Improved transmission line model for high-frequency modelling of through silicon vias

Accurate and reliable models can support Through Silicon Via (TSV) testing methods and improve the quality of 3D ICs. A model for expressing resistance and inductance of TSVs at frequencies up to 50 GHz is proposed. It is based on the two-parallel transmission cylindrical wires model, known also as the Transmission Line Model and improved through the fitting to ANSYS Q3D simulation results. The proximity effect between neighbouring TSVs that alters the paths through which current flows is empowered at high frequencies. The consideration of the dependence of the proximity effect on frequency for calculating TSV resistance and inductance is the main contribution of this work. Additionally, the modelling of resistance is extended to accurately correspond to a TSV in an array. The proposed models are in good agreement with the simulator results with an average error below 2% and 5.4% for the resistance and the inductance, respectively. The maximum error is 3% and 9.1%, respectively. In the case of the resistance of a TSV in an array, the maximum error is 4.7%. As long as the coefficients of the proposed equations have been extracted, the time for resistance and inductance calculation based on the presented models is negligible, compared to the time-consuming EM simulation.