Fast physics-based electromigration assessment by efficient solution of linear time-invariant (LTI) systems

Abstract

Electromigration (EM) is a key reliability concern in chip power/ ground (p/g) grids, which has been exacerbated by the high current levels and narrow metal lines in modern grids. EM checking is expensive due to the large sizes of modern p/g grids and is also inherently difficult due to the complex nature of the EM phenomenon. Traditional EM checking, based on empirical models, cannot capture the complexity of EM and better models are needed for accurate prediction. Thus, recent physics-based EM models have been proposed, which remain computationally expensive because they require solution of a system of partial differential equations (PDEs). In this paper, we propose a fast and scalable methodology for power grid EM verification, building on previous physics-based models. We first convert the PDE system to a succession of homogeneous linear time invariant (LTI) systems. Because these systems are found to be stiff, we numerically integrate them using optimized variable-step backward differentiation formulas (BDFs). Our method, for a number of IBM power grids and internal benchmarks, achieves an average speed-up of over 20x as compared to previously published work and has a runtime of only about 8 minutes for a 4 million node grid.