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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
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[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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