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近年来, 硅烯(单层硅)由于其独特的结构和电子性质以及在量子霍尔效应等领域的潜在应用而成为理论和实验研究的一个热点. 借助于四带次近邻紧束缚模型, 详细计算和研究了硅烯中受电场调制的体能隙和电子能级. 结果表明: 硅烯原胞中的两个子格处于不同的平面上, 可以通过外电场区分和控制这两个子格, 这将破坏在纯石墨烯中无法被破坏的K-K'对称性, 并消除由这一对称性导致的电子能级的二重简并; 外加电场还会引起硅烯中次近邻格点之间的Rashba自旋轨道耦合, 这一作用会在不同狄拉克点有选择地消除电子能级在部分电场点的简并, 相邻能级从交叉状态变为反交叉状态; 电子能级中除一些孤立的交叉点外, 每个能级都具有确定的自旋取向, 石墨烯中电子能级的四重简并在硅烯中被完全消除, 从而导致填充因子ν=0, ±1, ±2, ±3,…的量子霍尔平台.We investigate the electric field controlled energy gap and the Landau levels in silicene in detail. The energy gap at different Dirac points has different closing and reopening conditions and the 2-fold degeneracy induced by the K-K' symmetry is resolved. An externally applied electric field gives rise to two Rashba spin-orbit-couplings between the nearest neighbour and the next nearest neighbour in silicene. Both these couplings can resolve the spin degeneracy at some isolated values of the electric field, where the crossover of the successive Landau levels become anti-crossover. Except some special values of the electric field, the 4-fold degeneracy of energy levels associated with the K-K' symmetry and spin symmetry is completely resolved in silicene, each level has a definite spin polarization, which correspond to the quantum Hall plateaux with filling factor ν=0, ±1, ±2,….
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Keywords:
- silicene /
- Landau levels /
- degeneracy /
- spin-orbit-couplings
[1] Meyer J C, Geim A K, Katsnelson M I, Novoselov K S, Booth T J, Roth S 2007 Nature 446 60
[2] Novoselov K S, Geim A K, Morozov S V, Jiang D, Grigorieva M I K I V, Dubonos S V, Firsov A A 2005 Nature 438 197
[3] Rowlands D A, Zhang Y Z 2014 Chin. Phys. B 23 037101
[4] Liu C C, Feng W, Yao Y 2011 Phys. Rev. Lett. 107 076802
[5] Liu C C, Jiang H, Yao Y 2011 Phys. Rev. B 84 195430
[6] Wang S K, Tian H Y, Yang Y H, Wang J 2014 Chin. Phys. B 23 017203
[7] Vogt P, De Padova P, Quaresima C, Avila J, Frantzwskakis E, Asensio M C, Resta A, Ealet B, Lay G L 2012 Phys. Rev. Lett. 108 155501
[8] Lin C L, Arafune R, Kawahara K, Tsukahara N, Minamitani E, Kim Y, Takagi N, Kawai M 2012 Appl. Phys. Express 5 045802
[9] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
[10] Hasan M Z, Kane C 2010 Rev. Mod. Phys. 82 3045
[11] Onoda M, Nagaosa N 2003 Phys. Rev. Lett. 90 206601
[12] Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2008 Phys. Rev. Lett. 101 146802
[13] Qiao Z, Jiang H, Li X, Yao Y, Niu Q 2012 Phys. Rev. B 85 115439
[14] Prodan E 2009 Phys. Rev. B 80 125327
[15] Sheng D N, Weng Z Y, Sheng L, Haldane F D M 2006 Phys. Rev. Lett. 97 036808
[16] Yang Y, Xu Z, Sheng L, Wang B, Xing D Y, Sheng D N 2011 Phys. Rev. Lett. 107 066602
[17] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801
[18] Novoselov K S, Jiang Z, Zhang Y, Morozov S V, Stormer H L, Zeitler U, Maan J C, Boe-binger G S, Kim P, Geim A K 2007 Science 315 1379
[19] Ezawa M 2012 Phys. Rev. Lett. 109 055502
[20] Ezawa M 2013 Phys. Rev. B 87 155415
[21] Li Z J, Li Q, Cheng Z G, Li H B, Fang Y 2014 Chin. Phys. B 23 028102
[22] Fleurence A, Friedlein R, Ozaki T, Wang Y, Yamada-Takamura Y 2012 Phys. Rev. Lett. 108 245501
[23] Deacon R S, Chuang K C, Nicholas R J, Novoselov K S, Geim A K 2007 Phys. Rev. B 76 081406(R)
[24] Li G, Andrei E Y 2007 Nat. Phys. 3 623
[25] Matsui T, Kambara H, Niimi Y, Tagami K, Tsukada M, Fukuyama H 2005 Phys. Rev. Lett. 94 226403
[26] Pan H, Li Z S, Liu C C, Zhu G B, Qiao Z H, Yao Y G 2014 Phys. Rev. Lett. 112 106802
[27] Zhang Z Z, Chang K, Peeters F M 2008 Phys. Rev. B 77 235411
[28] Zheng Y, Ando T 2002 Phys. Rev. B 65 245420
[29] Krstajić P M, Vasilopoulos P 2012 Phys. Rev. B 86 115432
[30] Qiao Z, Yang S A, Feng W, Tse W K, Ding J, Yao Y, Wang J, Niu Q 2010 Phys. Rev. B 82 161414(R)
[31] Ezawa M 2012 New J. Phys. 14 033003
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[1] Meyer J C, Geim A K, Katsnelson M I, Novoselov K S, Booth T J, Roth S 2007 Nature 446 60
[2] Novoselov K S, Geim A K, Morozov S V, Jiang D, Grigorieva M I K I V, Dubonos S V, Firsov A A 2005 Nature 438 197
[3] Rowlands D A, Zhang Y Z 2014 Chin. Phys. B 23 037101
[4] Liu C C, Feng W, Yao Y 2011 Phys. Rev. Lett. 107 076802
[5] Liu C C, Jiang H, Yao Y 2011 Phys. Rev. B 84 195430
[6] Wang S K, Tian H Y, Yang Y H, Wang J 2014 Chin. Phys. B 23 017203
[7] Vogt P, De Padova P, Quaresima C, Avila J, Frantzwskakis E, Asensio M C, Resta A, Ealet B, Lay G L 2012 Phys. Rev. Lett. 108 155501
[8] Lin C L, Arafune R, Kawahara K, Tsukahara N, Minamitani E, Kim Y, Takagi N, Kawai M 2012 Appl. Phys. Express 5 045802
[9] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
[10] Hasan M Z, Kane C 2010 Rev. Mod. Phys. 82 3045
[11] Onoda M, Nagaosa N 2003 Phys. Rev. Lett. 90 206601
[12] Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2008 Phys. Rev. Lett. 101 146802
[13] Qiao Z, Jiang H, Li X, Yao Y, Niu Q 2012 Phys. Rev. B 85 115439
[14] Prodan E 2009 Phys. Rev. B 80 125327
[15] Sheng D N, Weng Z Y, Sheng L, Haldane F D M 2006 Phys. Rev. Lett. 97 036808
[16] Yang Y, Xu Z, Sheng L, Wang B, Xing D Y, Sheng D N 2011 Phys. Rev. Lett. 107 066602
[17] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801
[18] Novoselov K S, Jiang Z, Zhang Y, Morozov S V, Stormer H L, Zeitler U, Maan J C, Boe-binger G S, Kim P, Geim A K 2007 Science 315 1379
[19] Ezawa M 2012 Phys. Rev. Lett. 109 055502
[20] Ezawa M 2013 Phys. Rev. B 87 155415
[21] Li Z J, Li Q, Cheng Z G, Li H B, Fang Y 2014 Chin. Phys. B 23 028102
[22] Fleurence A, Friedlein R, Ozaki T, Wang Y, Yamada-Takamura Y 2012 Phys. Rev. Lett. 108 245501
[23] Deacon R S, Chuang K C, Nicholas R J, Novoselov K S, Geim A K 2007 Phys. Rev. B 76 081406(R)
[24] Li G, Andrei E Y 2007 Nat. Phys. 3 623
[25] Matsui T, Kambara H, Niimi Y, Tagami K, Tsukada M, Fukuyama H 2005 Phys. Rev. Lett. 94 226403
[26] Pan H, Li Z S, Liu C C, Zhu G B, Qiao Z H, Yao Y G 2014 Phys. Rev. Lett. 112 106802
[27] Zhang Z Z, Chang K, Peeters F M 2008 Phys. Rev. B 77 235411
[28] Zheng Y, Ando T 2002 Phys. Rev. B 65 245420
[29] Krstajić P M, Vasilopoulos P 2012 Phys. Rev. B 86 115432
[30] Qiao Z, Yang S A, Feng W, Tse W K, Ding J, Yao Y, Wang J, Niu Q 2010 Phys. Rev. B 82 161414(R)
[31] Ezawa M 2012 New J. Phys. 14 033003
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