Elektronische Zustände in Halbleiter-Nanostrukturen und Upscaling auf semiklassische Modelle

目前，像半导体激光器这样的最先进的光电子器件的设计是基于半导体纳米结构的。由于量子电子器件的出现，半导体纳米结构的作用在未来将会增加，而量子电子器件的出现得益于制造技术的日益成熟。为了理解和预测这种器件的物理性质，需要新的理论技术来计算它们的电子态。微观尺度上的多电子薛定谔方程的从头算解由于计算困难而失败。为了克服计算负担，利用问题固有的两个尺度--纳米结构中材料变化的介观尺度和每种材料原子的微观尺度--似乎是很自然的。在这个方向上最有希望的方法是由Burt提出的，他引入了包络函数，它描述了局部高振荡微观波函数的缓慢变化的介观部分。伯特指出了一种用微观势导出包络函数方程的方法。由此得到实空间中包络函数的伪微分方程组或傅里叶变换包络函数的积分方程组。这个项目的目的是对Burt的方法、它的精确公式以及对所涉及的算子的研究和用于求解所产生的方程的高效数值方法的发展进行数学上的理解。一旦成功，物理学家和工程师将可以使用一种高效的纳米结构建模和模拟工具。

Already at present time, the design of state of the art optoelectronic devices like semiconductor lasers is based on semiconductor nanostructures. The role of semiconductor nanostructures will increase in the future due to the advent of quantum electronic devices which is assisted by the increasing sophistication of fabrication technology. To unterstand and to predict the physical properties of such devices, new theoretical techniques for the computation of their electronic states are required. The ab initio solution of the many electron Schrödinger equations on the microscopic scale fails due to its computational effort. To overcome the computational burden, it appears to be natural to utilize the presence of two scales inherent to the problem - the mesoscopic scale of the variation of the materials in the nanostructure and the microscopic scale of the atoms of each material. The most promising approach in this direction has been proposed by Burt, who introduces envelope functions, which describe the slowly varying mesoscopic part of the locally highly oscillating microscopic wave functions. Burt points out a way to derive equations for the envelope functions in terms of the microscopic potential. In the result, one arrives at pseudodifferential equations for the envelope functions in real space or integral equations for Fourier transformed envelope functions. The aim of this project is a mathematically well understood presentation of Burt's approach, its precise formulation together with an investigation of the involved operators and the development of efficient numerical methods for solving the arising equations. On success, an efficient modeling and simulation tool for nanostructures will be available for physicists and engineers.