二維材料的性質經常與塊材不同，這篇論文使用第一原理研究許多二維材料。研究他們的電子結構來探求新的性質與可能的研究。許多二維材料引起研究者的廣泛興趣，量子自旋霍爾效應開啟了邊界態電流的研究。Z2拓撲絕緣體也一樣吸引了研究者的注意。目前更輕更薄的材料在應用上很有價值，二維材料在未來會越來越重要，而許多二維材料還沒被仔細研究，很值得去挖掘其中價值。

The properties of the 2D material are sometimes very different to the bulk. This thesis devotes to investigate some 2D material cases by the first-principles calculation. Calculate these material’s electronic structures to find novel properties and possible further research. Lots of 2D material has raised great interests among scientists. The quantum Hall Effect started the research of the 2D edge-states current. The Z2 topological insulator also caught researchers’ eye sight for a long time. Since the development of electronic industry, thinner and lighter materials are needed. The 2D material would be more and more important in the future. There are lots of 2D materials that have not been studied yet, and worth to be mined.

Contents

Abstract i

致謝 ii

1. Introduction 1

2. Theoretical background 2
2.1 Hohenberg-Kohn theorem …………………………………...…………… 2
2.2 Kohn-Sham equation ……………………………………………………... 3
2.3 The local density approximation (LDA) …………………………………. 5
2.4 The generalized gradient approximation (GGA) ………………………… 5
2.5 DFT +U …………………………………………………………………... 6
2.6 Phonon calculation ………………………………………………..……… 6

3. Possible 2D Ferromagnetic semiconductors, VX2 (X = Cl, Br, I) 9
3.1 Motivation ………………………………………………………………... 9
3.2 Crystal and magnetic structure ………………………………………...... 10
3.3 Computational condition …………………………………………………12
3.4 Exchange striction……………………………………………………….. 12
3.5 Approach Ferromagnetism ……………………………………………… 16
3.6 Possible approach ……………………………………………………….. 18

4. Phosphorene 20
4.1 The allotropes of phosphorene ……………………………………..…… 20
4.2 Blue (beta) phosphorene ………………………………………………... 21
4.3 Gamma phosphorene & delta phosphorene …………………………….. 24
4.4 Square-octagon phosphorene …………………………………………… 27
4.5 Bi/Sb in square-octagon lattice …………………………………………. 29
4.6 Bi/Sb in gamma-phosphorene lattice …………………………………… 30

5. Appendix – BaAu2 32
5.1 Motivation and structural information ………………………………...... 32
5.2 Computational condition and the results ………………….…………….. 33

Conclusion 35
Bibliography 36

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