Investigation on the Frequency Range of Validity of Electroquasistatic RC Models for Semiconductor Substrate Coupling Modeling

Abstract

Electroquasistatic analysis is currently the dominant approach for the modeling of semiconductor substrate noise coupling. The electroquasistatic (EQS) approximation is considered acceptable for frequencies such that the distances over which substrate interactions are considered are a small fraction of the wavelength. Yet, with clock bandwidths in state-of-the-art and future designs extending to multiple tens of gigahertz, it is necessary to accurately investigate and quantify the range of validity of the EQS approximation. This investigation is carried out in this paper by means of a rigorous electrodynamic model for the induced surface voltage due to an elementary dipole source in various types of commonly used semiconductor substrates for digital and radio-frequency integrated circuits. In addition to enabling a quantitative assessment of the frequency range of the validity of the electroquasistatic approximation and, hence, the RC models used commonly in computer-aided design tools for substrate noise coupling, the proposed electrodynamic model is used to investigate the prominence of inductive-like characteristics in noise propagation through the substrate. Through these investigations, it is demonstrated that the electrodynamic model used in this paper provides for a unified rigorous electromagnetic analysis of substrate noise coupling over the entire frequency bandwidth of interests to practical applications, from DC to multiple tens of gigahertz.