Spectral Element Method for the Schrödinger-Poisson System

A novel fast Spectral Element Method (SEM) with spectral accuracy for the self-consistent solution of the Schrödinger-Poisson system has been developed for the simulation of semiconductor nanodevices. The field variables in Schrödinger and Poisson equations are represented by high-order Gauss-Lobatto-Legendre (GLL) polynomials, and the stiffness and mass matrices of the system are obtained by GLL quadrature to achieve spectral accuracy. A diagonal mass matrix is obtained in the Schrödinger equation solver, and a regular eigenvalue solver can be used to find the eigenenergy. The predictor-corrector algorithm is applied to further improve the efficiency. The SEM allows arbitrary potential-energy and charge distributions. It can achieve high accuracy with an extremely low sampling density, thus significantly reducing the computer-memory requirements and lowering the computational time in comparison with conventional methods. Numerical results confirm the spectral accuracy and significant efficiency of this method, and indicate that the SEM is a highly efficient alternative method for semiconductor nanodevice simulation.