近年來，二維結構材料由於展現出獨特的物理特性，成為許多科學家的研究對象。石墨烯，這個最早理論提出和實驗驗證的二維材料，是許多研究二維材料的基礎。本碩士論文以緊束模型( tight-binding model ) 計算能帶結構，並且藉由隨機相位近似 ( Random Phase Approximation ) 方法，以模擬單層和多層有摻雜石墨烯系統的庫侖激發 ( Coulomb excitation )。

電子的庫侖激發必須符合動量和能量守恆以及費米-狄拉克分布，可分為單粒子的電子電洞激發和集體粒子的電漿子激發，而這些現象會在激發譜中顯現出來。激發譜的特性會受到轉移動量、激發頻率和費米能的大小等變數所影響。此外，對於多層的系統，堆疊層數和堆疊序列也會影響層間的原子相互作用和電子庫侖作用，造成激發譜的改變。這些預測可以由非彈性光學散射光譜儀和電子能量散逸光譜儀所驗證，而且對於多體物理現象的了解有所用處。

In recent years, the two-dimensional structure materials have become many scientists’ objectives of the study due to its unique physical properties. Graphene, the first two-dimensional condensed matter to be researched theoretically and verified experimentally, is the foundation for studying two-dimensional materials. In this thesis, I will use the tight-binding model to calculate band structures, and use the Random Phase Approximation method (RPA) to simulate the Coulomb excitations of monolayer and multilayer doped graphene systems.

The Coulomb excitations of electrons which obey the momentum and energy conservations and Fermi-Dirac distribution, can be classified as electron-hole single particle excitations and plasmon excitations. These phenomena would be shown in the excitation spectra. The properties of excitation spectra are influenced by the variables, such as transferred momentum, excitation frequency and Fermi energy. Besides, for multilayer systems, the number of layers and the stacking sequence have an impact on the interlayer atomic and electronic Coulomb interactions, and result in the changes in the excitation spectra. These predictions can be verified by the inelastic light scattering spectroscopy and the electron energy loss spectroscopy (EELS), and are useful for understanding many-body physics.

[1] P. R. Wallace, The Band Theory of Graphite. Phys. Rev. 71, 9 (622-634) (1947)

[2] K. W. –K. Shung, Dielectric function and plasmon structure of stage-1 intercalated graphite. Phys. Rev. B 34, 979 (1986)

[3] M. F. Lin, K. W. –K. Shung, Self-energy of electrons in graphite intercalation com-pounds. Phys. Rev. B 53, 1109 (1996)

[4] V. P. Gusynin, S. G. Sharapov, Unconventional Integer Quantum Hall Effect in Gra-phene. Phys. Rev. Lett. 95, 146801 (2005)

[5] Castro Neto, A. H., Guinea, et al., The electronic properties of graphene. Rev. Mod. Phys. 81, 109 (2009)

[6] Saito, R., Fujita, et al., Electronic structure of graphene tubules based on C60. Phys. Rev. B. 46, 1804 (1992)

[7] Tahir, M., Manchon, et al., Quantum spin/valley Hall effect and topological insulator phase transitions in silicone. Appl. Phys. Lett. 102, 162412 (2013)

[8] Lu, C. L., Chang, C. P., et al., Influence of an electric field on the optical properties of few-layer graphene with AB stacking. Phys. Rev. B 73, 144427 (2006)

[9] Koshino, M. & Ando, T., Magneto-optical properties of multilayer graphene. Phys. Rev. B 77, 115313 (2008)

[10] K. S. Novoselov, A. K. Geim, et al., Electric Field Effect in Atomically Thin Carbon Films. Science 306, 5696 (666-669) (2004)

[11] Yuanbo Zhang, Yan-Wen Tan, et al., Experimental observation of the quantum Hall effect and Berry's phase in graphene. Nature 438, 201-204 (2005)

[12] Kane, C. L. & Mele, E. J., Quantum Spin Hall Effect in Graphene. Phys. Rev. Lett. 95, 226801 (2005)

[13] Bolotin, K. I. et al., Ultrahigh electron mobility in suspended graphene. Solid State Commun 146, 351 (2008)

[14] Berger, C. et al., Electronic Confinement and Coherence in Patterned Epitaxial Gra-phene. Science 312, 5777, 1191 (2006)

[15] Hwang, E. H., Adam, et al., Carrier Transport in Two-Dimensional Graphene Layers. Phys. Rev. Lett. 98, 186806 (2007)

[16] Casiraghi, C. et al., Rayleigh Imaging of Graphene and Graphene Layers. Nano Lett. 7, 2711 (2011)

[17] Scalisea, E. et al., Vibrational properties of epitaxial silicene layers on (111) Ag. Appl. Surf. Sci. 291, 113 (2014)

[18] Yuanbo Zhang, Tsung-Ta Tang, et al., Direct observation of a widely tunable bandgap in bilayer graphene. Nature 459, 820–23 (2009)

[19] L. Zhang, P. Bampoulis, et al., Structural and Electronic Properties of Germanene on MoS2. Phys. Rev. Lett. 116, 256804 (2016)

[20] A. Acun, L. Zhang, et al., Germanene: the germanium analogue of graphene. J. Phys. Condens. Matter, Volume 27, Number 44 (2015)

[21] Jijun Zhao, Hongsheng Liu, et al., Rise of silicene: A competitive 2D material. Pro-gress in Metal Physics, Volume 83, Pages 24-151 (2016)

[22] M. Houssa, A. Dimoulas & A. Molle, Silicene: a review of recent experimental and theoretical investigations. J. Phys. Condens. Matter, Volume 27, Number 25 (2015)

[23] Apratim Khandelwal, Karthick Mani, et al., Phosphorene – The two-dimensional black phosphorous: Properties, synthesis and applications. Mater. Sci. Eng. B., Volume 221, Pages 17-34 (2017)

[24] Jelena Klinovaja & Daniel Loss, Spintronics in MoS2 monolayer quantum wires. Phys. Rev. B 88, 075404 (2013)

[25] Ming-Fa Lin & Feng-Lin Shyu, Temperature-Induced Plasmons in a Graphite Sheet. J. Phys. Soc. Jpn. Vol. 69, No. 2 (2000)

[26] C. W. Chiu, S. H. Lee, et al., Electronic excitations in doped monolayer graphenes. J. Appl. Phys 106, 113711 (2009)

[27] S. Y. Shin, N. D. Kim, et al., Control of the π plasmon in a single layer graphene by charge doping. Appl. Phys. Lett. 99, 082110 (2011)

[28] E. H. Hwang & S. Das Sarma, Exotic plasmon modes of double layer graphene. Phys. Rev. B 80, 205405 (2009)

[29] H. Ehrenreich & M. H. Cohen, Self-Consistent Field Approach to the Many-Electron Problem. Phys. Rev. 115, 786 (1959)

[30] J. H. Ho, C. P. Chang, M. F. Lin, Electronic excitations of the multilayered graphite. Phys. Lett. A 352, 446-450 (2006)

[31] M. I. Katsnelson, K. S. Novoselov & A. K. Geim, Chiral tunnelling and the Klein paradox in graphene. Nature Physics 2, 620 - 625 (2006)

[32] Andrea F. Young & Philip Kim, Quantum interference and Klein tunnelling in gra-phene heterojunctions. Nature Physics 5, 222 - 226 (2009)

[33] Wang-Kong Tse, Zhenhua Qiao, et al., Quantum anomalous Hall effect in single-layer and bilayer graphene. Phys. Rev. B 83, 155447 (2011)

[34] A. K. Geim & K. S. Novoselov, The rise of graphene. Nature Materials 6, 183 - 191 (2007)

[35] http://virtuallaboratory.colorado.edu

[36] Kai Yan, Hailin Peng, et al., Formation of Bilayer Bernal Graphene: Layer-by-Layer Epitaxy via Chemical Vapor Deposition. Nano Lett. 11 (3), 1106–1110 (2011)

[37] Zheng Liu, Kazu Suenaga, Peter J. F. Harris & Sumio Iijima, Open and Closed Edges of Graphene Layers. Phys. Rev. Lett. 102, 015501 (2009)

[38] N. W. Ashcroft & N. D. Mermin, Solid State Physics, Toronto: Thomson Learning (1976)

[39] T.O. Wehling, A.M. Black-Schaffer, A.V. Balatsky, Dirac materials. Adv.Phys 63 no.1, 1-76 (2014)

[40] J. H. Ho, C. L. Lu, et al., Coulomb excitations in AA- and AB-stacked bilayer graph-ites, Phys Rev. B 74, 8 (2006)

[41] Ying-Chih Chuang, Jhao-Ying Wu & Ming-Fa Lin, Electric Field Dependence of Ex-citation Spectra in AB-Stacked Bilayer Graphene. Sci Rep. 3, 1368 (2013)

[42] Changhua Bao, Wei Yao, et al., Stacking-Dependent Electronic Structure of Trilayer Graphene Resolved by Nanospot Angle-Resolved Photoemission Spectroscopy. Nano Lett 17 (3), pp 1564–1568 (2017)

[43] B. Partoens & F. M. Peeters, From graphene to graphite: Electronic structure around the K point. Phys. Rev. B 74, 075404 (2006)

[44] Jhao-Ying Wu, Szu-Chao Chen, et al., Plasma Excitations in Graphene: Their Spectral Intensity and Temperature Dependence in Magnetic Field. ACS Nano 2011, 5 (2), pp 1026–1032 (2011)

[45] Po-Hsin Shih, Yu-Huang Chiu, Jhao-Ying Wu, et al., Coulomb excitations of mono-layer germanene. Sci. Rep. 7, 40600 (2016)