多電子系統於半導體量子點上之數值研究

Abstract

本論文分為兩部份，在第一個部分，我們利用電流自旋密度 泛函理論，對於三個垂直排列成一行的半導體量子點提出了 一個理論模型，並利用一些數值方法來探究。此量子點分子 模型是由三度空間中的硬牆侷限位能以及外部磁場所組成， 由於非拋物能帶有效質量的採用，多電子的Hamiltonian在 有限差分法下，會導出一個三次特徵值問題。而我們利用自 恰法來獲得在量子點分子中六個電子的Kohn-Sham軌域及 能量，其中薛丁格方程與卜易松方程分別由Jacobi-Davidson 方法與GMRES所解出，從數值結果中，我們可以得知原本 在無磁場下，六個電子都位於中間的量子點，但在施加適當 之磁場後，每一個量子點都有兩個電子，Förster-Dexter共振 能量轉換效應也許因此可由兩個量子點分子所形成，並提供 量子計算中費米量子位元一個新的設計概念。

在第二個部份，我們對半導體量子點中多電子的Hamiltonian 之精確對角化提出了一個新的方法，由於我們的量子點模型 是由三度空間的硬牆侷限位能以及非拋物能帶有效質量所 組成，因此一些解析基底，例如拉格爾多項式，在此類模型 中是無法使用。在本方法中，我們利用單電子波函數所組成 的Slater行列式基底來建構多電子系統之波函數，其中單電 子Hamiltonian中使用了一個與能量及位置相關的有效質量， 並加入了適當的邊界條件，利用有限差分法，同樣也得到了 一個三次特徵值問題，並且亦透過Jacobi-Davidson方法來解 ，而多電子Hamiltonian中的庫倫矩陣元素則是透過解卜易 松問題來獲得，我們的數值結果呈現只須少許的單電子基底 便可得到一個良好的收歛狀態。

This thesis consists of two parts. In the first part, based on the current spin density functional theory, a theoretical model of three vertically aligned semiconductor quantum dots is proposed and numerically studied. This quantum dot molecule (QDM) model is treated with realistic hard-wall confinement potential and external magnetic field in three-dimensional setting. Using the effective-mass approximation with band nonparabolicity, the many-body Hamiltonian results in a cubic eigenvalue problem from a finite difference discretization. A self-consistent algorithm for solving the Schrödinger-Poisson system by using the Jacobi-Davidson method and GMRES is given to illustrate the Kohn-Sham orbitals and energies of six electrons in the molecule with some magnetic fields. It is shown that the six electrons residing in the central dot at zero magnetic field can be changed to such that each dot contains two electrons with some feasible magnetic field. The Förster-Dexter resonant energy transfer may therefore be generated by two individual QDMs. This may motivate a new paradigm of Fermionic qubits for quantum computing in solid-state systems.

In the second part, we propose a new approach to the exact diagonalization of many-electron Hamiltonian in semiconductor quantum dot (QD) structures. The QD model is based on realistic 3D finite hard-wall confinement potential and nonparabolic effective mass approximation that render analytical basis functions such as Laguerre polynomials inaccessible for the numerical treatment of this kind of models. In this approach, the many-electron wave function is expanded in a basis of Slater determinants constructed from numerical wave functions of the single-electron Hamiltonian with the energy and position dependent electron effective mass approximation and suitable boundary conditions which result in a cubic eigenvalue problem from a finite difference discretization. The nonlinear eigenvalue problem is also solved by using the Jacobi-Davidson method. The Coulomb matrix elements in the many-electron Hamiltonian are obtained by solving Poisson’s problems via GMRES. Numerical results reveal that a good convergence can be achieved by means of a few single-electron basis states.