由P. C. Donohue與P. E. Bierstedt於1969年所發表的論文中表明NbPS、TaPS、NbPSe的超導性質[1]，1999年由Ichimin Shirotani發表的實驗結果也探討了NbPS的電聲子交互作用係數[2]。我們以第一原理以及密度泛函微擾理論研究這些材料的電聲子結構與超導性質，在NbPS的研究中，我們計算所得到的超導臨界溫度為15K，與Donohue和Bierstedt的實驗結果12K相近，而電聲子交互作用係數的計算結果(λ=1.01)也與Ichimin Shirotani等人的實驗結果(λ=0.81)差異不大。在TaPS、NbPSe的研究中，我們分別得到11K、6K的超導臨界溫度，然而，在Donohue與Bierstedt的論文中提到溫度降至1.25K皆未出現超導現象。除超導的研究以外，我們從它們的能帶結構中看出疑似有能帶反轉的現象。NbPS、TaPS、NbPSe的晶體結構相同，電子結構也相近，我認為它們會有相似的超導性質，然而，近年只有NbPS有新的實驗證據，因此，我認為要有新的實驗確認TaPS與NbPSe是否有超導現象，並以其他的研究確認是否是拓樸材料。

Superconductivity of ternary compounds NbPS, TaPS and NbPSe have been studied with experiments in 1969 by P. C. Donohue and P. E. Bierstedt [1]. The electron-phonon constant of NbPS was also reported by Ichimin Shirotani in 1999 [2]. We use density functional perturbation theory to calculate electron-phonon interaction and superconducting properties of these compounds. On the study of NbPS, we found the electron-phonon constant λ=1.01 and the critical temperature T_c=15K, which are consistent with the result of experiments (λ=0.81, T_c=12K). In addition, the results of our calculation show that TaPS and NbPSe are also superconductors. However, they do not show superconductivity down to 1.25K in experiments done by Donohue and Bierstedt. We also found band inversion in the materials. We suggest more experience to check superconducting properties of these materials, and more research to confirm whether they are topological materials or not.

摘要 2
Abstract 3
第一章 簡介 6
第二章 理論與研究方法 7
2.1 凝體系統的哈密頓量 7
2.2布洛赫定理(Bloch’s theorem) 7
2.3 Hohenberg-Kohn原理 8
2.4 Kohn-Sham方程式 10
2.5 原子受力 11
2.6 晶格優化 12
2.7 晶格振動 12
2.8 密度泛函微擾理論 12
2.9 電聲子交互作用 13
2.10 研究方法 14
第三章 NbPS之電聲子結構計算 15
3.1 晶體結構 15
3.2 能帶結構 15
3.3 聲子能譜與電聲子交互作用 16
3.4 收斂測試 16
第三章圖區 18
第四章 TaPS之電聲子結構計算 25
4-1 能帶結構 25
4-2 聲子能譜與電聲子交互作用 25
4-3 收斂測試 25
第四章圖區 26
第五章 NbPSe之電聲子結構計算 31
5.1 能帶結構 31
5.2 聲子能譜與電聲子交互作用 31
5.3 收斂測試 31
第五章圖區 32
第六章 結論與展望 36
附錄A. Nb、Ta的電聲子計算 38
附錄B. NbSiAs的超導性質 42
參考文獻 45

[1] P. C. Donohue and P. E. Bierstedt, The Synthesis, Crystal Structure, and Superconducting Properties of Niobium Phosphorus Sulfide, Niobium Phosphorus Selenide, and Tantalum Phosphorus Sulfide, Inorg. Chem. 8, 2690, (1969).
[2] Ichimin Shirotani et al., Superconductivity of niobium phosphorous sulphide, NbPS, prepared at high pressure, J. Phys.: Condens. Matter 11 6231, (1999).
[3] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of Superconductivity, Phys. Rev. 108, 1175, (1957).
[4] F. Bloch, Über die Quantenmechanik der Elektronen in Kristallgittern, Z. Physik 52, 555, (1928).
[5] P. Hohenberg and W. Kohn, Inhomogeneous Electron Gas, Phys. Rev. 136, B864, (1964).
[6] M. Born and J. R. Oppenheimer, Zur Quantentheorie der Molekeln, Ann. Phys. 84, 457, (1927).
[7] W. Kohn and L. J. Sham, Self-Consistent Equations Including Exchange and Correlation Effects, Phys. Rev. 140, A1133, (1965).
[8] Feliciano Giustino, Electron-phonon Interaction from First Principles, Rev. Mod. Phys. 89, 015003, (2017).
[9] W. L. McMillan, Transition Temperature of Strong-Coupled Superconductors, Phys. Rev. 167, 331, (1968).
[10] P. B. Allen and R. C. Dynes, Transition Temperature of Strong-coupled Superconductors Reanalyzed, Phys. Rev. B 12, 905, (1975).
[11] G. Kresse and J. Hafner, Ab initio molecular dynamics for open-shell transition metals, Phys. Rev. B 48, 13115, (1993).
[12] G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mat. Sci. 6, 15, (1996).
[13] J. P. Perdew, K. Burke and M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett. 77, 3865, (1996).
[14] P. Giannozzi et al., QUANTUM ESPRESSO: a modular and opensource software project for quantum simulations of materials, J. Phys.: Condens. Matter 21 395502, (2009).
[15] M. Wierzbowska, S. de Gironcoli and P. Giannozzi, Origins of low- and high-pressure discontinuities of Tc in niobium, arXiv:cond-mat/0504077 [cond-mat.supr-con].
[16] H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188, (1976).
[17] W. Setyawan and S. Curtarolo, High-throughput electronic band structure calculations: Challenges and tools, Comp. Mater. Sci. 49, 299, (2010).
[18] Charles P. Poole Jr., Horacio A. Farach, Richard J. Creswick, Superconductivity, (Academic Press).
[19] Gihun Ryu et al, Superconductivity in a PbFCl-type pnictide: NbSiAs, EPL 99, 17002, (2012).