二十世紀量子力學的發展，使我們能夠以量子的角度去探討材料的特性。
在1970 年代，密度泛函數理論有了堅實的理論基礎，被廣泛應用於凝態材料計
算領域，造就不需要實驗提供任何參數的第一原理蓬勃發展。
本文是對真實存在的材料鉍化鎳與鉍化鉑進行拓樸與超導特性的研究。理論
預測鉍化鎳與鉍化鉑金屬為具有 Z2 拓樸的電子聲子超導體。兩材料在塊材電
子能帶上有狄拉克點，並於表面電子能帶上有拓樸的表面態，判別兩材料屬於
拓樸狄拉克半金屬。利用威爾遜迴圈方法，觀察瓦尼爾函數中心演變判別兩材
料屬於強拓樸狄拉克半金屬且Z2拓樸數為1。利用電子聲子耦合的一階近似計
算，由麥克米蘭-艾倫-戴恩斯超導溫度公式判別兩材料屬於弱耦合的電子聲子
超導體。懷疑兼具超導與拓樸的材料比想像更加豐富，展望新穎材料能運用於
未來尖端科技。

The development of quantum mechanics in the twentieth century allowed us to explore the properties of materials from a quantum perspective. In the 1970s, density functional theory (DFT) had a solid theoretical foundation and was widely used in the field of condensed material calculations, making the first principle to flourish. This thesis is mainly to study the topological and superconducting properties of real materials. It is predicted that the NiBi and PtBi metals are electron-phonon superconductors with Z2 topology. Using the Wilson loop method and the electronic band structure to distinguish that both belong to the strong topological Dirac semimetal materials and use the McMillan-Allen-Dynes superconducting temperature formula
to distinguish that both belong weakly coupled electron-phonon superconductors. I suspect that materials with both topology and superconductivity are richer than imagined. Looking forward novel materials can be applied to most advanced science and technology in the future.

摘要..............................................i
Abstract..........................................ii
致謝..............................................iii
目錄..............................................iv
緒論..............................................1
第一章 電子能帶與電子密度計算.......................3
1.1 第一原理計算真實固態系統........................3
1.2 密度泛函數理論與基態系統........................5
1.3 科恩-沈方程與科恩-沈位能........................8
1.4 布洛赫理論與第一布里淵區........................10
1.5 自洽場計算與電子能帶理論........................14
1.6 電子能帶結構與電子態密度........................18
第二章 聲子能帶與超導溫度計算.......................23
2.1 晶體動力學與聲子能帶理論 .......................23
2.2 聲子能帶結構與聲子態密度........................25
2.3 電子與聲子耦合的固態系統........................27
2.4 電子與聲子耦合的超導溫度 .......................32
第三章 Z2拓樸數與表面能帶計算.......................38
3.1 一維等效系統的威爾遜迴圈........................38
3.2 威爾遜迴圈法判別Z2拓樸數........................42
3.3 表面能帶上的塊材投影能帶........................47
3.4 拓樸材料具有的拓樸表面態........................55
第四章 強拓樸狄拉克半金屬之超導體-鉍化鎳(NiBi)........59
4.1 鉍化鎳塊材的原始晶格與倒空間晶格.................59
4.2 鉍化鎳塊材的電子能帶與電子態密度.................60
4.3 鉍化鎳的威爾遜迴圈與Z2拓樸數判別.................70
4.4 鉍化鎳表面的電子能帶與拓樸表面態.................73
4.5 鉍化鎳塊材的聲子能帶與聲子態密度.................83
4.6 鉍化鎳的艾利夏伯格函數與耦合係數.................85
4.7 鉍化鎳的超導溫度與超導量收斂測試.................89
第五章 強拓樸狄拉克半金屬之超導體-鉍化鉑(PtBi)........91
5.1 鉍化鉑塊材的原始晶格與倒空間晶格.................91
5.2 鉍化鉑塊材的電子能帶與電子態密度.................92
5.3 鉍化鉑的威爾遜迴圈與Z2拓樸數判別.................102
5.4 鉍化鉑表面的電子能帶與拓樸表面態.................105
5.5 鉍化鉑塊材的聲子能帶與聲子態密度.................116
5.6 鉍化鉑的艾利夏伯格函數與耦合係數.................118
5.7 鉍化鉑的超導溫度與超導量收斂測試.................121
附錄..............................................123
A 超導部份公式推導.................................123
B 拓樸部份公式推導.................................128
C 常用物理量單位表.................................131
參考..............................................132

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