Closed-form impedance model for annular through-silicon via pairs in three-dimensional integration

Annular through-silicon via (TSV) is one of the promising solutions for vertical interconnects in three-dimensional integration, which diminishes mismatch effects of the coefficients of thermal expansion between two different materials and saves cost, as compared with other TSV structures. In this study, an analytical impedance model is proposed. Closed-form formulas for calculations of the per-unit-height resistances and inductances are derived by calculating the electromagnetic field quantities in the cylindrically multilayered media, via solving the longitudinal current densities in the conducting layers. Combining with the conventional admittance model, a compact transmission line model is obtained, which can be used to model the electrical properties of annular TSVs. These formulas appropriately capture the real current distribution because of skin effects in metal conductors as well as the eddy current loss in semiconductor regions. The results from these analytical formulas have comparable accuracy as those obtained by the full-wave solver in a wideband frequency range, yet much higher computational efficiency than existing modelling methods.