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用数值方法研究了拓扑绝缘体薄膜体系在外加垂直磁场 作用下其边缘态的性质. 磁场的加入通过耦合k+eA, 即Peierls势替换关系和 该作用导致的Zeeman交换场体现在哈密顿量中. 考虑窄条圆环状结构的二维InAs/GaSb/AlSb薄膜量子阱材料, 当其处于拓扑非平庸状态, 即量子自旋霍尔态时, 会出现受时间反演对称性保护的两支简并边缘态, 而在垂直磁场的作用下, 时间反演对称性被破坏, 这时能带将形成一条条的朗道能级, 原来简并的两支边缘态也会分开到朗道能级谱线的两侧, 从电子态密度的空间分布情况则可以看到边缘态分别局域在材料的两个边界. 随着磁场的增大, 位于同一边界上的不同 自旋极化的边缘态将出现分离: 一支仍然局域在边缘, 另一支则随外加磁场的增加而有逐渐演化到材料内部的趋势. 文中还计算了同一边界上的两支边缘态之间的散射, 结果表明由于两个边缘态在空间发生分离, 相互之间的散射被很大的压制, 得到了其散射随磁场增加没有明显变化的结论, 所以磁场并不会增强散射过程, 也没有破坏体拓扑材料的性质, 说明了量子自旋霍尔态在没有时间反演对称的情况下也可以有较强的稳定性.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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