Numerical Methods for Wave Propagations in Inhomogeneous Media

在这项提案中，PI将发展数值方法及其数学分析，最终将其应用于研究纳米电子学中的波动现象，量子点的耦合阵列以及光刻中的相移掩模。经典电磁波和量子波的传播在这些物理和工程系统中起着关键作用。为了定量地了解这些系统中的波动现象，需要采用适当设计的数值算法进行精确、高效的数值模拟。针对这一目标，我们提出了以下三种数值方法：[1]纳米电子输运Wigner方程的自适应保守细胞平均谱法；层状介质中三维量子点量子波散射的快速积分解算器。基于本征振荡的平行谱元法求解复杂亥姆霍兹方程。这项研究的潜在技术影响是了解所涉及的物理学，并为纳米电子学（如纳米mosfet，相移掩模和量子点）提供设计指南。本研究中开发的数值方法将用于量子器件的工程设计，对保持美国在新型VLSI微芯片开发中的技术优势以及微芯片制造中的下一代x射线光刻技术具有重大影响。此外，在这个项目中训练的研究生将为竞争激烈的高科技就业市场提供熟练的劳动力以及潜在的学术研究人员。

In this proposal, the PI will develop numerical methods and their mathematical analysis, ultimately their implementations in studying wave phenomena in nano-electronics, coupled arrays of quantum dots, and phase shift masks in lithography. Propagation of classical electromagnetic and quantum waves plays a key role in these physical and engineering systems. In order to gain a quantitative understanding of the wave phenomena in those systems, accurate and efficient numerical simulations are needed with appropriately designed numerical algorithms. The targeted applications motivate our research with the following three proposed numerical methods: [1] An adaptive conservative cell average spectral method for Wigner equations in electron transport of nano-electronics; [2] A fast integral solver for quantum wave scattering in 3-D quantum dots in layered media [3] A parallel spectral element method based on eigen-oscillations for complex Helmholtz equations. The potential technology impact of this research is to understand the physics involved and provide design guidelines for nano-electronics such as nano-MOSFETs, phase shift masks, and quantum dots. The numerical methods developed in this research will be used for the engineering design of quantum devices with significant impact on maintaining US technology preeminence in the development of new VLSI microchips, and next generation X-ray lithography in microchip manufacturing. Also, graduate students trained in this project will provide skilled workforce in the competitive high technology job market as well as potential academic researchers.