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在紧束缚近似下, 提出有限系统的Bloch定理方法, 解析计算了Zigzag型石墨烯纳米带的电子态和能带.研究发现, 其电子态有两类, 分别是驻波态和边缘态; 驻波态的波矢为实数, 波函数是正弦函数形式; 边缘态的波矢主要是虚数, 实数部分为零或者π/2, 波函数是双曲正弦函数形式. Zigzag型石墨烯纳米带的能带由驻波态能量和边缘态能量组成, 我们推导了边缘态的关于无限长方向波矢和能量的精确取值范围. 讨论了边缘态和驻波态的过渡点, 发现两种电子态通过不同的方式在受限波矢趋于零时关于格点位置逼近线性关系. 当受限方向也变成无限长时, 可以得到与无限大石墨烯相同的能带关系.
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关键词:
- 紧束缚模型 /
- Zigzag型石墨烯纳米带 /
- 边缘态
Based on the tight-binding model, the electronic state and band of zigzag graphene nanoribbons are given analytically by a new method. The results show that there are only two kinds electronic states, i.e., the standing wave state and edge state. For the standing wave state, the wave function is sine function and the vector is real; for the edge state, the wave function is hyperbolic sine function and the vector is complex, whose real part is 0 or π/2. The energy band is composed of the energy of standing wave state and the energy of edge state. The accurate ranges of infinite direction wave vector and energy of the edge state are deduced. Then we discuss the transition point between the edge state and the standing wave state and find that the two kinds of electronic states tend to the linear relationship regarding the site of carbon lattice in different ways at the phase transition point. When the width of two restricted boundary goes to infinity, the result of the limited graphene tends to the infinite case.-
Keywords:
- the tight-binding model /
- zigzag graphene nanoribbons /
- edge state
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[17] Ren S Y 2001 Phys. Rev. B 64 035322
[18] Ren S Y 2002 Ann. Phys. (N. Y.) 301 22
[19] Ren S Y 2003 Europhys. Lett. 64 783
[20] Zhang S B, Yeh C Y, Zunger A 1993 Phys. Rev. B 48 11204
[21] Zhang S B, Zunger A 1993 Appl. Phys. Lett. 63 1399
[22] Ajoy A, Karmalkar S 2010 J. Phys. Condens. Matter 22 435502
[23] Jin Z F, Tong G P, Jiang Y J 2009 Acta Phys. Sin. 58 8537 (in Chinese) [金子飞, 童国平, 蒋永进 2009 58 8537]
[24] Wallace P R 1947 Phys. Rev. 71 622
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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] Das Sarma S, Adam S, Hwang E H 2011 Rev. Mod. Phys. 83 407
[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] Geim A K, Novoselov K S 2007 Nat. Mater. 6 183
[5] Katsnelson M I, Novoselov K S 2007 Solid State Commun. 143 3
[6] Katsnelson M I 2007 Mater. Today 10 20
[7] Novoselov K S, Jiang D, Schedin F, Booth T J, Khotkevich V V, Morozov S V, Geim A K 2005 Proc. Nat. Acad. Sci. USA 102 10451
[8] 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
[9] Liang X G, Fu Z L, Chou S Y 2007 Nano Lett. 7 3840
[10] Li D, Mueller M B, Gilje S, Kaner R B, Wallace G G 2008 Nat. Nanotechnol. 3 101
[11] Klein D J 1994 Chem. Phys. Lett. 217 261
[12] Son Y W, Cohen M L, Louie S G 2006 Phys. Rev. Lett. 97 216803
[13] Son Y W, Cohen M L, Louie S G 2006 Nature 444 347
[14] Sasaki K, Murakami S, Saito R 2006 Appl. Phys. Lett. 88 113110
[15] Wakabayashi K, Sasaki K, Nakanishi T, Enoki T 2010 Sci. Technol. Adv. Mater. 11 054504
[16] Ren S Y 2006 Electronic States in Crystals of Finite Size-Quantum Confinement of Bloch Waves (Beijing: Peking University Press) pp15-19 (in Chinese) [任尚元 2006 有限晶体中的电子态–-Bloch波的量子限域 (北京: 北京大学出版社) 第15–19页]
[17] Ren S Y 2001 Phys. Rev. B 64 035322
[18] Ren S Y 2002 Ann. Phys. (N. Y.) 301 22
[19] Ren S Y 2003 Europhys. Lett. 64 783
[20] Zhang S B, Yeh C Y, Zunger A 1993 Phys. Rev. B 48 11204
[21] Zhang S B, Zunger A 1993 Appl. Phys. Lett. 63 1399
[22] Ajoy A, Karmalkar S 2010 J. Phys. Condens. Matter 22 435502
[23] Jin Z F, Tong G P, Jiang Y J 2009 Acta Phys. Sin. 58 8537 (in Chinese) [金子飞, 童国平, 蒋永进 2009 58 8537]
[24] Wallace P R 1947 Phys. Rev. 71 622
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