534 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 21, NO. 5, MAY 2002 Electromagnetic Interconnects and Passives Modeling: Software Implementation Issues Wim Schoenmaker, Member, IEEE, and Peter Meuris Abstract—This is the second paper in a series on the simulation of on-chip high-frequency effects. A computer-aided approach in three dimensions is advocated, describing high-frequency effects such as current redistribution due to the skin-effect or eddy cur- rents and the occurrence of slow-wave modes. The electromagnetic environment is described by an electric scalar potential and a mag- netic vector potential as well as a ghost field. The latter one guar- antees a stable numerical implementation. This paper deals with the software implementation, the treatment of interfaces and do- main boundaries, scaling considerations, numbering schemes, and solver requirements. Some illustrative examples are shown. Index Terms—AC analysis, high frequency, interconnects, TCAD. I. INTRODUCTION I N A preceding paper [1], we presented an approach to ex- tract local information on the electromagnetic properties of passive structures on a chip using a three-dimensional field so- lution method. The basic idea is that critical regions (crossings, widenings, windings, turns) can be studied in detail and that the resulting parameters as functions of the frequency can be loaded in a SPICE net list. The detailed interconnect or passive element modeling directly relies upon the Maxwell equations that de- scribe the temporal and spatial evolution of the electromagnetic fields in media (1) (2) (3) (4) where , and denote the electrical induction, the electric field, the magnetic induction, the magnetic field, the current density, and the charge density. The following constitu- tive equations relate the inductances to the field strengths (5) The permittivity, , the permeability, , and the constitutive equation that relates the current to the electric field as well as Manuscript received February 2, 2001; revised September 26, 2001 and De- cember 15, 2001. This work was supported in part by the Flemish Institute of Science and Technology (IWT). This paper was recommended by Associate Ed- itor Z. Yu. The authors are with IMEC, B-3001 Heverlee, Belgium (e-mail: schoen@imec.be; meurisp@imec.be). Publisher Item Identifier S 0278-0070(02)02844-0. the carrier densities are determined by the medium under con- sideration. For a conductor the current is given by Ohm’s law (6) where the current density satisfies the current continuity equa- tion (7) We consider a dielectric medium whose lossy effects can be ne- glected. Therefore, no current continuity equations need to be solved in the dielectric materials. In the semiconducting regions, the current consists of negatively and positively charged car- rier currents obeying the current continuity equations (8) (9) In here, the charge and current densities are (10) (11) (12) and is the generation/recombination rate of charge car- riers. The current continuity equations provide the solution of the variables and . Note that the permittivity in (5) is real, whereas, for the applications envisaged, we may safely assume that the structure is nonmagnetic, i.e., . Whereas in [1] we have described the motivation for the in- troduction of a ghost field in order to obtain a stable solu- tion method for the discretized equations (see next section) and the mapping of the differential equations onto a discrete grid, we will show in this paper that the method performs well by showing detailed calculations on a series of specific structures. For that purpose it was required to develop a TCAD software en- vironment such that arbitrary Manhattan-like structures can be loaded (preprocessing), calculated, and that the results can be visualized (postprocessing). The development of the equation solver demands that a number of issues are addressed such as: 1) the solution strategy; 2) the boundary conditions; 3) the in- terface conditions; 4) scaling considerations; 5) node- and link numbering schemes; and 6) matrix solver requirements. Each of these problems is addressed in the subsequent sections. The aim of this paper is to provide the reader with the necessary in- formation on the solution method for solving the scalar poten- tial–vector potential–ghost field system. The small-signal and 0278-0070/02$17.00 © 2002 IEEE