Elektronische Zustände in Halbleiter-Nanostrukturen und Upscaling auf semiklassische Modelle

目前，最先进的光电器件如半导体激光器的设计已经基于半导体纳米结构。由于量子电子器件的出现，加上制造技术的日益成熟，半导体纳米结构的作用将在未来增加。为了理解和预测这些器件的物理性质，需要新的理论技术来计算它们的电子状态。许多电子Schrödinger方程在微观尺度上的从头解由于计算量大而失败。为了克服计算负担，似乎很自然地利用了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的原子的微观尺度。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中缓慢变化的介观部分。伯特指出了一种用微观势来推导包络函数方程的方法。在结果中，我们得到了实空间中包络函数的伪微分方程或傅里叶变换包络函数的积分方程。这个项目的目的是在数学上很好地理解Burt的方法，它的精确公式，以及对所涉及的算子的研究和求解产生的方程的有效数值方法的发展。如果成功，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仿真工具。

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.