Electrothermal Modeling for 3-D Nanoscale Circuit Substrates: Noise

Abstract

We apprehend, here, some insights into 3-D integration modeling, from a physical point of view, concerning electrical and temperature domains in transient regimes, applied to multilayered silicon substrates, as well as an insight into noise study. This paper is motivated by the need to develop more general analytical coupled solutions in these fields, manageable by the industry. It is also a track for a synthesis beginning. First of all, we calculate the spreading impedance. For this, our approach is twofold: compact Green kernels or a transmission line matrix method, with contacts over or into the multilayered substrate, is derived by solving Poisson’s equation analytically. All the resolution of the equation system is done in the reciprocal domain, extracting real value only at the end of the calculation process. It permits to extract 3-D impedances between any two embedded contacts of any shape, real or virtual; this is original, to the best of our knowledge. Indeed, we calculate 3-D time-dependent temperature maps, from heat equation, from any contact into the bulk. We propose not only to couple the two master equations, Poisson and heat, but to show that our methodology, applied to these two equations, can also be addressed to noise propagation. We investigate our models on both analytical and numerical methods. These extended models should enable extracting substrate impedances and parasitic elements, in 3-D.