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研究生:張偉庭
研究生(外文):Wei-Ting Chang
論文名稱:矽、鍺不同晶格面的熱傳導之分子動力學模擬
論文名稱(外文):Thermal Conductivities of Si and Ge at Different Index Planes by Molecular Dynamics Simulation
指導教授:楊照彥
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:應用力學研究所
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2008
畢業學年度:96
語文別:中文
論文頁數:66
中文關鍵詞:分子動力學漢彌爾頓方程式Tersoff位勢能函數自我相關函數Green-Kubo關係式
外文關鍵詞:molecular dynamics simulationHamiltonian equations of motionTersoff potentialautocorrelation functionGreen-Kubo formalism
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近年來在奈米碳管,奈米線,和微奈米尺度的超合金等問題上有廣泛被加以討論及模擬研究,本論文主要是在模擬研究奈米尺度下四價元素矽,鍺在不同晶格面的熱傳導量之比較,而所用方法為分子動力學法,其所運用到的運動方程式為古典力學中的牛頓力學,和漢彌爾頓方程式。分子初始位置是由鑽石結構單位晶包週期性排列,以有限的塊材來表示無限邊界的晶格排列,分子初始速度是由統計力學中麥克斯威爾分佈來加以給定。因為四價元素的鍵結主要為共價鍵屬於三體勢能且需要考慮到分子間的鍵角角度問題,所以選用Tersoff位勢能函數可以來計算共價鍵元素之間作用力和分子加速度,再由Velocity-Verlet積分方法可以得到每個分子新的位置和新的速度,因此在平衡狀態之下求得x,y,z三個方向的熱通量,經過不同時間的熱通量再來計算自我相關函數,最後再由Green-Kubo關係式得到熱傳導量。
In recent years, many researches on nanotube, nanowire, and superlattices have been carried out. In this thesis, we study the thermal conductivities of silicon and germanium at different index planes in the nanoscale. The simulation method used molecular dynamics simulation, and the governing equation is Hamiltonian equations of motion in classical Newtonian mechanics. The molecular initial position of a diamond unit cell structure is used for periodic boundary condition and expresses the infinite boundary crystal lattice arrangement by the limited bulk material. The molecular initial velocity is determined by Maxwell-Boltzmann distribution. Because silicon and germanium mainly belongs to three body potential for the covalent bonding and also the needs to consider the intermolecular bonding angle, we choose Tersoff potential to calculate covalent bonding intermolecular force and molecular acceleration. We obtain each molecular new position and the new velocity by the Velocity-Verlet integration method, and obtain the heat flux under the state of equilibrium. Through the different time evolution, we calculate the autocorrelation function from heat flux. Finally, we calculate the thermal conductivity by Green-Kubo formalism. We compare the thermal conductivities of silicon and germanium at different index planes.
目 錄

摘要………………………………………………………………………… Ⅰ
Abstract ……………………………………………………………………… Ⅱ
誌 謝………………………………………………………………………… Ⅲ
目 錄………………………………………………………………………… Ⅳ

第 一 章 概 論

1.1 研究背景………………………………………………………… 1
1.2 研究動機………………………………………………………… 2
1.3 文獻回顧………………………………………………………… 3

第 二 章 分 子 動 力 學 模 擬 法

2.1 分子動力學理論簡介………………………………………………5
2.2 初始條件設定………………………………………………………8
2.2.1 初始位置……………………………………………………8
2.2.2 初始速度……………………………………………………9
2.3 分子間位勢能的選擇……………………………………………10
2.3.2.1 Tersoff位勢能…………………………………………11
2.3.2.2 Tersoff位勢能的微分形式……………………………13
2.4 積分方法…………………………………………………………19
2.5 週期邊界條件……………………………………………………22
2.5.1最小映射法則……………………………………………22
2.6 原子間交互作用力計算……………………………………………24
2.7 無因次參數………………………………………………………26
2.8 溫度調節…………………………………………………………27

第 三 章 計算熱傳導量

3.1 簡介………………………………………………………………28
3.2 Green-Kubo關係式………………………………………………29
3.3 多體系統微觀熱流動表示式……………………………………30
3.4 自(我)相關函數推導到Green-Kubo關係式……………………33

第 四 章 結論與展望

4.1 矽、鍺晶體模擬結果………………………………………………35
4.1.1矽晶體研究方法……………………………………………35
4.1.2鍺晶體研究方法……………………………………………37
4.2 結論………………………………………………………………39
4.3 未來展望…………………………………………………………41
參考文獻………………………………………………………………………42
圖片與表格……………………………………………………………………45

表目錄

表1-1命定法與機率法方程式間的關係…………………………………………45
表3-1 Tersoff位勢能參數表…………………………………………………… 47
表3-3 Tersoff微分的關係表…………………………………………………… 48
表3-8無因次表格………………………………………………………………… 50
表4-15 矽晶體的溫度與熱傳導係數表格……………………………………… 65
表4-17 鍺晶體的溫度與熱傳導係數表格……………………………………… 66

圖目錄

圖2-1晶格常數晶格圖…………………………………………………… 46
圖2-2 不同晶格面的晶格圖………………………………………………………46
圖3-2 三體位勢能關係圖 ……………………………………………………47
圖3-4最小映射法則 ………………………………………………… 49
圖3-5最小映射法則 ……………………………………………… 49
圖3-6最小映射法則 …………………………………………………49
圖3-7鄰近儲列法列表圖…………………………………………………………50
4-1a溫度1000K晶格面(100)的normalized heat current autocorrelation…51
4-1b溫度1000K晶格面(111)的normalized heat current autocorrelation…51
4-2a溫度700K晶格面(100)的normalized heat current autocorrelation…52
4-2b溫度700K晶格面(111)的normalized heat current autocorrelation…52
4-3a溫度500K晶格面(100)的normalized heat current autocorrelation…53
4-3b溫度500K晶格面(111)的normalized heat current autocorrelation…53
4-4a溫度1000K晶格面(100)的normalized heat current autocorrelation…54
4-4b溫度1000K晶格面(111)的normalized heat current autocorrelation…54
4-5a溫度900K晶格面(100)的normalized heat current autocorrelation…55
4-5b溫度900K晶格面(111)的normalized heat current autocorrelation…55
4-6a溫度800K晶格面(100)的normalized heat current autocorrelation…56
4-6b溫度800K晶格面(111)的normalized heat current autocorrelation…56
4-7a溫度700K晶格面(100)的normalized heat current autocorrelation…57
4-7b溫度700K晶格面(111)的normalized heat current autocorrelation…57
4-8a溫度600K晶格面(100)的normalized heat current autocorrelation…58
4-8b溫度600K晶格面(111)的normalized heat current autocorrelation…58
4-9a溫度500K晶格面(100)的normalized heat current autocorrelation…59
4-9b溫度500K晶格面(111)的normalized heat current autocorrelation…59
4-10a溫度400K晶格面(100)的normalized heat current autocorrelation…60
4-10b溫度400K晶格面(111)的normalized heat current autocorrelation…60
4-11a溫度300K晶格面(100)的normalized heat current autocorrelation…61
4-11b溫度300K晶格面(111)的normalized heat current autocorrelation…61
4-12a溫度200K晶格面(100)的normalized heat current autocorrelation…62
4-12b溫度200K晶格面(111)的normalized heat current autocorrelation…62
4-13a溫度100K晶格面(100)的normalized heat current autocorrelation…63
4-13b溫度100K晶格面(111)的normalized heat current autocorrelation…63
圖4-14 NVE模擬下的動能、位勢能及總動能…………………………………… 64
圖4-16 矽晶體的溫度與熱傳導係數關係圖…………………………………… 65
圖4-18 鍺晶體的溫度與熱傳導關係圖………………………………………… 66
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