外加磁場下多電子量子點之能量和束縛電子數目的關係

Abstract

在這個報告中，我們提出了在外加磁場下三維InAs/GaAs 量子點的addition energy，在此我們利用exact diagonalization 方法來求得多電子量子點的束縛能階隨外加磁場的變化，這方法首先是找出單電子量子點的能階和對應的波函數，在列式的過程中我們將以下三點考慮進去：(1)電子的有效質量和Lande’ factor 是隨著總能和位置改變的(2)有限的hard wall 限制位能(4)外加磁場跟磁動量的交互作 用產生的Zeeman effect。我們利用非線性遞迴方法來解決這個題。由於電子間的交互作用對多電子量子點磁變能階，在我們求出單電子量子點的能階和對應的波函數後，我們利用這些結果去計算電子間的庫倫作用力，並利用以上所有相關數據來建構多電子量子點的 Hamiltonian，然後就可以算出Addition energy 了。所有的結果在這個報告中都會呈現出來，我們發現Addition energy 跟束縛電子的 數量之關係符合Shell 結構。然而，有一點值得注意的是，當外加磁場為零時某些addition energy 的值會微微小於零，這是由於我們在計算過程中假設不同的單電子量子點的波函數之間是彼此正交的，但這只是個估計，與實際情形有微小差距，這導致我們用來創造多電子量子點Hamiltonian 的基底並非如我們假設般彼此正交，然而，當我們將這些基底正交化後，預料便能將addition energy 往上提升。

In this report, we present a study on the addition energy of interacting electrons for a realistic three-dimensional (3D) model of semiconductor nano-scale InAs/GaAs quantum dots under an external magnetic field. We supposed the interaction between electrons are totally caused by Coulomb effect. Here we use the exact diagonalization method to obtain the confined energy states of few electrons that exists in a quantum dot as a function of external magnetic fields. The first step is to find the eigenstates of a single electron. The model formulation includes (i) the energy and position dependent effective mass and Lande’ factor for electrons (ii) finite hard wall confinement potential (iii) the Zeeman effect caused by interaction between the mesoscopic angular momentum and the magnetic field. We apply the non-linear iterative method to obtain self-consistent solutions of this 3D problem. The interactionbetween electrons has important effects on the magnetic field dependence of the energy spectrum. After finding out the energy states of single electron confined in the dot and corresponding wave function dependence on an external magnetic field, we make use of these results to evaluate the Coulomb potential between electrons and construct the Hamiltonian of multi-electrons quantum dot. Then we obtain the energy states of few electrons confined in a dot and take all these data to get the addition energy for different applied magnetic field. All the results will be presented. We found that there are some values of addition energy are slightly negative when the external magnetic field is zero. However, the addition energy in a quantum dot should be always positive. This is because the states which are used to create the Hamiltonian of multi-electrons quantum dot are not diagonal to each other. We expect that the addition energy will shift up if we diagonalize the states.