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系統識別號 U0026-1707201910252500
論文名稱(中文) 石墨烯中的edge state和pn interface state
論文名稱(英文) Edge and pn interface states in graphene
校院名稱 成功大學
系所名稱(中) 物理學系
系所名稱(英) Department of Physics
學年度 107
學期 2
出版年 108
研究生(中文) 游希平
研究生(英文) Hsi-Ping Yu
學號 L26061066
學位類別 碩士
語文別 中文
論文頁數 72頁
口試委員 指導教授-劉明豪
口試委員-張泰榕
口試委員-張景皓
中文關鍵字 2DEG  石墨烯  Parabolic cylinder function  Bulk states  Edge states  Interface states 
英文關鍵字 2DEG  Graphene  Parabolic cylinder function  Bulk states  Edge states  Interface states 
學科別分類
中文摘要 在2017年時, Overweg et. al在石墨烯的實驗中發現電導出現一個新的振盪。在此實驗中,在石墨烯上構建top gate和back gate並利用調整此二電閘的電壓改變石墨烯上不同區域的載子濃度,最後形成不同的接面型態。當磁場上升到0.5(T)時,此團隊發現在bipolar的區域中,電導出現新的振盪,而此振盪無法被snake states等會影響電導的效應所解釋。然而此振盪可以使用tight-binding model等數值模擬得到。

因為可以使用數值模擬得到,此團隊決定利用數值模擬討論更多不同種樣品參數,例如樣品的寬度等等,但是此振盪都存在。不過當此團隊將contacts (電路中與石墨烯相接的點的材料)的能量固定時發現此振盪消失了。所以此團隊認為這個新的振盪跟bulk Landau level有關。雖然這個新的振盪可以利用tight-binding model得到,但是物理意義仍然不明確,所以此團隊決定利用解析解,從公式上找出此振盪出現的原因。

本研究主要是以上討論的前置作業,將利用解析解找出石墨烯的bulk Landau level。求出bulk Landau level的同時,也會求出edge states和interface states,所以在此研究中也會同時討論。此外,因為此問題並不是非常簡單,所以本研究將先討論較容易的二維電子氣(2DEG)系統並利用其結果討論石墨烯上。
英文摘要 In 2017, Overweg et. al found new oscillation pattern in conductance when magnetic field rise up to 0.5(T). Although this pattern can be simulated by tight-binding model, physical mechanism in this pattern have not understood. So they want to find analytical solution and check conductance changing by this solution. After they used tight-binding model to simulate many different set up, oscillation pattern disappear when energy in leads are fixed. According to this result, they thought that oscillating pattern have relation to bulk Landau level. In this research, I try to solve edge states, interface states and bulk states in graphene. 2DEG is easier system than graphene. I will solve 2DEG first and use conclusion from 2DEG to describe graphene situation. First, I will introduce what edge states, interface states and bulk states are. And how these three different states are distinguished. I suppose that two different potential model using to describe material boundary and junction, respectively. After some mathematical process, band structure and eigenfucntions can be found. I pick 4 different points in different situations to plot current densities and check these 4 points whether edge states, interface states or bulk states. In addition to analytical solutions, I also use KWANT to calculate numerical solutions. After compare analytical and numerical solutions, we find that they have good overlap. It means analytical solutions can be used to find correct band structure and quantum states.
論文目次 摘要--i
英文延伸摘要--ii
誌謝--viii
目錄--ix
表格--xi
圖片--xii
Nomenclature--xiv
Chapter1. 研究動機--1
Chapter2. 二維電子氣、石墨烯和Parabolic cylinder function介紹--7
2.1 二維電子氣介紹--7
2.2 石墨烯介紹--9
2.3 Parabolic cylinder function (PCF)介紹--13
Chapter3. 2DEG和石墨烯之Edge States--25
3.1 Edge States和Bulk States介紹--25
3.2 Edge States模型--26
3.3 2DEG--27
3.4 石墨烯--33
Chapter4. 2DEG和石墨烯之Interface States--39
4.1 Interface States和Snake State簡介--39
4.2 Interface States模型--43
4.3 2DEG--43
4.4 石墨烯--49
Chapter5. 結論和未來展望--56
5.1 Edge and Interface States模型總結--56
5.2 穿透率解析解目前進度--56
5.3 未來方向--58
References----59
Appendix A. 有磁場狀況下薛丁格和狄拉克方程--60
Appendix B. 位能為拋物線形式時2DEG討論--64
Appendix C. Klein tunneling--68
參考文獻 [1] Hiske Overweg, Hannah Eggimann, Ming­Hao Liu, Anastasia Varlet, Marius Eich,
Pauline Simonet, Yongjin Lee, Kenji Watanabe, Takashi Taniguchi, Klaus Richter,
Vladimir I. Fal’ko, Klaus Ensslin, and Thomas Ihn. Oscillating magnetoresistance in
graphene p­n junctions at intermediate magnetic fields. Nano Letters, 17(5):2852–2857,
2017.
[2] Milhail I. Katsnelson. Graphene Carbon in Two Dimensions. Cambridge University
Press, 1st edition, 2012.
[3] Frank W. J. Olver. NIST Handbook of Mathematical Functions. Cambridge University
Press, 1st edition, 2010
[4] I Bartos and B Rosenstein. Edge states in a strong magnetic field. J. Phys. A: Math.
Gen., 27(2), 1993
[5] J. Tworzydło, I. Snyman, A. R. Akhmerov, and C. W. J. Beenakker. Valley­isospin dependence of the quantum hall effect in a graphene p−n junction. Phys. Rev. B, 76:035411,
Jul 2007
[6] Christoph W Groth, Michael Wimmer, Anton R Akhmerov, and Xavier Waintal. Kwant:
a software package for quantum transport. New Journal of Physics, 16(6):063065, jun
2014
[7] W Häusler R Egger L Cohnitz, A De Martino. Chiral interface states in graphene p­n
junctions. Phys. Rev. B, 94:165443, 2016
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