簡易檢索 / 詳目顯示

回結果列表
研究生： 吳侑恩
Wu, Yu-En
論文名稱： 石墨烯中透過針孔－拋物線接面整束之電子透鏡
Electron Lensing in Graphene through Pinhole-Parabolic Collimation
指導教授： 劉明豪
Liu, Ming-Hao
學位類別： 碩士
Master
系所名稱： 理學院 - 物理學系
Department of Physics
論文出版年： 2026
畢業學年度： 114
語文別： 英文
論文頁數： 57
中文關鍵詞： 石墨烯 、電子光學 、克萊因穿隧 、負折射 、量子傳輸 、偽磁場
外文關鍵詞： Graphene, electron optics, Klein tunneling, negative refraction, quantum transport, pseudomagnetic field
相關次數： 點閱：27  下載：0 
分享至:
查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報

    • 單層石墨烯因其線性色散關係與無質量狄拉克費米子之特性，可展現負折射與克萊因穿隧等獨特的彈道傳輸現象，為研究電子光學提供了絕佳的平台。本論文利用可擴展之緊束縛模型進行數值模擬，系統性地探討石墨烯中電子束的量子傳輸行為。我們首先驗證了拋物線 p-n 接面的電子聚焦能力。為進一步提升電子束品質，本研究提出了一種結合針孔式注入器與拋物線 p-n 接面的雙重準直架構。透過半高寬分析證實，此混合架構能產生高度準直且不發散的電子束，表現顯著優於單一注入源。基於此窄電子束，我們進一步探討其在不同位能分佈下的進階電子光學應用。研究發現在圓環接面中會出現軌跡偏折、全反射與共振態；同時，透過平滑梯度閘極成功實現了可調控的電子束分流與橫向漂移現象。最後，我們探討了電子束與應變引發之偽磁場的交互作用，揭示了顯著的谷自由度傳輸行為。本研究結果有助於加深對石墨烯系統中電子光學行為的理解，並為未來奈米電子元件的設計提供理論參考。

    Single-layer graphene, characterized by its linear energy dispersion and massless Dirac fermions, exhibits unique ballistic transport phenomena such as negative refraction and Klein tunneling, making it an ideal platform for electron optics. In this thesis, we systematically investigate the quantum transport simulation of electron beams in graphene using scalable tight-binding model. We first validate the focusing capabilities of a parabolic pn junction. To further enhance beam quality, we propose a novel double collimation architecture that integrates a pinhole injector with a parabolic pn junction. Our full-width at half-maximum (FWHM) analysis demonstrates that this hybrid configuration produces a highly collimated, non-dispersing electron beam that is significantly superior to standalone point contacts or pinhole injectors. Utilizing this optimized narrow beam, we explore advanced electron-optical applications across various potential landscapes. We observe trajectory deflection, total reflection, and resonant states within circular ring junctions, and we demonstrate tunable beam splitting and transverse drift using smooth gradient gates. Finally, we investigate the beam's interaction with strain-induced pseudomagnetic fields, revealing distinct valley-dependent transport behaviors. These findings provide valuable theoretical insights into graphene-based electron optics and into the design of future nanoelectronic devices.

    摘要 i Abstract ii 誌謝 iii Acknowledgements iv Contents v List of Figures vii List of Symbols viii 1 Introduction 1 1.1 Graphene 1 1.2 Klein Tunneling 2 1.3 Negative Refraction 4 1.4 Motivation 6 1.5 Simulation Methods 7 1.5.1 Tight-Binding Model in Single-Layer Graphene 7 1.5.2 Scalable Tight-Binding Model 10 1.5.3 Transport Theory 11 1.5.4 matQT 14 2 Electron Lensing Methods 15 2.1 Parabolic p-n Junction 15 2.1.1 Simulation 15 2.1.2 Experiment 17 2.2 Pinhole Collimation 18 2.3 Pinhole-Parabolic Collimation 21 2.4 Discussion 21 3 Application of the Electron Beam 25 3.1 Circular Ring Junction 26 3.2 Gradient Junction 28 3.2.1 Dual Circle Gradient Junction 28 3.2.2 Y-Direction Gradient Junction 31 3.3 Pseudomagnetic Field 33 3.3.1 Theoretical Background and Scalable Model 34 3.3.2 Local Pseudomagnetic Field 35 3.3.3 Uniform Pseudomagnetic Field 38 4 Conclusion and Outlook 40 4.1 Conclusion 40 4.2 Outlook 41 References 42

    [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov. Electric field effect in atomically thin carbon films. Science, 306(5696):666–669, 2004.
    [2] A. K. Geim and K. S. Novoselov. The rise of graphene. Nature Materials, 6(3):183–191, March 2007.
    [3] MI Katsnelson and KS Novoselov. Graphene: New bridge between condensed matter physics and quantum electrodynamics. Solid State Communications, 143(1-2):3–13, 2007.
    [4] Antonio H Castro Neto, Francisco Guinea, Nuno MR Peres, Kostya S Novoselov, and Andre K Geim. The electronic properties of graphene. Reviews of modern physics, 81(1):109–162, 2009.
    [5] Himadri Chakraborti, Cosimo Gorini, Angelika Knothe, Ming-Hao Liu, Péter Makk, François D Parmentier, David Perconte, Klaus Richter, Preden Roulleau, Benjamin Sacépé, Christian Schönenberger, and Wenmin Yang. Electron wave and quantum optics in graphene. Journal of Physics: Condensed Matter, 36(39):393001, July 2024.
    [6] Cory R Dean, Andrea F Young, Inanc Meric, Chris Lee, Lei Wang, Sebastian Sorgenfrei, Kenji Watanabe, Takashi Taniguchi, Phillip Kim, Kenneth L Shepard, et al. Boron nitride substrates for high-quality graphene electronics. Nature nanotechnology, 5(10):722–726, 2010.
    [7] Lei Wang, I Meric, PY Huang, Q Gao, Y Gao, H Tran, T Taniguchi, Kenji Watanabe, LM Campos, DA Muller, et al. One-dimensional electrical contact to a two-dimensional material. Science, 342(6158):614–617, 2013.
    [8] Luca Banszerus, Michael Schmitz, Stephan Engels, Matthias Goldsche, Kenji Watanabe, Takashi Taniguchi, Bernd Beschoten, and Christoph Stampfer. Ballistic Transport Exceeding 28 μm in CVD Grown Graphene. Nano Lett., 16(2):1387–1391, February 2016.
    [9] Peter Rickhaus, Romain Maurand, Ming-Hao Liu, Markus Weiss, Klaus Richter, and Christian Schönenberger. Ballistic interferences in suspended graphene. Nature Communications, 4(1):2342, August 2013.
    [10] Clevin Handschin, Péter Makk, Peter Rickhaus, Ming-Hao Liu, K. Watanabe, T. Taniguchi, Klaus Richter, and Christian Schönenberger. Fabry–pérot resonances in a graphene/hbn moiré superlattice. Nano Letters, 17(1):328–333, January 2017.
    [11] Anastasia Varlet, Ming-Hao Liu, Viktor Krueckl, Dominik Bischoff, Pauline Simonet, Kenji Watanabe, Takashi Taniguchi, Klaus Richter, Klaus Ensslin, and Thomas Ihn. Fabry-pérot interference in gapped bilayer graphene with broken anti-klein tunneling. Phys. Rev. Lett., 113:116601, Sep 2014.
    [12] Corentin Déprez, Louis Veyrat, Hadrien Vignaud, Goutham Nayak, Kenji Watanabe, Takashi Taniguchi, Frédéric Gay, Hermann Sellier, and Benjamin Sacépé. A tunable fabry–pérot quantum hall interferometer in graphene. Nature Nanotechnology, 16(5):555–562, May 2021.
    [13] Yuval Ronen, Thomas Werkmeister, Danial Haie Najafabadi, Andrew T. Pierce, Laurel E. Anderson, Young Jae Shin, Si Young Lee, Young Hee Lee, Bobae Johnson, Kenji Watanabe, Takashi Taniguchi, Amir Yacoby, and Philip Kim. Aharonov–bohm effect in graphene-based fabry–pérot quantum hall interferometers. Nature Nanotechnology, 16(5):563–569, May 2021.
    [14] L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim. Cloning of dirac fermions in graphene superlattices. Nature, 497(7451):594–597, May 2013.
    [15] C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim. Hofstadter's butterfly and the fractal quantum Hall effect in moiré superlattices. Nature, 497(7451):598–602, May 2013.
    [16] Vadim V. Cheianov, Vladimir Fal’ko, and B. L. Altshuler. The focusing of electron flow and a veselago lens in graphene p-n junctions. Science, 315(5816):1252–1255, 2007.
    [17] Gil-Ho Lee, Geon-Hyoung Park, and Hu-Jong Lee. Observation of negative refraction of Dirac fermions in graphene. Nature Physics, 11(11):925–929, November 2015.
    [18] Shaowen Chen, Zheng Han, Mirza M. Elahi, K. M. Masum Habib, Lei Wang, Bo Wen, Yuanda Gao, Takashi Taniguchi, Kenji Watanabe, James Hone, Avik W. Ghosh, and Cory R. Dean. Electron optics with p-n junctions in ballistic graphene. Science, 353(6307):1522–1525, 2016.
    [19] J. R. Williams and C. M. Marcus. Snake states along graphene 𝑝−𝑛junctions. Phys. Rev. Lett., 107:046602, Jul 2011.
    [20] Thiti Taychatanapat, Jun You Tan, Yuting Yeo, Kenji Watanabe, Takashi Taniguchi, and Barbaros Özyilmaz. Conductance oscillations induced by ballistic snake states in a graphene heterojunction. Nature Communications, 6(1):6093, February 2015.
    [21] Peter Rickhaus, Péter Makk, Ming-Hao Liu, Endre Tóvári, Markus Weiss, Romain Maurand, Klaus Richter, and Christian Schönenberger. Snake trajectories in ultraclean graphene p–n junctions. Nature Communications, 6(1):6470, 2015.
    [22] Thiti Taychatanapat, Kenji Watanabe, Takashi Taniguchi, and Pablo Jarillo-Herrero. Electrically tunable transverse magnetic focusing in graphene. Nature Physics, 9(4):225–229, April 2013.
    [23] Sei Morikawa, Ziwei Dou, Shu-Wei Wang, Charles G. Smith, Kenji Watanabe, Takashi Taniguchi, Satoru Masubuchi, Tomoki Machida, and Malcolm R. Connolly. Imaging ballistic carrier trajectories in graphene using scanning gate microscopy. Applied Physics Letters, 107(24):243102, 12 2015.
    [24] Sagar Bhandari, Gil-Ho Lee, Anna Klales, Kenji Watanabe, Takashi Taniguchi, Eric Heller, Philip Kim, and Robert M. Westervelt. Imaging cyclotron orbits of electrons in graphene. Nano Lett., 16(3):1690–1694, March 2016.
    [25] O. Klein. Die reflexion von elektronen an einem potentialsprung nach der relativistischen dynamik von dirac. Zeitschrift für Physik, 53(3):157–165, 1929.
    [26] Vadim V. Cheianov and Vladimir I. Fal’ko. Selective transmission of dirac electrons and ballistic magnetoresistance of 𝑛−𝑝junctions in graphene. Phys. Rev. B, 74:041403(R), Jul 2006.
    [27] M. I. Katsnelson, K. S. Novoselov, and A. K. Geim. Chiral tunnelling and the Klein paradox in graphene. Nature Physics, 2(9):620–625, September 2006.
    [28] Andrei V. Shytov, Mark S. Rudner, and Leonid S. Levitov. Klein backscattering and fabry-pérot interference in graphene heterojunctions. Phys. Rev. Lett., 101:156804, Oct 2008.
    [29] Andrea F. Young and Philip Kim. Quantum interference and Klein tunnelling in graphene heterojunctions. Nature Physics, 5(3):222–226, March 2009.
    [30] N. Stander, B. Huard, and D. Goldhaber-Gordon. Evidence for klein tunneling in graphene 𝑝−𝑛junctions. Phys. Rev. Lett., 102:026807, Jan 2009.
    [31] Ming-Hao Liu, Jan Bundesmann, and Klaus Richter. Spin-dependent klein tunneling in graphene: Role of rashba spin-orbit coupling. Phys. Rev. B, 85:085406, Feb 2012.
    [32] P. E. Allain and J. N. Fuchs. Klein tunneling in graphene: Optics with massless electrons. The European Physical Journal B, 83(3):301–317, October 2011.
    [33] Ming-Hao Liu, Cosimo Gorini, and Klaus Richter. Creating and steering highly directional electron beams in graphene. Phys. Rev. Lett., 118:066801, Feb 2017.
    [34] Arthur W. Barnard, Alex Hughes, Aaron L. Sharpe, Kenji Watanabe, Takashi Taniguchi, and David Goldhaber-Gordon. Absorptive pinhole collimators for ballistic dirac fermions in graphene. Nat. Commun., 8:15418, 2017.
    [35] Peter Bøggild, José M. Caridad, Christoph Stampfer, Gaetano Calogero, Nick Rübner Papior, and Mads Brandbyge. A two-dimensional dirac fermion microscope. Nat. Commun., 8:15783, 2017.
    [36] F. Martins, B. Hackens, M. G. Pala, T. Ouisse, H. Sellier, X. Wallart, S. Bollaert, A. Cappy, J. Chevrier, V. Bayot, and S. Huant. Imaging electron wave functions inside open quantum rings. Phys. Rev. Lett., 99:136807, Sep 2007.
    [37] Peter Rickhaus, Péter Makk, Ming-Hao Liu, Klaus Richter, and Christian Schönenberger. Gate tuneable beamsplitter in ballistic graphene. Applied Physics Letters, 107(25):251901, 12 2015.
    [38] Peter Rickhaus, Ming-Hao Liu, Péter Makk, Romain Maurand, Samuel Hess, Simon Zihlmann, Markus Weiss, Klaus Richter, and Christian Schönenberger. Guiding of Electrons in a Few-Mode Ballistic Graphene Channel. Nano Lett., 15(9):5819–5825, September 2015.
    [39] Emmanuel Paredes-Rocha, Yonatan Betancur-Ocampo, Nikodem Szpak, and Thomas Stegmann. Gradient-index electron optics in graphene 𝑝− 𝑛junctions. Phys. Rev. B, 103:045404, Jan 2021.
    [40] Jule-Katharina Schrepfer, Szu-Chao Chen, Ming-Hao Liu, Klaus Richter, and Martina Hentschel. Dirac fermion optics and directed emission from single- and bilayer graphene cavities. Phys. Rev. B, 104:155436, Oct 2021.
    [41] Shiang-Bin Chiu, Alina Mreńca-Kolasińska, Ka Long Lei, Ching-Hung Chiu, Wun-Hao Kang, Szu-Chao Chen, and Ming-Hao Liu. Manipulating electron waves in graphene using carbon nanotube gating. Phys. Rev. B, 105:195416, May 2022.
    [42] Zachary J. Krebs, Wyatt A. Behn, Keenan J. Smith, Margaret A. Fortman, Kenji Watanabe, Takashi Taniguchi, Pathak S. Parashar, Michael M. Fogler, and Victor W. Brar. Nanoscale imaging of magnetotransport around a circular 𝑝−𝑛 junction in graphene. Phys. Rev. Lett., 136:136301, Apr 2026.
    [43] Ming-Hao Liu, Peter Rickhaus, Péter Makk, Endre Tóvári, Romain Maurand, Fedor Tkatschenko, Markus Weiss, Christian Schönenberger, and Klaus Richter. Scalable tight-binding model for graphene. Phys. Rev. Lett., 114:036601, Jan 2015.
    [44] Supriyo Datta. Electronic Transport in Mesoscopic Systems. Cambridge Studies in Semiconductor Physics and Microelectronic Engineering. Cambridge University Press, 1995.
    [45] Supriyo Datta. Exclusion principle and the landauer-büttiker formalism. Phys. Rev. B, 45:1347–1362, Jan 1992.
    [46] Clevin Handschin, Bálint Fülöp, Péter Makk, Sofya I. Blanter, Markus Weiss, Kenji Watanabe, Takashi Taniguchi, Szabolcs Csonka, and Christian Schönenberger. Point contacts in encapsulated graphene. Applied Physics Letters, 107:183108, 2015.
    [47] Alina Mreńca-Kolasińska, Christophe De Beule, Jia-Tong Shi, Aitor Garcia-Ruiz, Denis Kochan, Klaus Richter, and Ming-Hao Liu. Pseudomagnetotransport in strained graphene. arXiv preprint arXiv:2505.21056, 2025. [48] N. Levy, S. A. Burke, K. L. Meaker, M. Panlasigui, A. Zettl, F. Guinea, A. H. Castro Neto, and M. F. Crommie. Strain-induced pseudo–magnetic fields greater than 300 tesla in graphene nanobubbles. Science, 329(5991):544–547, 2010.
    [49] Walter Ortiz, Nikodem Szpak, and Thomas Stegmann. Graphene nanoelectromechanical systems as valleytronic devices. Physical Review B, 106(3):035416, 2022.
    [50] SP Milovanović and FM Peeters. Strain controlled valley filtering in multi-terminal graphene structures. Applied Physics Letters, 109(20), 2016.
    [51] Francisco Guinea, Mikhail I Katsnelson, and AK Geim. Energy gaps and a zero-field quantum hall effect in graphene by strain engineering. Nature Physics, 6(1):30–33, 2010.
    [52] Maria AH Vozmediano, MI Katsnelson, and Francisco Guinea. Gauge fields in graphene. Physics Reports, 496(4-5):109–148, 2010.
    [53] P Nigge, AC Qu, É Lantagne-Hurtubise, Erik Mårsell, S Link, G Tom, M Zonno, M Michiardi, M Schneider, S Zhdanovich, et al. Room temperature strain-induced landau levels in graphene on a wafer-scale platform. Science advances, 5(11):eaaw5593, 2019.
    [54] Ming-Hao Liu, Christophe De Beule, Alina Mreńca-Kolasińska, Hsin-You Wu, Aitor Garcia-Ruiz, Denis Kochan, and Klaus Richter. Scalable tight-binding model for strained graphene. Phys. Rev. B, 113:195429, May 2026.
    [55] Vitor M. Pereira, A. H. Castro Neto, and N. M. R. Peres. Tight-binding approach to uniaxial strain in graphene. Phys. Rev. B, 80:045401, Jul 2009.
    [56] Mikkel Settnes, Stephen R Power, and Antti-Pekka Jauho. Pseudomagnetic fields and triaxial strain in graphene. Phys. Rev. B, 93(3):035456, 2016.
    [57] Manlin Luo, Hao Sun, Zhipeng Qi, Kunze Lu, Melvina Chen, Dongho Kang, Youngmin Kim, Daniel Burt, Xuechao Yu, Chongwu Wang, et al. Triaxially strained suspended graphene for large-area pseudo-magnetic fields. Optics letters, 47(9):2174–2177, 2022.

    QR CODE