Analytical Expressions for High-Frequency VLSI Interconnect Impedance Extraction in the Presence of a Multilayer Conductive Substrate

We propose an efficient method to accurately compute the frequency-dependent impedance of VLSI interconnects in the presence of multilayer conductive substrates. The resulting accuracy (errors less than 3%) and CPU time reduction (more than an order of magnitude) emerge from three different ingredients: a 2-D Green's function approach with the correct quasi-static limit, a modified discrete complex images approximation to the Green's function, and a continuous dipole expansion to evaluate the magnetic vector potential at the short distances that are relevant to VLSI interconnects. This approach permits the evaluation of the self-impedance and mutual-impedance of multi-conductor current loops, including substrate effects, in terms of easily computable analytical expressions that involve their relative separations and the electromagnetic parameters of the multilayer substrate.