Accelerating Physics-Based Electromigration Analysis via Rational Krylov Subspaces

TL;DR

Electromigration reliability analysis in nanometer-scale interconnects requires solving Korhonen's EM PDEs on large interconnect trees, which is computationally demanding with finite-difference methods. The authors propose two rational Krylov-based frameworks: ExtRaKrylovEM, a frequency-domain reduction, and EiRaKrylovEM, a time-domain exponential-integration approach, both guided by shift times tied to critical dynamics such as nucleation $t_{\mathrm{nuc}}$ and post-void steady state. A gradient-descent–style coordinate-descent optimization automatically selects reduction orders and shift times to minimize nucleation-time and resistance-change errors, with residual estimators aiding accuracy control. Experimental results on synthetic and industry-scale grids show 20–500× speedups with sub-0.1% errors using only 4–6 Krylov orders, while conventional extended Krylov methods require 50+ orders and incur larger nucleation-time errors, enabling scalable EM-aware optimization and stochastic analysis.

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.

Figures (5)

• Figure 1: The nucleation times at different reduction order using standard Krylov subspace method for a 100 segment wire (critical stress is $10^8$ Pa)
• Figure 2: The stress evolution in 4 segments of 5-segment wire including nucleation and post-void phases
• Figure 3: The calculated nucleation time vs shift time used (critical stress is $10^8$ Pa)
• Figure 4: Comparison between our proposed ExtRaKrylovEM and EiRaKrylovEM with Back Euler based FDM under different analysis time steps for 500 segment wire.
• Figure 5: Accuracy comparison with the finite difference method SunYu:TDMR'20 at different times. Red dot indicates the nucleated cathode node