Self-consistent solution of the Poisson and Schrödinger equations in accumulated semiconductor-insulator interfaces

Abstract

A general method for the study of quantum effects in accumulation layers is presented. The Schrödinger and Poisson equations are self-consistently solved in a finite quantum box which includes the whole metal-insulator-semiconductor structure. An appropriate choice of the boundary conditions allows the achievement of box-independent results. For the first time, the electrostatical potential and quantum energy levels of an accumulated n-type semiconductor are fully self-consistently calculated without considering the electric-quantum limit approximation. Hence, being able to treat the problem even at room temperature, we report results in the whole range from liquid-helium temperature to room temperature and beyond. This has been possible because our method allows the calculation of both bound and mobile electron states and their introduction into the Poisson equation on equal footing. The effect of the penetration of the wave functions into the oxide has been determined, and it has been demonstrated that the consideration of an infinite semiconductor-insulator interface barrier leads to more serious errors than previously estimated by other authors. Having included the oxide-metal interface into the quantum box, we also propose a simple method to calculate the tunnel current which flows through the insulator. Although the contribution of many subbands has to be added up to obtain the total current, oscillations in the Fowler–Nordheim current-voltage characteristic, which are due to reflection resonances at the insulator-anode interface, are clearly observed. Initially conceived for the accumulation layer problem, the presented method is obviously valid for treating inversion layers as well.