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根据π电子的紧束缚模型, 通过有限系统的Bloch定理方法, 解析计算了有限尺寸石墨烯的电子态和能带. 研究发现, 其电子态有且只有两类, 分别是驻波态和边缘态.驻波态时, 波函数形式是两个方向都是正弦函数; 边缘态时, 波函数形式是Armchair边界的方向是双曲正弦函数, Zigzag边界的方向是正弦函数. 其能带由总碳原子数N个离散的本征值组成, 推导了定量计算边缘态的本征值个数的表达式, 并通过态密度来分析边缘态的存在和与无限大情况的一致性. 所有的分析中数值结果与解析理论都完全一致, 当两个受限方向都变成无限长时, 可以得到与无限大石墨烯相同的结果.The limited graphene means that two directions of graphene are limited, one is zigzag type boundary and the other is armchair type boundary. Based on the tight-binding model, the electronic state and band of the limited graphene are given analytically. The results show that there are only two kinds of electronic states, i.e., the standing wave state and edge state. For the standing wave state, the wave function is in the form of sine function in two directions; for the edge state, the wave function is in the form of hyperbolic sine function in the direction of armchair boundary and in the form of sine function in the direction of zigzag boundary. The band is composited of total carbon atom number N discrete eigenvalues. The expression of quantitativly calculating the number of eigenvalues of edge state is deduced. Through the density of states of the limited graphene we analyze the existence of the edge state and the consistency in the infinity case. The results from the analitical method are the same as the numerical resullts. When the width of two restricted boundary goes into infinity, the result of the limited graphene tends to that in the infinity case.
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Keywords:
- the tight-binding model /
- graphene /
- edge state /
- density of states
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[2] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109
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[11] Ezawa M 2006 Phys. Rev. B 73 045432
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[13] Jiang L W, Zheng Y S, Yi C S, Li H D, Lue T Q 2009 Phys. Rev. B 80 155454
[14] Wakabayashi K, Sasaki K, Nakanishi T, Enoki T 2010 Sci. Technol. Adv. Mater. 11 054504
[15] Sasaki K, Murakami S, Saito R 2006 J. Phys. Soc. Jpn. 75 074713
[16] Sasaki K, Murakami S, Saito R 2006 Appl. Phys. Lett. 88 113110
[17] Zheng H X, Wang Z F, Luo T, Shi Q W, Chen J 2007 Phys. Rev. B 75 165414
[18] Brey L, Fertig H A 2006 Phys. Rev. B 73 235411
[19] Fujita M, Wakabayashi K, Nakada K, Kusakabe K 1996 J. Phys. Soc. Jpn. 65 1920
[20] Zhu R, Chen H M 2009 Appl. Phys. Lett. 95 122111
[21] Zhu R, Guo Y 2007 Appl. Phys. Lett. 91 252113
[22] Guo X X, Liu D, Li Y X 2011 Appl. Phys. Lett. 98 242101
[23] Wallace P R 1947 Phys. Rev. 71 622
[24] Bena C, Kivelson S A 2005 Phys. Rev. B 72 125432
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[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666
[2] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109
[3] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, Firsov A A 2005 Nature 438 197
[4] Zhang Y B, Tan Y W, Stormer H L, Kim P 2005 Nature 438 201
[5] Nomura K, MacDonald A H 2006 Phys. Rev. Lett. 96 256602
[6] Brey L, Fertig H A 2006 Phys. Rev. B 73 195408
[7] Katsnelson M I, Novoselov K S, Geim A K 2006 Nat. Phys. 2 620
[8] Rusin T M, Zawadzki W 2008 Phys. Rev. B 78 125419
[9] Rusin T M, Zawadzki W 2009 Phys. Rev. B 80 045416
[10] Berger C, Song Z M, Li X B, Wu X S, Brown N, Naud C, Mayou D, Li T B, Hass J, Marchenkov A N, Conrad E H, First P N, de Heer W A 2006 Science 312 1191
[11] Ezawa M 2006 Phys. Rev. B 73 045432
[12] Klein D J 1994 Chem. Phys. Lett. 217 261
[13] Jiang L W, Zheng Y S, Yi C S, Li H D, Lue T Q 2009 Phys. Rev. B 80 155454
[14] Wakabayashi K, Sasaki K, Nakanishi T, Enoki T 2010 Sci. Technol. Adv. Mater. 11 054504
[15] Sasaki K, Murakami S, Saito R 2006 J. Phys. Soc. Jpn. 75 074713
[16] Sasaki K, Murakami S, Saito R 2006 Appl. Phys. Lett. 88 113110
[17] Zheng H X, Wang Z F, Luo T, Shi Q W, Chen J 2007 Phys. Rev. B 75 165414
[18] Brey L, Fertig H A 2006 Phys. Rev. B 73 235411
[19] Fujita M, Wakabayashi K, Nakada K, Kusakabe K 1996 J. Phys. Soc. Jpn. 65 1920
[20] Zhu R, Chen H M 2009 Appl. Phys. Lett. 95 122111
[21] Zhu R, Guo Y 2007 Appl. Phys. Lett. 91 252113
[22] Guo X X, Liu D, Li Y X 2011 Appl. Phys. Lett. 98 242101
[23] Wallace P R 1947 Phys. Rev. 71 622
[24] Bena C, Kivelson S A 2005 Phys. Rev. B 72 125432
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