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.

I. Introduction

At Nanoscale, where electronic wave properties are significant, the analysis of electron transport in nanodevices is intricately linked to solving the Schrödinger equation. This equation is particularly crucial for understanding key phenomena such as ballistic transport, the interference effect, and the tunneling effect. Meanwhile, considering the wide frequency range of nanodevices, simulating of the 3-D time-dependent Schrödinger equation is indispensable [1]. Nanoscale devices such as carbon nanotubes, quantum interference transistors, and metal oxide semiconductor field effect transistors (MOSFETs), can be investigated through the Schrödinger equation to obtain their transient behavior [2], [3], [4].