Low-Frequency Analysis of Lossy Interconnect Structures Based on Two-Region Augmented Volume-Surface Integral Equations

Abstract

Real conductors of interconnect structures are lossy and their skin depth becomes large at low frequencies. The traditional one-region formation with the approximation of perfectly electric conductors (PECs) or surface impedances may not be valid anymore, and two-region integral equation formations are needed in the integral equation approach. Also, the electric field integral equation (EFIE) tends to break down at low frequencies, and augmented electric field integral equation (AEFIE) has been proposed to remedy the problem. In this work, we treat lossy conductors as penetrable objects and propose two-region augmented hybrid field integral equations (AHFIEs) for low-frequency analysis. The HFIEs consist of the EFIE of describing the exterior of a conductor and the magnetic field integral equation (MFIE) of describing its interior. Since the magnetic current density appears in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal{ L}}$ </tex-math></inline-formula> operator in the HFIEs, we select the magnetic charge density as a new unknown function to be solved and introduce the continuity equation of magnetic current density as an extra equation. By incorporating the volume integral equations (VIEs) of describing the substrate with arbitrary penetrable media in the interconnect structures, the two-region augmented volume-surface integral equations (AVSIEs) are formulated for entire structures. The AVSIEs are solved by the method of moments (MoM) where the Rao–Wilton–Glisson (RWG) and Schaubert–Wilton–Glisson (SWG) basis functions are used to represent the surface current densities of AHFIEs and volume current densities of VIEs, respectively, while a pulse basis function is employed to represent the charge densities of AHFIEs. Numerical examples are presented to illustrate the approach and good results have been obtained.