利用有限差分法計算半導體量子點電子結構

Abstract

本篇論文主要是在探討如何利用有限差分法計算半導體量子點的電子結構，並應用於以下三種量子點：1. hierarchical量子點 2.droplet epitaxy量子點 3. InAs/GaAs自組式(self-assembled)量子點。在文章中利用了多能帶 理論以及波包近似法計算量子點電子結構。在使用三維均勻格點的有限差分法中，在單一維度上至少需使用70格點以上對於基態的計算才可以達到較佳收斂性。在計算上使用的電腦配備為CPU 2.27GHz與linux作業系統，要達到收斂的計算時間約15個小時，記憶體的使用大小為14GB。 在三種量子點中以hierarchical量子點的高度以及長度都是最大，其導電帶的能階量化約5meV，價電帶能階量化約1.5meV。droplet epitaxy量子點，高度與hierarchical量子點接近，但是長度略小一些。在能量上導電帶的能階量化約10meV，價電帶能階量化約3meV。InAs/GaAs自組式量子點，高度與長度都小於其他兩種量子點，所以在能階量化都比較大。導電帶能階量化約70meV，價電帶能階量化約25meV。

We present finite difference method simulation for the electronic structures of semiconductor quantum dots in the framework of multi-band k□p theory and envelope function approximation (EFA). By using the numerical techniques, the electronic structures of three kinds of quantum dots, i.e. hierarchical quantum dots[17], droplet epitaxy quantum dots[18] and InAs/GaAs self-assembled quantum dots[19] are computed. In the three-dimensional finite difference method with uniform grids, it is found that more than 70 grids in a dimension is necessary to get satisfactory convergence consequences. With the grid number, the numerical time more than 15 hours and 15GB RAM size are needed to execute a code on a machine of CPU 2.27GHz and linux O.S.. Among the three types of quantum dots under consideration, the hierarchical quantum dots have greater sizes than others with height ~7nm and length ~ 70nm. As a result, the lateral quantization of hierarchical quantum dots is about 5meV for an electron and about 1.5meV for a valence hole. For droplet epitaxy quantum dots, whose heights are close to the hierarchical quantum dots but lengths are smaller, the quantization energy are about 10meV for a conduction electron and about 3meV for a valance hole. Self-assembled quantum dots usually have the smallest sizes than others. It turns out that the quantization is about 70meV for a conduction electron and about 25meV for a valance hole confined in a self-assembled dot.