Electromagnetic Analysis for Inhomogeneous Interconnect and Packaging Structures Based on Volume-Surface Integral Equations

Electromagnetic analysis for interconnect and packaging structures usually relies on the solutions of surface integral equations (SIEs) in integral equation solvers. Though the SIEs are necessary for the conductors in the structures, one has to assume a homogeneity of material for each layer of a substrate if SIEs are used for the substrate. When the inhomogeneity of materials in the substrate has to be taken into account, then volume integral equations (VIEs) are indispensable. In this paper, we consider the inhomogeneous materials of substrate and replace the SIEs with the VIEs to form volume-surface integral equations (VSIEs) for the entire structures. Also, the use of VIEs could alleviate low-frequency effects, remove the need of selecting a basis function for magnetic current, and facilitate geometric discretization in some scenarios. The VSIEs are solved with the method of moments by using the Rao-Wilton-Glisson basis function to represent the surface current on the conductors and Schaubert-Wilton-Glisson basis function to expand the volume current inside the substrate. To avoid the inconvenience of charge density in the traditional implementation of the VIEs, we suggest that the dyadic Green's function be kept in its original form without moving the gradient operator onto the basis and testing functions. Numerical examples are presented to demonstrate the effectiveness of the approach.