本文探討碳奈米卷軸（Carbon Nanoscrolls, CNSs）的電子能帶、傳輸行為與拓樸性質。碳奈米卷軸是由單層石墨烯螺旋捲曲而成的徑向超晶格(Radial superlattices)，展現出與平面石墨烯顯著不同的量子輸運特性。本研究以緊束縛模型為基礎，搭配 Kwant 套件進行數值模擬，建立由鋸齒邊石墨烯奈米帶捲曲形成的模型，系統性分析其幾何結構對能帶與導電特性的影響。
本研究特別關注捲軸圈數對能帶結構與電導的調控效果，並比較施加與未施加軸向磁通量時的差異。模擬結果顯示，隨著捲軸圈數增加，低能區的能帶結構出現顯著變化：整數圈數會產生能隙，而位於能隙邊緣的反摺能帶則貢獻額外的傳輸態。在較高能區，無論是整數或非整數圈數的捲軸，皆會出現能帶於動量空間中的些微左右平移。此外，非整數圈數的捲軸亦會出現接近零能的平坦能帶。當系統施加軸向磁通量後，較高能量特徵與未加磁場時大致相似，但在零能量附近會出現能帶交叉點，且其數量與捲軸圈數呈現對應關係。
為進一步挖掘系統的拓樸特性，我們計算哈密頓量非對角塊所對應的拓樸纏繞數，並分析其隨磁通量變化的行為。結果顯示，纏繞數會在不同的動量區域反映出不同的拓樸相，且區分不同相的相變點為零能量的能帶交叉點。透過 Landauer 公式和 S-Matrix 方法進行電導計算，可以比較發現外加磁通後的電導增益與幾何圈數正相關。
綜合以上結果，本研究證實碳奈米卷軸的幾何圈數可以調控拓樸不變量，增加幾何圈數即可獲得更高的拓樸纏繞數，且反映在外加磁通量後的電導，實現可控拓樸磁電導增益，成功系統性的建立了石墨烯一維拓樸量子傳輸的新效應，同時還為調控拓樸不變量的拓樸材料系統提供全新的方向。

We develop a tight-binding model for carbon nanoscrolls (CNSs) formed from zigzag graphene ribbons and explore how their geometry governs electronic and topological behavior. Integer-turn scrolls open energy gaps, while non-integer turns yield zero-energy flat bands. Under an axial magnetic flux of 0.5, ϕ₀, band crossings emerge, increasing in number with scroll turns and signaling topological transitions characterized by winding numbers from chiral symmetry. The number of turns sets both the total band crossings and the maximal winding number. Conductance simulations reveal flux-induced enhancements correlated with these crossings, establishing CNSs as a geometry-tunable platform for topological transport.

[1] F. Schwierz, J. Pezoldt, and R. Granzner. Two-dimensional materials and their prospects in transistor electronics. Nanoscale, 7:8261–8283, 2015.
[2] C. Berger, Z. Song, T. Li, X. Li, A. Y. Ogbazghi, R. Feng, Z. Dai, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer. Ultrathin epitaxial graphite: 2d electron gas properties and a route toward graphene-based nanoelectronics. The Journal of Physical Chemistry B, 108(52):19912–19916, 2004.
[3] H. O. Pierson. Handbook of Carbon, Graphite, Diamonds and Fullerenes: Processing, Properties and Applications. Materials science and process technology series. William Andrew, 2012.
[4] S. Iijima and T. Ichihashi. Single-shell carbon nanotubes of 1-nm diameter. Nature, 363(6430):603–605, Jun 1993.
[5] W. Krätschmer, L. D. Lamb, K. Fostiropoulos, and D. R. Huffman. Solid C₆₀: a new form of carbon. Nature, 347(6291):354–358, Sep 1990.
[6] R. C. Haddon et al. Conducting films of C₆₀ and C₇₀ by alkali-metal doping. Nature, 350(6316):320–322, Mar 1991.
[7] L. Degiorgi et al. Optical properties of the alkali-metal-doped superconducting fullerenes: K₃C₆₀ and Rb₃C₆₀. Phys. Rev. B, 49:7012–7025, Mar 1994.
[8] A. Y. Ganin et al. Bulk superconductivity at 38 K in a molecular system. Nature Materials, 7(5):367–371, May 2008.
[9] T. W. Odom, J.-L. Huang, P. Kim, and C. M. Lieber. Structure and electronic properties of carbon nanotubes. The Journal of Physical Chemistry B, 104(13):2794–2809, 2000.
[10] T. Ando. The electronic properties of graphene and carbon nanotubes. NPG Asia Materials, 1(1):17–21, Oct 2009.
[11] H. Ajiki and T. Ando. Aharonov–Bohm effect in carbon nanotubes. Physica B: Condensed Matter, 201:349–352, 1994.
[12] A. K. Geim and A. H. MacDonald. Graphene: Exploring carbon flatland. Physics Today, 60(8):35–41, Aug 2007.
[13] K. S. Novoselov et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature, 438(7065):197–200, Nov 2005.
[14] G. P. Mikitik and Y. V. Sharlai. Manifestation of Berry’s phase in metal physics. Phys. Rev. Lett., 82:2147–2150, Mar 1999.
[15] I. A. Luk’yanchuk and Y. Kopelevich. Phase analysis of quantum oscillations in graphite. Phys. Rev. Lett., 93:166402, Oct 2004.
[16] Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature, 438(7065):201–204, Nov 2005.
[17] K. S. Novoselov et al. Room-temperature quantum Hall effect in graphene. Science, 315(5817):1379–1379, 2007.
[18] R. Bacon. Growth, structure, and properties of graphite whiskers. Journal of Applied Physics, 31(2):283–290, Feb 1960.
[19] V. P. Dravid et al. Buckytubes and derivatives: Their growth and implications for buckyball formation. Science, 259(5101):1601–1604, 1993.
[20] S. Iijima. Helical microtubules of graphitic carbon. Nature, 354(6348):56–58, Nov 1991.
[21] S. Amelinckx et al. A structure model and growth mechanism for multishell carbon nanotubes. Science, 267(5202):1334–1338, 1995.
[22] L. M. Viculis, J. J. Mack, and R. B. Kaner. A chemical route to carbon nanoscrolls. Science, 299(5611):1361–1361, 2003.
[23] X. Xie et al. Controlled fabrication of high-quality carbon nanoscrolls from monolayer graphene. Nano Letters, 9(7):2565–2570, 2009.
[24] X. Cui et al. Rolling up transition metal dichalcogenide nanoscrolls via one drop of ethanol. Nature Communications, 9(1):1301, Apr 2018.
[25] A. K. Schaper et al. Comparative studies on the electrical and mechanical behavior of catalytically grown multiwalled carbon nanotubes and scrolled graphene. Nano Letters, 11(8):3295–3300, 2011.
[26] X. Chen, Q. Zhou, J. Wang, and Q. Chen. Formation of graphene nanoscrolls and their electronic structures based on ab initio calculations. The Journal of Physical Chemistry Letters, 13(11):2500–2506, Mar 2022.
[27] Y.-J. Zhong et al. Magnetoconductance modulations due to interlayer tunneling in radial superlattices. Nanoscale Horizons, 7:168–173, 2022.
[28] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett., 49:405–408, Aug 1982.
[29] Q. Niu, D. J. Thouless, and Y.-S. Wu. Quantized Hall conductance as a topological invariant. Phys. Rev. B, 31:3372–3377, Mar 1985.
[30] C.-Z. Chang et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science, 340(6129):167–170, 2013.
[31] Y.-F. Zhao et al. Tuning the Chern number in quantum anomalous Hall insulators. Nature, 588(7838):419–423, Dec 2020.
[32] Z. Li, Y. Han, and Z. Qiao. Chern number tunable quantum anomalous Hall effect in monolayer transitional metal oxides via manipulating magnetization orientation. Phys. Rev. Lett., 129:036801, Jul 2022.
[33] Y. Shi et al. Electronic phase separation in multilayer rhombohedral graphite. Nature, 584(7820):210–214, Aug 2020.
[34] Y. Sha et al. Observation of a Chern insulator in crystalline ABCA-tetralayer graphene with spin-orbit coupling. Science, 384(6694):414–419, 2024.
[35] T. Han et al. Correlated insulator and Chern insulators in pentalayer rhombohedral-stacked graphene. Nature Nanotechnology, 19(2):181–187, Feb 2024.
[36] J. E. Lennard-Jones. The electronic structure of some diatomic molecules. Trans. Faraday Soc., 25:668–686, 1929.
[37] F. Bloch. Über die Quantenmechanik der Elektronen in Kristallgittern. Zeitschrift für Physik, 52(7):555–600, Jul 1929.
[38] C. Kittel. Kittel’s Introduction to Solid State Physics. Wiley, 2018.
[39] J. C. Slater and G. F. Koster. Simplified LCAO method for the periodic potential problem. Phys. Rev., 94:1498–1524, Jun 1954.
[40] M. J. Allen, V. C. Tung, and R. B. Kaner. Honeycomb carbon: A review of graphene. Chemical Reviews, 110(1):132–145, 2010.
[41] L. E. F. F. Torres, S. Roche, and J. C. Charlier. Introduction to Graphene-Based Nanomaterials: From Electronic Structure to Quantum Transport. Cambridge University Press, 2020.
[42] A. H. Castro Neto et al. The electronic properties of graphene. Rev. Mod. Phys., 81:109–162, Jan 2009.
[43] Y. Aharonov and D. Bohm. Significance of electromagnetic potentials in the quantum theory. Phys. Rev., 115:485–491, Aug 1959.
[44] R. G. Chambers. Shift of an electron interference pattern by enclosed magnetic flux. Phys. Rev. Lett., 5:3–5, Jul 1960.
[45] A. Tonomura et al. Evidence for Aharonov–Bohm effect with magnetic field completely shielded from electron wave. Phys. Rev. Lett., 56:792–795, Feb 1986.
[46] W. P. Su, J. R. Schrieffer, and A. J. Heeger. Solitons in polyacetylene. Phys. Rev. Lett., 42:1698–1701, Jun 1979.
[47] A. J. Heeger et al. Solitons in conducting polymers. Rev. Mod. Phys., 60:781–850, Jul 1988.
[48] J. Zak. Berry’s phase for energy bands in solids. Phys. Rev. Lett., 62:2747–2750, Jun 1989.
[49] S. Ryu and Y. Hatsugai. Topological origin of zero-energy edge states in particle-hole symmetric systems. Phys. Rev. Lett., 89:077002, Jul 2002.
[50] J. K. Asbóth, L. Oroszlány, and A. Palyi. A short course on topological insulators: band structure and edge states in one and two dimensions, 2016.
[51] C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal. Kwant: a software package for quantum transport. New Journal of Physics, 16(6):063065, 2014.
[52] R. Peierls. Zur Theorie des Diamagnetismus von Leitungselektronen. Zeitschrift für Physik, 80(11):763–791, Nov 1933.
[53] J. Luttinger. The effect of a magnetic field on electrons in a periodic potential. Phys. Rev., 84(4):814, 1951.
[54] G. H. Wannier. Dynamics of band electrons in electric and magnetic fields. Rev. Mod. Phys., 34:645–655, Oct 1962.
[55] D. R. Hofstadter. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B, 14:2239–2249, Sep 1976.
[56] S.-Y. Lin et al. Feature-rich geometric and electronic properties of carbon nanoscrolls. Nanomaterials, 11(6), 2021.
[57] S. Cho et al. Aharonov–Bohm oscillations in a quasi-ballistic three-dimensional topological insulator nanowire. Nature Communications, 6(1):7634, Jul 2015.
[58] S. S. Hong et al. One-dimensional helical transport in topological insulator nanowire interferometers. Nano Letters, 14(5):2815–2821, May 2014.
[59] E. Zhang et al. Graphene rolls with tunable chirality. Nature Materials, 24(3):377–383, Mar 2025.
[60] S. Qiao et al. One-dimensional MoS₂ nanoscrolls as miniaturized memories. Nano Letters, 24(15):4498–4504, 2024.
[61] S. Yu et al. Transition metal dichalcogenides nanoscrolls: Preparation and applications. Nanomaterials, 13(17), 2023.
[62] J. Shi, K. Cai, L.-N. Liu, and Q.-H. Qin. Self-assembly of a parallelogram black phosphorus ribbon into a nanotube. Scientific Reports, 7(1):12951, Oct 2017.
[63] Y. Wang et al. Highly promoted carrier mobility and intrinsic stability by rolling up monolayer black phosphorus into nanoscrolls. The Journal of Physical Chemistry Letters, 9(23):6847–6852, 2018.
[64] Y. Wang et al. Blue phosphorus nanoscrolls. Phys. Rev. B, 102:165428, Oct 2020.
[65] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu. Classification of topological quantum matter with symmetries. Rev. Mod. Phys., 88:035005, Aug 2016.
[66] S. Carr et al. Twistronics: Manipulating the electronic properties of two-dimensional layered structures through their twist angle. Phys. Rev. B, 95:075420, Feb 2017.