郭茂坤臺灣大學：應用力學研究所廖柏亭2007-11-292018-06-292007-11-292018-06-292004http://ntur.lib.ntu.edu.tw//handle/246246/62449摘 要 本研究旨在研究自聚式量子點的應變效應，並以砷化銦(鎵)/砷化鎵(In(Ga)As/GaAs)為基礎的自聚式量子點為例，分析其機械及光學特性。文中首先描述目前文獻中模擬異質材料接合問題所常見的三種不同的「等效熱應力理論」，並以理論證明此三種模擬方法，對於裸露型量子點，都將得到相同的結果。 本研究並以線彈性力學及等效熱應力理論，配合有限元素法套裝軟體，估算量子點內因異質材料間的晶格不匹配所引致的應變場分佈；同時將模擬所得之應變場，與參考文獻利用「高解析影像處理法」所量測得之量子點應變場，相互比較。本研究澄清該系列文獻對應變的定義，同時發現考慮砷化銦(鎵)量子點中的銦之濃度後，模擬的應變場與實驗結果十分吻合；亦即對於應變場模擬而言，量子點中的銦之濃度為不可忽略的因素。 最後，本研究將此應變場效應，藉由變形勢能的方式，加入薛丁格方程式中，而同樣以有限元素法予以分析，藉此評估應變效應對於導電帶、價電帶的特徵能量與電子、電洞機率密度函數分佈之影響，進而得到能帶間的躍遷能量與發光波長。Abstract This research investigates the strain effects and the optical properties of In(Ga)As/GaAs self-assembled quantum dots. There are three different models in the literatures using thermal stress theories to investigate the strain fields of heterojunction problems. In this work, it is shown analytically that these three different models lead to the identical result, at least for unburied quantum dots. A Model based on linear elasticity and thermal stress theory is then developed to analyze the strain field induced by lattice-mismatch between quantum dot and substrate. Some obtained numerical results are then compared against to the experimental data reported by others using high resolution image processing. The misinterpretation of strain in above-mentioned data is pointed out and the experimental data are then re-interpreted. It is found that the numerical results and the re-interpreted data have excellent agreement as long as the concentration of In is taken into account. Finally, the induced strain field in the quantum dot is incorporated, with the aid of the Pikus-Bir Hamiltonian and Luttinger-Kohn formalism, into the three-dimensional steady state effective mass Schrödinger equation. The solutions of the steady state Schrödinger equations are solved numerically again by using of a commercial finite element package. The energy levels as well as the wave functions of both conduction and valence bands of quantum dot are calculated. Energies and wavelengths of interband optical transitions are then obtained numerically.目錄 目錄 i 表目錄 iii 圖目錄 v 第一章 導論 1 1-1研究動機 1 1-2量子點製程 2 1-3文獻回顧 3 1-4分析架構 6 1-5內容大綱 8 第二章 彈性力學理論架構 9 2-1晶格不匹配的形成 10 2-2統御方程式與邊界條件 11 2-3量子點應變場模擬 12 2-3-1材料的組成律 12 2-3-2等效熱應力理論 15 2-3-3三種等效熱應力理論之驗證 18 2-4等效等向性材料探討 32 第三章 量子力學理論架構 34 3-1薛丁格方程式 34 3-2有效質量理論 35 3-3異質結構的位能函數與變形勢能 40 第四章 模擬結果與分析 45 4-1量子點應變場模擬 45 4-1-1砷化銦鎵量子點應變場模擬 46 4-1-2砷化銦量子點應變場模擬 48 4-2量子點應變場模擬結果 51 4-2-1砷化銦鎵量子點應變場模擬結果 51 4-2-2砷化銦量子點應變場模擬結果 52 4-3砷化銦量子點光學特性模擬 55 4-4砷化銦量子點光學特性模擬結果 57 第五章 結論與未來展望 61 5-1結論 61 5-2未來展望 62 參考文獻 63 附錄A：FEMLAB係數設定 67en-US有限元素法薛丁格方程式自聚式量子點應變strainfinite element methodSchrodinger equationself-assembled quantum dotthesis