載子雜質在柱形量子點下隨著穩定電場與磁場的影響

Abstract

由於半導體製程的進步, 電子在材料內的量子行為越來越重要. 當電子在材料內受到一維的束縛時, 我們稱之為量子井系統. 同理, 量子線及量子點系統內, 電子受到二維/三維束縛所產生的效應, 亦極為重要. 此時古典的方式無法完善地描述電子的行為而必須用量子力學來解釋. 本文主要描述雜質在量子點下隨著電磁場變化的行為. 電子在量子點內所受到的束縛位能跟距離的平方成正比的, 可用簡諧位能來代表. 如果是空間各向同性的話, 我們便稱為球形點; 如果位能在平面方向跟垂直方向是各不相同的話, 我們便稱之為柱形點. 我們主要是想討論柱形點在外加電磁場下的情形. 另外半導體都會牽扯到雜質的問題. 因為背景材料的影響, 在有效質量近似下, 一個雜質系統相當於電子環繞一正電荷, 相當於似氫原子系統加上邊界. 由於整個系統無法得到解析解, 故我們用數值分析的方式來模擬此系統在外場變化下的情形. 我們採用 Legendre-Guass-Lobatto 方法. 因為這種方法對於庫侖問題特別有效. 最後的結論中我們將結果與微擾方式算能階來做比較, 並畫出基態以及激發態能量的分佈, 以期望對實驗上測量光譜有直接上的幫助.

The progress of semiconductor technology fabricates devices in the nano-scales successfully. The scale is close to the atomic size that the quantum behaviors of electrons are very important. Scales, temperatures, external field now have subtle influences on the system. And we realized that investigations often can only be explained with quantum mechanics. A semiconductor quantum dot is a structure of three dimensinal confinements. In this article we will discuss the behaviors of an impurity in the vertical quantum dot with static electromagnetic field. The concentration of dopants allows us to simplify the many-body problem to be tractable. Since the effective mass approximation remains valid for band structures of GaAs with shallow donors, the problem can be solved as a hydrogenic atom confined in a well-described harmonic potential. The Schrodinger equation of the system can not be solved analytically, therefore efficient numerical simulations are important. We use the Gauss-Legendre-Lobatto pseudospectral method to solve the problem. The well-located grid points and well-behaved wave functions on the boundaries greatly improve the realiability and accuracy of our results. Finally, our results are compared with published perturbation treatment. We also present the ground- and excited state energy levels. The outcomes would be helpful on optical spectral measurements.