Spectral element method for the Schrodinger-Poisson system

A novel fast spectral element method (SEM) with exponential accuracy for the self-consistent solution of the Schrodinger-Poisson system has been developed for the simulation of semiconductor nanodevices. Gauss-Lobatto-Legendre polynomials were used to represent the unknown fields in the Schrodinger and Poisson equations. The model allows arbitrary potential energy and charge distributions. The predictor-corrector algorithm is applied to solve the outer loop of the self-consistent iteration. The nonlinear Poisson equation is solved by Newton´s method to increase the efficiency of the system. In this paper, the spectral element method is first applied on an infinite quantum well under an external bias to solve the Schrodinger equation. Numerical results confirm the spectral accuracy of this method. The method can achieve high accuracy with a low sampling density, thus significantly lowering the computer memory and computational time compared to conventional methods.