High-Frequency Electrical Model of Through-Silicon Vias for 3-D Integrated Circuits Considering Eddy Current and Proximity Effects

A novel pi-type equivalent-circuit model of through-silicon vias (TSVs) for 3-D integrated circuits (3-D ICs) considering eddy current and proximity effects is proposed in this paper. The numerical models for the serial resistance and inductance of TSVs are derived by computing the longitudinal electrical field in Cu and the longitudinal magnetic vector potentials in SiO2 and Si substrate with Fourier-Bessel series. Furthermore, the closed-form formulas for the parallel capacitance and conductance of TSVs are derived by using the conformal mapping method. The proposed pi-type equivalent-circuit model is verified using 3-D full-wave field solver High-Frequency Structural Simulator, showing that it is highly accurate up to 100 GHz. Using the proposed model, the electrical performance of TSVs with high density can be evaluated accurately and quickly in a wideband frequency range.