A rigorous Fermi-Dirac statistics-based MOSFET channel surface potential equation using polylogarithms

Abstract

The MOSFET channel Surface Potential Equation (SPE), being the fundamental building block of all Surface Potential (SP)-based MOSFET compact models, must be as physically faithful as possible. To that end, in this article we propose a novel alternative form of the SPE, which is valid under many circumstances and operating conditions (e.g. degenerate doping, incomplete dopant ionization, strong carrier inversion and accumulation, cryogenic operation) where the occupation probabilities of charges cannot be well described by the Maxwell-Boltzmann (MB) statistics approximation. Thus, to more accurately describe free charge carrier densities they are defined here by their corresponding Fermi-Dirac Integrals (FDI). But, unlike previous formulations of the SPE directly expressed in terms of FDIs or some analytic approximation thereof, in our formulation we substitute from the beginning the FDIs by their equivalent polylogarithm representations, before integrating Poisson’s equation to obtain the proposed SPE. Since polylogarithms are functions that may be analytically differentiated and integrated, and because there already exist well established algorithms and software routines for their efficient numerical computation, the proposed polylogarithmic form of the MOSFET channel SPE constitutes a more accurate alternative to existing SPE approximate forms.