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Using the tight binding Kane-Mele model including the self-consistent on-site Coulomb interactions (O-CIs), we study the influence of transverse electric field in the narrow zigzag graphene nanoribbon (ZGNR) plane on the edge band structure in order to investigate the way to control the type of quantum spin Hall (QSH) system in the ZGNR. The theoretical results show that when applying weak electric field intensity, the direction of electric field can adjust these two spin-down edge bands moving along the different directions in one-dimensional q space, which leads to the two different types of degenerative breakdown of two pure spin-down edge states at q=0.5. When applying positive electric field the energy of spin-down edge band at edge site 1 is higher than that at edge site 8. On the contrary, when applying negative electric field the energy of spin-down edge band at edge site 8 is higher than that at edge site 1. It shows that we can use the direction of electric field to control the two spin-down edge currents occurring at two different energies. Further, when the electric field intensity increases above 0.69 V/nm, the increased large band gap between the two spin-down edge bands leads to the inversion of these two spin-down edge bands. That is to say, there is a spin-down band gap, however, there is not a band gap for spin-up edge band in the region of spin-down band gap. Thus the system becomes half-metallic, and the QSH does not belong in the type B any longer. Specially, when the electric field intensity reaches 1.17 V/nm in the region of spin-down band gap, the pure spin-up edge state appears at q=0.5, which shows that the strong pure spin-up edge current along the edge site 8 can occur. With increasing the intensity of electric field, the QSH system undergoes three processes from the type B to the type C. When the electric field intensity is more than 1.42 V/nm, the two spin-up edge bands also present band inversion and turn into the conduction band and the valence band, respectively. Thus the system becomes semiconducting and the QSH system does not belong in the type C, ordinary quantum Hall system. Finally, according to the results discussed above, we can expect that using the direction and the intensity of the transverse electric field in plane we can adjust the properties of edge current, and control the type of QSH system varying from the type B to the type C.
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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] 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
[3] Wallace P R 1947 Phys. Rev. 71 622
[4] Neto A H C, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109
[5] Fujita M, Wakabayashi K, Nakada K, Kusakabe K 1996 J. Phys. Soc. Jpn. 65 1920
[6] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801
[7] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802
[8] Dresselhaus G F, Dresselhaus M S, Mavroides J G 1966 Carbon 4 433
[9] Min H, Hill J E, Sinitsyn N A, Sahu B R, Kleinman L, MacDonald A H 2006 Phys. Rev. B 74 165310
[10] Qiao Z, Wang J 2007 Nanotechnology 18 435402
[11] Zarea M, Sandler N 2009 Phys. Rev. B. 79 165442
[12] Zarea M, Sandler N 2009 Phys. B: Condens. Matter 404 2694
[13] Hatsugai Y 1993 Phys. Rev. Lett. 71 3697
[14] Xu X Z, Yu J C, Zhang Z H, Liu K H 2017 Chin. Sci. Bull. 62 2220(in Chinese) [徐小志, 余佳晨, 张智宏, 刘开辉 2017 科学通报 62 2220]
[15] Zhang Y B, Tang T T, Girit C, Hao Z, Martin M C, Zettl A, Crommie M F, Shen Y R, Wang F 2009 Nature 459 820
[16] Son Y W, Cohen M L, Louie S G 2006 Nature 444 347
[17] Wang Z G, Hao N N, Zhang P 2009 Phys. Rev. B 80 115420
[18] Li H C, Sheng L, Xing D Y 2012 Phys. Rev. Lett. 108 196806
[19] Guo J, Gunlycke D, White C T 2008 Appl. Phys. Lett. 92 163109
[20] Gunlycke D, Areshkin D A, Li J, Mintmire J W, White C T 2007 Nano Lett. 7 3608
[21] Hu B, Liu N, Liu H 2018 Journal of Nanjing Normal University 41 42 [胡边, 刘娜, 刘红 2018 南京师范大学学报 41 42]
[22] Liu H, Hu B, Liu N 2016 Phys. Lett. A 380 3738
[23] Sheng L, Sheng D N, Ting C S, Haldane F D M 2005 Phys. Rev. Lett. 95 136602
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