Simulation of Optical Properties ofGaN/AlN Quantum Dots

由於氮化鎵量子點具有介穩態(Metastable-state) 的閃鋅結構 (Zinc-Blende Structure)和穩態的烏采結構(Wurtzite Structure)再加上其 本身就擁有的壓電性質(Piezoelectric Effect)，以往『不』考慮壓電效 應下閃鋅結構的砷化銦鎵量子點(InGaAs Quantum dots)的計算已經無 法再適用。因此參考安得列夫(A. D. Andreev)和奧瑞立(E. P. O’Reilly) 於公元兩千年提出的理論。我們進行了氮化鎵量子點光學性質的理論 計算，關於計算的方法和要點簡述如下 首先由傅氏轉換求得單一量子點幾何形狀的特徵函數(Geometric Characteristic Function)，再根據傳統的彈性力學中的格林函數(Green’s Function)和特徵函數的關係經過一連串推導得到形變量和特徵函數 對於波數向量的關係。接著由求得的形變量可以推得因形變引起的壓 電位能，再將得到的壓電位勢(Piezoelectric Potential)，與形變量分別 帶入八能帶k．P 模型(Eight-Band k．P strain-dependent Hamiltonian ) 中的應變和動量交互作用矩陣與動量相關矩陣，並對兩個矩陣相加所 得到全部的漢米爾頓矩陣求其特徵值和特徵向量，將可以得到該量子 點的能階分佈，波函數和可能的光學躍遷態。而這些參數的計算對於 將來設計氮化鎵量子點的半導體元件，如量子點雷射和量子點記憶 體，甚至量子計算是非常重要的。 最後我們將由計算所得到的能態與之前的研究團隊計算或實驗 所得到的結果做詳盡的比較，除此也將討論壓電場的存在對於量子點 內量子侷限史塔克效應的影響。Due to the intrinsic built-in piezoelectric effect, the conventional calculation of InAs quantum dot without considering the piezoelectricity is not suitable. Referring to the study proposed by A. D. Andreev et al. in 2000, we perform the optical property calculation about the GaN quantum dots. First, we obtain the characteristic function of the QD shape, then applied it to obtain the strain distribution according to the classical elasticity. Second from obtained strain tensors, we can derived the piezoelectric potential and substitute it into strain-moment interaction element in the k·p matrix. Then solve the eigen-value problem to obtain the eigen energy and eigen vector which will be utilized in the calculation of the optical transition. The parameters such as eigen-energy and electron wave function are very important in the design of the GaN quantum dot device especially in the quantum dot laser, quantum memory and quantum computation. Finally we will compare the calculated eigen-energy, wave-function, and the optical properties with theoretical calculation or the experiment data from other researchers. Moreover, we will discuss how dose the existence of the piezoelectric field affect the quantum confined stark effect of the zinc-blende and wurtzite quantum dot.Content 致謝 i 摘要 ii Abstract iii Content iv List of Figures vi List of Tables viii Chapter 1 Introduction 1 Chapter 2 Theory 6 2.1 Eight-Band k•p Theory 6 2.1.1 Envelop Function Effective Mass Theory 6 2.1.2 Eight-Band k•p Model A General Description 6 2.1.3 Hamiltonian of Zinc-Blende Semiconductors 9 2.1.4 Hamiltonian of Wurtzite Semiconductor 9 2.2 Strain Calculation in QDs 10 2.2.1 Continuum Mechanical Model 12 2.2.2 Valence Force Field Model 13 2.2.3 Analytical Solution for strain in single pyramidal QDs 13 2.2.4 Fourier Transform Method Based on Green’s Function 14 2.3 Piezoelectric Effect in QDs 16 2.3.1 Piezoelectric Field in Real Space 16 2.3.2 Piezoelectric Field in Fourier Space 18 2.4 Exciton , Oscillator Strengths, and Radiative Lifetime 20 Chapter 3 Assumption and Plane Wave Expansion Method 22 3.1 Assumptions 22 3.2 Plane-Wave Expansion and the Resulting Hamiltonian 23 3.2.1 Plane-Wave Expansion Method 23 3.2.2 Hamiltonian for Wurtzite Structure 25 3.2.3 Hamiltonian for Zinc-Blende Structure 27 3.3 Material Parameters 29 3.4 Flow Chart of Programming 33 Chapter 4 Results and Discussion 34 4.1 Strain, Fitting Experiment and Wave Function in QDs 34 4.1.1 Strain Distribution of the WZ and ZB QDs 34 4.1.2 Experimental and Calculated Eigen Energy of the WZ Type Dots 36 4.1.3 Wave Function 38 4.2 Piezoelectric Effect in the ZB and WZ QDs 39 4.3 Radiative Lifetime due to Size and Piezoelectric Effect 44 4.4 Stark Effect in Single and Stacked GaN/AlN QDs 46 Chapter 5 Conclusion 50 Appendices 52 APPENDIX A: QD’s Characteristic Function of Various Geometries 52 APPENDIX B: Strain Tensors of Various Lattice Structure 56 APPENDIX C: Strain Tensors in Plane X-Z (y=0) 59 References 6