A Unified Finite-Element Solution From Zero Frequency to Microwave Frequencies for Full-Wave Modeling of Large-Scale Three-Dimensional On-Chip Interconnect Structures

It has been observed that a full-wave finite-element-based solution breaks down at low frequencies. This hinders its application to on-chip problems in which broadband modeling from direct current to microwave frequencies is required. Although a static formulation and a full-wave formulation can be stitched together to solve this problem, it is cumbersome to implement both static and full-wave solvers and make transitions between these two when necessary. In this work, a unified finite-element solution from zero frequency to microwave frequencies is developed for full-wave modeling of large-scale three-dimensional on-chip interconnect structures. In this solution, a single full-wave formulation is used. No switching to a static formulation is needed at low frequencies. This is achieved by first identifying the reason why a full-wave eigenvalue-based solution breaks down at low frequencies, and then developing an approach to eliminate the reason. The low frequency breakdown problem was found to be attributed to the discrepant frequency dependence of the real part and the imaginary part of the eigenvalues, which leads to an ill-conditioned eigenvalue system at low frequencies. The discrepant frequency dependence of the real part and the imaginary part is further attributed to the different scaling of the transverse and longitudinal fields with respect to frequency in a transmission-line type structure. By extracting transverse and longitudinal fields separately in the framework of a full-wave formulation, we avoid the numerical difficulty of solving an ill-conditioned eigen-system at low frequencies. The validity of the proposed approach is demonstrated by numerical and experimental results.