Accelerating Physics-Based Electromigration Analysis via Rational Krylov Subspaces

2026-01-30

AI Summary

This study addresses the high computational cost of simulating electromigration (EM) stress evolution in nanoscale VLSI interconnects, a challenge exacerbated by the inefficiency of conventional finite difference methods in handling large-scale structures. The authors introduce, for the first time, rational Krylov subspace model order reduction into EM analysis, proposing two efficient solvers: ExtRaKrylovEM in the frequency domain and EiRaKrylovEM in the time domain. These methods integrate exponential integration with a coordinate descent optimization strategy to automatically select optimal expansion points and reduction orders. Validated on both synthetic and industrial power grids, the approach achieves sub-0.1% prediction errors for void nucleation time and resistance change using only 4–6th-order models, delivering speedups of 20–500× over state-of-the-art techniques.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Sampling/Simulation-based Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

Electromigration (EM) induced stress evolution is a major reliability challenge in nanometer-scale VLSI interconnects. Accurate EM analysis requires solving stress-governing partial differential equations over large interconnect trees, which is computationally expensive using conventional finite-difference methods. This work proposes two fast EM stress analysis techniques based on rational Krylov subspace reduction. Unlike traditional Krylov methods that expand around zero frequency, rational Krylov methods enable expansion at selected time constants, aligning directly with metrics such as nucleation and steady-state times and producing compact reduced models with minimal accuracy loss. Two complementary frameworks are developed: a frequency-domain extended rational Krylov method, ExtRaKrylovEM, and a time-domain rational Krylov exponential integration method, EiRaKrylovEM. We show that the accuracy of both methods depends strongly on the choice of expansion point, or shift time, and demonstrate that effective shift times are typically close to times of interest such as nucleation or post-void steady state. Based on this observation, a coordinate descent optimization strategy is introduced to automatically determine optimal reduction orders and shift times for both nucleation and post-void phases. Experimental results on synthesized structures and industry-scale power grids show that the proposed methods achieve orders-of-magnitude improvements in efficiency and accuracy over finite-difference solutions. Using only 4 to 6 Krylov orders, the methods achieve sub-0.1 percent error in nucleation time and resistance change predictions while delivering 20 to 500 times speedup. In contrast, standard extended Krylov methods require more than 50 orders and still incur 10 to 20 percent nucleation time error, limiting their practicality for EM-aware optimization and stochastic EM analysis.

Problem

• Electromigration
• Stress evolution
• VLSI interconnects
• Partial differential equations
• Computational efficiency

Innovation

• rational Krylov subspace
• electromigration analysis
• model order reduction
• time-constant expansion
• coordinate descent optimization