Dephasing effects in topological insulators
Junjie Qi, Haiwen Liu, Hua Jiang, X. C. Xie
Dephasing effects in topological insulators
Topological insulators, a class of typical topological materials in both two dimensions and three dimensions,are insulating in bulk and metallic at surface. The spin-momentum locked surface states and peculiar transport properties exhibit promising potential applications on quantum devices, which generate extensive interest in the last decade. Dephasing is the process of the loss of phase coherence, which inevitably exists in a realistic sample. In this review, we focus on recent progress in dephasing effects on the topological insulators. In general, there are two types of dephasing processes: normal dephasing and spin dephasing. In two-dimensional topological insulators, the phenomenologically numerical investigation shows that the longitudinal resistance plateaus is robust against normal dephasing but fragile with spin dephasing. Several microscopic mechanisms of spin dephasing are then discussed. In three-dimensional topological insulators, the helical surface states exhibit a helical spin texture due to the spin-momentum locking mechanism. Thus, normal dephasing has close connection to spin dephasing in this case, and gives rise to anomalous “gap-like” feature. Dephasing effects on properties of helical surface states are investigated.
dephasing effects / topological insulators / backscattering
[1] |
D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall conductance in a twodimensional periodic potential, Phys. Rev. Lett. 49(6), 405 (1982)
CrossRef
ADS
Google scholar
|
[2] |
K. von Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45(6), 494 (1980)
CrossRef
ADS
Google scholar
|
[3] |
K. von Klitzing, 25 years of quantum Hall effects: A personal view on the discovery, physics and applications of this quantum effect, Séminaire Poincaré 2, 1 (2004)
CrossRef
ADS
Google scholar
|
[4] |
D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall conductance in a twodimensional periodic potential, Phys. Rev. Lett. 49(6), 405 (1982)
CrossRef
ADS
Google scholar
|
[5] |
M. Kohmoto, Topological invariant and the quantization of the Hall conductance, Ann. Phys. 160(2), 343 (1985)
CrossRef
ADS
Google scholar
|
[6] |
F. D. M. Haldane, Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett. 61(18), 2015 (1988)
CrossRef
ADS
Google scholar
|
[7] |
C. Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L. L. Wang, Z. Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S. C. Zhang, K. He, Y. Wang, L. Lu, X. C. Ma, and Q. K. Xue, Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator, Science 340(6129), 167 (2013)
CrossRef
ADS
Google scholar
|
[8] |
C. L. Kane and E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett. 95(22), 226801 (2005)
CrossRef
ADS
Google scholar
|
[9] |
C. L. Kane and E. J. Mele, Z2 topological order and the quantum spin Hall effect, Phys. Rev. Lett. 95(14), 146802 (2005)
CrossRef
ADS
Google scholar
|
[10] |
H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Kleinman, and A. H. MacDonald, Intrinsic and Rashba spin– orbit interactions in graphene sheets, Phys. Rev. B 74, 165310 (2006)
CrossRef
ADS
Google scholar
|
[11] |
D. Huertas-Hernando, F. Guinea, and A. Brataas, Spinorbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps, Phys. Rev. B 74(15), 155426 (2006)
CrossRef
ADS
Google scholar
|
[12] |
Y. G. Yao, F. Ye, X.-L. Qi, S.-C. Zhang, and Z. Fang, Spin-orbit gap of graphene: First-principles calculations, Phys. Rev. B 75, 041401(R) (2007)
CrossRef
ADS
Google scholar
|
[13] |
J. C. Boettger and S. B. Trickey, First-principles calculation of the spin–orbit splitting in graphene, Phys. Rev. B 75, 121402(R) (2007)
CrossRef
ADS
Google scholar
|
[14] |
M. Gmitra, S. Konschuh, C. Ertler, C. Ambrosch-Draxl, and J. Fabian, Band-structure topologies of graphene: Spin–orbit coupling effects from first principles, Phys. Rev. B 80(23), 235431 (2009)
CrossRef
ADS
Google scholar
|
[15] |
B. A. Bernevig and S. C. Zhang, Quantum spin Hall effect, Phys. Rev. Lett. 96(10), 106802 (2006)
CrossRef
ADS
Google scholar
|
[16] |
B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314(5806), 1757 (2006)
CrossRef
ADS
Google scholar
|
[17] |
M. König, S. Wiedmann, C. Brune, A. Roth, H. Buhmann, L. W. Molenkamp, X. L. Qi, and S. C. Zhang, Quantum spin Hall insulator state in HgTe quantum wells, Science 318(5851), 766 (2007)
CrossRef
ADS
Google scholar
|
[18] |
I. Knez, R. R. Du, and G. Sullivan, Evidence for helical edge modes in inverted InAs/GaSb quantum wells, Phys. Rev. Lett. 107(13), 136603 (2011)
CrossRef
ADS
Google scholar
|
[19] |
S. Murakami, Quantum spin Hall effect and enhanced magnetic response by spin–orbit coupling, Phys. Rev. Lett. 97(23), 236805 (2006)
CrossRef
ADS
Google scholar
|
[20] |
C. C. Liu, W. Feng, and Y. Yao, Quantum spin Hall effect in silicene and two-dimensional germanium, Phys. Rev. Lett. 107(7), 076802 (2011)
CrossRef
ADS
Google scholar
|
[21] |
F. C. Chuang, L. Z. Yao, Z. Q. Huang, Y. T. Liu, C. H. Hsu, T. Das, H. Lin, and A. Bansil, Prediction of largegap two-dimensional topological insulators consisting of bilayers of group III elements with Bi, Nano Lett. 14(5), 2505 (2014)
CrossRef
ADS
Google scholar
|
[22] |
J. J. Zhou, W. Feng, C. C. Liu, S. Guan, and Y. Yao, Large-Gap Quantum Spin Hall insulator in single layer bismuth monobromide Bi4Br4, Nano Lett. 14(8), 4767 (2014)
CrossRef
ADS
Google scholar
|
[23] |
W. Luo and H. J. Xiang, Room temperature quantum spin Hall insulators with a buckled square lattice, Nano Lett. 15(5), 3230 (2015)
CrossRef
ADS
Google scholar
|
[24] |
Y. D. Ma, L. Kou, A. Du, and T. Heine, Group 14 element-based noncentrosymmetric quantum spin Hall insulators with large bulk gap, Nano Res. 8(10), 3412 (2015)
CrossRef
ADS
Google scholar
|
[25] |
C. Si, J. Liu, Y. Xu, J. Wu, B. L. Gu, and W. Duan, Functionalized germanene as a prototype of large-gap twodimensional topological insulators, Phys. Rev. B 89(11), 115429 (2014)
CrossRef
ADS
Google scholar
|
[26] |
Y. Xu, B. Yan, H. J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, Large-gap quantum spin Hall insulators in tin films, Phys. Rev. Lett. 111(13), 136804 (2013)
CrossRef
ADS
Google scholar
|
[27] |
Y. D. Ma, Y. Dai, L. Kou, T. Frauenheim, and T. Heine, Robust two-dimensional topological insulators in methylfunctionalized bismuth, antimony, and lead bilayer films, Nano Lett. 15(2), 1083 (2015)
CrossRef
ADS
Google scholar
|
[28] |
Z. G. Song, C. C. Liu, J. Yang, J. Han, M. Ye, B. Fu, Y. Yang, Q. Niu, J. Lu, and Y. Yao, Quantum spin Hall insulators and quantum valley Hall insulators of BiX/SbX(X= H, F, Cl and Br) monolayers with a record bulk band gap, NPG Asia Mater. 6(12), e147 (2014)
CrossRef
ADS
Google scholar
|
[29] |
H. M. Weng, X. Dai, and Z. Fang, Transition-metal pentatelluride ZrTe5 and HfTe5: A paradigm for large-gap quantum spin Hall insulators, Phys. Rev. X 4(1), 011002 (2014)
CrossRef
ADS
Google scholar
|
[30] |
Y. D. Ma, L. Kou, X. Li, Y. Dai, S. C. Smith, and T. Heine, Quantum spin Hall effect and topological phase transition in two-dimensional square transitionmetal dichalcogenides, Phys. Rev. B 92(8), 085427 (2015)
CrossRef
ADS
Google scholar
|
[31] |
Y. D. Ma, L. Kou, X. Li, Y. Dai, and T. Heine, Two-dimensional transition metal dichalcogenides with a hexagonal lattice: Room-temperature quantum spin Hall insulators, Phys. Rev. B 93(3), 035442 (2016)
CrossRef
ADS
Google scholar
|
[32] |
S. M. Nie, Z. Song, H. Weng, and Z. Fang, Quantum spin Hall effect in two-dimensional transition-metal dichalcogenide haeckelites, Phys. Rev. B 91(23), 235434 (2015)
CrossRef
ADS
Google scholar
|
[33] |
X. F. Qian, J. Liu, L. Fu, and J. Li, Quantum spin Hall effect in two-dimensional transition metal dichalcogenides, Science 346(6215), 1344 (2014)
CrossRef
ADS
Google scholar
|
[34] |
Y. D. Ma, L. Kou, X. Li, Y. Dai, and T. Heine, Room temperature quantum spin Hall states in two-dimensional crystals composed of pentagonal rings and their quantum wells, NPG Asia Mater. 8(4), e264 (2016)
CrossRef
ADS
Google scholar
|
[35] |
Z. F. Wang, N. Su, and F. Liu, Prediction of a twodimensional organic topological insulator, Nano Lett. 13(6), 2842 (2013)
CrossRef
ADS
Google scholar
|
[36] |
B. Zhao, J. Zhang, W. Feng, Y. Yao, and Z. Yang, Quantum spin Hall and Z2 metallic states in an organic material, Phys. Rev. B 90(20), 201403 (2014)
CrossRef
ADS
Google scholar
|
[37] |
E. M. Spanton, K. C. Nowack, L. J. Du, G. Sullivan, R. R. Du, and K. A. Moler, Images of edge current in InAs/GaSb quantum wells, Phys. Rev. Lett. 113(2), 026804 (2014)
CrossRef
ADS
Google scholar
|
[38] |
L. J. Du, I. Knez, G. Sullivan, and R. R. Du, Robust helical edge transport in gated InAs/GaSb bilayers, Phys. Rev. Lett. 114(9), 096802 (2015)
CrossRef
ADS
Google scholar
|
[39] |
S. F. Wu, V. Fatemi, Q. D. Gibson, K. Watanabe, T. Taniguchi, R. J. Cava, and P. Jarillo-Herrero, Observation of the quantum spin Hall efft up to 100 kelvin in a monolayer crystal, Science 359(6371), 76 (2018)
CrossRef
ADS
Google scholar
|
[40] |
F. Reis, G. Li, L. Dudy, M. Bauernfeind, S. Glass, W. Hanke, R. Thomale, J. Schafer, and R. Claessen, Bismuthene on a SiC substrate: A candidate for a high-temperature quantum spin Hall material, Science 357(6348), 287 (2017)
CrossRef
ADS
Google scholar
|
[41] |
Z. Y. Fei, T. Palomaki, S. Wu, W. Zhao, X. Cai, B. Sun, P. Nguyen, J. Finney, X. Xu, and D. H. Cobden, Edge conduction in monolayer WTe2, Nat. Phys. 13(7), 677 (2017)
CrossRef
ADS
Google scholar
|
[42] |
S. J. Tang, C. Zhang, D. Wong, Z. Pedramrazi, H. Z. Tsai, C. Jia, B. Moritz, M. Claassen, H. Ryu, S. Kahn, J. Jiang, H. Yan, M. Hashimoto, D. Lu, R. G. Moore, C. C. Hwang, C. Hwang, Z. Hussain, Y. Chen, M. M. Ugeda, Z. Liu, X. Xie, T. P. Devereaux, M. F. Crommie, S. K. Mo, and Z. X. Shen, Quantum spin Hall state in monolayer 1T′-WTe2, Nat. Phys. 13(7), 683 (2017)
CrossRef
ADS
Google scholar
|
[43] |
A. Roth, C. Brne, H. Buhmann, L. W. Molenkamp, J. Maciejko, X. L. Qi, and S. C. Zhang, Nonlocal transport in the quantum spin Hall state, Science 325(5938), 294 (2009)
CrossRef
ADS
Google scholar
|
[44] |
I. Knez, C. T. Rettner, S. H. Yang, S. S. P. Parkin, L. J. Du, R. R. Du, and G. Sullivan, Observation of edge transport in the disordered regime of topologically insulating InAs/GaSb quantum wells, Phys. Rev. Lett. 112(2), 026602 (2014)
CrossRef
ADS
Google scholar
|
[45] |
J. J. Zhou, T. Zhou, S.-G. Cheng, H. Jiang, and Z. Q. Yang, Engineering topological quantum dot through planar magnetization in bismuthene, arxiv: 1812.11514 (2018)
|
[46] |
L. Fu and C. L. Kane, Topological insulators with inversion symmetry, Phys. Rev. B 76(4), 045302 (2007)
CrossRef
ADS
Google scholar
|
[47] |
D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, A topological Dirac insulator in a quantum spin Hall phase, Nature 452(7190), 970 (2008)
CrossRef
ADS
Google scholar
|
[48] |
H. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang, Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface, Nat. Phys. 5(6), 438 (2009)
CrossRef
ADS
Google scholar
|
[49] |
Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Observation of a large-gap topological-insulator class with a single Dirac cone on the surface, Nat. Phys. 5(6), 398 (2009)
CrossRef
ADS
Google scholar
|
[50] |
D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Observation of unconventional quantum spin textures in topological insulators, Science 323(5916), 919 (2009)
CrossRef
ADS
Google scholar
|
[51] |
Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S. K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen, Experimental realization of a three-dimensional topological insulator Bi2Te3, Science 325(5937), 178 (2009)
CrossRef
ADS
Google scholar
|
[52] |
J. S. Zhang, C. Z. Chang, Z. C. Zhang, J. Wen, X. Feng, K. Li, M. H. Liu, K. He, L. L. Wang, X. Chen, Q. K. Xue, X. C. Ma, and Y. Y. Wang, Band structure engineering in (Bi1–xSbx)2Te3 ternary topological insulators, Nat. Commun. 2(1), 574 (2011)
CrossRef
ADS
Google scholar
|
[53] |
D. Kim, S. Cho, N. P. Butch, P. Syers, K. Kirshenbaum, S. Adam, J. Paglione, and M. S. Fuhrer, Surface conduction of topological Dirac electrons in bulk insulating Bi Bi2Se3, Nat. Phys. 8(6), 459 (2012)
CrossRef
ADS
Google scholar
|
[54] |
Y. Xu, I. Miotkowski, C. Liu, J. F. Tian, H. Nam, N. Alidoust, J. N. Hu, C. K. Shih, M. Z. Hasan, and Y. P. Chen, Observation of topological surface state quantum Hall effect in an intrinsic three-dimensional topological insulator, Nat. Phys. 10(12), 956 (2014)
CrossRef
ADS
Google scholar
|
[55] |
D. A. Kozlov, Z. D. Kvon, E. B. Olshanetsky, N. N. Mikhailov, S. A. Dvoretsky, and D. Weiss, Transport properties of a 3D topological insulator based on a strained high-mobility HgTe film, Phys. Rev. Lett. 112(19), 196801 (2014)
CrossRef
ADS
Google scholar
|
[56] |
J. Liao, Y. B. Ou, X. Feng, S. Yang, C. J. Lin, W. M. Yang, K. H. Wu, K. He, X. C. Ma, Q. K. Xue, and Y. Q. Li, Observation of Anderson localization in ultrathin films of three-dimensional topological insulators, Phys. Rev. Lett. 114(21), 216601 (2015)
CrossRef
ADS
Google scholar
|
[57] |
H. C. Wang, H. W. Liu, C. Z. Chang, H. K. Zuo, Y. F. Zhao, Y. Sun, Z. C. Xia, K. He, X. C. Ma, X. C. Xie, Q. K. Xue, and J. Wang, Crossover between weak antilocalization and weak localization of bulk states in ultrathin Bi2Se3 films, Sci. Rep. 4(1), 5817 (2015)
CrossRef
ADS
Google scholar
|
[58] |
D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, A tunable topological insulator in the spin helical Dirac transport regime, Nature 460(7259), 1101 (2009)
CrossRef
ADS
Google scholar
|
[59] |
L. Fu, Topological crystalline insulators, Phys. Rev. Lett. 106(10), 106802 (2011)
CrossRef
ADS
Google scholar
|
[60] |
C. K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88(3), 035005 (2016)
CrossRef
ADS
Google scholar
|
[61] |
S. Chakravarty and A. Schmid, Weak localization: The quasiclassical theory of electrons in a random potential, Phys. Rep. 140(4), 193 (1986)
CrossRef
ADS
Google scholar
|
[62] |
A. Stern, Y. Aharonov, and Y. Imry, Phase uncertainty and loss of interference: A general picture, Phys. Rev. A 41(7), 3436 (1990)
CrossRef
ADS
Google scholar
|
[63] |
H. Jiang, S. Cheng, Q. F. Sun, and X. C. Xie, Topological insulator: A new quantized spin Hall resistance robust to dephasing, Phys. Rev. Lett. 103(3), 036803 (2009)
CrossRef
ADS
Google scholar
|
[64] |
I. Żutić, J. Fabian, and S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys. 76(2), 323 (2004)
CrossRef
ADS
Google scholar
|
[65] |
J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Zutic, Semiconductor spintronics, Acta Phys. Slovaca 57(4–5), 565 (2007)
CrossRef
ADS
Google scholar
|
[66] |
T. L. Schmidt, S. Rachel, F. von Oppen, and L. I. Glazman, Inelastic electron backscattering in a generic helical edge channel, Phys. Rev. Lett. 108(15), 156402 (2012)
CrossRef
ADS
Google scholar
|
[67] |
J. I. Väyrynen, M. Goldstein, and L. I. Glazman, Helical edge resistance introduced by charge puddles, Phys. Rev. Lett. 110(21), 216402 (2013)
CrossRef
ADS
Google scholar
|
[68] |
J. C. Budich, F. Dolcini, P. Recher, and B. Trauzettel, Phonon-induced backscattering in helical edge states, Phys. Rev. Lett. 108(8), 086602 (2012)
CrossRef
ADS
Google scholar
|
[69] |
J. J. Qi, H. W. Liu, H. Jiang, and X. C. Xie, Effective spin dephasing mechanism in confined two-dimensional topological insulators, Sci. China Phys. Mech. Astron. 59(7), 677811 (2016)
CrossRef
ADS
Google scholar
|
[70] |
H. W. Liu, H. Jiang, Q. F. Sun, and X. C. Xie, Dephasing effect on backscattering of helical surface states in 3D topological insulators, Phys. Rev. Lett. 113(4), 046805 (2014)
CrossRef
ADS
Google scholar
|
[71] |
J. Liao, Y. B. Ou, H. W. Liu, K. He, X. C. Ma, Q. K. Xue, and Y. Q. Li, Enhanced electron dephasing in threedimensional topological insulators, Nat. Commun. 8(1), 16071 (2017)
CrossRef
ADS
Google scholar
|
[72] |
C. Wu, B. A. Bernevig, and S. C. Zhang, Helical liquid and the edge of quantum spin Hall systems, Phys. Rev. Lett. 96(10), 106401 (2006)
CrossRef
ADS
Google scholar
|
[73] |
Q. F. Sun, J. Wang, and H. Guo, Quantum transport theory for nanostructures with Rashba spin–orbital interaction, Phys. Rev. B 71(16), 165310 (2005)
CrossRef
ADS
Google scholar
|
[74] |
Y. Meir and N. S. Wingreen, Landauer formula for the current through an interacting electron region, Phys. Rev. Lett. 68(16), 2512 (1992)
CrossRef
ADS
Google scholar
|
[75] |
A. P. Jauho, N. S. Wingreen, and Y. Meir, Timedependent transport in interacting and noninteracting resonant-tunneling systems, Phys. Rev. B 50(8), 5528 (1994)
CrossRef
ADS
Google scholar
|
[76] |
S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge: Cambridge University Press, 1995
CrossRef
ADS
Google scholar
|
[77] |
M. Büttiker, Role of quantum coherence in series resistors, Phys. Rev. B 33(5), 3020 (1986)
CrossRef
ADS
Google scholar
|
[78] |
Y. X. Xing, Q. F. Sun, and J. Wang, Inuence of dephasing on the quantum Hall effect and the spin Hall effect, Phys. Rev. B 77(11), 115346 (2008)
CrossRef
ADS
Google scholar
|
[79] |
E. J. Koop, B. J. van Wees, D. Reuter, A. D. Wieck, and C. H. van der Wal, Spin accumulation and spin relaxation in a large open quantum dot, Phys. Rev. Lett. 101(5), 056602 (2008)
CrossRef
ADS
Google scholar
|
[80] |
S. M. Frolov, A. Venkatesan, W. Yu, J. A. Folk, and W. Wegscheider, Electrical generation of pure spin currents in a two-dimensional electron gas, Phys. Rev. Lett. 102(11), 116802 (2009)
CrossRef
ADS
Google scholar
|
[81] |
Q. F. Sun, Y. X. Xing, and S. Q. Shen, Double quantum dot as detector of spin bias, Phys. Rev. B 77(19), 195313 (2008)
CrossRef
ADS
Google scholar
|
[82] |
Y. X. Xing, Q. F. Sun, and J. Wang, Spin bias measurement based on a quantum point contact, Appl. Phys. Lett. 93(14), 142107 (2008)
CrossRef
ADS
Google scholar
|
[83] |
Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Observation of the spin Hall effect in Semiconductors, Science 306(5703), 1910 (2004)
CrossRef
ADS
Google scholar
|
[84] |
V. Sih, W. H. Lau, R. C. Myers, V. R. Horowitz, A. C. Gossard, and D. D. Awschalom, Generating spin currents in semiconductors with the spin Hall effect, Phys. Rev. Lett. 97(9), 096605 (2006)
CrossRef
ADS
Google scholar
|
[85] |
M. König, M. Baenninger, A. G. F. Garcia, N. Harjee, B. L. Pruitt, C. Ames, P. Leubner, C. Brüne, H. Buhmann, L. W. Molenkamp, and D. Goldhaber-Gordon, Spatially resolved study of backscattering in the quantum spin Hall state, Phys. Rev. X 3, 021003 (2013)
CrossRef
ADS
Google scholar
|
[86] |
R. Jackiw and C. Rebbi, Solitons with fermion number 1/2, Phys. Rev. D 13(12), 3398 (1976)
CrossRef
ADS
Google scholar
|
[87] |
F. Zhang, C. L. Kane, and E. J. Mele, Surface states of topological insulators, Phys. Rev. B 86(8), 081303 (2012)
CrossRef
ADS
Google scholar
|
[88] |
W. Y. Shan, J. Lu, H. Z. Lu, and S. Q. Shen, Vacancyinduced bound states in topological insulators, Phys. Rev. B 84(3), 035307 (2011)
CrossRef
ADS
Google scholar
|
[89] |
A. Ström, H. Johannesson, and G. I. Japaridze, Edge dynamics in a quantum spin Hall state: Effects from Rashba spin–orbit interaction, Phys. Rev. Lett. 104(25), 256804 (2010)
CrossRef
ADS
Google scholar
|
[90] |
J. Fabian and S. Das Sarma, Phonon-induced spin relaxation of conduction electrons in aluminum, Phys. Rev. Lett. 83(6), 1211 (1999)
CrossRef
ADS
Google scholar
|
[91] |
G. Grimvall, Electron–Phonon Interaction in Metals, North-Holland Pub, 1981
|
[92] |
K. Saha and I. Garate, Phonon-induced topological insulation, Phys. Rev. B 89(20), 205103 (2014)
CrossRef
ADS
Google scholar
|
[93] |
E. Lhuillier, S. Keuleyan, and P. Guyot-Sionnest, Optical properties of HgTe colloidal quantum dots, Nanotechnology 23(17), 175705 (2012)
CrossRef
ADS
Google scholar
|
[94] |
A. Fasolino, E. Molinari, and J. C. Maan, Calculated superlattice and interface phonons of InAs/GaSb superlattices, Phys. Rev. B 33(12), 8889 (1986)
CrossRef
ADS
Google scholar
|
[95] |
D. Hernangómez-Pérez, J. Ulrich, S. Florens, and T. Champel, Spectral properties and local density of states of disordered quantum Hall systems with Rashba spin– orbit coupling, Phys. Rev. B 88(24), 245433 (2013)
CrossRef
ADS
Google scholar
|
[96] |
S. Y. Xu, Y. Xia, L. A. Wray, S. Jia, F. Meier, J. H. Dil, J. Osterwalder, B. Slomski, A. Bansil, H. Lin, R. J. Cava, and M. Z. Hasan, Topological phase transition and texture inversion in a tunable topological insulator, Science 332(6029), 560 (2011)
CrossRef
ADS
Google scholar
|
[97] |
T. Sato, K. Segawa, K. Kosaka, S. Souma, K. Nakayama, K. Eto, T. Minami, Y. Ando, and T. Takahashi, Unexpected mass acquisition of Dirac fermions at the quantum phase transition of a topological insulator, Nat. Phys. 7(11), 840 (2011)
CrossRef
ADS
Google scholar
|
[98] |
S. Y. Xu, M. Neupane, C. Liu, D. M. Zhang, A. Richardella, L. A. Wray, N. Alidoust, M. Leandersson, T. Balasubramanian, J. Snchez-Barriga, O. Rader, G. Landolt, B. Slomski, J. H. Dil, J. Osterwalder, T. R. Chang, H. T. Jeng, H. Lin, A. Bansil, N. Samarth, and M. Z. Hasan, Hedgehog spin texture and Berrys phase tuning in a magnetic topological insulator, Nat. Phys. 8(8), 616 (2012)
CrossRef
ADS
Google scholar
|
[99] |
S. Souma, M. Komatsu, M. Nomura, T. Sato, A. Takayama, T. Takahashi, K. Eto, K. Segawa, and Y. Ando, Spin polarization of gapped Dirac surface states near the topological phase transition in TlBi(S1–xSex)2, Phys. Rev. Lett. 109(18), 186804 (2012)
CrossRef
ADS
Google scholar
|
[100] |
B. L. Altshuler and A. G. Aronov, Electron–Electron Interactions in Disordered Systems, edited by A. L. Efros and M. Pollak, Elsevier, Amsterdam, 1985
CrossRef
ADS
Google scholar
|
[101] |
D. Belitz and S. Das Sarma, Inelastic phase-coherence time in thin metal films, Phys. Rev. B 36(14), 7701 (1987)
CrossRef
ADS
Google scholar
|
[102] |
V. B. Berestetskii, E. M. Lifshits, and L. P. Pitaevskii, Quantum Electrodynamics, Elsevier, Oxford, 1971
|
[103] |
Y. Imry, Introduction to Mesoscopic Physics, Oxford University Press, 2008
|
[104] |
B. L. Altshuler, A. G. Aronov, and D. E. Khmelnitsky, Effects of electron–electron collisions with small energy transfers on quantum localisation, J. Phys. C Solid State Phys. 15(36), 7367 (1982)
CrossRef
ADS
Google scholar
|
[105] |
B. I. Shklovskii and A. L. Efros, Electron Properties of Doped Semiconductors, Springer Science and Business Media, 2013
|
[106] |
S. Malzard, C. Poli, and H. Schomerus, Topologically protected defect states in open photonic systems with non- Hermitian charge-conjugation and parity-time symmetry, Phys. Rev. Lett. 115(20), 200402 (2015)
CrossRef
ADS
Google scholar
|
[107] |
P. San-Jose, J. Cayao, E. Prada, and R. Aguado, Majorana bound states from exceptional points in nontopological superconductors, Sci. Rep. 6(1), 21427 (2016)
CrossRef
ADS
Google scholar
|
[108] |
T. E. Lee, Anomalous edge state in a non-Hermitian lattice, Phys. Rev. Lett. 116(13), 133903 (2016)
CrossRef
ADS
Google scholar
|
[109] |
D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, Edge modes, degeneracies, and topological numbers in non-Hermitian systems, Phys. Rev. Lett. 118(4), 040401 (2017)
CrossRef
ADS
Google scholar
|
[110] |
Y. Xu, S. T. Wang, and L. M. Duan, Weyl exceptional rings in a three-dimensional dissipative cold atomic gas, Phys. Rev. Lett. 118(4), 045701 (2017)
CrossRef
ADS
Google scholar
|
[111] |
Z. P. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Topological phases of non- Hermitian systems, Phys. Rev. X 8(3), 031079 (2018)
CrossRef
ADS
Google scholar
|
[112] |
H. Jiang, C. Yang, and S. Chen, Topological invariants and phase diagrams for one-dimensional two-band non- Hermitian systems without chiral symmetry, Phys. Rev. A 98(5), 052116 (2018)
CrossRef
ADS
Google scholar
|
[113] |
S. Y. Yao and Z. Wang, Edge states and topological invariants of non-Hermitian systems, Phys. Rev. Lett. 121(8), 086803 (2018)
CrossRef
ADS
Google scholar
|
[114] |
S. Y. Yao, F. Song, and Z. Wang, Non-Hermitian Chern bands, Phys. Rev. Lett. 121(13), 136802 (2018)
CrossRef
ADS
Google scholar
|
[115] |
Private communication with Y. Y. Wang.
|
[116] |
F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80(4), 1355 (2008)
CrossRef
ADS
Google scholar
|
[117] |
J. J. Qi, H. W. Liu, C. Z. Chen, H. Jiang, and X. C. Xie, Quantum to classical crossover under dephasing effects in a two-dimensional percolation model, arxiv: 1903.01764 (2019)
|
[118] |
H. Jiang, L. Wang, Q. F. Sun, and X. C. Xie, Numerical study of the topological Anderson insulator in HgTe/CdTe quantum wells, Phys. Rev. B 80(16), 165316 (2009)
CrossRef
ADS
Google scholar
|
[119] |
D. W. Xu, J. J. Qi, J. Liu, X. C. Sacksteder, X. C. Xie, and H. Jiang, Phase structure of the topological Anderson insulator, Phys. Rev. B 85(19), 195140 (2012)
CrossRef
ADS
Google scholar
|
[120] |
C. Z. Chen, H. W. Liu, H. Jiang, Q. F. Sun, Z. Q. Wang, and X. C. Xie, Tunable Anderson metal-insulator transition in quantum spin-Hall insulators, Phys. Rev. B 91(21), 214202 (2015)
CrossRef
ADS
Google scholar
|
[121] |
C. Z. Chen, J. T. Song, H. Jiang, Q. F. Sun, Z. Q. Wang, and X. C. Xie, Disorder and metal-insulator transitions in Weyl semimetals, Phys. Rev. Lett. 115(24), 246603 (2015)
CrossRef
ADS
Google scholar
|
[122] |
C. Z. Chen, H. Liu, and X. C. Xie, Effects of random domains on the zero Hall plateau in the quantum anomalous Hall effect, Phys. Rev. Lett. 122(2), 026601 (2019)
CrossRef
ADS
Google scholar
|
[123] |
Y. Ando and L. Fu, Topological crystalline insulators and topological superconductors: From concepts to materials, Annu. Rev. Condens. Matter Phys. 6(1), 361 (2015)
CrossRef
ADS
Google scholar
|
[124] |
B. H. Yan and C. Felser, Topological materials: Weyl semimetals, Annu. Rev. Condens. Matter Phys. 8(1), 337 (2017)
CrossRef
ADS
Google scholar
|
[125] |
W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Quantized electric multipole insulators, Science 357(6346), 61 (2017)
CrossRef
ADS
Google scholar
|
/
〈 | 〉 |