過去研究已經證實，單獨的Bi 與Sb 雙層結構(bilayer)在應力(strain)下是二維(2D)拓樸絕緣體(Topological Insulator)材料。在本篇論文中，透過考慮自旋軌道耦合(spin-orbit coupling)的第一原理計算，研究受應力下的Bi(111)與Sb(111)雙層結構(bilayer)在不同半導體基板上的電子結構(electronic structures)與拓樸能帶(bandtopology)。我們發現當二維拓樸絕緣體放在Si(111)與Ge(111)基板，介面上會有電子的轉移，而使系統變為金屬。而二維拓樸絕緣體放在六角晶格結構氮化硼(hexagonal-BN)基板上，Bi(111)與Sb(111)可以保留應力下的反常拓樸態(non-trivial)與絕緣體性質。其中在六角晶格結構氮化硼 h-BN-√3×√3 重構系統穩定性高於六角晶格結構-BN-2×2 重構。因此我們認為六角晶格結構氮化硼為最佳候選基板，而以六角晶格結構氮化硼(h-BN)-√3×√3 來維持二維拓樸絕緣體較為穩定。

Recent studies have shown that the single Bi and Sb bilayers are two dimensional topological insulators (2D TIs) under strains. In this study, electronic structures and band topology of the strained Bi(111) and Sb(111) single bilayers on three semiconducting or insulating substrates are investigated using first-principle calculations with spin-orbit coupling. First, charge transfer is found to be from the Si(111) and Ge(111) substrates which causes the system becomes meteallic. Moreover, hexagonal-BN can retain nontrivial topological insulating phase for the Bi(111) and Sb(111) single bilayer thin film at two different coverages. Finally, our stability analysis shows that 2D TIs on hexagonal-BN-√3×√3 is more stable than those on the hexagnal-BN-2×2. Therefore, hexagonal-BN-√3×√3 is an appropriate surperlattice structure to support the two-dimensional topological insulators.

摘要 .................................................................................................................................. i
Abstract ............................................................................................................................ iii
Figures ............................................................................................................................. vi
Tables .............................................................................................................................. ix
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 spin density approximation (LSDA) and
generalized gradient approximation (GGA) .............................................................. 8
2.1.4 Spin-orbit coupling ......................................................................................... 10
2.2 The pseudopotential method ................................................................................. 13
2.2.1 Norm-conserving pseudopotential ................................................................. 13
2.2.2 Projector augmented waves (PAW) ............................................................... 15
2.3 Hellmann-Feynman theorem ................................................................................ 17
2.4 Van der Waals fouce density functionals.............................................................. 18
2.5 Topological insulator ............................................................................................ 18
2.6 Computational methods ........................................................................................ 21
3. Review of Bi and Sb bilayers ..................................................................................... 22
3.1 Atomic structures of Bi(111)/Sb(111) bi-layer thin films .................................... 22
3.2 Band structures of Bi(111) bi-layer thin films ...................................................... 23
3.3 Band structures of Sb(111) bi-layer thin films ..................................................... 27
4. Result and discussions ： Bi bilayer on substrates .................................................. 30
4.1 Bi(111) bi-layer on h-BN √3×√3 substrates ......................................................... 32
4.2 Bi(111) bi-layer on h-BN 2×2 substrates .............................................................. 36
4.3 Bi(111) bi-layer on Si(111) 1×1 substrates ........................................................... 40
4.4 Bi(111) bi-layer on Ge(111) 1×1 substrates ......................................................... 44
5. Result and discussions：Sb bilayer on substrate ....................................................... 48
5.1 Sb(111) bi-layer on h-BN √3×√3 substrates ......................................................... 48
5.2 Sb(111) bi-layer on h-BN 2×2 substrates ............................................................. 52
5.3 Sb(111) bi-layer on Si(111) 1×1 substrates .......................................................... 56
5.4 Sb(111) bi-layer on Ge(111) 1×1 substrates ......................................................... 60
6. Discussion: Stability ................................................................................................... 64
7. Conclusion .................................................................................................................. 68
Bibliography ................................................................................................................... 69

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