近代材料科學中，調控材料性質(例如 :電性、磁性、光學性質...等等)最有效方法之一，即在具備週期條件之晶格結構中，引入晶格缺陷(包括 :空缺、摻雜、晶格錯位...等等)。相較於實驗方法(透過新穎技術及濃度控制以製備摻雜晶體)，理論計算則是利用超晶胞(supercell)方法，建立原子層級之摻雜晶體模型。然而，進行超晶胞計算，將隨所對應之布里淵區(Brillouin Zone)縮小所致之能帶摺疊效應，而失去計算所得之能帶結構與無摻雜晶體能帶之關聯性。因此，我們的研究工作是結合基底轉換與倒空間展開方法，將超晶胞的能帶結構展開。展開後的能帶結構圖，可以讓我們直接看出摻雜對於能帶結構的影響(如：對稱性破壞、能帶寬度變化…等等)，而計算結果可直接與ARPES實驗比較。此工作將探討三類不同的結構 : IV族半導體摻雜結構、III-V族缺陷結構與二維(2-D)石墨烯結構。

In modern materials science, introducing the impurities (such as, vacancies, dopants, lattice dislocations, etc.) inside a periodic lattice structure is the most effective approach to manipulate the fundamental characteristics  of materials, including electronic , magnetic and optical properties.  
Corresponding to the progress of modern experiments by means of novel techniques and sample preparation of doping crystals, theoretical atomic-level simulation of doping structures can be performed within a supercell scheme.  However, the connection between the corresponding structures of doped and undoped compounds will be obscured by zone-folding induced reduction of Brillouin zone in a supercell calculation..  Thus, it is our task to unfold the supercell band structure by combining the basis transformation and momentum space unfolding methods.  The unfolded band structure  enables us to observe directly the impacts of doping on the band structure such as symmetry breaking and band width modification.  A direct comparison   between the first-principles unfolded band structure calculations and  ARPES measurements can be achieved..   In this thesis, we will study  the following three systems: group IV semiconductor doping, group III-V defects and 2D graphene structures.

目錄
第一章  緒論 ................................ ................................ ................................ ............... 1
1-1 1 研究動機 ................................ ................................ ................................ .......... 1
1-2 2 晶格缺陷的理論與實驗研究 ................................ ................................ .......... 2
第二章  計算方法及理論 ................................ ................................ ........................... 6
2-1 1 基本假設 ................................ ................................ ................................ .......... 6
2-2 密度泛函理論 (Density Function Theory) ................................ ................. 10
2-2-1  Hohenberg -Kohn Kohn理論 ................................ ................................ .......... 10
2-2-2n Kohn-Sham 理論 ................................ ................................ ................... 11
2-2-3 交換相干能  ................................ ................................ ............................. 13
2-2-4 K-points sampling................................ ................................ ................. 15
2-2-5 有限平面波基底 ............................... ................................ ..................... 16
2-2-6  Kohn-Sham 方程式 ................................ ................................ .............. 17
第三章   Bloch functions 與 Wannier functions 之比較 ................................ ....... 19
3-1 Bloch functions................................ ................................ .............................. 19
3-2 Wannier Functions ................................ ................................ ....................... 20
第四章  能帶展開理論  ................................ ................................ ............................... 25
4-1 1 展開定理 ................................ ................................ ................................ ........ 25
4-2 2 展開基底 ................................ ................................ ................................ ........ 28
4-3 3 收斂性質測試  ................................ ................................ ................................ 28
第五章  點缺陷對材料能帶結構之影響 ................................ ................................ . 32
5-1 IV  半導體 ................................ ................................ ................................ ....... 32
5-2  III -V族半導體 族半導體 ................................ ................................ .............................. 38
5-3  二維類石墨烯材料 ................................ ................................ ........................ 44
第六章 總結 …………… ………………………………………………………..…………… ..48
參考文獻  ................................ ................................ ................................ ..................... 50
V
圖表目錄
圖 1-2.1 ARPES  的示意圖 ................................ ................................ ........................ 5
圖 2-2.1 LDA 示意圖 ................................ ................................ .............................. 15
圖 2-2.2 K-points sampling  示意圖 ................................ ................................ ........ 16
圖 3-1.1 Bloch functions 結果之示意圖 ................................ ................................ . 19
圖 3-2.1 Wannier functions 轉換結果之示意圖 ............................... ..................... 21
圖 4-3.1 層狀氮化硼展開能帶結構圖 ................................ ................................ ... 30
圖 4-3.2 上圖為矽能帶展開結構之像，左所用 dis_froz_win 為 8 eV 左 .. 31
圖 5-1.1 矽之能帶結構圖 與晶體................................ ............................... 32
圖 5-1.2 矽(摻雜碳 )之能帶結構圖 與晶體................................ ................. 32
圖 5-1.3 上圖為純矽的單元晶胞 (黑線 )，與矽摻雜碳的超晶胞 ，與矽摻雜碳的超晶胞 (色線 )。 ...... 34
圖 5-1.4 矽(摻雜碳 )之能帶結構圖 、晶體與弛豫的矽偏移量............. 35
圖 5-1.5 矽(摻雜鍺 )之能帶結構圖 與晶體................................ ................. 36
圖 5-1.6 矽(摻雜鍺 )之能帶結構圖 與晶體................................ ................. 37
圖 5-1.7 能帶結構圖比較 ................................ ................................ ....................... 37
圖 5-2.1 氮化硼之能帶結構圖 、晶體與布里淵區....................... 38
圖 5-2.2 氮化硼 (硼空缺 )之能帶結構圖 與晶體................................ ........ 39
圖 5-2.3 氮化硼 (氮空缺 )之能帶結構圖 與晶體................................ ........ 40
圖 5-2.4 氮化硼 Zincblende Zincblende 結構 (硼空缺 )之能帶結構圖與晶體。  ……... …… .42
圖 5-2.5 氮化硼 Zincblende Zincblende 結構 (氮空缺 )之能帶結構圖與晶體。  …… .…43
圖 5-3.1 石墨烯之能帶結構圖 、晶體與布里淵區石墨烯之能帶結構圖 、…………… …… .. …44
圖 5-3.2 Silicene Silicene 之能帶結構圖 、晶體結構與布里淵區結構圖 ..................... 45
圖 5-3.3  Silicene Silicene 移除一顆矽原子以碳取代  之能帶結構圖。 …………………………… .46
圖 5-3.4  Silicene p －能帶結構投 －能帶結構投 ………………………………………………………………………………………… …47

參考文獻
1.	Andrea Damascelli, Phys. Scripta T109, 66 (2004).
2.	Felix Bloch, Z. Physik 52, 555-600 (1928).
3.	Nicola Marzari, Arash A. Mostofi, Jonathan R. Yates, Ivo Souza and David Vanderbilt, Rev. Mod. Phys. 84,1419 (2012).
4.	I. Deretzis, G. Calogero, G. G. N. Angilella and A. La Magna, Euro Phys. Lett. 107 (2014).
5.	Wei Ku, Tom Berlijn and Chi-Cheng Lee, Phys. Rev. Lett. 104, 216401 (2010).
6.	Chi-Cheng Lee, Yukiko Yamada-Takamura and Taisuke Ozaki, J. Phys.:Conden. Matt. 25 (2013).
7.	Shuang Li, Yifeng Wu, Yi Tu, Yonghui Wang, Tong Jiang, Wei Liu and Yonghao Zhao, Sci. Rep. 5 (2015).
8.	K. S. Thygesen, L. B. Hansen and K.W. Jacobsen, Phys. Rev. Lett. 94, 026405 (2005).
9.	Milan Tomi´c, Harald O. Jeschke and Roser Valent´, Phys. Rev. B 90, 195121 (2014).
10.	Runzhi Wang, Emanuel A. Lazar, Hyowon Park, Andrew J. Millis and Chris A. Marianetti, Phys. Rev. B 90, 165125 (2014).
11.	Ivo Souza, Nicola Marzari and David Vanderbilt, Phys. Rev. B 65, 035109 (2001).
12.	Nicola Marzari and David Vanderbilt, Phys. Rev. B 56, 12847 (1997).
13.	Michael P. Marder, Condensed Matter Physics (2010).
14.	Fawei Zheng, Ping Zhang and Wenhui Duan, Comp. Phys. Commun. 189, 213 (2014).
15.	Timothy B. Boykin, Arvind Ajoy, Hesameddin Ilatikhameneh, Michael Povolotskyi and Gerhard Klimeck, Phys. B 491, 22(2016).
16.	Jungwook Woo, Kyung-Han Yun, Sung Beom Cho and Yong-Chae Chung, Phys. Chem. Chem. Phys. 16, 13477 (2014).
17.	Won Seok Yun and J. D. Lee, Phys. Chem. C 5, 2822 (2015).
18.	P. B. Allen, T. Berlijn, D. A. Casavant and J. M. Soler, Phys. Rev. B 87, 085322 (2013).
19.	Junfeng Gao, Junfeng Zhang, Hongsheng Liu, Qinfang Zhangc and Jijun Zhao, Nanoscale 5, 9785 (2013).