本論文使用Big-Bang方法結合Basin-hopping方法，搭配Stillinger-Weber勢能函數預測極細鍺奈米線結構，並使用密度泛函理論驗證鍺奈米線的穩定性。得到穩定的奈米線結構後，在利用量子力學與分子力學計算探討其各項性質，例如: 熱性質、機械性質與電子性質等。除了鍺奈米線本身的物理性質之外，也探討鋰離子在鍺奈米線上的吸附與擴散行為。在熱性質方面，我們計算了鍺奈米線的聲子態密度 (Density of state, DOS) 圖譜並與塊材做比較。透過分子動力學計算預測了四條奈米線的熔點，有三條鍺奈米線 (helix, pentagon, hexagon) 的預估熔點大約是385~450 K，其中7-1鍺奈米熔點為230 K，表示此奈米線無法在室溫下穩定。由於7-1奈米線的熱穩定較差，在探討機械性質以及鋰離子吸附與擴散行為時，就不將此線列入考慮。在機械性質方面，極細鍺奈米線的降伏應力與楊氏模數皆大於塊材鍺。而且也發現溫度對於機械性質會有影響，但是溫度越高影響力就越小，其中機械性質最佳者為Pentagon奈米線。電子性質方面，藉由分析鍺奈米線的電子態密度，可以瞭解奈米線的軌域分佈情形。結果顯示，由於奈米線的特殊結構導致庫倫遮罩效應較少，而且又結合量子效應，最後奈米線的能帶結構橫跨費米能級變成導體。而且軌域上的電子受到較大的庫倫作用力，所以有嚴重的能帶分裂現象。在鋰離子於鍺奈米線上的吸附與擴散行為方面，發現極細鍺奈米線在吸附鋰離子時能比吸附在石墨或石墨烯上穩定。此外，也發現奈米線在吸附鋰離子時比較不會有膨脹現象，這表示當鋰離子吸附與脫附時，奈米線的結構變化不會太嚴重。而鋰離子在pentagon奈米線表面上的H3-H3吸附位置間的擴散擁有最低的能障，大約為0.526 eV，與其他鋰電池的電極材料相近。

In this study, the structure of four ultrathin germanium nanowires (GeNWs) were predicted by big-bang method combined with basin-hopping method and Stillinger-Weber potential. The thermal, mechanical and electronic properties of the GeNWs were further examined by density functional theory (DFT) and molecular mechanics. Not only the physical properties of GeNWs but also the absorption and diffusion behaviors of Li ion on GeNWs were investigated. For thermal properties, the phonon densities of states (PDOS) of four GeNWs were calculated, and the melting points of GeNWs were also predicted by molecular dynamics (MD) simulation. There are three types of GeNWs (helix, pentagon, hexagon) can remain about 400 K, but the 7-1 GeNW is melted about 230 K, which indicates that 7-1 GeNW is unstable at room temperature. Therefore, the 7-1 GeNW was excluded from post simulations and disccustions. In mechanical properties of GeNWs, the yielding stress and Young’s modules of GeNWs are greater than that of bulk Ge. The thermal effect on mechanical properties of GeNWs are also investigated, and the pentagon GeNW exhibits the best mechanical properties among these three GeNWs. In the electronic properties, the electron density of state (DOS) of GeNWs was calculated by DFT to understand orbital hybridization. The result shows that the energy bands of GeNWs shift up to cross the Fermi-level and resulting in conducting nanowires due to the less Coulomb screening effect and quantum confinement from nanowire structure. In addition, the energy band of GeNWs will exhibit significant splitting if which orbital electrons suffer from large Coulomb interactions. For the adsorption and diffustion behavior of Li ion on GeNWs, the various of adsorption sites and which corresponding adsorption energy have been calculated. The absorption energy of Li ion absorption on GeNWs are more stable than that on graphit or graphene. The barrier and transition states of Li ion diffusion on GeNWs were studied by the nudged elastic band (NEB) method, and the results show that the diffusion barrier of Li ion on GeNWsare similar with other material of Li ion battery electrode. The Li ion diffusion on the surface of pentagon GeNW between H3 and H3 adsorption sites with the lowest barrier, which only around 0.526 eV.

論文審定書 i
論文公開授權書 ii
致謝 iii
中文摘要 iv
英文摘要 v
目錄 vii
圖次 ix
表次 xii
第一章 緒論 1
1.1 研究目的與動機 1
1.2 鍺奈米線文獻回顧 3
1.3 本文架構 9
第二章 模擬方法與理論介紹 10
2.1 分子靜力學方法 10
2.1.1 Big-bang方法 10
2.1.2 Basin-hopping方法 11
2.1.3 勢能函數 13
2.1.4 虛擬壁面 15
2.2 密度泛函理論 (Density functional theory, DFT) 17
2.2.1 多粒子系統薛丁格方程式 18
2.2.2 波恩-歐本海默近似 (Born-Oppenheimer approximation) 19
2.2.3 電子密度 20
2.2.4 托瑪斯-費米模型 (Thomas-Fermi model) 21
2.2.5 霍恩貝格-科恩理論 (Hohenberg-Kohn model) 22
2.2.6 科恩-軒姆方程式 (Kohn-Sham equation) 24
2.2.7 交換相關函數 (Exchange-Correlation Function) 26
2.3 分子動力學理論基礎及方法 29
2.3.1 積分法則 30
2.3.2 時間步階選取 (Time step) 31
2.3.3 系綜 (Ensemble) 32
2.3.4 諾斯-胡佛恆溫法 (Nosé-Hoover) 33
2.4 原子級應力分析 35
2.5 NEB方法 (Nudged elastic band method) 39
2.6 分析方法 41
2.6.1 原子距離變化量統計 (∆R) 41
2.6.2 吸附能 (Adsorption energy) 42
第三章 結果與討論 43
3.1 物理模型之建構 44
3.2 鍺奈米線之熱穩定性質分析 49
3.3 鍺奈米線之機械性質分析 54
3.4 鍺奈米線之電子性質分析 68
3.5 鍺奈米線之鋰離子吸附性質分析 75
3.6 鍺奈米線之鋰離子擴散性質分析 85
第四章 結論與建議 91
4.1 結論與建議 91
4.2 未來展望 93
參考文獻 94

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