Development of Numerical Methods for Semiconductor Device Simulation and Electron Microscopy

这项研究将集中于不连续问题的两种数值技术及其物理应用。将特别注意数值方法的发展及其对完善物理系统数学模型的影响。陡峭的梯度和不连续性是半导体器件模拟的特征。虽然已经开发了数值方法来处理这些特征，但这些方法需要改进和定制，以捕捉半导体器件模拟中载流子流所表现出的特征。另一方面，用于描述半导体中载流子输运的数学模型不断地被评估和改变。稳态加权基本无振荡方法将被改进并用于确定半导体器件中电流传输和沉积的宏观模型的有效性。在用电子显微镜确定蛋白质结构时，间断也是一个问题。物理结构的内在不连续性质，以及假设网格状结构中的结构重复是周期函数，导致傅立叶系数的缓慢衰减。傅里叶系数外推和Gegenbauer多项式方法将进一步发展并应用于电子显微镜领域，以获得更好的蛋白质结构分辨率。这有可能被添加到任何电子显微镜软件中作为后处理步骤，并提供更好的分辨率结构。半导体器件模拟模型的数值方法允许高效且廉价地模拟半导体器件生产中涉及的工艺。然而，这些过程有许多不连续性，需要敏感的数值方法来捕捉密度和压力的急剧变化，而不会影响它们。这种方法被称为加权基本无振荡方法，已经被开发用于类似的问题，但对于半导体模拟所需的长时间尺度来说并不有效。本项目的目的是进一步发展这些数值方法，使它们在计算机上有效地用于半导体器件的模拟。有效的数值方法还将用于比较不同的模型，这些模型试图描述物理问题，并评估哪些模型最接近现实。这个项目的另一个方面涉及用电子显微镜研究蛋白质结构中不连续的影响。最近发展了一些数学方法，通过增加一个平滑步骤来解决潜在的问题，该步骤在保持真实不连续性的同时平滑了数值伪影。这些方法从未在蛋白质结构上使用过，需要针对它量身定做。这些方法可以提高由电子显微镜确定的蛋白质结构的分辨率。

This study will focus on two numerical techniques for discontinuous problems and their physical applications. Particular attention will be paid to the development of a numerical method and its impact on refining the mathematical model of the physical system. Sharp gradients and discontinuities are characteristic of semiconductor device simulations. While numerical methods have been developed to handle these characteristics, these methods need to be refined and tailored to capture the features exhibited by carrier flow in semiconductor device simulations. On the other hand, the mathematical models that are used to describe carrier transport in semiconductors are constantly evaluated and changed. Steady-state weighted essentially non oscillatory methods will be refined and used to determine the validity of macroscopic models of current transport and deposition in semiconductor devices. Discontinuities are also a problem in the determination of protein structure by electron microscopy. The inherently discontinuous nature of physical structures, and the assumption that repetition of the structure in a gridlike formation is a periodic function leads to slow decay of the Fourier coefficients. Fourier coefficient extrapolation and Gegenbauer polynomial methods will be further developed and applied to the field of electron microscopy to achieve better resolution protein structures. This has the potential to be added to any electron microscopy software as a postprocessing step, and provide better resolution structures.Numerical methods for semiconductor device simulation models allow efficient and inexpensive simulation of the processes involved in semiconductor device production. However, these processes have many discontinuities that require sensitive numerical methods to capture the sharp changes in density and pressure without smearing them. Such methods, known as Weighted Essentially Non-Oscillatory methods, have been developed for use in similar problems, but are not efficient for the long time scales necessary for semiconductor simulations. The aim of this project is to further develop these numerical methods, and make them efficient for semiconductor device simulation on computers. Efficient numerical methods will also serve to compare different models, which attempt to describe the physical problem, and to evaluate which models best compare to reality. Another aspect of this project deals with the effects of discontinuities in protein structures studied by electron microscopy. Mathematical methods have been recently developed to solve the underlying problem by adding a smoothing step, which smoothes away numerical artifacts while keeping the real discontinuities. These methods have never been used on protein structures, and need to be tailored to it. These methods may improve the resolution of protein structures determined by electron microscopy.