由Andre Geim和他的學生-2010年諾貝爾物理獎得主-發現了獨特的石墨單原子層結構：石墨烯而建立新奈米時代里程碑。基於石墨烯基礎的了解與應用上的考量的廣大收穫，對於此類低維度原子層狀結構的研究已蓬勃發展。在目前的工作，依據密度泛函微擾理論，我們透過Abinit第一原理計算軟體，探討單層與數層六角晶格層狀氮化硼的組成結構、電子結構與晶格振動特性。
    首先，藉由總能量的計算，決定單層與數層層狀氮化硼的結構；其次，我們的結果顯示:層狀氮化硼的能隙，可以透過層數的增加來加以調整；最後，計算數層層狀氮化硼的拉曼頻率與強度，進而探討層之間的交互作用勤行，同時這項工作也討論特殊的低頻層狀剛體振動模式。

The milestone of new nano era was established from the discovery of  unique mono atomic layered structure of graphite:graphene by Andre Geim and his student who were awarded Nobel Laureate in 2010. Based on tremendous triumph on the fundamental understanding and application consideration of graphene, research on such low-dimensional atomic layer structure has been extended rapidly. In the present work, we are going to explore the structural, electronic structure, and lattice vibrational properties of mono- and few-layer hexagonal BN sheet by performing first-principles calculations in an ab-initio package, Abinit within the Density Functional Perturbation Theory.

First of all, the table structures of mono- and few-layer BN sheets can be determined by calculated total energies. Secondly, our results show bandgap of BN sheets can be tuned by increasing the number of BN layers. Finally, Raman frequencies and intensities with respect to few-layer BN sheets were also calculated to explore the interlayer interaction. Meanwhile, significant low-frequency rigid-layer modes have also been discussed in this work.

第一章 導論...............................................1
1.1 :研究動機......................................1
1.2 :六角晶格氮化硼層狀結構........................2
1.3:論文架構.......................................4
第二章 第一原理理論.......................................5
2.1:密度泛函理論 (DFT).............................6
2.1.1: Kohn-Sham理論.............................7
2.1.2:交換相干能................................11
2.1.3:虛位勢....................................13
2.2:密度泛函微擾理論..............................14
2.2.1:晶格動力學................................14
2.2.2:線性響應..................................16
2.2.3:2n+1定理..................................20
2.2.4:高階微擾..................................22
第三章 聲子頻率與拉曼張量計算............................28
3.1:聲子計算......................................30
3.2:LO-TO splitting 計算..........................32
3.3:拉曼張量與拉曼強度計算........................34
第四章 層狀與塊材六角晶格結構之氮化硼計算................37
4.1:氮化硼的結構總能計算..........................37
4.2:層數對氮化硼電子結構的影響....................40
第五章 層狀與塊材六角晶格結構氮化硼之聲子計算............46
5.1: 塊材之聲子計算...............................46
5.1.1:晶格常數對聲子頻率的影響..................46
5.1.2:聲子色散關係..............................48
5.1.3:低頻特徵振動模式..........................48
5.2: 拉曼頻譜.....................................53
5.3: 拉曼強度與層數的關係.........................54
第六章 結論..............................................60
參考文獻.................................................62
 
圖目錄
圖1-1 ：石墨烯不同的堆疊方式...............................2
圖1-2 ：氮化硼堆疊方式.....................................4
圖2-1 ：虛位勢............................................13
圖3-1 ：拉曼張量與聲子計算流程............................30
圖3-2 ：晶體位置示意圖....................................31
圖3-3 ：聲子散射模型......................................36
圖4-1 ：總能與層數關係圖..................................38
圖4-2 ：雙層結構下，層與層之間的距離與能量關係圖..........39
圖4-3 ：第一布里淵區能帶結構之路徑........................40
圖4-4 ：單層氮化硼之能帶結構(direct band gap).............41
圖4-5 ：雙層氮化硼之能帶結構(indirect band gap) ..........41
圖4-6 ：三層氮化硼之能帶結構(indirect band gap)...........42
圖4-7 ：四層氮化硼之能帶結構(indirect band gap)...........42
圖4-8 ：五層氮化硼之能帶結構(indirect band gap)...........43
圖4-9 ：氮化硼塊材之能帶結構(indirect band gap)...........43
圖4-10：能隙與層數關係(紅線為塊材)........................44
圖4-11：價帶的鍵結(σ、π鍵)..............................45
圖5-1 ：單層結構聲子分布..................................48
圖5-2 ：雙層結構中會產生虛頻率的振動模式..................49
圖5-3 ：層距與頻率關係圖..................................49
圖5-4 ：不同層數在低頻率部份之shear mode..................51
圖5-5 ：不同層數在低頻率部份之compressive mode............52
圖5-6 ：單層到塊材之拉曼頻譜計算..........................54
圖5-7 ：單層狀態下，真空層的距離與強度的關係..............55
圖5-8 ：修正過後的拉曼頻譜................................55
圖5-9 ：拉曼光譜及強度與層數關係..........................56
圖5-10：強度與層數之關係圖(經過歸一化結果) ...............56
圖5-11：Shear mode頻率與層數倒數關係圖....................57
圖5-12：Shear mode強度與層數關係圖....................... 57
圖5-13：Compressive Ag mode頻率與層數倒數關係圖...........58
圖5-14：Compressive Ag mode強度與層數關係圖...............58
 
表目錄
表3-1 能量泛函對位移、電場微分代表的意義.................28
表4-1 氮化硼總能計算....................................37
表4-2 雙層結構下的距離與能量關係........................39
表5-1 不同晶格常數對應到聲子頻率........................47
表5-2 層之間的振動所對應之頻率..........................53

[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, Science 306, 666–669. (2004).
[2] Haibo Zeng, Chunyi Zhi, Zhuhua Zhang, Xianlong Wei, Xuebin Wang, Wanlin Guo, Yoshio Bando, and Dmitri Golberg. Nano. Lett. 2010, 10, pp 5049–5055
[3] Jae-Kap Lee, So-Hyung Lee, Jae-Pyoung Ahn, Seung-Cheol Lee and Wook-Seong Lee, United States Application US20100028573
[4] J. L. Yin, M. L. Hu, Zhizhou Yu, C. X. Zhang, L. Z. Sun, J. X. Zhong, Physica B 406 , 2293–2297, (2011)
[5] “Principles of Quantum Mechanics”, 2nd edition edited by R. Shankar (New York : Plenum Press, 1994)
[6] Stefano Baroni, Stefano de Gironcoli, Andrea Dal Corso , Rev. Mod Phys. 73, 515-562 (2001)
[7] W. Kohn, L. Sham, Phys. Rev. A 140, 1133 (1965)
[8] P. Hohenberg, W. Kohn, Phys. Rev. B 136, 864 (1964)
[9] “Atomic and Electronic Structure of Solids” edited by Efthimios Kaxiras, (Cambridge : Cambridge University Press, 2003)
[10]X. Gonze, Phys. Rev. B 55, 10337 (1997)
[11] “First-principles study of the nonlinear responses of insulators to electric fields:applications to ferroelectric oxides” edited by Veithen Marek, (PhD. Thesis, 2005)
[12]W. Kohn and L. J. Sham, Phys. Rev. 140, 1133 (1965)
[13] “Electronic structure” edited by Richard M. Martin,
     (New York : Cambridge University Press, 2004)
[14]John P. Perdew and Wang Yue, Phys. Rev. B 33, 8800 (1986)
[15]M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J.D.Joannopouios, Rev. Mod Phys. 64, 1045 (1992)
[16]V. Milman, B. Winkler, J. A. White, C. J. Pickard, M. C. Payne, E. V. Akhmatskaya and R. H. Nobes, J. Quantum Chem. 77, 8950 (2000)
[17]D. Vanderbilt. Phys. Rev. B 41, 7892 (1990)
[18]P. Giannozzi, S. de Gironcoli, P. Pavone, S. Baroni. Phys. Rev. B 43, 9 (1991)
[19]X. Gonze, Phys. Rev. A 52, 1096 (1995)
[20]X. Gonze, Phys. Rev. B 55, 10355 (1997)
[21]X. Gonze, Phys. Rev. B 52, 1086 (1995)
[22]Bo Zhou and Deyan He, J. Raman Spectrosc. 39, p.1475–1481 (2008)
[23] “Solid-state spectroscopy: an introduction / Hans Kuzmany.” edited by Hans Kuzmany, (Berlin : Springer, 1998)
[24]G. Deinzer and D. Strauch, Phys. Rev. B 66, 100301(2002)
[25]Roman V. Gorbachev, Ibtsam Riaz, Rahul R. Nair, Rashid Jalil, Liam Britnell,Branson D. Belle, Ernie W. Hill, Kostya S. Novoselov, Kenji Watanabe, Takashi Taniguchi, Andre K. Geim, and Peter Blake, small, 7, 4, (2011)
[26]P. H. Tan, W. P. Han, W. J. Zhao, Z. H. Wu, K. Chang, H. Wang, Y. F. Wang, N. Bonini, N. Marzari, G. Savini, A. Lomobardo, and A.C. Ferrari, arXiv:1106. 1146v1 (2011)
[27]Noa Marom, Jonathan Bernstein, Jonathan Garel, Alexandre Tkatchenko, Ernesto Joselevich, Leeor Kronik, and Oded Hod, Phys. Rev. Lett. 105, 046801 (2010)
[28] http://www.abinit.org