石墨烯是最廣為人知的二維材料，其新穎的物理與化學特性被學界與業界認為 是具有高度潛力的新穎材料，基於石墨烯的各類衍生結構研究也被如火如荼的發展 當中，奈米卷軸作為徑向超晶格結構的一種分類，而本篇將著重於石墨烯奈米捲軸 的理論模型開發，並基於此模型計算，展現奈米卷軸的新穎物理性質。

在計算材料物理領域中，緊束縛模型是一種常被使用的方法，許多學者基於緊 束縛模型得出與實驗高度吻合的結果。而隨著計算機科學的進展，Python 此語言 在簡潔度與開源性都被大眾所認可，而成為一種廣泛的接口，許多應用領域都將 Python 作為主要編寫的介面。本篇將以 Python 上的緊束縛模型套件「Kwant」為基 礎，建構石墨烯奈米卷軸的模型，並計算其電導特性。

基於新穎的設計理念，在此模型的設計中，我們將準二維的奈米結構重新設計， 轉至二維平面上重新建模。得益於此方法，我們避開了建模初期的大規模的參數設 置，也使此模型未來的拓展更具有彈性。在此模型中，我們不但能對模型施加位能 與向量勢以外，我們亦能在此模型中施加雜質，期待能確認量子態對材料不完美的 堅韌性的同時，也能給予實驗學者能對應的結果。我們設計了不同種類的雜質分佈 方法，例如:普通隨機雜質、層間耦合強度不均、介面上雜質等多種方法。在我們 的計算中，主要採取普通隨機與介面兩種方法，以利後人實驗證實奈米卷軸的新穎特性。

我們也比較了石墨烯奈米帶與奈米卷軸的差異，而我們也發現在奈米帶過渡至 奈米卷軸時，基態量子電導會急劇上升(至少 200%)，此結果僅僅需要改變石墨烯 的幾何結構便能得出，被我們視為是非常具有潛力的未來材料。同時我們也發現， 在緊束縛模型中代表層間交互作用的變量，在我們給予不同的正負號時，其波函數 與色散關係皆等價於施加了半個 Aharonov-Bohm effect 週期的磁通量量值。且在施加 半週期磁通量後，奈米卷軸的傳輸能力更加驚人，可承受相當強度的雜質且仍保持極好的傳輸特性，我們認為此一特性可能與材料的拓樸性質高度相關。

本研究基於 Kwant 給出相當具有彈性的量子傳輸模型，可應用於大多數的情況， 亦能依據實驗學者要求進行不同的增減，後人也能擴充成為其他相似徑向超晶格結 構。後續延伸研究亦能基於這些計算結果的引導，發掘奈米卷軸的眾多可能。

Graphene is the most well-known two-dimensional material, and its novel physical and chem- ical properties are considered by academia and industry to have high potential as an innova- tive material. Various research on graphene-based derivative structures is also actively being developed, with nanoscrolls being a category of radial superlattice structures. We focus on the theoretical model development of graphene nanoscrolls and, based on this model calcu- lation, demonstrates the novel physical properties of nanoscrolls.

In the field of computational materials physics, the tight-binding model is a commonly used method, and many scholars obtain results that are highly consistent with experiments based on the tight-binding model. With the advancement of computer science, Python is widely recognized for its conciseness and open-source nature, becoming a widely used interface in many application fields. In this article, we build a model of graphene nanoscrolls based on the Python tight-binding package “Kwant"and calculate their conductance properties.

Based on a novel design concept, we redesign the quasi-two-dimensional nanostructure to be modeled on a two-dimensional plane. This method avoids large-scale parameter settings at the initial stage of modeling and makes the model more flexible for future expansion. In this model, we can apply potential and vector potential, as well as introduce impurities, to con- firm the robustness of the quantum state to material imperfections and provide corresponding results for experimental scholars. We design various types of impurity distribution methods, such as ordinary random impurities, uneven interlayer coupling strength, and interface im- purities. In our calculations, we mainly adopt ordinary random and interface methods to facilitate the experimental verification of the novel properties of nanoscrolls.

We also compare the differences between graphene nanoribbons and nanoscrolls and find that when nanoribbons transition to nanoscrolls, the ground state quantum conductance in- creases sharply (at least 200%). This result can be obtained simply by changing the geo- metric structure of graphene, and we consider it a very promising future material. At the same time, we find that in the tight-binding model, the variable representing interlayer in- teraction, when given different positive and negative signs, is equivalent to applying half of an Aharonov-Bohm effect period of magnetic flux value to the wave function and dispersion relation. Moreover, after applying half a period of magnetic flux, the transmission capability of the nanoscroll becomes even more astonishing, capable of withstanding considerable im- purity strength while maintaining excellent transmission characteristics. We believe that this feature may be highly related to the topological properties of the material.

This study is based on the Kwant quantum transport model, which is quite flexible and can be applied in most situations. It can also be modified according to the requirements of exper- imental scholars, and we look forward that future researchers can expand it into other similar radial superlattice structures. Subsequent research can also be guided by these calculation results to explore the many possibilities of nanoscrolls.

[1] Andre Konstantin Geim and Konstantin Novoselov. The rise of graphene. Nature Materials, 6(3):183–191, 2007.

[2] Sumio Iijima. Helical microtubules of graphitic carbon. Nature, 354(6348):56–58, 1991.

[3] Singh Rohitkumar Shailendra and Valayathoor Narayanan Ramakrishnan. Gaa-cntfet based single/dual-channel and single/dual-chirality digital gates for high speed and low power application. In 2018 4th International Conference on Devices, Circuits and Sys- tems (ICDCS), pages 174–180, 2018.

[4] Xu Xie, Long Ju, Xiaofeng Feng, Yinghui Sun, Ruifeng Zhou, Kai Liu, Shoushan Fan, Qunqing Li, and Kaili Jiang. Controlled fabrication of high-quality carbon nanoscrolls from monolayer graphene. Nano Letters, 9(7):2565–2570, 07 2009.

[5] Anton R Akhmerov Christoph W Groth, Michael Wimmer and Xavier Waintal. Kwant: a software package for quantum transport. New J. Phys., 16:063065, June 2014.

[6] Daniel S. Fisher and Patrick A. Lee. Relation between conductivity and transmission matrix. Phys. Rev. B, 23:6851–6854, Jun 1981.

[7] Philip Russell Wallace. The band theory of graphite. Phys. Rev., 71:622–634, May 1947.

[8] Sheng-Chin Ho, Ching-Hao Chang, Yu-Chiang Hsieh, Shun-Tsung Lo, Botsz Huang, Thi-Hai-Yen Vu, Carmine Ortix, and Tse-Ming Chen. Hall effects in artificially corru- gated bilayer graphene without breaking time-reversal symmetry. Nature Electronics, 4(2):116–125, 2021.

[9] Yitian Wang, Chenghuan Jiang, Qian Chen, Qionghua Zhou, Haowei Wang, Jianguo Wan, Liang Ma, and Jinlan Wang. Highly promoted carrier mobility and intrinsic stabil- ity by rolling up monolayer black phosphorus into nanoscrolls. The Journal of Physical Chemistry Letters, 9(23):6847–6852, 12 2018.

[10] Xueping Cui, Zhizhi Kong, Enlai Gao, Dazhen Huang, Yang Hao, Hongguang Shen, Chong-an Di, Zhiping Xu, Jian Zheng, and Daoben Zhu. Rolling up transition metal dichalcogenide nanoscrolls via one drop of ethanol. Nature Communications, 9(1):1301, 2018.

[11] Philip Warren Anderson. Absence of diffusion in certain random lattices. Phys. Rev., 109:1492–1505, Mar 1958.

[12] Tongcang Li and Shao-Ping Lu. Quantum conductance of graphene nanoribbons with edge defects. Phys. Rev. B, 77:085408, Feb 2008.

[13] Supriyo Datta. Electronic Transport in Mesoscopic Systems. Cambridge Studies in Semiconductor Physics and Microelectronic Engineering. Cambridge University Press, 1995.

[14] Charles R. Harris, K. Jarrod Millman, Stéfan J. van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, Robert Kern, Matti Picus, Stephan Hoyer, Marten H. van Kerkwijk, Matthew Brett, Allan Haldane, Jaime Fernández del Río, Mark Wiebe, Pearu Peterson, Pierre Gérard-Marchant, Kevin Sheppard, Tyler Reddy, Warren Weckesser, Hameer Abbasi, Christoph Gohlke, and Travis E. Oliphant. Array programming with numpy. Nature, 585(7825):357–362, 2020.

[15] Riichiro Saito, Gene Dresselhaus, and Mildred S Dresselhaus. Physical Properties of Carbon Nanotubes. PUBLISHED BY IMPERIAL COLLEGE PRESS AND DIS- TRIBUTED BY WORLD SCIENTIFIC PUBLISHING CO., 1998.

[16] Dean Moldovan, Miša Anđelković, and Francois Peeters. pybinding v0.9.5: a Python package for tight- binding calculations, August 2020. This work was supported by the Flemish Science Foundation (FWO-Vl) and the Methusalem Funding of the Flemish Government.

[17] Yu-Jie Zhong, Angus Huang, Hui Liu, Xuan-Fu Huang, Horng-Tay Jeng, Jhih-Shih You, Carmine Ortix, and Ching-Hao Chang. Magnetoconductance modulations due to interlayer tunneling in radial superlattices. Nanoscale Horiz., 7:168–173, 2022.

[18] Edward McCann and Mikito Koshino. The electronic properties of bilayer graphene. Reports on Progress in Physics, 76(5):056503, apr 2013.

[19] Georg Kresse and Jürgen Furthmüller. Efficient iterative schemes for ab initio total- energy calculations using a plane-wave basis set. Phys. Rev. B, 54:11169–11186, Oct 1996.

[20] GeorgKresseandJürgenHafner.Abinitiomoleculardynamicsforopen-shelltransition metals. Phys. Rev. B, 48:13115–13118, Nov 1993.

[21] David M Ceperley and Berni J Alder. Ground state of the electron gas by a stochastic method. Phys. Rev. Lett., 45:566–569, Aug 1980.

[22] NguyenH.Le,AndrewJ.Fisher,NeilJ.Curson,andEranGinossar.Topologicalphases of a dimerized fermi–hubbard model for semiconductor nano-lattices. npj Quantum Information, 6(1):24, 2020.

[23] Jens H Bardarson, Piet Brouwer, and Joel E Moore. Aharonov-bohm oscillations in disordered topological insulator nanowires. Phys. Rev. Lett., 105:156803, Oct 2010.

[24] Rudolf Peierls. Zur theorie des diamagnetismus von leitungselektronen. Zeitschrift für Physik, 80(11):763–791, 1933.

[25] Akin Akturk and Neil Goldsman. Electron transport and full-band electron-phonon interactions in graphene. Journal of Applied Physics, 103(5), 03 2008. 053702.

[26] Jian-Hao Chen, Chaun Jang, Shudong Xiao, Masa Ishigami, and Michael S. Fuhrer. Intrinsic and extrinsic performance limits of graphene devices on sio2. Nature Nan- otechnology, 3(4):206–209, 2008.

[27] Andre Konstantin Geim. Graphene: Status and prospects. Science, 324(5934):1530– 1534, 2009.

[28] Edward McCann and Mikito Koshino. The electronic properties of bilayer graphene. Rep. Prog. Phys., 76:056503, 2013.