A 3-D Spectral Element Time-Domain Method With Perfectly Matched Layers for Transient Schrödinger Equation

A spectral element time-domain (SETD) method with perfectly matched layers (PML) is proposed to simulate the behavior of electron waves, interference effects and tunneling effects, in three-dimensional (3-D) devices by solving Schrödinger equation. The proposed method employs Gauss-Lobatto-Legendre (GLL) polynomials to represent the wave function. Easy construction of higher-order element makes refinement straightforward and spectral accuracy can be obtained from the SETD. Meanwhile, by utilizing the GLL quadrature, a diagonal mass matrix is obtained which is meaningful in the time-stepping process. Numerical experiments confirm that, for open boundary problems, employing PML yields results characterized by high numerical efficiency, remarkable flexibility and ease of implementation. These findings underscore the effectiveness of SETD-PML in addressing the challenges posed by open boundary conditions, making it a reliable choice for numerical simulations. Some illustrative numerical examples are presented to demonstrate the performance of the proposed method.