High-Frequency Modeling of TSVs for 3-D Chip Integration and Silicon Interposers Considering Skin-Effect, Dielectric Quasi-TEM and Slow-Wave Modes

TLDR

The authors model TSVs in silicon of varying resistivity from 100 MHz to 130 GHz, predicting slow‑wave, quasi‑TEM, and skin‑effect mode transitions with resistivity‑frequency charts and extending conventional per‑unit‑length expressions to analytically capture the mixed‑dielectric, frequency‑dependent behavior around the vias. The study quantifies mode‑induced signal‑integrity degradation, proposes three coaxial TSV designs to mitigate it, and demonstrates that the extended analytical expressions agree closely with EM simulations up to 130 GHz, enabling direct implementation in EDA tools.

Abstract

Through-silicon vias (TSVs) in low, medium and high resistivity silicon for 3-D chip integration and interposers are modeled and thoroughly characterized from 100 MHz to 130 GHz, considering the slow-wave, dielectric quasi-TEM and skin-effect modes. The frequency ranges of these modes and their transitions are predicted using resistivity-frequency domain charts. The impact of the modes on signal integrity is quantified, and three coaxial TSV configurations are proposed to minimize this impact. Finally, conventional expressions for calculating the per-unit-length circuit parameters of transmission lines are extended and used to analytically capture the frequency dependent behavior of TSVs, considering the impact of the mixed dielectric (silicon dioxide-silicon-silicon dioxide) around the TSVs. Excellent correlation is obtained between the analytical calculations using the extended expressions and electromagnetic field simulations up to 130 GHz. These extended expressions can be implemented directly in electronic design automation tools to facilitate performance evaluation of TSVs, prior to system design.