A novel method to solve analytically the non-linear Poisson equation in the inversion layer of a MOSFET

Abstract

Despite the enormous importance that the metal oxide semiconductors (MOS) and the field effect transistors (MOSFETs) have in the actual semiconductor technology, the task of finding the analytical solution of the Poisson equation in the inversion layer was not fully accomplished for more than half a century. It is a well-known fact that due to the non-linear nature of this equation, the attempts to solve it got stuck after the first integration. Nevertheless, experimental and applied researchers found ways to characterize, control, and develop MOSFET devices based on approximate models, and numerical calculations. Here I present a new method to analytically solve the nonlinear Poisson equation, in principle, for any charge distribution in the inversion layer of a MOS, and for the charge density of the original Shockley model, in particular. To this purpose, a physical argument related to the charge and field energies in the transient population-inversion process is introduced and a new nonlinear {\it but solvable} second order differential equation is obtained, whose solution also solves the original Poisson equation. The analytical results presented here allow us to derive explicit expressions for the electrical potential distribution and, very importantly, for the charge distribution, the inversion layer width and the effective impurity concentration in the depletion layer. The quantum Hall effect and other physical phenomena were discovered in the inversion layer of a MOS. The Hall effect was explained under the assumption that the charge distribution is a two-dimensional electron gas, we show here however that the 2D limit is reached only at high gate voltages. When applied to MOSFET structures, we obtain new expressions for the drain currents in the inversion and depletion layers, as functions of the impurity concentration and the gate voltage. This method can be used directly, as a building block, for multi-gate MOSFETs.