Accuracy-improved coupling capacitance model for through-silicon via (TSV) arrays using dimensional analysis

Abstract

In this paper, we show that using the relation between the inductance matrix and the capacitance matrix in a homogeneous medium to extract the coupling capacitance in a through silicon via (TSV) array is inaccurate. This is because this relation assumes a lossless, homogeneous surrounding medium. We show that this model can cause an error up to 70% in coupling capacitance compared to Q3D extractor simulations. Instead of using high accuracy, time consuming numerical electromagnetic techniques, we suggest a correction methodology for the inverse inductance model so that it can account for the non-homogeneous nature of the TSVs surrounding medium and the lossy nature of the silicon substrate. Dimensional analysis is used to understand the correction function dependencies on TSV dimensions and to reduce the number of independent variables needed for regression analysis. Once the independent variables are reduced, multiple regression techniques are used to estimate the correction function for TSV arrays. Q3D extractor simulations are used to show that the corrected model reduces the coupling capacitance error significantly. Then, the correction function behavior vs. frequency is discussed.