Statistical compact model extraction for skew‐normal distributions

Abstract

A technique to extract statistical model parameters for skewed Gaussian process variations is proposed. Statistical compact model extraction traditionally assumes that underlying process variations are Gaussian in nature. ON currents in certain high voltage technologies, which are linear in process deviations, show skew in their distribution and hence is indicative of skew in the underlying process variations. The use of skew-normal random variables is proposed to model such variations. Artificial neural networks (ANNs) are used to empirically model the functional relation of performance on process deviations and a framework to propagate skew-normal random variables through ANNs is proposed. A non-linear optimisation problem is formulated to extract the parameters that characterise the skew-normal process variations, with constraints imposed on the objective function to penalise any deviation from Gaussian variations. Results show that the extracted parameters, when simulated, match the performance parameter targets to within 3% for both Gaussian and skewed process variations. SIM , I D, 2 SIM , I D, 3 SIM …) whose statistics (Σ Sim) matches Σ Tar. This is known as the statistical compact model extraction (SCME) problem. Central to any BPV problem is an empirical model relating performance to process deviations. The model should (a) capture the functional dependence of performance parameters on process deviations and (b) enable analytical propagation of RVs. The methodology in [1-3] uses a linear model, Stevanovic and McAndrew [4] use a quadratic model, Kovac et al. [5] use a cubic model while Viraraghavan et al. [6] use an artificial neural network (ANN) model. A common underlying assumption that worked very well for generations of technologies, is that the probability distribution of process variations is Gaussian in nature, which made analytical propagation tractable. With Gaussian process variations, performance variations can be non-Gaussian only through non-linear transformations like in the case of leakage (off) current. Linear transformations should necessarily result in Gaussian variations in performance parameters as is the case with ON currents and threshold voltages [6]. However, when linear performance parameters show skew in measured data, there is a need to revisit the Gaussian process variation assumption. Certain devices in high voltage technologies have shown non-Gaussian variations in ON currents, which are otherwise known to be linear in process deviations [7]. Viraraghavan et al. [7] propose the use of a non-standard skew-Gaussian RV-based methodology to handle skew but is limited to linear performance parameters. In reality, SCME needs to deal with a combination of linear and non-linear ones. In this paper, we look at a generalised SCME problem where process variations are modelled as skew-normal (SN) RVs whose parameters are estimated through an optimisation formulation.