs are being designed for 0.5 V. At these low voltages, random dopant fluctuations (RDFs) result in a stochastic component of logic delay that can be comparable to the global corner delay. Moreover, the probability density function (PDF) of this stochastic delay can be highly non-Gaussian. In order to predict the statistical impact of RDF-induced local variations on logic timing, it is necessary to incorporate these effects into a timing closure methodology. This paper presents a computationally efficient methodology for stochastic characterization of standard cell libraries at low voltage, where the cell delay is a nonlinear function of the transistor random variables (RVs), and the resulting cell delay has a non-Gaussian PDF. It also presents a computationally efficient methodology for computing any point on the PDF of a timing path (TP) delay, in the case where cell delays are non-Gaussian. The method is called nonlinear operating point analysis of local variation (NLOPALV). The general NLOPALV theory is developed. It is applied to cell library characterization, and the accuracy of the NLOPALV approach is validated by comparison to Monte Carlo simulation. NLOPALV is also applied to timing path analysis on a 28 nm DSP IC. The approach has been implemented using commercial CAD tools, and integrated into a commercial IC design flow. The NLOPALV approach gives timing results that are within 5% accuracy compared to Monte Carlo analysis at 0.5 V. This compares to errors on the order of 50% when the Gaussian approximation is used.

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