在1911年，物理學家Heike Kamerlingh Onnes發現在4K的極低溫環境下，汞的電阻會降到零，也因為這個發現及低溫技術的應用，他獲得了1913年的諾貝爾物理獎。自此，物理學家展開了對超導材料的研究。
1957年John Bardeen、Leon Cooper、和John Robert Schrieffer三人提出了超導的微觀理論，解釋的常規超導體的超導現象，獲得了1972年的諾貝爾物理獎。
而到了2016年，諾貝爾物理獎頒給David James Thouless、Frederick Duncan Michael Haldane、和John Kosterlitz，因為他們三人在拓樸領域和拓樸相變理論的貢獻，打開了一扇通往新研究方向的大門。
此研究以密度泛函理論為基礎，運用VASP和Quantum Espresso，計算六角形晶格結構氮化鉬的電子與聲子能帶結構，以McMillan公式和Allen與Dynes修正的McMillan公式預測其超導溫度，並計算MoN的拓樸不變量Z2及拓樸表面態。
計算的結果顯示，MoN是超導相變溫度達到27K的BCS超導體，非平庸的Z2拓樸不變量說明它是個拓樸超導體。目前的實驗中所發現的拓樸超導體相變溫度均在10K以下，MoN相較之下展現了遠高於它們的相變溫度。

In 1911, physicist Heike Kamerlingh Onnes found the electric resistance of mercury would sharply drop to zero under the environment of 4K low temperature, and he was awarded by Nobel prize of physics. From then on, physicists started the research on the superconducting materials.
1957, John Bardeen, Leon Cooper, and John Robert Schrieffer proposed the superconductivity theory, explain the superconducting mechanism and get the Nobel prize of physics in 1972.
2016 Nobel prize of physics award David James Thouless, Frederich Duncan Michael Haldane, and John Kosterlitz for opening a new direction for research.
This study based on density functional theory, VASP and Quantum Espresso are used to estimate material properties of MoN and calculate its band structure of electron and phonon, predict its superconducting transition temperature by McMillan formula and Allen-Dynes modified McMillan formula, and calculate the topological invariant and surface state.
The result of this research shows MoN is a BCS superconductor with transition temperature 27K and non-trivial topological Z2 invariant indicate that it is a good candidate for topological superconductor.
For most of the topological superconductors have transition temperature lower than 10K, MoN shows a high transition temperature in comparison with topological superconductors found in experiments now.

1 Introduction 8
1.1 Topological materials, superconductors, and topological superconductors . . 8
1.1.1 Topological materials . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.2 Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.3 Topological superconductors . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Density functional theory (DFT) . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Hohenberg­Kohn theorem . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Kohn­Sham ansatz and Kohn­Sham equation . . . . . . . . . . . . . 14
1.2.3 Exchange and correlation energy, Pseudo­potential approximation . . 17
1.3 Bloch theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Wannier functions and Z2 invariants . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Density functional perturbation theory (DFPT) . . . . . . . . . . . . . . . . 23
1.5.1 Linear response theory . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.6 Eliashberg theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Investigation of Molybdenum Nitride 31
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Computational detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Electronic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Investigation of topological properties 36
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Computational detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Wannier function and Wilson loop . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 Surface state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Investigation of superconductivity 45
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Computational detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Phonon structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Electron­phonon coupling and superconductivity . . . . . . . . . . . . . . . 47
4.4.1 Electron­phonon coupling . . . . . . . . . . . . . . . . . . . . . . . 47
4.4.2 Convergence test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5.1 MoN rock­salt structure under pressure . . . . . . . . . . . . . . . . 53
5 Conclusion 54
6 Bibliography and References 54

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