運用自旋軌道耦合(spin-orbit coupling)的第一原理方法研究重金屬銻(Sb)及α-錫(Sn)超薄薄膜的能帶計算。並藉由調變參數來改變電子結構。從一系列調變的能帶結構圖，可以看出在材料的布理淵區高對稱點上有電子軌道能帶反轉現象(band inversion)，即所謂的拓樸相變(Topological phase transition)。研究結果發現在銻(111) 1層雙分子層(single bi-layer)與α-錫(111) 1層雙分子層(single bilayer)、α-錫(111) 2層雙分子層(two bilayers)皆成功的看出布理淵區高對稱點Γ點上能帶反轉，而導致拓樸相變。銻(111) 1層雙分子層(single bilayer)可以從一般絕緣體(Trivial insulator)轉換成二維的拓樸絕緣體(Topological insulator)。α-錫(111) 1層雙分子層(single bilayer)從一般半金屬(Trivial semi-metal)轉換成拓樸半金屬(Topological semi-metal)。α-錫(111) 2層雙分子層(two bilayers)從拓樸半金屬(topological semi-metal)轉換成一般半金屬(Trivial metal)。

Band structures of ultrathin films of heavy elements, α-Sn and Sb, were investigated using first-principle calculations with the inclusion of spin-orbit coupling. The band structures were gradually varied as the physical parameters were adjusted. The band inversion was obtained at the high symmetry point in Brillouin zone, making a topological phase transition. In this study, the band inversion at Γ point of the Brillouin zone was predicted in single bi-layer of Sb(111) and single bi-layer and two bi-layers of α-Sn(111). The topological phase transition is from trivial insulator to topological insulator for single bi-layer of Sb(111). Finally, the topological phase transition is from trivial semi-metal to topological semi-metal for single bi-layer of α-Sn(111), whereas as it is from topological semi-metal to trivial metal for two bi-layers of α-Sn(111).

Contents
Abstract iii
摘要 iv
1. Introduction 1
2. Theory and Computational Methods 5
2.1 Density functional theory (DFT) 5
2.1.1 Thomas-Fermi model 5
2.1.2 The Hohenberg-Kohn Theorem 6
2.1.3 The Kohn-Sham equation with local density approximation (LDA) 9
2.1.4 Spin orbit coupling 11
2.2 The pseudopotential method 12
2.2.1 Norm-conserving pseudopotential 12
2.2.2 Projector augmented waves (PAW) 14
2.3 Hellmann-Feynman Theorem 16
2.4 Computational Package and Methods 17
3. Result and discussions : Sb 18
3.1 Atomic structures of Sb bulk 18
3.2 Band structures of Sb(111) multi-bilayer thin films 21
3.3 Sb(111) 1 bi-layer 23
3.4 Sb(111) 2 bi-layers 29
3.5 Sb(111) 3 bi-layers 33
4. Result and discussions : α-Sn 38
4.1 Atomic structures of α-Sn bulk 38
4.2 Band structures of α-Sn(111) multi thin films 41
4.3α-Sn(111) 1 bi-layer 43
4.4α-Sn(111) 2 bi-layers 48
5. Conclusions 55
Bibliography 56

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