Frequency-Domain Transient Electromigration Analysis Using Circuit Theory

The analysis of transient stress buildup in on-chip interconnects due to electromigration (EM) requires the solution of partial differential equations (PDEs) with appropriate boundary conditions, but prior approaches have been computationally expensive. This paper uses a stress-electrical equivalence to map the solution of the system of PDEs for a general multisegment interconnect to an RC network. For tree structures, this system is solved in linear time in the frequency domain using model order reduction (MOR) techniques. We present two MOR approaches: one that is not guaranteed to provide a stable approximant due to the presence of the mass-conservation equation, but empirically does so for a large fraction of testcases; and another that is guaranteed-stable. To achieve a guaranteed-stable solution, the approach approximates the RC circuit in a Krylov space and captures the impact of mass conservation in the form of a mass conservation excitation. However, the latter is observed to be slightly less accurate than the first approach when it does provide a solution. The method demonstrates excellent accuracy against a commercial numerical solver, and is scalable, solving transient EM analysis problems on large power grid interconnect benchmarks.