Statistical interconnect metrics for physical-design optimization

In this paper, statistical models for the efficient analysis of interconnect delay and crosstalk noise in the presence of back-end process variations are developed. The proposed models enable closed-form computation of means and variances of interconnect-delay, crosstalk-noise peak, and coupling-induced-delay change for given magnitudes of variation in relevant process parameters, such as linewidth, metal thickness, metal spacing, and interlayer dielectric (ILD) thickness. The proposed approach is based on the observation that if the variations in different physical dimensions are assumed to be independent normal random variables, then the interconnect behavior also tends to have a Gaussian distribution. In the proposed statistical models, delay and noise are expressed directly as functions of changes in physical parameters. This formulation allows us to preserve all correlations and can be very useful in evaluating delay and noise sensitivities due to changes in various physical dimensions. For interconnect-delay computation, the authors express the resistance and capacitance of a line as a linear function of random variables and then use these to compute circuit moments. They show that ignoring higher order terms in the resulting variational moments does not result in a loss of accuracy. Finally, these variability-aware moments are used in known closed-form delay and slew metrics to compute interconnect-delay probability density functions (pdfs). Similarly for coupling noise and dynamic-delay analysis, the authors rely on the linearity (Gaussian) assumption, allowing us to truncate nonlinear terms and express noise and dynamic-delay pdfs as linear functions of variations in relevant geometric dimensions. They compare their approach to SPICE-based Monte Carlo simulations and report the error in mean and standard deviation of interconnect delay to be 1% and 4% on average, respectively