論 文 摘 要 本篇論文主要目的是利用等效質量理論和瑞利里茲變分法來解量子點的能階與波函數，而所得到的結果可以讓我們初步分析量子點元件的特性以及給予實驗上在製作前的參考和特性預測。 首先我們先利用量子井解析解研究量子井的電性與光性，包括在均勻電場下電子的穿隧速率、動態行為分析以及次能帶的吸收光譜探討，藉由這些探討，可以對能階與波函數的物理以及影響它們的因素更加的了解，這將有助於我們對量子點的研究。 等效質量理論在計算上不但簡易迅速，且要加入新的場作用與新的結構也是很容易的。我們先將等效質量理論和瑞利里茲變分法應用在一維量子井中，計算的結果與解析解比較可證明此數值方法的可靠性，之後將其推廣應用到三維的量子點中。 由於量子點的形狀會影響到電子結構，所以我們會探討不同形狀量子點的能階與波函數分佈，並考慮外加電場的影響，接著探討垂直耦合量子點系統，包括其鍵結與反鍵結能階和波函數，以及影響耦合強度的因素。 最後將理論應用在分析量子點雷射上。這邊分為三個部份來探討。第一部份，由於量子點的均勻性對雷射特性有很大的影響，所以我們引入高斯分佈函數來模擬量子點大小的分佈，並利用費米黃金規則來計算量子點雷射的發光光譜與雷射增益。第二部份探討不同形狀的應力緩衝層對於能階的影響。第三部份，由於加入銻元素會縮小應力緩衝層的能隙，當銻成份達到某一比例時會使得砷化鎵與應力緩衝層之間的能帶形成Type II的結構，我們將探討這對量子點雷射發光波 長的影響。 Abstract The main goal of this dissertation is to calculate the energy levels and wave functions of quantum dots using the Raleigh-Ritz variation method in the framework of effective mass method. Based on the calculated energy levels and wave functions, the optical and transport properties of devices made from individual QDs can be fully analyzed. To clarify the Raleigh-Ritz variation method, we use a single quantum well system as an example due to one-dimensional Schrödinger equation with an exact solution. The optical and transport properties of quantum wells such as tunneling rate, emission and absorption spectrum and dynamic property are investigated. The results obtained using the Raleigh-Ritz variation method are in a good agreement with those obtained by the exact solution. Subsequently, the Raleigh-Ritz variation method is extended into a quantum dot system. The energy levels and wave functions of pyramidal, conical and disk QDs under a uniform electric field are calculated. In addition, we also calculate the energy levels of double quantum dots. The coupling strength between two quantum dots is readily obtained. In the framework of effective mass method, the energy levels of QDs are mainly attributed to volume effect rather than shape effect. Finally, the optical properties of quantum dot laser are theoretically studied. We have employed the Fermi golden rule to calculate the emission spectrum and modal gain of quantum dot laser in which dot size fluctuations are included using the Gaussian distribution function. We found that effects of the different shape of strain reducing layers significantly influence the emission spectrum. Due to the type-II band alignment between InAs quantum dots and antimony strain reducing layers, how the type-II structure of band alignment influences the emission spectrum of QDs is discussed.
