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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.
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Keywords:
- the tight-binding model /
- zigzag graphene nanoribbons /
- edge state
[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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[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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