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The properties of the edge states in the topological insulator InAs/GaSb/AlSb quantum well in the preflence of a perpendicular magnetic field are studied numerically. The effect of the magnetic field is included in our model by adding an on-site Zeeman term and a vector potential to the electron wave vector: k+eA. When the material is in the topologically nontrivial state, a pair of degenerate counter-propagating spin-polarized edge states exist in the bulk band gap on each edge of the sample, which are gapless in the absence of the magnetic field due to the protection of the time reflersal symmetry. #br#Nonzero magnetic field breaks the time reflersal symmetry, and leads to Landau levels in the electron energy spectrum. However, one can still find a pair of counter-propagating spin-polarized edge states in the bulk energy gap near each sample boundary.The edge states are gapped, and their distributions relative the sample edge depend on the strength of the magnetic field. With the increase of the magnetic field, one edge state remains located near the sample boundary, but the other tends to evolve into the bulk gradually. Furthermore, we study the scattering between the two edge states caused by impurities. We show that the scattering rate is suppressed because of the spatial separation of two edge states, and shows no significant enhancement when the magnetic field increases, which suggests that even though the time reflersal symmetry is broken, the quantum spin Hall state remains to be relatively robust.
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Keywords:
- topological insulator /
- quantum spin hall effect /
- landau levels /
- edge state
[1] Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494
[2] Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405
[3] Haldane F D M 1988 Phys. Rev. Lett. 45 61
[4] C L Kane, E J Mele 2005 Phys. Rev. Lett. 95 226801
[5] Bernevig B A, Mele E J 2005 Phys. Rev. Lett. 96 106802
[6] Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757
[7] König M, Wiedmann S, Brne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766
[8] Wu C, Bernevig B A, Zhang S C 2006 Phys. Rev. Lett. 96 106401
[9] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801
[10] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802
[11] Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757
[12] Sheng L, Sheng D N, Ting C S, Haldane F D M 2005 Phys. Rev. Lett. 95 136602
[13] Sheng D N, Weng Z Y, Sheng L, Haldane F D M 2006 Phys. Rev. Lett. 97 036808
[14] Prodan E 2009 Phys. Rev. B 80 125327
[15] Li H C, Sheng L, Sheng D N, Xing D Y 2010 Phys. Rev. B 82 165104
[16] Prodan E 2010 New J. Phys. 12 065003
[17] Du L J, Knez I, Sullivan G, Du R R 2013 arXiv:1306.1925. http:arxiv.orgabs1306.1925
[18] Liu C X, Hughes T L, Qi X L, Wang K, Zhang S C 2008 Phys. Rev. Lett. 100 236601
[19] Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2008 Phys. Rev. Lett. 101 146802
[20] Yang Y, Xu Z, Sheng L, Wang B G, Xing D Y 2011 Phys. Rev. Lett. 107 066602
[21] Li H C, Sheng L, Xing D Y 2012 Phys. Rev. Lett. 108 196806
[22] Li H C, Sheng L, Shen R, Wang B G, Sheng D N, Xing D Y 2013 Phys. Rev. Lett. 110 266802
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[1] Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494
[2] Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405
[3] Haldane F D M 1988 Phys. Rev. Lett. 45 61
[4] C L Kane, E J Mele 2005 Phys. Rev. Lett. 95 226801
[5] Bernevig B A, Mele E J 2005 Phys. Rev. Lett. 96 106802
[6] Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757
[7] König M, Wiedmann S, Brne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766
[8] Wu C, Bernevig B A, Zhang S C 2006 Phys. Rev. Lett. 96 106401
[9] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801
[10] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802
[11] Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757
[12] Sheng L, Sheng D N, Ting C S, Haldane F D M 2005 Phys. Rev. Lett. 95 136602
[13] Sheng D N, Weng Z Y, Sheng L, Haldane F D M 2006 Phys. Rev. Lett. 97 036808
[14] Prodan E 2009 Phys. Rev. B 80 125327
[15] Li H C, Sheng L, Sheng D N, Xing D Y 2010 Phys. Rev. B 82 165104
[16] Prodan E 2010 New J. Phys. 12 065003
[17] Du L J, Knez I, Sullivan G, Du R R 2013 arXiv:1306.1925. http:arxiv.orgabs1306.1925
[18] Liu C X, Hughes T L, Qi X L, Wang K, Zhang S C 2008 Phys. Rev. Lett. 100 236601
[19] Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2008 Phys. Rev. Lett. 101 146802
[20] Yang Y, Xu Z, Sheng L, Wang B G, Xing D Y 2011 Phys. Rev. Lett. 107 066602
[21] Li H C, Sheng L, Xing D Y 2012 Phys. Rev. Lett. 108 196806
[22] Li H C, Sheng L, Shen R, Wang B G, Sheng D N, Xing D Y 2013 Phys. Rev. Lett. 110 266802
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