Linn Tobias, Bittner Kai, Brachtendorf Hans Georg, Jungemann Christoph
Institute of Electromagnetic Theory, RWTH Aachen University, Kackertstr. 15-17, 52072 Aachen, Germany.
University of Applied Sciences of Upper Austria, 4232 Hagenberg, Austria.
J Sci Comput. 2020;85(1):6. doi: 10.1007/s10915-020-01311-z. Epub 2020 Sep 22.
Instabilities of electron plasma waves in high-mobility semiconductor devices have recently attracted a lot of attention as a possible candidate for closing the THz gap. Conventional moments-based transport models usually neglect time derivatives in the constitutive equations for vectorial quantities, resulting in parabolic systems of partial differential equations (PDE). To describe plasma waves however, such time derivatives need to be included, resulting in hyperbolic rather than parabolic systems of PDEs; thus the fundamental nature of these equation systems is changed completely. Additional nonlinear terms render the existing numerical stabilization methods for semiconductor simulation practically useless. On the other hand there are plenty of numerical methods for hyperbolic systems of PDEs in the form of conservation laws. Standard numerical schemes for conservation laws, however, are often either incapable of correctly handling the large source terms present in semiconductor devices due to built-in electric fields, or rely heavily on variable transformations which are specific to the equation system at hand (e.g. the shallow water equations), and can not be generalized easily to different equations. In this paper we develop a novel well-balanced numerical scheme for hyperbolic systems of PDEs with source terms and apply it to a simple yet non-linear electron transport model.
高迁移率半导体器件中电子等离子体波的不稳定性最近作为填补太赫兹间隙的一种可能候选方案引起了广泛关注。传统的基于矩的输运模型通常在矢量量的本构方程中忽略时间导数，从而导致偏微分方程（PDE）的抛物型系统。然而，为了描述等离子体波，需要包含这些时间导数，这就导致了PDE的双曲型系统而非抛物型系统；因此，这些方程组的基本性质完全改变了。额外的非线性项使得现有的用于半导体模拟的数值稳定方法实际上毫无用处。另一方面，对于以守恒律形式存在的双曲型PDE系统有大量的数值方法。然而，守恒律的标准数值格式通常要么无法正确处理由于内置电场而在半导体器件中存在的大源项，要么严重依赖于特定于手头方程组（例如浅水方程）的变量变换，并且不容易推广到不同的方程。在本文中，我们为带有源项的双曲型PDE系统开发了一种新颖的保持平衡的数值格式，并将其应用于一个简单但非线性的电子输运模型。
